A Cookbook for Multivariate Statistics

Dennis Wu

2022-10-16

Preface

This is a sample book written in Markdown. You can use anything that Pandoc’s Markdown supports, e.g., a math equation \(a^2 + b^2 = c^2\).

The bookdown package can be installed from CRAN or Github:

install.packages("bookdown")
# or the development version
# devtools::install_github("rstudio/bookdown")

Remember each Rmd file contains one and only one chapter, and a chapter is defined by the first-level heading #.

To compile this example to PDF, you need XeLaTeX. You are recommended to install TinyTeX (which includes XeLaTeX): https://yihui.org/tinytex/.

This cookbook contains data and code regarding different methods for multivariate analysis.

Chapter 1 Principal Component Analysis

Principal Component Analysis (PCA): A multivariate technique in analyzing quantitative data. The main goal is perform data reduction to reduce noise and extract the most important information from a large size data. The principal component is the line of best fit. This line maximizes the inertia (variance) of an array of data points (very similar to regression but different!).

PCA gives a map for the rows (factor scores) and another based on columns (loadings). These two maps are both described by the same components, but maps different information (interpreted differently).

Factors scores are based on row observation; therefore, interpret the distance between participants if each row is a participant. Loadings describes the column variables. Loadings are interpreted by the angle between them (180 angle = opposite correlation/direction, 90 degree = no correlation, less than 90 degree = correlated)

The distance from the origin of the map is important; they represent the squared distance from the mean which is also called the inertia (variance/information/sum of squares). With Pythagorean theorem, the total information contributed by a data point (squared distance to the origin) equals the sum of squared factor scores.

Step by step process:

1. PCA finds the new origin of the data by taking average of horizontal and vertical range of all data points (then projects onto a 2D plane)

2. PCA sets up a new graph with new axis that maximizes the inertia(variance) from the data points. This new graph really is to show the largest combined data points (sum of squares found by Pythagorean theorem).

3. PCA sets up subsequent axis orthogonal to the first one and also maximizes the remaining inertia.

Interpretation of maps:

• On the maps, PCA allows to explore relationship of the data among both rows and columns. Rows are interpreted differently compared to columns.

• Factor scores (rows): We can interpret factor score map by looking at the distance between row data points using group means, tolerance level (spread out), bootstrap intervals (how confident we are), and clustering (grouping of the data).

1.1 The Data

This data was collected from a great number of individuals. Data was looking at how different personalities is related to different types of drug consumption. Description of participants includes gender, age, ethnicity, and country.

load("DrugConsumption.RData")
drugconsumption <- data

personality <- data[,c(6:12)]
drugs <- data[,c(13:31)]

Let’s see what the data looks like

# a <- knitr::kable(data, format = "html")
# kableExtra::scroll_box(a, width = "500px", height = "500px", fixed_thead = T)

a <- data
glimpse(a)

## Rows: 1,885
## Columns: 31
## $ AGE         <dbl> 3, 2, 3, 1, 3, 6, 4, 3, 3, 5, 2, 4, 5, 5, 5, 5, 3, 4, 5, 3…
## $ GENDER      <fct> F, M, M, F, F, F, M, M, F, M, F, M, F, F, F, M, F, M, M, M…
## $ EDUCATION   <dbl> 6, 9, 6, 8, 9, 4, 8, 2, 6, 8, 7, 5, 7, 6, 6, 7, 5, 2, 7, 6…
## $ COUNTRY     <fct> UK, UK, UK, UK, UK, CA, US, UK, CA, UK, UK, OT, UK, CA, UK…
## $ ETHNICITY   <fct> WA, WH, WH, WH, WH, WH, WH, WH, WH, WH, WH, WH, WH, WH, WH…
## $ NSCORE      <dbl> 28, 18, 20, 23, 32, 18, 20, 13, 31, 22, 15, 13, 45, 17, 16…
## $ ESCORE      <dbl> 20, 36, 29, 18, 12, 22, 16, 36, 39, 24, 29, 24, 25, 29, 33…
## $ OSCORE      <dbl> 17, 30, 15, 21, 18, 10, 18, 15, 14, 11, 13, 22, 24, 21, 24…
## $ ASCORE      <dbl> 18, 29, 13, 28, 22, 36, 22, 22, 29, 28, 19, 11, 13, 30, 20…
## $ CSCORE      <dbl> 25, 24, 17, 29, 33, 35, 31, 35, 32, 26, 36, 21, 19, 41, 35…
## $ IMPULSIVITY <dbl> 4, 3, 2, 2, 4, 2, 4, 5, 2, 2, 5, 6, 8, 3, 8, 3, 4, 2, 3, 2…
## $ SS          <dbl> 3, 6, 8, 3, 6, 2, 7, 5, 2, 4, 7, 10, 7, 4, 9, 5, 1, 4, 6, …
## $ ALCOHOL     <dbl> 6, 6, 7, 5, 5, 3, 7, 6, 5, 7, 6, 6, 6, 2, 7, 6, 7, 7, 7, 5…
## $ AMPHET      <dbl> 3, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 3, 1, 2, 3, 2…
## $ AMYL        <dbl> 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1…
## $ BENZOS      <dbl> 3, 1, 1, 4, 1, 1, 1, 1, 1, 2, 1, 1, 5, 1, 1, 1, 2, 1, 3, 1…
## $ CAFF        <dbl> 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7…
## $ CANNABIS    <dbl> 1, 5, 4, 3, 4, 1, 2, 1, 1, 2, 3, 5, 4, 1, 1, 2, 4, 7, 4, 2…
## $ CHOC        <dbl> 6, 7, 5, 5, 7, 5, 6, 5, 7, 7, 6, 6, 6, 1, 7, 6, 6, 5, 7, 7…
## $ COKE        <dbl> 1, 4, 1, 3, 1, 1, 1, 1, 1, 1, 3, 3, 2, 1, 1, 3, 1, 2, 3, 1…
## $ CRACK       <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
## $ ECSTASY     <dbl> 1, 5, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 2, 3, 2…
## $ HEROIN      <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
## $ KETAMINE    <dbl> 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1…
## $ LEGALH      <dbl> 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1…
## $ LSD         <dbl> 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1…
## $ METH        <dbl> 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 7…
## $ MUSHROOMS   <dbl> 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2, 2, 1…
## $ NICOTINE    <dbl> 3, 5, 1, 3, 3, 7, 7, 1, 7, 7, 3, 7, 7, 2, 7, 1, 7, 7, 1, 2…
## $ SEMER       <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
## $ VSA         <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1…

The data has 1885 observations, with 31 measures (columns). The columns contains descriptive information (Age, gender, education, country, ethnicity) and scale quantity for drug consumption ranging from 1 = Never used, 2 = Used over a decade ago, 3 = used in the last decade, 4 = used in last year, 5 = used in last month, 6 = used in last week, 7 = yesterday There are also scoring not based on scale for personality. Some are nominal like gender, country, ethnicity but mostly factors Others are ordinal like scoring for the drugs and education Personality score is different and are raw number scoring, not on a scale Other columns contain quantitative information like Escore/nScore are quantitative - Escore = extravert, Ascore = Agreebleness, Oscore = openness to experience, Nscore = neuroticism SS = impulse score SEmer= fake drug

Drop the non numerical columns and factors

drop <- c("EDUCATION", "COUNTRY", "GENDER", "ETHNICITY", "AGE", "GenderAge")
data_2 <- data[,!names(data) %in% drop]

1.1.1 Correlation Plot

Let’s check to see what the data looks like and how the variables are related.

corr_plot <- corrplot(cor.res, is.cor = FALSE, tl.cex = 1, cl.cex = .8, method = "color")

cor.plot <- recordPlot() #save to the environment for PPT

Certain drugs are more correlated than others: Mushrooms and LSD (party drugs).Addictive drugs are highly correlated (crack & heroin).Some of the personalities are related to most of the drugs. For example, sensation seeking is positively correlated to most drugs.

1.2 PCA Starts!

library(InPosition)

## 
## Attaching package: 'InPosition'

## The following object is masked from 'package:PTCA4CATA':
## 
##     boot.ratio.test

Set the seed so that your analysis is reproducible.

#same seed make it easier for reproducibility
set.seed(42)

This is the PCA function. Run inference PCA with the Gender as design. We will see how Gender connects with the PCA

#Running permutation on each component and then bootstrapping the data and tests
res_pcaInf <- epPCA.inference.battery(data_2, center = TRUE, scale = "SS1",
                                      DESIGN = data$GENDER, graphs = FALSE,
                                      test.iters = 999) #kept this the same just incase
##the bigger the test.iter number, the higher the accuracy

res_pcaInf$Fixed.Data gives the same thing as epPCA:

#checking to see what it looks like
res_pcaInf$Fixed.Data #fixed data from the actual dataset

## **ExPosition output data**
## *Contains the following objects:
## 
##   name              
## 1 "$ExPosition.Data"
## 2 "$Plotting.Data"  
##   description                                                                 
## 1 "All ExPosition classes output (data, factor scores, contributions, etc...)"
## 2 "All ExPosition & prettyGraphs plotting data (constraints, colors, etc...)"

res_pcaInf$Inference.Data #permutation/bootstrapping data

## **Results for Principal Component Analysis Inference Battery**
## Permutation was performed on  26 components and bootstrap performed on  26  variables
## *The results are available in the following objects:
## 
##   name         
## 1 "$components"
## 2 "$fj.boots"  
##   description                                                           
## 1 "p-values ($p.vals) and permutations ($eigs.perm) for each component."
## 2 "A list with bootstrap data and tests."

The inference is really to test how reliable the variables are. This function allows us to see the barplots later on for reliability of the variables.

1.2.1 Testing the eigenvalues

This graph shows behind the scenes of the permutation test

zeDim = 1
pH1 <- prettyHist(
  distribution = res_pcaInf$Inference.Data$components$eigs.perm[,zeDim], 
           observed = res_pcaInf$Fixed.Data$ExPosition.Data$eigs[zeDim], 
           xlim = c(0, 7), # needs to be set by hand, 7.5 wont work
           breaks = 5,
           border = "white", 
           main = paste0("Permutation Test for Eigenvalue ",zeDim),
           xlab = paste0("Eigenvalue ",zeDim), 
           ylab = "", 
           counts = FALSE, 
           cutoffs = c(0.975))

eigs1 <- recordPlot()
zeDim = 2
pH2 <- prettyHist(
  distribution = res_pcaInf$Inference.Data$components$eigs.perm[,zeDim], 
           observed = res_pcaInf$Fixed.Data$ExPosition.Data$eigs[zeDim], 
           xlim = c(0, 5), # needs to be set by hand
           breaks = 5,
           border = "white", 
           main = paste0("Permutation Test for Eigenvalue ",zeDim),
           xlab = paste0("Eigenvalue ",zeDim), 
           ylab = "", 
           counts = FALSE, 
           cutoffs = c(0.975))

zeDim = 3
pH2 <- prettyHist(
  distribution = res_pcaInf$Inference.Data$components$eigs.perm[,zeDim], 
           observed = res_pcaInf$Fixed.Data$ExPosition.Data$eigs[zeDim], 
           xlim = c(0, 2), # needs to be set by hand
           breaks = 5,
           border = "white", 
           main = paste0("Permutation Test for Eigenvalue ",zeDim),
           xlab = paste0("Eigenvalue ",zeDim), 
           ylab = "", 
           counts = FALSE, 
           cutoffs = c(0.975))

eigs3 <- recordPlot()

This shows the significance of the dimensions based on permutation test. The further away, the more significant it is. p<.05 or less if the observed value is further away from the histogram.

Scree Plot: Show the results from permutation with Scree plot (i.e., color the significant components) by adding the estimated p-values to the PlotScree function.

my.scree <- PlotScree(ev = res_pcaInf$Fixed.Data$ExPosition.Data$eigs,
                      p.ev = res_pcaInf$Inference.Data$components$p.vals,
                      plotKaiser = T)

my.scree <- recordPlot() # you need this line to be able to save them in the end

The purple dots indicate the dimensions that contribute to the most variance. The first 5 dimensions contribute the most. We will only look at the first 2.

1.2.2 Biplot of all data

Let’s see what the colors are for the different genders

Let’s recode the colors so that female is pink and male is blue

##"305ABF" = blue = female
#"#84BF30" = green = male
res_pcaInf$Fixed.Data$Plotting.Data$fi.col <- dplyr::recode(res_pcaInf$Fixed.Data$Plotting.Data$fi.col, "#305ABF" =  "pink",
                                                           "#84BF30" = "blue" )

Let’s get the biplot of all of the data colored by the gender

my.fi.plot <- createFactorMap(res_pcaInf$Fixed.Data$ExPosition.Data$fi, # data
                            title = "Participant Row Factor Scores based on Country", # title of the plot
                            axis1 = 1, axis2 = 2, # which component for x and y axes
                            # 1 = age, 2 = Escore
                            pch = 19, # the shape of the dots (google `pch`)
                            cex = 2, # the size of the dots
                            text.cex = 2.5, # the size of the text
                            alpha.points = 0.1,
                            col.points = res_pcaInf$Fixed.Data$Plotting.Data$fi.col, # color of the dots
                            col.labels = res_pcaInf$Fixed.Data$Plotting.Data$fi.col, # color for labels of dots
                            display.labels = F)

fi.labels <- createxyLabels.gen(1,2,
                             lambda = res_pcaInf$Fixed.Data$ExPosition.Data$eigs,
                             tau = round(res_pcaInf$Fixed.Data$ExPosition.Data$t),
                             axisName = "Component "
                             )
fi.plot <- my.fi.plot$zeMap + fi.labels # you need this line to be able to save them in the end
fi.plot

Blue is male, pink is female. Really difficult to interpret as the dots are all over the place. Might need to look at the means to really understand what is going on.

Now get the color for each group (Gender levels):

# get index for the first row of each group
grp.ind <- order(data$GENDER)[!duplicated(sort(data$GENDER))]
grp.col <- res_pcaInf$Fixed.Data$Plotting.Data$fi.col[grp.ind] # get the color
grp.name <- data$GENDER[grp.ind] # get the corresponding groups
names(grp.col) <- grp.name

1.2.3 Group means

Getting the group means grouping by Gender for each column (Each level of Gender gets the group mean by combining the columns)

group.mean <- aggregate(res_pcaInf$Fixed.Data$ExPosition.Data$fi, by = list(data$GENDER),
                     mean)
group.mean

##   Group.1          V1           V2           V3            V4           V5
## 1       F  0.02080090  0.001134979  0.001449019  0.0003652334  0.001879331
## 2       M -0.02077884 -0.001133775 -0.001447482 -0.0003648461 -0.001877338
##             V6           V7            V8            V9          V10
## 1  0.001612014 -0.001312873  0.0008346671  0.0009116603  0.003382835
## 2 -0.001610304  0.001311481 -0.0008337820 -0.0009106936 -0.003379248
##             V11           V12           V13           V14          V15
## 1  0.0004535038 -0.0009736137 -0.0007131315 -0.0007362386  0.002582911
## 2 -0.0004530229  0.0009725812  0.0007123752  0.0007354579 -0.002580172
##            V16           V17           V18           V19           V20
## 1  0.002494894  0.0002310316  9.205714e-05 -0.0003633237 -0.0005959691
## 2 -0.002492248 -0.0002307866 -9.195952e-05  0.0003629384  0.0005953371
##             V21           V22           V23           V24           V25
## 1 -0.0004441831  1.439406e-05 -0.0004981096 -0.0004336957 -0.0003845784
## 2  0.0004437120 -1.437880e-05  0.0004975814  0.0004332358  0.0003841706
##             V26
## 1 -0.0003998093
## 2  0.0003993853

# need to format the results from `aggregate` correctly
rownames(group.mean) <- group.mean[,1] # Use the first column as row names
fi.mean <- group.mean[,-1] # Exclude the first column
fi.mean

##            V1           V2           V3            V4           V5           V6
## F  0.02080090  0.001134979  0.001449019  0.0003652334  0.001879331  0.001612014
## M -0.02077884 -0.001133775 -0.001447482 -0.0003648461 -0.001877338 -0.001610304
##             V7            V8            V9          V10           V11
## F -0.001312873  0.0008346671  0.0009116603  0.003382835  0.0004535038
## M  0.001311481 -0.0008337820 -0.0009106936 -0.003379248 -0.0004530229
##             V12           V13           V14          V15          V16
## F -0.0009736137 -0.0007131315 -0.0007362386  0.002582911  0.002494894
## M  0.0009725812  0.0007123752  0.0007354579 -0.002580172 -0.002492248
##             V17           V18           V19           V20           V21
## F  0.0002310316  9.205714e-05 -0.0003633237 -0.0005959691 -0.0004441831
## M -0.0002307866 -9.195952e-05  0.0003629384  0.0005953371  0.0004437120
##             V22           V23           V24           V25           V26
## F  1.439406e-05 -0.0004981096 -0.0004336957 -0.0003845784 -0.0003998093
## M -1.437880e-05  0.0004975814  0.0004332358  0.0003841706  0.0003993853

Plotting the group means for gender

fi.mean.plot <- createFactorMap(fi.mean,
                                title = "Participant Row Factor Scores based on Gender",
                                alpha.points = 0.9,
                                col.points = grp.col[rownames(fi.mean)],
                                col.labels = grp.col[rownames(fi.mean)],
                                pch = 17, 
                                cex = 4,
                                text.cex = 4)
fi.WithMean <- my.fi.plot$zeMap_background+ my.fi.plot$zeMap_dots +
  fi.mean.plot$zeMap_dots + fi.mean.plot$zeMap_text + fi.labels
fi.WithMean

#my.fi.plot$zeMap_dots I took the dots out because it was just too messy 

Based on the group means of gender, we can see that on component 1 differs on gender. Particularly, there is a difference between males and females.

1.2.4 Tolerance interval

We can plot the tolerance interval. This graph will show the range of the data, group means and individual dots based on Gender

TIplot <- MakeToleranceIntervals(res_pcaInf$Fixed.Data$ExPosition.Data$fi,
                            design = as.factor(data$GENDER),
                            col = grp.col[rownames(fi.mean)],
                            line.size = 1, 
                            line.type = 10,
                            alpha.ellipse = .1,
                            alpha.line    = .4,
                            p.level       = .95)
# If you get some errors with this function, check the names.of.factors argument in the help.

fi.WithMeanTI <- my.fi.plot$zeMap_background + my.fi.plot$zeMap_dots + 
  fi.mean.plot$zeMap_dots + fi.mean.plot$zeMap_text + TIplot + fi.labels

fi.WithMeanTI

Males are more spread out further to left while females are more to the right. However, males and females overlap a lot.

1.2.5 Bootstrap interval

We can also add the bootstrap interval for the group means to see if these group means are significantly different than our actual data. (Sanity check to make sure our collected data is reliable)

First step: bootstrap the group means

# Depend on the size of your data, this might take a while
#computes bootstrapping estimates for th means of the groups of observation described by factor scores (the group is based on GenderxAge)
fi.boot <- Boot4Mean(res_pcaInf$Fixed.Data$ExPosition.Data$fi,
                     design = data$GENDER,
                     niter = 1000)
# Check what you have
fi.boot

Second step: plot it! Let’s plot the bootstrap for the group (gender) means.

# Check other parameters you can change for this function
bootCI4mean <- MakeCIEllipses(fi.boot$BootCube[,c(1:2),], # get the first two components
                              col = grp.col[rownames(fi.mean)],
                              line.size = 2,
                              line.type = 1,
                              alpha.ellipse = .2)

fi.WithMeanCI <- my.fi.plot$zeMap_background + bootCI4mean + 
  my.fi.plot$zeMap_dots + fi.mean.plot$zeMap_dots + 
  fi.mean.plot$zeMap_text + fi.labels

fi.WithMeanCI

You might not be able to see it, but there is a small eclipse going around the triangles. Bootstrapping suggest it is not that much different. The eclipses circles the group means for each level of Gender and the size of the eclipses is very small; therefore, the bootstrapping for the group means is reliable.

1.2.6 Loadings

Loadings are based on columns of variance.

Let’s change the colors for personality and drugs

col4X <- prettyGraphsColorSelection(
                   n.colors = ncol(personality),
                   starting.color = 42)

col4Y <- prettyGraphsColorSelection(
                    n.colors = ncol(drugs),
                    starting.color = 13)
col4Var = c(col4X,col4Y)

This chunk of code is to see which column/loading are correlated.

cor.loading <- cor(data_2, res_pcaInf$Fixed.Data$ExPosition.Data$fi)
rownames(cor.loading) <- rownames(cor.loading)

loading.plot <- createFactorMap(cor.loading,
                                constraints = list(minx = -1, miny = -1,
                                                   maxx = 1, maxy = 1),
                                col.points = col4Var)
LoadingMapWithCircles <- loading.plot$zeMap + 
  addArrows(cor.loading, color = col4Var) + 
  addCircleOfCor() + xlab("Component 1") + ylab("Component 2")

LoadingMapWithCircles

Again, PCA compares the data points based on the center, not directly to each other. We can compare drugs to drugs and how related they are. We can compare personalities to personality. We can also compare drugs to personality to see which personality is more likely to partake in which drugs. For example, we can see people with neuroticism and sensation seeking is more likely to participate in drugs. Neuroticism almost has a 90 degree angle to Sensation Seeking, suggesting that they are not related.

• Component 1: which personality is more likely to participate in drugs.

• Component 2: breaks down which personality is like to partake in which drugs.

You can also include the variance of each component and plot the factor scores for the columns (i.e., the variables):

my.fj.plot <- createFactorMap(res_pcaInf$Fixed.Data$ExPosition.Data$fj, # data
                            title = "DrugConsumption Column Factor Scores", # title of the plot
                            axis1 = 1, axis2 = 2, # which component for x and y axes
                            # 1 = Alcohol, 2 = Amphet
                            pch = 19, # the shape of the dots (google `pch`)
                            cex = 5, # the size of the dots
                            text.cex = 5, # the size of the text
                            col.points = col4Var, # color of the dots
                            col.labels = col4Var, # color for labels of dots
                            )

fj.plot <- my.fj.plot$zeMap + fi.labels # you need this line to be able to save them in the end
fj.plot

We can use this graph to compare the drugs and personalities with the gender. It seems like men are more likely to participate in drugs compared to women.

1.2.7 Bootstrap Ratio of columns

This technique is to check if our components are reliable and similar if we do a bootstrapping Note: This is not the same as the contribution bars

1.2.7.0.1 Component 1 & 2

signed.ctrJ <- res_pcaInf$Fixed.Data$ExPosition.Data$cj * sign(res_pcaInf$Fixed.Data$ExPosition.Data$fj)

# plot contributions for component 1 from our actual data
ctrJ.1 <- PrettyBarPlot2(signed.ctrJ[,1],
                         threshold = 1 / NROW(signed.ctrJ),
                         font.size = 2,
                         color4bar = gplots::col2hex(res_pcaInf$Fixed.Data$Plotting.Data$fj.col), # we need hex code
                         ylab = 'Contributions',
                         ylim = c(1.2*min(signed.ctrJ[,1]), 1.2*max(signed.ctrJ[,1])),
                         horizontal = FALSE
) + ggtitle("Contribution barplots", subtitle = 'Component 1: Variable Contributions (Signed)')

# plot contributions for component 2 from our actual data
ctrJ.2 <- PrettyBarPlot2(signed.ctrJ[,2],
                         threshold = 1 / NROW(signed.ctrJ),
                         font.size = 2,
                         color4bar = gplots::col2hex(res_pcaInf$Fixed.Data$Plotting.Data$fj.col), # we need hex code
                         ylab = 'Contributions',
                         ylim = c(1.2*min(signed.ctrJ[,2]), 1.2*max(signed.ctrJ[,2])),
                         horizontal = FALSE
) + ggtitle("",subtitle = 'Component 2: Variable Contributions (Signed)')


BR <- res_pcaInf$Inference.Data$fj.boots$tests$boot.ratios
laDim = 1

# Plot the bootstrap ratios for Dimension 1
ba001.BR1 <- PrettyBarPlot2(BR[,laDim],
                        threshold = 2,
                        font.size = 2,
                   color4bar = gplots::col2hex(res_pcaInf$Fixed.Data$Plotting.Data$fj.col), # we need hex code
                  ylab = 'Bootstrap ratios',
                  horizontal = FALSE
                  #ylim = c(1.2*min(BR[,laDim]), 1.2*max(BR[,laDim]))
) + ggtitle("Bootstrap ratios", subtitle = paste0('Component ', laDim))

# Plot the bootstrap ratios for Dimension 2
laDim = 2
ba002.BR2 <- PrettyBarPlot2(BR[,laDim],
                        threshold = 2,
                        font.size = 2,
                   color4bar = gplots::col2hex(res_pcaInf$Fixed.Data$Plotting.Data$fj.col), # we need hex code
                  ylab = 'Bootstrap ratios',
                  horizontal = FALSE,
                  #ylim = c(1.2*min(BR[,laDim]), 1.2*max(BR[,laDim]))
) + ggtitle("",subtitle = paste0('Component ', laDim))

We then use the next line of code to put barplots and bootstraps side to side:

  grid.arrange(
    as.grob(ctrJ.1),
    as.grob(ctrJ.2),
    as.grob(ba001.BR1),
    as.grob(ba002.BR2),
    ncol = 2,nrow = 2,
    top = textGrob("Barplots for variables", gp = gpar(fontsize = 25, font = 3))
  )

BothCtrJ <- recordPlot() # you need this line to be able to save them in the end

Bootstrapping makes it more intense because we have a large sample size (amplifies it). But with bootstrapping, we can see whether the variables are reliable. It looks like our contributions are reliable and stable as everything is similar to the bootstrap.

Breakdown of the barplots:

Component 1: which personality is likely to participate in drugs

Component 2: which personality is likely to participate in which drugs.

1.3 Summary

When we interpret the factor scores and loadings together, the PCA revealed:

• Component 1: Which personality is likely to participate in drugs

• Component 2: Breaks down which personality is likely to participate in what kind drugs.

Conclusion: The difference of gender is based on the variables(columns). We can see that men are more likely to participate in drugs. Furthermore, we can also see that individuals with high sensation seeking is likely to participate in drugs.

The following chunk can give you a .pptx file with all your figures saved in the directory.

# Save figures to PPT. 

# savedList <- saveGraph2pptx(file2Save.pptx = 'AllFigures_infPCA',
#                             title = 'All Figures for inference',
#                             addGraphNames = TRUE)

Chapter 2 Correspondence Analysis

Corresponding Analysis (CA)

CA is a multivariate analysis method that is built upon PCA (Very similar). However, the only difference is that instead of using elucidation distance to find the best-fit line for inertia, chi-square is used to find measurement of the distance among each data point. CA is best when use to analyze nominal data or qualitative data (as opposed to quantitative).

Note: Qualitative means data that is numeric numerically but doesn’t have actual value (Likert scale,random numbers assigned as ID). Quantitative means data that has actual value meaning (how many, how much).

CA takes Contingency table as its Input. Besides being able to compare rows to rows and column to column, CA is especially powerful in comparing rows and columns directly with each others.

Contingency Table:

Contingency table is a two-way table that summarize the relationship between several categorical variables. A contingency table is a special type of frequency distribution table, where two variables are shown simultaneously.(StatisticHowTo) The table is transformed into a probability matrix and then multipled by the mass(rows) and weights(columns).

Step by step process:

1. Perform PCA to get new components and eigen values

2. Calculate chi-square distance between data points so we can get rows and columns to have similar variance and to map them together. We can also create factor scores from rows (mass) and columns (weights).

3. Finds Inertia by using Generalized SVD to decompose the chi-square X^2 into orthogonal components. A note to know:rows are multiplied by mass while columns are multipled by weights.

4. Projects factor scores and loadings onto a new plane.

Other info: Generally, we can look at symmetric and aysmmetric plots to see similarities of the variables. This is based on row to row/ col to col or row to col. 

Asymmetric Map: Either row or column factor scores are normalized.I s a simplex. Can interpret all ways round (col-col, row-row, col-row)

Symmetric Map: Both row or column factor scores are normalized. Is NOT a simplex. Cannot interpret row to column but can interpret row to row/ col to col. 

Final Thoughts:

CA is great for nominal/qualitative data. This analysis is great in that we can compare rows to columns and see the relationship between these two through symmetric plots. In other words, we can explore the relationship between observations and variables by summarizing the data through multi-dimension reduction into 1-3 important dimensions.

2.1 Turkey Data

We will be using Turkey data. This data contains what ar people’s opinion on different brands of turkey. This was collected in a Spanish speaking country. Some meaning of words are lost during translations process.

This is a dataset looking at how females feel to turkey ham. Age ranges from 25-35 (50%) to 26-45 (50%). Original dataset contains 8 observations and 24 columns of emotions for turkey. The data is not subsetted at all. All original data was used for analysis.

Load the data

data <- readxl::read_xlsx("7. Turkey ham emotions.xlsx")
data <- as.data.frame(data)

#remove the first column
data_2 <- data[,-1]

#add the first column values as row names
rownames(data_2) <- data[,1]

X <- as.matrix(data_2)

View the main data

# a <- knitr::kable(data, format = "html")
# kableExtra::scroll_box(a, width = "500px", height = "500px", fixed_thead = T)

a <- data
dplyr::glimpse(a)

## Rows: 8
## Columns: 24
## $ Product                  <chr> "San Rafael", "Fud", "Zwan natural", "Capistr…
## $ Happy                    <dbl> 9, 9, 4, 7, 10, 8, 11, 4
## $ `Pleasantly surprised`   <dbl> 9, 4, 8, 5, 3, 7, 7, 4
## $ `Unpleasantly surprised` <dbl> 1, 4, 6, 3, 2, 6, 2, 2
## $ Salivating               <dbl> 8, 7, 6, 9, 4, 3, 5, 5
## $ Famished                 <dbl> 11, 12, 16, 15, 11, 13, 14, 13
## $ Refreshed                <dbl> 27, 18, 27, 31, 25, 30, 31, 26
## $ Desired                  <dbl> 16, 8, 9, 8, 11, 11, 13, 7
## $ Soothed                  <dbl> 12, 16, 18, 12, 17, 16, 10, 17
## $ Comforted                <dbl> 4, 2, 2, 4, 6, 8, 4, 4
## $ Disgusted                <dbl> 5, 5, 5, 8, 3, 12, 11, 6
## $ Energetic                <dbl> 9, 5, 11, 4, 4, 8, 5, 5
## $ Joy                      <dbl> 11, 21, 20, 6, 23, 13, 20, 17
## $ Impressed                <dbl> 5, 5, 9, 3, 7, 6, 5, 2
## $ Interested               <dbl> 15, 10, 17, 16, 18, 16, 20, 15
## $ Irritated                <dbl> 3, 5, 1, 3, 3, 3, 3, 2
## $ Melancholic              <dbl> 0, 0, 2, 1, 1, 0, 0, 0
## $ Nostalgic                <dbl> 2, 2, 0, 2, 0, 2, 1, 0
## $ Relaxed                  <dbl> 6, 13, 8, 14, 13, 10, 11, 15
## $ Revitalized              <dbl> 3, 2, 5, 1, 4, 3, 2, 4
## $ Romantic                 <dbl> 2, 4, 1, 0, 2, 3, 4, 0
## $ Sad                      <dbl> 6, 4, 6, 5, 2, 3, 2, 3
## $ Sensual                  <dbl> 2, 1, 5, 5, 4, 2, 4, 2
## $ `Weel-being`             <dbl> 12, 10, 11, 10, 12, 7, 6, 12

The data pattern for CA is slightly different. In CA, we analyze the chi-square of the contingency table. As a result, let’s try to plot the chi-square data pattern instead of the correlation matrix.

Chi-square test for the contingency table

# get Chi2 -- we can use the available package to get the Chi2
chi2    <-  chisq.test(X)

Chi-square is in counts, but CA analyzed probabilities (i.e., the profiles). So, we need to divide the chi-square statistics by the total sum of the data to get the probabilities. Also, the chi-square statistic adds the chi-squares in all cells and give one number. In CA, however, we keep the pattern of chi-squares instead of adding all of them up.

Now let’s get the probabilities

# Components of chi2: the chi-squares for each cell before we add them up to compute the chi2
Inertia.cells <- chi2$residuals / sqrt(sum(X))
# To be Plotted

# You can always compute it directly from the data
Z <- X / sum(X) # observed 
r <- as.matrix(rowSums(Z)) # expected for each row
c <- as.matrix(colSums(Z)) # expected for each column
# Inertia.cells 
test.Inertia.cells <- diag( as.vector(r^(-1/2)) ) %*% 
                     (Z - r%*%t(c) ) %*% diag(as.vector(c^(-1/2)))

Plot this residual:

corrplot((Inertia.cells), method = "color", is.cor = FALSE) # I only used the transposed matrix because I prefer to present it in landscape

## You can also do it without transposing it (the commented code). The plot will be the same but just in portreit.
# corrplot(Inertia.cells, is.cor = FALSE)
a0.residuals <- recordPlot()

This is saying how variables are associated with each other. For example, Melancholic and Zwan natural are strongly associated with each other. Capistrano and Joy are strongly not associated with each other.

2.2 Analysis

There are two different ways to present the CA analysis: symmetric and asymmetric.

Something to note:   Asymmetry we can compare rows and columns. Asymmetry uses the columns (weights) as the axis for the simplex and the rows (masses) are typically contained within. We can interpret the rows and columns together   Symmetry we can only compare rows to rows and columns to columns.

Function for the CA

# run CA
resCA.sym  <- epCA(X, symmetric = TRUE, graphs = FALSE)
resCAinf.sym4bootJ  <- epCA.inference.battery(X, symmetric = TRUE, graphs = FALSE, test.iters = 25)
resCAinf.sym4bootI  <- epCA.inference.battery(t(X), symmetric = TRUE, graphs = FALSE, test.iters = 25)

# to run a plain CA but asymmetric
# this is using the columns as the simplex (you can also use rows by running epCA with t(X))
resCA.asym <- epCA(X, symmetric = FALSE, graphs = FALSE)

Inference battery

For the inference battery, you might find that it takes a very long time to run. In order to make it not run that long, let’s set the seed for reproducibility and make the iterations not that high.

set.seed(22)

res_fast_perm <- data4PCCAR::fastPerm4CA(X, nIter = 20, compact = FALSE)
res_fast_boot <- data4PCCAR::fastBoot4CA(X)

The inference is to allow us to see how reliable the variables are for the barplots.

2.2.1 Scree Plot/Eigen value test

Show the results from permutation with Scree plot (i.e., color the significant components) by adding the estimated p-values to the PlotScree function.

my.scree <- PlotScree(ev = resCA.sym$ExPosition.Data$eigs,
                      p.ev = resCAinf.sym4bootJ$Inference.Data$components$p.vals, 
                      plotKaiser = TRUE)

my.scree <- recordPlot() # you need this line to be able to save them in the end

It seems like none of the dimensions are significant. However, the kaiser line indicates the first 3 dimensions explains the most variance. However, we will only look at the first two.

Here are the things you will need to create the plot. You can decide where you want to put these code and don’t necessarily need to organize the script this way (i.e., list all these information in a chunk before the plotting functions).

# Here are the factor scores you need
Fj.a <- resCA.asym$ExPosition.Data$fj
Fi   <- resCA.sym$ExPosition.Data$fi
Fj   <- resCA.sym$ExPosition.Data$fj

# constraints -----
# first get the constraints correct
constraints.sym  <- minmaxHelper(mat1 = Fi, mat2  = Fj)
constraints.asym <- minmaxHelper(mat1 = Fi, mat2  = Fj.a)

# Get some colors ----
color4Authors <- prettyGraphsColorSelection(n.colors = nrow(Fi))
# baseMaps ----
colnames(Fi) <- paste("Dimension ", 1:ncol(Fi))
colnames(Fj) <- paste("Dimension ", 1:ncol(Fj))
colnames(Fj.a) <- paste("Dimension ", 1:ncol(Fj.a))

2.2.2 Asymmetric Factor Scores Plot

First show an asymmetric plot without the labels for the rows/columns you project inside the simplex.

Fi.a <- resCA.asym$ExPosition.Data$fj

map.I.sup.asym <- asymMap$baseMap + 
                          asymMap$I_points +
                          asymMap$J_labels + asymMap$J_points +
                          labels4CA + ggtitle('Asymmetric Map with Supplementary Observation and Simplex') 

theSimplex.i <- ggConvexHull(Fi.a, 
              percentage = 1,
              col.hull = ggplot2::alpha('lightblue',.2),
              col.line = ggplot2::alpha('lightblue',.7),
              line.size = .4,
              alpha.hull = .2,
              names.of.factors = "Dimension ")

map.IJ.asym.i.simplex <- asymMap$baseMap +
                          asymMap$I_points +
                          asymMap$J_labels + asymMap$J_points +
                          labels4CA +  theSimplex.i
map.IJ.asym.i.simplex

Something to remember: We are comparing the distance between the plots, not to the barycenter!

Interpretation:

In this asymmetric plot, we can interpret rows and columns together. However, it is difficult to interpret the columns and rows together as we can see that the rows are bunched together in the middle, not showing much difference.

But if we were to interpret the rows and columns together, the green dots (columns) that are closest to the purple represents that they are very similar. However based on the columns, we can see the melancholic is very far away from the other columns. This means that melancholic is very different compared to the others as it is very far away to all other plots.

Eigen value for dimension 1 is bigger than dimension 2, suggesting it contributes to more variance.

2.2.3 Plot the symmetric plot

Next, show the symmetric plot with all labels printed.

Thing to remember: symmetric plot means you can compare row to row or column to column only!

Function for the base map

# factor scores
symMap  <- createFactorMapIJ(Fi,Fj,
                    col.points.i = 'Blue',
                    col.labels.i = 'Blue')

# plot the row factor scores with confidence intervals
map.sepI.sym <- symMap$baseMap  + 
  symMap$I_labels + symMap$I_points +
  ggtitle('Symmetric: Row') + 
  labels4CA


# plot the columns factor scores with confidence intervals
map.sepJ.sym <- symMap$baseMap +
  symMap$J_labels + symMap$J_points +
  ggtitle('Symmetric: Column') + 
  labels4CA

grid.arrange(
    map.sepI.sym, map.sepJ.sym,
    ncol = 2,nrow = 1,
    top = textGrob("Factor scores", gp = gpar(fontsize = 18, font = 3))
  )

BothmapIJ <- recordPlot() # you need this line to be able to save them in the end

The general idea is that we can only compare rows to rows or columns to columns. Dots that are closer to together are very similar. Dots that are further away are different.

We can definitely interpret that melancholic is an outlier or is very different compared to the rest because it is very far away from all other plots.

Dimension 1 is Melancholic vs. Disgusted

Dimension 2 is Romantic vs Melancholic

2.2.4 Contributions and bootstrap ratios barplots

For CA, we plot the contributions for both rows and columns. As noted a few times, please feel free to change the colors to help you tell the story, or flip the bars to horizontal if that makes it look better.

signed.ctrI <- resCA.sym$ExPosition.Data$ci * sign(resCA.sym$ExPosition.Data$fi)
signed.ctrJ <- resCA.sym$ExPosition.Data$cj * sign(resCA.sym$ExPosition.Data$fj)

Bootstrap ratios

BR.I <- resCAinf.sym4bootI$Inference.Data$fj.boots$tests$boot.ratios
BR.J <- resCAinf.sym4bootJ$Inference.Data$fj.boots$tests$boot.ratios

This bootstrap barplot is to see whether the variabile are reliable, meaning if we see the same variables of significance as the previous barplot.

I arrange these plots to put contribution and bootstrap ratio plots side by side.

grid.arrange(
     as.grob(ctrI.1),as.grob(ctrJ.1),as.grob(ctrI.2),as.grob(ctrJ.2),as.grob(ba001.BR1.I),as.grob(ba002.BR1.J),as.grob(ba003.BR2.I),as.grob(ba004.BR2.J),
    ncol = 4,nrow = 2,
    top = textGrob("Contribution   &   Bootstrap ratios", gp = gpar(fontsize = 18, font = 3))
  )

Both.IJ <- recordPlot() # you need this line to be able to save them in the end

With the barplots, we can see which rows/columns are similar based on the dimensions. General idea is that if they are going the same direction, they are similar. If they are going opposite direction, they are different.

For Dimension 1:

• Rows: we can see that Zwan virginia and Zwan natural are similar.

• Columns: We can see nostalgic, disgusted, desired and happy are similar.

Dimension 2:

• Rows: Capistrano and Zwan natural are similar.

• Columns: Sad, melancholic, energetic, and pleasantly surprised are similar.

The brand of turkey makes it difficult to interpret because these are translated brands from Spanish. What is interested is that for component 1 of columns, Disgusted and Happy are contribution the same direction. For component 2 of columns, melancholic and energetic is contributing to the same direction.

The bootstrapping causes some variables to be non-significance like zwan virginia in component 1 of rows, suggesting that this is not reliable. Same for some of the columns in component 1. For component 2, some of the columns and rows did not survive the bootstrap. Overally, the bootstrap shows that some of the variables are reliable while some are not.

2.3 Summary

When we interpret the factor scores and loadings together, the CA revealed:

• Component 1: Individuals differ based on Energetic & Unpleasantly surprised vs Happy & Comfort

• Component 2: Individuals differ based on Disgust, comfort, and salivating vs desired and unpleasantly surprised

• Both: Difficult to interpret but individuals differ based on how they feel about the turkey. The bootstrap ratio makes the contributions more conservative and accurate. Some survived the bootstrap, some did not.

• Do you prefer symmetric or asymmetric plot for your data?

I prefer the symmetric plot because comparing the rows to columns didn’t make sense for my data as we did not know what the rows actually means. Furthermore, most of the rows were centered in the middle making it difficult to interpret.

Chapter 3 Barycentric Discriminant Analysis (BADA) on Drug Consumption

BADA is similar to PCA, except we are using group/category means (barycenters) of quantitative data set. It is very similar to unsupervised machine learning, which classifies observations. BADA creates new space called group means of each category (groups must be more than 2), then projects it into the same space to see which classification best fit.

The goal of BADA is to combine the measurements to create new variables (called components or discriminant variables) that best separate the categories. Then, classify based on a-priori categories (pre-assgiend groups).

EX: assign subjects to a given diagnostic group (ASD groups) on the bases on brain imaging data or psychological data (a-priori categories will be clinical groups).

The basic idea is we use PCA on a subset of data to see how the observation can be categorized based on the provided data. Then we use the rest of the data and perform PCA on it and see the quality of the categorization based on the first function. Basically train a subset of data to see which categorization and then use the rest of the data to see quality of the categorization.

More info on Jackknife info:

Jackknife: using all observation except for one (keep one out) and repeat process

Fixed effect: train data and test to see how well it performs

Random effect: after training with sample data, input new data to see how well it performs

Quick overview:

• each category of interest is computed to get group means (barycenter)

• generalized PCA is performed on the category, gives us factor scores for categories and factor scores for variables

• original observations are then projected onto the factor scores map to see which observation is closest to a category based (this provides us the how close the observations are based on this map which contains the factor scores as well)

• then compute whether the a priori and a posterioir category assignments are correct (check quality)

Great to use when there are more variables (columns) than observations aka N << P or when measurements are qualitative barycenter = weighted average; center of gravity of the observations of a given category

Final Note: BADA is essential PCA but instead of using individual variables of the columns, BADA uses the group means of whatever pre-assigned group you choose.

More info can be found Barycentric Discriminant Analysis (BADIA) - Herve Arbi & Lynne J. Williams

We will be using the Drug consumption dataset. Design is based on country. The data looks at whether personality and drugs are correlated.

3.1 Load Data

load data

load("DrugConsumption.RData")
qualitative <- data[,1:5]
personality <- data[,6:12]
drugs <- data[,13:31]
personalityXdrug <- data[,6:31]

# a <- knitr::kable(data, format = "html")
# kableExtra::scroll_box(a, width = "500px", height = "500px", fixed_thead = T)

a <- data
dplyr::glimpse(a)

## Rows: 1,885
## Columns: 31
## $ AGE         <dbl> 3, 2, 3, 1, 3, 6, 4, 3, 3, 5, 2, 4, 5, 5, 5, 5, 3, 4, 5, 3…
## $ GENDER      <fct> F, M, M, F, F, F, M, M, F, M, F, M, F, F, F, M, F, M, M, M…
## $ EDUCATION   <dbl> 6, 9, 6, 8, 9, 4, 8, 2, 6, 8, 7, 5, 7, 6, 6, 7, 5, 2, 7, 6…
## $ COUNTRY     <fct> UK, UK, UK, UK, UK, CA, US, UK, CA, UK, UK, OT, UK, CA, UK…
## $ ETHNICITY   <fct> WA, WH, WH, WH, WH, WH, WH, WH, WH, WH, WH, WH, WH, WH, WH…
## $ NSCORE      <dbl> 28, 18, 20, 23, 32, 18, 20, 13, 31, 22, 15, 13, 45, 17, 16…
## $ ESCORE      <dbl> 20, 36, 29, 18, 12, 22, 16, 36, 39, 24, 29, 24, 25, 29, 33…
## $ OSCORE      <dbl> 17, 30, 15, 21, 18, 10, 18, 15, 14, 11, 13, 22, 24, 21, 24…
## $ ASCORE      <dbl> 18, 29, 13, 28, 22, 36, 22, 22, 29, 28, 19, 11, 13, 30, 20…
## $ CSCORE      <dbl> 25, 24, 17, 29, 33, 35, 31, 35, 32, 26, 36, 21, 19, 41, 35…
## $ IMPULSIVITY <dbl> 4, 3, 2, 2, 4, 2, 4, 5, 2, 2, 5, 6, 8, 3, 8, 3, 4, 2, 3, 2…
## $ SS          <dbl> 3, 6, 8, 3, 6, 2, 7, 5, 2, 4, 7, 10, 7, 4, 9, 5, 1, 4, 6, …
## $ ALCOHOL     <dbl> 6, 6, 7, 5, 5, 3, 7, 6, 5, 7, 6, 6, 6, 2, 7, 6, 7, 7, 7, 5…
## $ AMPHET      <dbl> 3, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 3, 1, 2, 3, 2…
## $ AMYL        <dbl> 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1…
## $ BENZOS      <dbl> 3, 1, 1, 4, 1, 1, 1, 1, 1, 2, 1, 1, 5, 1, 1, 1, 2, 1, 3, 1…
## $ CAFF        <dbl> 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7…
## $ CANNABIS    <dbl> 1, 5, 4, 3, 4, 1, 2, 1, 1, 2, 3, 5, 4, 1, 1, 2, 4, 7, 4, 2…
## $ CHOC        <dbl> 6, 7, 5, 5, 7, 5, 6, 5, 7, 7, 6, 6, 6, 1, 7, 6, 6, 5, 7, 7…
## $ COKE        <dbl> 1, 4, 1, 3, 1, 1, 1, 1, 1, 1, 3, 3, 2, 1, 1, 3, 1, 2, 3, 1…
## $ CRACK       <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
## $ ECSTASY     <dbl> 1, 5, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 2, 3, 2…
## $ HEROIN      <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
## $ KETAMINE    <dbl> 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1…
## $ LEGALH      <dbl> 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1…
## $ LSD         <dbl> 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1…
## $ METH        <dbl> 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 7…
## $ MUSHROOMS   <dbl> 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2, 2, 1…
## $ NICOTINE    <dbl> 3, 5, 1, 3, 3, 7, 7, 1, 7, 7, 3, 7, 7, 2, 7, 1, 7, 7, 1, 2…
## $ SEMER       <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
## $ VSA         <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1…

3.2 Start of BADA

Bada function with design as country

#group by country, getting groups means by country
resBADA <- tepBADA(personalityXdrug, scale = T, center = T, DESIGN = qualitative$COUNTRY,
                   graphs = FALSE)

Inference for bootstrapping

set.seed(25)
nIter = 25
resBADA.inf <- tepBADA.inference.battery(personalityXdrug, scale = T, center = T,
                   DESIGN = qualitative$COUNTRY,
                  test.iters = nIter,
                  graphs = FALSE)

## [1] "It is estimated that your iterations will take 0.67 minutes."
## [1] "R is not in interactive() mode. Resample-based tests will be conducted. Please take note of the progress bar."
## ================================================================================

Group Means based on country

Drugconsumption_means<- PTCA4CATA::getMeans(
               resBADA$TExPosition.Data$fii, 
               qualitative$COUNTRY)

Drugconsumption_means

##            V1          V2          V3         V4          V5          V6
## AU -1.5174925 -0.49810195  0.39138583 -0.8343027  0.30486443 -0.06099346
## CA -0.4322262 -0.84764892 -0.37091403  0.3366642  0.16532515 -0.52846880
## IE -0.9564754 -0.03539351  1.29561657  0.4757515 -0.01789525 -0.04765731
## NZ -2.7068000  1.48006324 -0.34795815  0.1058350  0.02662981 -0.07700708
## OT -0.8582760 -0.36610117  0.03622381 -0.2987558 -0.67825067 -0.15391034
## UK  1.1548979  0.67831003  0.16263026 -0.1169208  0.03637940 -0.11378940
## US -1.7095602 -1.02514402 -0.33590351  0.2927053  0.02052473  0.33674356

Group means with row names as country and column names as different variables

#getting group means 
scale_drug_means <- scale(personalityXdrug, center = T, scale = T)

Drugconsumption_means_2 <- PTCA4CATA::getMeans(
               scale_drug_means, 
               qualitative$COUNTRY)

Drugconsumption_means_2

##         NSCORE     ESCORE      OSCORE      ASCORE      CSCORE   IMPULSIVITY
## AU -0.08667547  0.1065528  0.18782943  0.01368664 -0.15053440  0.2947968529
## CA  0.10924574 -0.1853636 -0.09398204 -0.28212579  0.02966078  0.0777709098
## IE -0.10086456  0.3214777  0.02793205  0.05851728 -0.23510234 -0.2359384354
## NZ  0.38075354 -0.6764856  0.21828607  0.14441857 -0.12021759 -0.0002500672
## OT -0.06561516 -0.1114116  0.41238264 -0.29951508 -0.34998709  0.0820411257
## UK -0.11963189  0.1025404 -0.28958720  0.14573507  0.19211156 -0.1981018736
## US  0.22967308 -0.1554398  0.44892542 -0.17036123 -0.26645329  0.3216740091
##             SS     ALCOHOL      AMPHET        AMYL      BENZOS         CAFF
## AU  0.43761125  0.00986656  0.68119597  0.61300193  0.39547022  0.031129913
## CA  0.03928271 -0.35612598  0.29882466 -0.20305389  0.12017682 -0.093759914
## IE  0.25561631 -0.36432843  0.42578053  0.41636848  0.04538592  0.328516654
## NZ  0.38541647 -0.02630099 -0.30308792  1.12111669  0.71481056  0.283659436
## OT  0.32633069  0.01316738  0.03711404  0.16234060  0.02768588 -0.190762665
## UK -0.30386715  0.08205775 -0.35582509  0.05796397 -0.34406019  0.008500793
## US  0.43921413 -0.08860646  0.53378713 -0.19576291  0.57385882  0.021764153
##      CANNABIS        CHOC        COKE       CRACK     ECSTASY      HEROIN
## AU  0.4580003 -0.01288765  0.15043686 -0.02369617  0.60741456 -0.03930498
## CA  0.2106612 -0.43554663  0.35681761  0.50955577  0.15820553  0.20506947
## IE  0.4199503  0.17751407  0.28996544 -0.11661475  0.50737932  0.02512101
## NZ  0.6166769  0.63651820  0.02559550 -0.35554824 -0.06922834  0.41167697
## OT  0.4306943 -0.02787084  0.06144227 -0.03157063  0.22087578 -0.08298362
## UK -0.4857126  0.12106183 -0.25156245 -0.21021992 -0.30398753 -0.25498395
## US  0.7212211 -0.16381360  0.37753502  0.33079571  0.42178444  0.46268570
##       KETAMINE     LEGALH          LSD        METH  MUSHROOMS   NICOTINE
## AU  0.26201159  0.3495177  0.890180426  0.03787348  0.4153388  0.1470733
## CA  0.27771393  0.1671862  0.166860651  0.16817542  0.3818065  0.1550094
## IE -0.05674583  0.4157422 -0.007738207 -0.07683877 -0.2300064  0.7040253
## NZ  0.35308514  1.2538962  0.897632033  0.22681129  0.6906525 -0.3316292
## OT  0.20026681  0.1988639  0.515705604 -0.06448690  0.3750971  0.3347004
## UK -0.11327424 -0.3733857 -0.444042640 -0.34779700 -0.4340747 -0.2525068
## US  0.09997589  0.5715355  0.602885528  0.63633001  0.6362908  0.3416020
##          SEMER        VSA
## AU  0.52120019  0.3000916
## CA -0.05993216 -0.1278888
## IE -0.05993216  0.7965488
## NZ -0.05993216  1.6278259
## OT -0.05993216  0.1220166
## UK -0.05993216 -0.2304050
## US  0.08655075  0.3536735

3.2.1 Heat Map

corrplot3 <- corrplot::corrplot((as.matrix(Drugconsumption_means_2)), method = 'color',
                            tl.pos = "lt", 
                            tl.col = "black",
                            tl.cex = 0.7,
                            is.corr = FALSE,
                            addCoefasPercent = TRUE,
                            number.cex=0.5,
                            col=colorRampPalette(c("white","brown"))(200))

heat_map <- recordPlot()

This is the group mean based on country and correlation of the variables/columns. For example, NZ has greater VSA, benzos, and legal H than other countries.

3.2.2 Scree plot

# The ScreePlot. Fixed Effects. ----
# Get the ScreePlot
# scree for ev ----
PlotScree(ev = resBADA$TExPosition.Data$eigs,
          p.ev = resBADA.inf$Inference.Data$components$p.vals,
   title = 'BADA Drug Consumption: Scree Plot with P-Values',
   plotKaiser = T, 
   color4Kaiser = ggplot2::alpha('darkorchid4', .5),
   lwd4Kaiser  = 2)

# Save the plot for pptx
a0002.Scree.sv <- recordPlot()

Everything is significant, but we will only look at the first two dimensions are they contribute the most variance and are above the kaiser line. Eigen 1 and eigen 2 both show that the null hypothesis should be rejected, the first 2 dimensions should are true in the sense that they contribute to the most variance.

3.2.3 Loadings & Factor Scores

Change color of the countries

3.2.3.1 Symmetric Map

On this map, we can only compare rows to rows and columns to columns

# I-set map ----
# a graph of the observations
options(ggrepel.max.overlaps = Inf)

Imap <- PTCA4CATA::createFactorMap(
  resBADA$TExPosition.Data$fii,
  col.points = countryColors,
  col.labels = countryColors,
  alpha.points = .08,
  display.labels = F
)

# make labels ----
label4Map <- createxyLabels.gen(1,2,
          lambda = resBADA$TExPosition.Data$eigs,
          tau = resBADA$TExPosition.Data$t)

Recode the group means of country to match colors

# .UK "#305ABF"
# .CA "#84BF30"
# .US "#BF30AD"
# .OT "#30BFA7"
# .AU "#BF7D30"
# .IE "#5430BF"
# .NZ "#36BF30"

# "#305ABF"= 'indianred4', 
# "#84BF30"= 'gold', 
# "#BF30AD" = 'lightpink2',
# "#30BFA7" = "orange",
# "#BF7D30" = "Blue",
# "#5430BF" = "darkred",
# "#36BF30" =  "purple"

col4Mean_recode <- recode(rownames(Drugconsumption_means),
                     AU = "Blue",
                     CA = 'gold',
                     IE = "green",
                     NZ =  "purple",
                      OT = "orange",
                     UK = 'indianred4',
                     US = 'lightpink2'
                     )
names(col4Mean_recode) <- rownames(Drugconsumption_means)

3.2.3.2 Group Means Map

# the map
options(ggrepel.max.overlaps = Inf)
MapGroup <- PTCA4CATA::createFactorMap(Drugconsumption_means,
             # use the constraint from the main map
             constraints = Imap$constraints,
             col.points = col4Mean_recode,
             cex = 7,  # size of the dot (bigger)
             col.labels = col4Mean_recode,
             text.cex = 6,
             display.label = T,
             display.points = T
             )
# The map with observations and group means
a003.bada <- Imap$zeMap + 
               label4Map + 
               MapGroup$zeMap_dots + 
               MapGroup$zeMap_text


print(a003.bada)

Here we can see that NZ and UK group means are very different to the other countries. However, we can probably test if this is true based on confidence intervals.

3.2.3.3 Confidence Interval Map

# Confidence intervals
# Bootstrapped CI ----
#_________________________________________________
# Create Confidence Interval Plots
fi.boot <- resBADA.inf$Inference.Data$boot.data$fi.boot.data$boots
# We want to use the rownames of fi.boot 
# as reference to get the correct
# color. 
# However, the original rownames include "." 
# and don't match with 
# the original row names. 
# So, the `sub` function was used to get rid of 
# the "." by replacing all "." 
# in the rownames of fi.boot as an empty 
# string.
rownames(fi.boot) <- sub("[[:punct:]]","",
                          rownames(fi.boot))
# use function MakeCIEllipses 
# from package PTCA4CATA
GraphElli <- PTCA4CATA::MakeCIEllipses(
    resBADA.inf$Inference.Data$boot.data$fi.boot.data$boots,
    col = col4Mean_recode[rownames(fi.boot)], 
    # col = col4Means
    # use rownames as reference to pick the color
    p.level = .95
)

# create the I-map with Observations, 
# means and confidence intervals
#
a004.bada.withCI <-  Imap$zeMap_background  + 
                        Imap$zeMap_dots     + 
                        MapGroup$zeMap_dots + 
                        MapGroup$zeMap_text +
                      GraphElli + label4Map +
  ggtitle('BADA: Group Centers with CI and Observations')

# plot it!
print(a004.bada.withCI)

It seems like the New Zealand has a big confidence interval. Confidence intervals that overlap suggest that they are not different. For example, NZ is similar to IE. UK is not similar to any of the other countries.

3.2.3.4 Tolerance Level Map

Tolerance Level for rows means/country. This graph will show the overall spread of the data

# with Hull ----
Fii <- resBADA$TExPosition.Data$fii
colnames(Fii) <- paste0('Dimension ', 1:ncol(Fii))
# getting the color correct: an ugly trick
col4Hull <- col4Mean_recode[match(names(col4Mean_recode), 
                     levels(qualitative$COUNTRY))]
GraphHull <- PTCA4CATA::MakeToleranceIntervals(
                      Fii,
                      design = qualitative$COUNTRY,
                      col = col4Hull,
                # the next line is required 
                # for some strange unknown reasons
          names.of.factors =  c("Dim1","Dim2"),
          p.level = 1.00,
          alpha.ellipse = .02)
#
a006.bada.withHull <-  Imap$zeMap_background  + 
                          Imap$zeMap_dots     + 
                          MapGroup$zeMap_dots + 
                          MapGroup$zeMap_text +
                          GraphHull           + 
                          label4Map           +
    ggtitle('BADA: Group Centers with Hulls and Observations')
a006.bada.withHull

Tolerance Level is all over the place with similar areas/overlapping. This graph doesn’t show much.

3.2.3.5 Loading Map

Loadings/Columns

# J-set ----
# gt colors
col4X <- prettyGraphsColorSelection(
                   n.colors = ncol(personality),
                   starting.color = 42)

col4Y <- prettyGraphsColorSelection(
                    n.colors = ncol(drugs),
                    starting.color = 13)
col4Var = c(col4X,col4Y)
#_________________________________________________
Fj <- resBADA$TExPosition.Data$fj
baseMap.j <- PTCA4CATA::createFactorMap(
                        Fj,
                        col.points   = col4Var,
                        alpha.points =  .1,
                        col.labels   = col4Var)
#_________________________________________________
# arrows
zeArrows <- addArrows(Fj, color = col4Var)
# A graph for the J-set
# A graph for the J-set
b001.aggMap.j <- baseMap.j$zeMap_background + # background layer
                      baseMap.j$zeMap_dots + 
                      baseMap.j$zeMap_text +  # dots & labels
                      label4Map 
b002.aggMap.j <- b001.aggMap.j + zeArrows
# We print this Map with the following code
#dev.new()
print(b002.aggMap.j)

print(b001.aggMap.j)

Neuroticism, Sensation Seeking, and impulsivity is correlated to a lot of the drugs. It is very likely that these personality traits are correlated to drug consumption. This graph also shows which personality is related to which types of drugs. For example, we can see that impulsivity is related to meth, coke, nicotine, and ecstasy.

However, the main thing we need to keep in mind is that sensation seeking is further out from neuroticism and impulsivity, suggesting that sensation seeking contributes the most. We will test this with the barplots.

3.2.4 Contribution Plots

3.2.4.1 Barplots

How related the columns are

# Ctr J-set 
###### 1 ====
# 
ctrj <- resBADA$TExPosition.Data$cj
signed.ctrj <- ctrj * sign(Fj)
# BR1
c001.plotCtrj.1 <- PrettyBarPlot2(
           bootratio = round(100*signed.ctrj[,1]), 
           threshold = 100 / nrow(signed.ctrj), 
           ylim = NULL, 
           color4bar = gplots::col2hex(col4Var),
           color4ns = "gray75", 
           plotnames = TRUE, 
    main = 'Important Contributions Variables. Dim 1.', 
           ylab = "Signed Contributions",
    horizontal = F,
    signifOnly = T)
# print(c001.plotCtrj.1)

###### 2 ====
# 
c002.plotCtrj.2 <- PrettyBarPlot2(
  bootratio = round(100*signed.ctrj[,2]), 
  threshold = 100 / nrow(signed.ctrj), 
  ylim = NULL, 
  color4bar = gplots::col2hex(col4Var),
  color4ns = "gray75", 
  plotnames = TRUE, 
  main = 'Important Contributions Variables. Dim 2.', 
  ylab = "Signed Contributions",
  horizontal = F,
  signifOnly = T)
# print(c002.plotCtrj.2)

Bootstrap 1

BRj <- resBADA.inf$Inference.Data$boot.data$fj.boot.data$tests$boot.ratios
# BR1
d001.plotBRj.1 <- PrettyBarPlot2(
  bootratio = BRj[,1], 
  threshold = 2, 
  ylim = NULL, 
  color4bar = gplots::col2hex(col4Var),
  color4ns = "gray75", 
  plotnames = TRUE, 
  main = 'Bootstrap Ratios Variables. Dim 1.', 
  ylab = "Bootstrap Ratios",
  horizontal = F,
  signifOnly = T)
# print(d001.plotBRj.1)

Bootstrap 2

d003.plotBRj.2 <- PrettyBarPlot2(
  bootratio = BRj[,2], 
  threshold = 2, 
  ylim = NULL, 
  color4bar = gplots::col2hex(col4Var),
  color4ns = "gray75", 
  plotnames = TRUE, 
  main = 'Bootstrap Ratios Variables. Dim 2.', 
  ylab = "Bootstrap Ratios",
  horizontal = F,
  signifOnly = T)
# print(d003.plotBRj.2)

 grid.arrange(
    as.grob(c001.plotCtrj.1),
    as.grob(c002.plotCtrj.2),
    as.grob(d001.plotBRj.1),
    as.grob(d003.plotBRj.2),
    ncol = 2,nrow = 2,
    top = textGrob("Barplots for variables", gp = gpar(fontsize = 25, font = 3))
  )

BothCtrJ <- recordPlot() # you need this line to be able to save them in the end

Barplot 1: Sensation seeking individuals are more likely to partake in drugs, particularly amyl, benzos, cannabis, legalh, lsd, mushrooms, VSA.

barplot 2: tells us the different types of drugs.AMYL, Chocolate, VSA are similar. Amphet, Coke, Crack, Ecstasy, nicotine are similar.

After bootstrapping, everything is more significant. Bootstrapp is to test whether the barplots are stable/reliable. We can see that all the variables from the contribution plot survived the bootstrapping.

Bootstrap for dimension 2 some variables did not survive the bootstrapp.

3.3 Accuracy Testing

Accuracy (Prediction & Confusion Matrix)

Fixed Confusion Matrix

We will now look at how good our prediction on new observations. We will first look at the fixed effects.

Fixed: train data and test to see how well it performs on our original data

Fixed Accuracy Table/ Data confusion matrix

#Fixed CM
fixed_cm <- as.data.frame(resBADA.inf$Inference.Data$loo.data$fixed.confuse)
a <- knitr::kable(fixed_cm, format = 'html')
kableExtra::scroll_box(a, width = "910px", height = "500px", fixed_thead = T)

	.UK	.CA	.US	.OT	.AU	.IE	.NZ
.UK	764	32	64	28	10	5	1
.CA	31	13	61	4	1	0	0
.US	30	21	235	19	9	1	0
.OT	64	10	94	35	9	0	0
.AU	71	5	51	16	16	4	0
.IE	69	3	24	10	2	8	0
.NZ	15	3	28	6	7	2	4

#save in ppt
# p<-tableGrob(fixed_cm)
# grid.arrange(p)
# fixed_cm_df <- recordPlot()

Fixed Accuracy Percentage

resBADA.inf$Inference.Data$loo.data$fixed.acc

## [1] 0.5702918

Random Confusion Matrix: After training with sample data, input new data to see how well it performs

Predict something we don’t know

#random cm
random_cm <- as.data.frame(resBADA.inf$Inference.Data$loo.data$loo.confuse)
a <- knitr::kable(random_cm, format = 'html')
kableExtra::scroll_box(a, width = "910px", height = "500px")

	.UK.actual	.CA.actual	.US.actual	.OT.actual	.AU.actual	.IE.actual	.NZ.actual
.UK.predicted	763	33	63	29	10	5	1
.CA.predicted	32	9	64	6	1	1	0
.US.predicted	30	23	230	19	10	1	1
.OT.predicted	64	11	94	30	10	0	2
.AU.predicted	71	5	52	17	14	5	0
.IE.predicted	69	3	24	11	2	6	0
.NZ.predicted	15	3	30	6	7	2	1

#save in ppt
# p<-tableGrob(random_cm)
# grid.arrange(p)
# random_cm_df <- recordPlot()

Random Accuracy Percentage

resBADA.inf$Inference.Data$loo.data$loo.acc

## [1] 0.5586207

55% is not that bad considered how many observations we have. The random accuracy is very close to the fixed accuracy.

3.4 Conclusion

Component 1: Which personality are more likely to partake in drugs rows = Countries Cols = Sensation seeking, amyl, benzos, cannabis, legalh, lst, mushrooms, VSA

Component 2: Different types of drugs Row = Countries Col = amyl, chocolate, VSA vs Amphet, coke, Crack, Ecstasy, Nicotine

Performance: Both seems stable and are around the same percentage. We should use this model because Fixed effect= around 57% Random effect = around 56%

Conclusion DiCA vs BADA are the same in terms of the accuracy of prediction. however, I would go with BADA due to the visualization of the graphs are easier to interpret.

Chapter 4 Multiple Correspondence Analysis

What is Multiple Correspondence Analysis(MCA)?

MCA is built upon principal component analysis, and an extension of the corresponding analysis. MCA is similar to PCA in such that we are looking at how each variable contributes to how much variance based on into nominal categorical (qualitative) data, particularly the mathematics side of it. MCA also uses binning and mass(row)/weight(column) similar to CA. Ultimately, we want to tease apart the data structure for the data and see which variables and how much are contributing to the variance.

MCA can look at qualitative or nominal data but can look at quantitative if needed. Qualitative variables are transformed into nominal data somewhat similar structure to dummy code (0/1). For example in gender, one column for male(1/0) another column for female(1/0). If there is quantitative data, we can transform it by binning it (look at the histogram and choose how much we should bin). Then, we put it in a Burt table (inner product of a design or indicator matrix, google to see picture) ie, individuals × variables matrix, where the rows represent individuals and the columns are dummy variables representing categories of the variables. Keep in mind that input data must be a contingency table.

Contingency table is a two-way table that summarize the relationship between several categorical variables. A contingency table is a special type of frequency distribution table, where two variables are shown simultaneously (look at infoStatisticHowTo for more info).

We then look at the chi-square distance between the different variables and the participants. We then maximize the variance to look at the dimensions. Specifically, uses chi-square distance to measure the distance among each data point (unlike the Elucidian distance from the barycenter from the PCA aka Pythagorean distance to best-fit line). This will change the inertia into being orthogonal components.

Chi-square has 2 important goals: - Variance between row and columns will be similar and will be both mapped together - Create factor scores for both columns and rows

Overall outline of MCA: - Follow PCA to find new components - Calculate chi-square based on CA to find distance between data points. - CA finds the inertia using generalized SVD to decompose chi-squared into orthogonal compononets. - CA projects row/column factor scores onto a new plane symmetric/aysmmetric plane plots. (These plots can help find whether rows/columns can be compared and how strong the association is)

Difference between MCA & CA:

• CA only looks at 2-dimension data while MCA looks at multiple dimension.

• CA looks at relative table(each column/variable not different levels within the column) while MCA looks at categorical data (all levels for each categorical variable)

• CA can normalize the different variables for interpretation while MCA we cannot and have to look at the raw data(This means we can compare row to row/col to col and row to column)

• MCA only uses symmetric maps while CA uses symmetric and asymmetric (only row to row/col to col. 

• MCA tends to over-estimate the eigen value compared to CA due to the awkward of binning the quantiative data.

For more info: Multiple Correspondence Analysis - Hervé Abdi & Dominique Valentin 2007

4.1 Data

Important to note: MCA only looks at qualitative data!

This data was collected from a great number of individuals. Data was looking at how different personalities is related to different types of drug consumption. Description of participants includes gender, age, ethnicity, and country.

Loading the drug data for MCA

#load drug data
load("DrugConsumption.RData")
df <- as.data.frame(data, stringsAsFactors = T)

#change age into a factor
df$AGE <- as.factor(df$AGE)
df$EDUCATION <- as.factor(df$EDUCATION)
df$COUNTRY <- as.factor(df$COUNTRY)
df$ETHNICITY <- as.factor(df$ETHNICITY)

#subset and select only qualitative data
qualitative_df <- df[,1:5]

#subset quant
quant_data <- data[,(6:31)]

personality <- data[,c(6:12)]
drugs <- data[,c(13:31)]

View the data

# a <- knitr::kable(df, format = "html")
# kableExtra::scroll_box(a, width = "500px", height = "500px", fixed_thead = T)

a <- df
dplyr::glimpse(a)

## Rows: 1,885
## Columns: 31
## $ AGE         <fct> 3, 2, 3, 1, 3, 6, 4, 3, 3, 5, 2, 4, 5, 5, 5, 5, 3, 4, 5, 3…
## $ GENDER      <fct> F, M, M, F, F, F, M, M, F, M, F, M, F, F, F, M, F, M, M, M…
## $ EDUCATION   <fct> 6, 9, 6, 8, 9, 4, 8, 2, 6, 8, 7, 5, 7, 6, 6, 7, 5, 2, 7, 6…
## $ COUNTRY     <fct> UK, UK, UK, UK, UK, CA, US, UK, CA, UK, UK, OT, UK, CA, UK…
## $ ETHNICITY   <fct> WA, WH, WH, WH, WH, WH, WH, WH, WH, WH, WH, WH, WH, WH, WH…
## $ NSCORE      <dbl> 28, 18, 20, 23, 32, 18, 20, 13, 31, 22, 15, 13, 45, 17, 16…
## $ ESCORE      <dbl> 20, 36, 29, 18, 12, 22, 16, 36, 39, 24, 29, 24, 25, 29, 33…
## $ OSCORE      <dbl> 17, 30, 15, 21, 18, 10, 18, 15, 14, 11, 13, 22, 24, 21, 24…
## $ ASCORE      <dbl> 18, 29, 13, 28, 22, 36, 22, 22, 29, 28, 19, 11, 13, 30, 20…
## $ CSCORE      <dbl> 25, 24, 17, 29, 33, 35, 31, 35, 32, 26, 36, 21, 19, 41, 35…
## $ IMPULSIVITY <dbl> 4, 3, 2, 2, 4, 2, 4, 5, 2, 2, 5, 6, 8, 3, 8, 3, 4, 2, 3, 2…
## $ SS          <dbl> 3, 6, 8, 3, 6, 2, 7, 5, 2, 4, 7, 10, 7, 4, 9, 5, 1, 4, 6, …
## $ ALCOHOL     <dbl> 6, 6, 7, 5, 5, 3, 7, 6, 5, 7, 6, 6, 6, 2, 7, 6, 7, 7, 7, 5…
## $ AMPHET      <dbl> 3, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 3, 1, 2, 3, 2…
## $ AMYL        <dbl> 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1…
## $ BENZOS      <dbl> 3, 1, 1, 4, 1, 1, 1, 1, 1, 2, 1, 1, 5, 1, 1, 1, 2, 1, 3, 1…
## $ CAFF        <dbl> 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7…
## $ CANNABIS    <dbl> 1, 5, 4, 3, 4, 1, 2, 1, 1, 2, 3, 5, 4, 1, 1, 2, 4, 7, 4, 2…
## $ CHOC        <dbl> 6, 7, 5, 5, 7, 5, 6, 5, 7, 7, 6, 6, 6, 1, 7, 6, 6, 5, 7, 7…
## $ COKE        <dbl> 1, 4, 1, 3, 1, 1, 1, 1, 1, 1, 3, 3, 2, 1, 1, 3, 1, 2, 3, 1…
## $ CRACK       <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
## $ ECSTASY     <dbl> 1, 5, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 2, 3, 2…
## $ HEROIN      <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
## $ KETAMINE    <dbl> 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1…
## $ LEGALH      <dbl> 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1…
## $ LSD         <dbl> 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1…
## $ METH        <dbl> 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 7…
## $ MUSHROOMS   <dbl> 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2, 2, 1…
## $ NICOTINE    <dbl> 3, 5, 1, 3, 3, 7, 7, 1, 7, 7, 3, 7, 7, 2, 7, 1, 7, 7, 1, 2…
## $ SEMER       <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
## $ VSA         <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1…

This data contains qualitative information such as gender,age, education, country, ethnicity. Also contains quantitative data such as personality score and how often they partake in drug. Each row is a candidate and each column is qualitative or quantitative scores. This data looks how how personality and drugs are related or drugs and drugs/ personality and personality are related. For this MCA, we will only look at the the nominal data such as the personality and the drugs

4.1.1 histogram binning

#histogram to know how much to bin
ggplot <- ggplot(melt(quant_data), aes(value)) + geom_histogram() + facet_wrap(~variable)

## No id variables; using all as measure variables

ggplot

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

We can bin based on eyeballing the histogram. For example, histogram that goes from low to high and then low, we bin as 3. For histogram that goes high and then low, we bin as 2.

Binning the data to fit to the MCA

#recode/bin quantitative data
## cluser from column 1 to 5 and bin at 4 based on histogram
data[, c(6:10)] <- data.frame(
    lapply(data[, c(6:10)], 
    function(x) Ckmedian.1d.dp(x, k = 5)$cluster), 
    row.names = rownames(data)) 

data[, c(11:12)] <- data.frame(
    lapply(data[, c(11:12)], 
    function(x) Ckmedian.1d.dp(x, k = 3)$cluster), 
    row.names = rownames(data))

data[, c(13:31)] <- data.frame(
    lapply(data[, c(13:31)], 
    function(x) Ckmedian.1d.dp(x, k = 2)$cluster), 
    row.names = rownames(data))

4.2 MCA analysis begins!

Selecting the numeric data only

data_2 <- data[,c(6:31)]

MCA function

resMCA <- epMCA(data_2,DESIGN = data$GENDER, graphs = FALSE) 

Let’s take a lookt a the nominal data

#recode and make it 0/1
nominal_data <- makeNominalData(data_2)

b <- knitr::kable(nominal_data, format = 'html')
kableExtra::scroll_box(b, width = "500px", height = "500px")

NSCORE.3	NSCORE.2	NSCORE.4	NSCORE.1	NSCORE.5	ESCORE.2	ESCORE.5	ESCORE.4	ESCORE.1	ESCORE.3	OSCORE.2	OSCORE.5	OSCORE.3	OSCORE.1	OSCORE.4	ASCORE.2	ASCORE.4	ASCORE.1	ASCORE.3	ASCORE.5	CSCORE.3	CSCORE.2	CSCORE.4	CSCORE.5	CSCORE.1	IMPULSIVITY.2	IMPULSIVITY.1	IMPULSIVITY.3	SS.1	SS.2	SS.3	ALCOHOL.2	ALCOHOL.1	AMPHET.2	AMPHET.1	AMYL.1	AMYL.2	BENZOS.2	BENZOS.1	CAFF.2	CAFF.1	CANNABIS.1	CANNABIS.2	CHOC.1	CHOC.2	COKE.1	COKE.2	CRACK.1	CRACK.2	ECSTASY.1	ECSTASY.2	HEROIN.1	HEROIN.2	KETAMINE.1	KETAMINE.2	LEGALH.1	LEGALH.2	LSD.1	LSD.2	METH.1	METH.2	MUSHROOMS.1	MUSHROOMS.2	NICOTINE.1	NICOTINE.2	SEMER.1	SEMER.2	VSA.1	VSA.2
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0	1	0	0	0	1	0	0	0	0	0	0	1	0	0	0	0	1	0	0	1	0	0	0	0	1	0	0	0	1	0	1	0	0	1	1	0	1	0	1	0	1	0	1	0	0	1	1	0	0	1	1	0	1	0	0	1	1	0	1	0	1	0	0	1	1	0	1	0
0	1	0	0	0	0	0	1	0	0	0	1	0	0	0	0	0	0	0	1	0	0	1	0	0	1	0	0	0	0	1	1	0	1	0	1	0	1	0	1	0	0	1	1	0	0	1	1	0	1	0	1	0	1	0	0	1	0	1	0	1	1	0	1	0	1	0	0	1
1	0	0	0	0	0	0	1	0	0	0	0	0	0	1	0	1	0	0	0	0	0	0	1	0	1	0	0	0	1	0	1	0	0	1	1	0	1	0	1	0	0	1	0	1	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0
0	0	0	1	0	0	1	0	0	0	0	1	0	0	0	0	1	0	0	0	0	0	0	1	0	0	0	1	0	0	1	1	0	0	1	1	0	0	1	0	1	1	0	1	0	1	0	1	0	1	0	1	0	1	0	0	1	0	1	1	0	0	1	0	1	1	0	1	0
0	0	0	0	1	1	0	0	0	0	0	1	0	0	0	1	0	0	0	0	0	0	0	0	1	0	0	1	0	0	1	1	0	1	0	1	0	1	0	0	1	0	1	0	1	0	1	1	0	0	1	1	0	0	1	0	1	0	1	0	1	0	1	0	1	1	0	1	0
0	0	1	0	0	1	0	0	0	0	0	0	0	0	1	0	1	0	0	0	0	0	0	1	0	0	0	1	0	0	1	0	1	1	0	1	0	1	0	1	0	0	1	1	0	0	1	0	1	0	1	0	1	1	0	0	1	1	0	1	0	0	1	0	1	1	0	1	0
0	0	0	1	0	1	0	0	0	0	1	0	0	0	0	1	0	0	0	0	0	0	1	0	0	0	1	0	0	1	0	0	1	1	0	1	0	1	0	1	0	0	1	0	1	1	0	1	0	1	0	1	0	1	0	0	1	1	0	1	0	1	0	0	1	1	0	1	0
0	0	0	0	1	0	0	0	1	0	0	1	0	0	0	0	1	0	0	0	0	1	0	0	0	1	0	0	1	0	0	0	1	1	0	1	0	1	0	1	0	1	0	0	1	1	0	1	0	1	0	1	0	1	0	0	1	0	1	0	1	1	0	1	0	1	0	1	0
1	0	0	0	0	0	0	1	0	0	0	0	1	0	0	0	0	0	1	0	0	0	0	0	1	0	0	1	0	0	1	1	0	1	0	1	0	0	1	1	0	1	0	1	0	0	1	0	1	0	1	1	0	1	0	1	0	0	1	1	0	0	1	0	1	1	0	0	1
0	0	1	0	0	0	0	1	0	0	0	0	1	0	0	1	0	0	0	0	0	0	1	0	0	1	0	0	0	1	0	1	0	1	0	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1	0	1	0	1	0	0	1	0	1	1	0	1	0
1	0	0	0	0	0	0	0	0	1	1	0	0	0	0	0	0	0	1	0	0	0	1	0	0	1	0	0	0	0	1	1	0	0	1	1	0	0	1	0	1	0	1	1	0	1	0	1	0	0	1	1	0	0	1	0	1	0	1	1	0	0	1	0	1	1	0	1	0
0	1	0	0	0	0	0	0	1	0	1	0	0	0	0	0	1	0	0	0	1	0	0	0	0	1	0	0	1	0	0	0	1	1	0	1	0	0	1	1	0	1	0	1	0	0	1	0	1	0	1	1	0	1	0	0	1	0	1	0	1	1	0	0	1	1	0	1	0
1	0	0	0	0	0	1	0	0	0	0	0	1	0	0	0	0	1	0	0	0	0	1	0	0	1	0	0	0	0	1	1	0	0	1	1	0	0	1	0	1	0	1	1	0	1	0	1	0	0	1	1	0	1	0	0	1	0	1	1	0	1	0	1	0	1	0	1	0
0	0	0	1	0	0	0	0	0	1	0	1	0	0	0	0	0	0	1	0	1	0	0	0	0	0	0	1	0	0	1	0	1	1	0	1	0	1	0	1	0	0	1	1	0	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	1	0	1	0
1	0	0	0	0	0	0	0	0	1	0	0	0	0	1	0	0	0	1	0	0	0	0	0	1	0	0	1	0	0	1	1	0	1	0	1	0	0	1	1	0	1	0	0	1	0	1	0	1	0	1	1	0	1	0	1	0	0	1	1	0	0	1	0	1	1	0	0	1
0	0	0	1	0	0	1	0	0	0	0	0	0	0	1	0	0	0	1	0	1	0	0	0	0	0	1	0	0	0	1	0	1	0	1	1	0	0	1	0	1	0	1	0	1	1	0	1	0	0	1	1	0	1	0	1	0	0	1	1	0	0	1	1	0	1	0	1	0
1	0	0	0	0	1	0	0	0	0	0	0	1	0	0	0	0	1	0	0	0	1	0	0	0	0	1	0	0	0	1	0	1	1	0	1	0	1	0	1	0	0	1	1	0	0	1	1	0	0	1	1	0	1	0	0	1	0	1	1	0	1	0	0	1	1	0	0	1
0	0	0	0	1	0	0	1	0	0	0	1	0	0	0	0	0	0	1	0	0	0	0	0	1	0	0	1	0	0	1	1	0	1	0	0	1	1	0	1	0	0	1	1	0	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	1	0	1	0	0	1
0	1	0	0	0	0	0	1	0	0	0	0	0	0	1	0	0	0	1	0	1	0	0	0	0	0	0	1	0	0	1	0	1	1	0	1	0	0	1	1	0	0	1	0	1	1	0	1	0	0	1	1	0	0	1	1	0	0	1	0	1	0	1	0	1	1	0	1	0
0	1	0	0	0	0	0	0	0	1	0	0	0	0	1	0	0	0	1	0	1	0	0	0	0	1	0	0	0	0	1	1	0	1	0	1	0	1	0	1	0	0	1	1	0	0	1	0	1	0	1	0	1	1	0	0	1	0	1	0	1	0	1	0	1	1	0	1	0
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0	1	0	0	0	0	0	0	0	1	0	1	0	0	0	0	0	0	1	0	0	0	1	0	0	0	0	1	0	0	1	1	0	1	0	1	0	1	0	1	0	1	0	0	1	1	0	1	0	1	0	1	0	1	0	0	1	1	0	0	1	0	1	0	1	1	0	1	0
0	0	1	0	0	0	0	0	0	1	0	0	0	0	1	1	0	0	0	0	0	0	1	0	0	0	0	1	0	0	1	1	0	0	1	1	0	1	0	1	0	0	1	0	1	0	1	0	1	1	0	0	1	1	0	1	0	1	0	0	1	1	0	0	1	1	0	1	0
0	0	1	0	0	0	0	1	0	0	0	1	0	0	0	1	0	0	0	0	0	0	1	0	0	0	0	1	0	0	1	1	0	1	0	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1	0	1	0	0	1	0	1	0	1	0	1	0	1	1	0	1	0
0	0	0	1	0	0	0	0	0	1	0	1	0	0	0	1	0	0	0	0	0	0	1	0	0	1	0	0	0	0	1	1	0	1	0	1	0	1	0	0	1	0	1	1	0	0	1	0	1	0	1	0	1	0	1	0	1	0	1	1	0	0	1	0	1	1	0	0	1
1	0	0	0	0	0	0	0	0	1	0	1	0	0	0	1	0	0	0	0	0	0	0	0	1	0	0	1	0	0	1	1	0	0	1	1	0	1	0	1	0	0	1	1	0	0	1	1	0	0	1	1	0	1	0	0	1	0	1	0	1	0	1	0	1	1	0	1	0
1	0	0	0	0	1	0	0	0	0	0	1	0	0	0	0	0	0	1	0	0	0	0	1	0	0	1	0	0	1	0	1	0	0	1	1	0	0	1	1	0	0	1	0	1	1	0	1	0	0	1	1	0	1	0	0	1	0	1	1	0	0	1	0	1	1	0	1	0
0	0	0	0	1	0	0	0	0	1	0	0	0	0	1	1	0	0	0	0	1	0	0	0	0	0	0	1	0	1	0	1	0	0	1	0	1	1	0	1	0	0	1	1	0	0	1	0	1	0	1	0	1	1	0	0	1	0	1	1	0	0	1	0	1	1	0	1	0
0	1	0	0	0	0	0	0	0	1	0	0	0	0	1	0	0	1	0	0	0	0	0	0	1	1	0	0	0	0	1	0	1	1	0	0	1	0	1	1	0	0	1	1	0	1	0	1	0	0	1	1	0	1	0	0	1	1	0	1	0	0	1	0	1	1	0	1	0
1	0	0	0	0	0	1	0	0	0	0	1	0	0	0	0	0	0	1	0	1	0	0	0	0	1	0	0	0	0	1	0	1	1	0	0	1	1	0	0	1	0	1	1	0	0	1	1	0	0	1	1	0	1	0	0	1	0	1	0	1	0	1	0	1	1	0	1	0
0	0	0	1	0	0	0	1	0	0	0	1	0	0	0	0	0	0	0	1	0	0	0	1	0	0	1	0	0	1	0	1	0	1	0	1	0	0	1	1	0	0	1	1	0	1	0	1	0	0	1	1	0	1	0	1	0	0	1	1	0	1	0	0	1	1	0	1	0
0	0	0	0	1	0	0	0	0	1	0	1	0	0	0	1	0	0	0	0	1	0	0	0	0	0	1	0	0	0	1	1	0	1	0	1	0	1	0	1	0	0	1	1	0	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	1	0	0	1	1	0	0	1
0	1	0	0	0	0	0	0	0	1	0	1	0	0	0	0	1	0	0	0	0	0	0	0	1	0	0	1	0	0	1	1	0	1	0	1	0	0	1	0	1	0	1	1	0	1	0	1	0	0	1	1	0	0	1	0	1	1	0	1	0	1	0	1	0	1	0	1	0
0	0	1	0	0	0	0	0	1	0	0	0	0	0	1	1	0	0	0	0	1	0	0	0	0	0	0	1	0	0	1	1	0	1	0	1	0	1	0	1	0	0	1	1	0	0	1	0	1	0	1	0	1	1	0	0	1	0	1	0	1	0	1	0	1	1	0	0	1
0	0	0	0	1	0	0	0	0	1	0	1	0	0	0	0	1	0	0	0	0	1	0	0	0	1	0	0	0	1	0	1	0	0	1	1	0	1	0	1	0	1	0	0	1	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0
1	0	0	0	0	0	0	0	1	0	0	0	0	0	1	1	0	0	0	0	1	0	0	0	0	0	0	1	0	0	1	1	0	0	1	1	0	0	1	0	1	0	1	1	0	1	0	1	0	0	1	1	0	1	0	0	1	0	1	1	0	0	1	0	1	1	0	0	1
1	0	0	0	0	1	0	0	0	0	0	1	0	0	0	1	0	0	0	0	0	1	0	0	0	0	0	1	0	0	1	1	0	0	1	1	0	0	1	0	1	0	1	1	0	0	1	1	0	0	1	1	0	0	1	0	1	0	1	1	0	0	1	1	0	1	0	0	1
0	0	0	0	1	0	0	0	1	0	1	0	0	0	0	0	0	1	0	0	0	0	1	0	0	1	0	0	0	0	1	1	0	1	0	1	0	1	0	1	0	0	1	1	0	0	1	1	0	0	1	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1
1	0	0	0	0	0	0	0	1	0	0	1	0	0	0	1	0	0	0	0	0	0	0	1	0	0	1	0	0	1	0	0	1	0	1	1	0	0	1	0	1	0	1	1	0	1	0	1	0	1	0	1	0	1	0	0	1	1	0	1	0	0	1	1	0	1	0	1	0
0	1	0	0	0	0	0	1	0	0	0	0	0	0	1	0	0	0	1	0	1	0	0	0	0	0	0	1	0	0	1	0	1	1	0	1	0	1	0	0	1	0	1	1	0	1	0	1	0	0	1	0	1	1	0	0	1	0	1	0	1	1	0	0	1	1	0	1	0
0	0	0	1	0	0	1	0	0	0	0	1	0	0	0	0	1	0	0	0	0	0	0	0	1	0	0	1	0	0	1	1	0	0	1	1	0	0	1	0	1	0	1	1	0	1	0	1	0	1	0	1	0	1	0	0	1	0	1	1	0	1	0	1	0	1	0	0	1
0	1	0	0	0	0	1	0	0	0	0	0	0	0	1	0	1	0	0	0	0	0	0	0	1	0	0	1	0	0	1	1	0	0	1	1	0	0	1	0	1	1	0	1	0	1	0	1	0	0	1	1	0	1	0	0	1	0	1	0	1	0	1	0	1	1	0	1	0
0	0	0	0	1	0	0	0	1	0	0	0	0	1	0	0	0	1	0	0	0	0	0	0	1	0	0	1	0	1	0	0	1	1	0	0	1	1	0	1	0	0	1	0	1	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	0	1	1	0	1	0
0	0	1	0	0	0	0	0	1	0	0	0	0	0	1	0	0	1	0	0	0	0	0	0	1	0	0	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1	0	1	0	1	0	0	1	1	0	1	0	0	1	0	1	1	0	0	1	0	1	1	0	1	0
0	1	0	0	0	0	1	0	0	0	0	1	0	0	0	0	1	0	0	0	0	0	1	0	0	0	0	1	0	0	1	0	1	1	0	1	0	1	0	1	0	1	0	0	1	0	1	1	0	0	1	1	0	1	0	0	1	0	1	1	0	0	1	0	1	1	0	0	1

inference MCA

resMCA.inf <- epMCA.inference.battery(data_2, DESIGN = data$GENDER, 
                                      graphs = FALSE)

## [1] "It is estimated that your iterations will take 0.09 minutes."
## [1] "R is not in interactive() mode. Resample-based tests will be conducted. Please take note of the progress bar."
## ================================================================================

4.2.1 Scree plot

Scree plot with significance

scree.mca <- PlotScree(ev = resMCA$ExPosition.Data$eigs, 
                       p.ev = resMCA.inf$Inference.Data$components$p.vals,
                       plotKaiser = T,
            title = "MCA. Explained Variance per Dimension")

b0001a.Scree <- recordPlot() # Save the plot

The first 7 dimensions are significant. Later on I will show all 2 dimensions but really the first 1 is the most important because it contributes to the most variance. Anything after the 1st dimensions contributes very little and it’s very likely that dimension 2 and afters show the same information.

4.2.2 Correlation/heat map plot

Computed and extracted from the Burt table. Burt table is where the columns have 1/0 for each variable and then computed based on the barycenter

#getting colors
cJ <- resMCA$ExPosition.Data$cj
color4Var <- prettyGraphs::prettyGraphsColorSelection(
                                          ncol(data), starting.color = 15)

# Pseudo Heat Map. Correlation ----
# We need correlation to compare with PCA
corrMatBurt.list <- phi2Mat4BurtTable(data_2)
col <- colorRampPalette(
  c("#BB4444", "#EE9988", "#FFFFFF", "#77AADD", "#4477AA"))

corr4MCA.r <- corrplot::corrplot(
         as.matrix(corrMatBurt.list$phi2.mat^(1/2)), 
         method="color", col=col(200),  
         type="upper", 
         tl.col = color4Var, 
         tl.cex = .9,
         tl.srt = 45, #Text label color and rotation
         number.cex = .8,
         diag = F # needed to have the color of variables correct
         )

# dev.new()
a0000.corMat.phi <- recordPlot()

It seems like Education and age have somewhat high correlation compared to others. Same thing for Gender and Education. Furthermore, a lot of the drugs are correlated like Coke and Ecstasy.

Getting the color of the plots

col4X <- prettyGraphsColorSelection(
                   n.colors = ncol(personality),
                   starting.color = 42)

col4Y <- prettyGraphsColorSelection(
                    n.colors = ncol(drugs),
                    starting.color = 13)
col4Var = rbind(col4X,col4Y)

4.2.3 Dimension plots for each column

#get the dimension data for each cell
varCtr <- data4PCCAR::ctr4Variables(cJ) 
rownames(col4Var) <- rownames(varCtr)

Dimension 1 plot

nFact <- min(5, ncol(cJ) - 1)

laTable <- round(varCtr[,1:nFact]*1000)

col4Levels <- data4PCCAR::coloringLevels(
           rownames(resMCA$ExPosition.Data$fj), col4Var)
col4Labels <- col4Levels$color4Levels

varCtr1 <- varCtr[,1]
names(varCtr1) <- rownames(varCtr)
a0005.Var.ctr1  <- PrettyBarPlot2(varCtr1,
              main = 'Variable Contributions: Dimension 1',
              ylim = c(-.05, 1.2*max(varCtr1)),
              font.size = 5,
              threshold = 1 / nrow(varCtr),
                                color4bar = gplots::col2hex(col4Var),
              horizontal = F
)
print(a0005.Var.ctr1)

It seems like Sensation seeking is related to most of the drugs for dimension 1.

Dimension 2

varCtr2 <- varCtr[,2]
names(varCtr2) <- rownames(varCtr)
a0006.Var.ctr2  <- PrettyBarPlot2(varCtr2,
                    main = 'Variable Contributions: Dimension 2',
                    ylim = c(-.05, 1.2*max(varCtr2)),
                    threshold = 1 / nrow(varCtr),
                    font.size = 5,
                    color4bar = gplots::col2hex(col4Var),
                     horizontal = F
)
print(a0006.Var.ctr2)

It seems like for dimension 2, neuroticism extravert openness agreeableness and conscientiousness is relate to heroin and meth. This plot seems to say that if these personalities were to participate in drugs, they are likely to participate in heroin and meth.

#combine all graphs
 grid.arrange(
    as.grob(a0005.Var.ctr1),
    as.grob(a0006.Var.ctr2),
    ncol = 1,nrow = 2,
    top = textGrob("Barplots for variables", gp = gpar(fontsize = 25, font = 3))
  )

barplot_contri_barplot<- recordPlot() # you need this line to be able to save them in the end

It seems like heroin and ecstasy are drugs that every personality will partake in.

4.2.4 Contribution plots for each column

4.2.4.1 Contribution plot 1 and 2 for Columns

ctrV12 <- PTCA4CATA::createFactorMap(X =  varCtr, 
                        title = "Variable Contributions", 
                        col.points = col4Var,
                        col.labels = col4Var,
                        alpha.points = 0.5,
                        cex = 2.5, 
                        alpha.labels = 1, 
                        text.cex = 4,
                        font.face = "plain", 
                        font.family = "sans")

ctr.labels <- createxyLabels.gen(
  1,2, lambda = resMCA$ExPosition.Data$eigs,
  tau = resMCA$ExPosition.Data$t)

a0007.Var.ctr12  <- ctrV12$zeMap  + ctr.labels
#
print(a0007.Var.ctr12)

This is very messy and difficult to interpret. Maybe we should only look at significant variables.

4.2.4.2 Contribution plot significance dimension 1 and 2 for Columns

absCtrVar <- as.matrix(varCtr) %*% 
                    diag(resMCA$ExPosition.Data$eigs)
varCtr12  <- (absCtrVar[,1] + absCtrVar[,2]) / 
                (resMCA$ExPosition.Data$eigs[1] + 
                resMCA$ExPosition.Data$eigs[2])
importantVar <- (varCtr12 >=  1 / length(varCtr12))
col4ImportantVar <- col4Var
col4NS <- 'gray90' 
col4ImportantVar[!importantVar] <- col4NS

ctrV12.imp <- PTCA4CATA::createFactorMap(X =  varCtr, 
             title = "Important Variables: Contributions", 
             col.points = col4ImportantVar,
             col.labels = col4ImportantVar,
             alpha.points = 0.5,
                        cex = 2.5, 
                        alpha.labels = 1, 
                        text.cex = 4,
                        font.face = "plain", 
                        font.family = "sans")
a0008.Var.ctr12.imp  <- ctrV12.imp$zeMap  + ctr.labels
#
print(a0008.Var.ctr12.imp)

This looks way batter. We can see that most if not all party drugs like LSD & Mushroom are related to sensation seeking.

4.2.4.3 Contribution signifiance plot dimension 3 and 2

varCtr23  <- (absCtrVar[,3] + absCtrVar[,2]) / 
   (resMCA$ExPosition.Data$eigs[3] + resMCA$ExPosition.Data$eigs[2])
importantVar23 <- (varCtr23 >=  1 / length(varCtr23))
col4ImportantVar23 <- col4Var
col4NS <- 'gray90' 
col4ImportantVar23[!importantVar23] <- col4NS

## ----ctrV23.ns-----------------------------------------------------------
ctrV23.imp <- PTCA4CATA::createFactorMap(X =  varCtr,
                                         axis1 = 3, axis2 = 2,
                        title = "Important Variables: Contributions 3 * 2", 
                        col.points = col4ImportantVar23,
                        col.labels = col4ImportantVar23,
                        alpha.points = 0.5,
                        cex = 2.5, 
                        alpha.labels = 1, 
                        text.cex = 4,
                        font.face = "plain", 
                        font.family = "sans")
ctr.labels23 <- createxyLabels.gen(
  3,2, lambda = resMCA$ExPosition.Data$eigs,
  tau = resMCA$ExPosition.Data$t
)
a0009.Var.ctr23.imp  <- ctrV23.imp$zeMap  + ctr.labels23
#
print(a0009.Var.ctr23.imp)

## Warning: ggrepel: 16 unlabeled data points (too many overlaps). Consider
## increasing max.overlaps

For dimensions 3 and 2, we can see that heroine is related to all personality. Heroine is a type of drug everyone will likely to partake in.

4.2.5 Bootstrap ratio for each columns

set.seed(25)

The bootstrap ratio plot currently does not work right now

4.2.5.1 Bootstrap ratio for dimension 1,2,3

# Get the pseudo Bootstrap Ratios
## currently does not work 
BrLevels <- resMCA.inf$Inference.Data$fj.boots$tests$boot.ratios
wJ       <- 1 / resMCA.inf$Fixed.Data$ExPosition.Data$W
wJnIter    <- 100
Br4Variables <- data4PCCAR::BR4varMCA(BrLevels, wJ, nIter) 

## ----BR41----------------------------------------------------------------
VarBR1 <- Br4Variables$pseudoBR.pos[,1]
c0010.Var.br1  <- PrettyBarPlot2(VarBR1,
    main = 'Variable Pseudo Bootstrap Ratios: Dimension 1',
    ylim = 2,
    threshold = 2,
    font.size = 2,
     color4bar = gplots::col2hex(col4Var),
    horizontal = F,
)

VarBR2 <- Br4Variables$pseudoBR.pos[,2]
c0011.Var.br2  <- PrettyBarPlot2(VarBR2,
   main = 'Variable Pseudo Bootstrap Ratios: Dimension 2',
   ylim = 2,
  threshold = 2,
  font.size = 2,
  color4bar = gplots::col2hex(col4Var),
  horizontal = F
)

VarBR3 <- Br4Variables$pseudoBR.pos[,3]
c0012.Var.br3  <- PrettyBarPlot2(VarBR3,
  main = 'Variable Pseudo Bootstrap Ratios: Dimension 3',
  ylim = 2,
  threshold = 2,
  font.size = 2,
  color4bar = gplots::col2hex(col4Var),
  horizontal = F
)

#combine all graphs
 grid.arrange(
    as.grob(c0010.Var.br1),
    as.grob(c0011.Var.br2),
    as.grob(c0012.Var.br3),
    ncol = 1,nrow = 3,
    top = textGrob("Barplots for variables", gp = gpar(fontsize = 25, font = 3))
  )

barplot_latent_1<- recordPlot() # you need this line to be able to save them in the end

This test is to see whether the barplots was reliable and it seem like it is. It seems like it is based on our contribution barplots.

4.2.6 Biplot for different levels per category

4.2.6.1 Biplot for dimension 1 and 2

axis1 = 1
axis2 = 2
Fj <- resMCA$ExPosition.Data$fj
# generate the set of maps
BaseMap.Fj <- createFactorMap(X = Fj , # J- Factor scores,
                              axis1 = axis1, axis2 = axis2,
                              title = 'MCA. Variables', 
                              col.points = col4Labels, 
                              cex = 1,
                              col.labels = col4Labels, 
                              text.cex = 2.5,
                              force = 2)
# add labels
labels4MCA <- createxyLabels.gen(x_axis = axis1, 
                                 y_axis = axis2,
               lambda = resMCA$ExPosition.Data$eigs,
               tau = resMCA$ExPosition.Data$t)
# make the maps
b0002.BaseMap.Fj <- BaseMap.Fj$zeMap + labels4MCA 
b0003.BaseMapNoDot.Fj  <- BaseMap.Fj$zeMap_background +
                         BaseMap.Fj$zeMap_text + labels4MCA 

## ----basemap Fj ----
# print(b0002.BaseMap.Fj)

col4Levels.imp <- data4PCCAR::coloringLevels(rownames(Fj),
                       col4ImportantVar)
BaseMap.Fj.imp <- createFactorMap(X = Fj, # Fj
                  axis1 = axis1, axis2 = axis2,
                  title = 'MCA. Important Variables', 
                  col.points = col4Levels.imp$color4Levels, 
                  cex = 1,
                  col.labels = col4Levels.imp$color4Levels, 
                  text.cex = 2.5,
                  force = 2)

b0010.BaseMap.Fj <- BaseMap.Fj.imp$zeMap + labels4MCA 
print(b0010.BaseMap.Fj)

This map shows the different levels of each categorical/nominal data.This significance only graph shows that for dimension 1 and 2, It seems like all levels of sensation seeking is involved in all levels of coke, legal h, meth, ecstasy, ketamine , lsd and heroin.

lines4J <- addLines4MCA(Fj, 
                  col4Var = col4Levels.imp$color4Variables, 
                  size = .7,
                  linetype = 1,
                  alpha = .4)
 b0020.BaseMap.Fj <-  b0010.BaseMap.Fj + lines4J
 print( b0020.BaseMap.Fj)

Graphs showing the directions of the columns per category. Very messing, but I appreciate the graph.

zeNames          <- getVarNames(rownames(Fj)) 
importantsLabels <- zeNames$stripedNames %in% zeNames$variableNames[importantVar]
Fj.imp <- Fj[importantsLabels,]
lines4J.imp <- addLines4MCA(Fj.imp, 
            col4Var = col4Levels$color4Variables[which(importantVar)], 
            size = .9, linetype = 3, alpha = .5)
 b0021.BaseMap.Fj <-  b0020.BaseMap.Fj + lines4J.imp
 print( b0021.BaseMap.Fj)

Same graph as above but the lines connecting are dotted now. But still messy.

4.2.6.2 Biplot for dimension 3 and 2

col4Levels23.imp <- data4PCCAR::coloringLevels(rownames(Fj),
                                     col4ImportantVar23)
axis3 = 3
BaseMap.Fj23.imp <- createFactorMap(X = Fj , # resMCA$ExPosition.Data$fj,
                              axis1 = axis3, axis2 = axis2,
                              title = 'MCA. Important Variables. Dimensions 2 & 3', 
                      col.points = col4Levels23.imp$color4Levels, 
                              cex = 1,
                      col.labels = col4Levels23.imp$color4Levels, 
                              text.cex = 2.5,
                              force = 2)
labels4MCA23 <- createxyLabels.gen(x_axis = axis1, y_axis = axis2,
               lambda = resMCA$ExPosition.Data$eigs,
               tau = resMCA$ExPosition.Data$t)
b0030.BaseMap.Fj23 <- BaseMap.Fj23.imp$zeMap + labels4MCA23 

# zeNames          <- getVarNames(rownames(Fj)) 
importantsLabels23 <- zeNames$stripedNames %in% zeNames$variableNames[importantVar23]
Fj23.imp <- Fj[importantsLabels23,]
lines4J23.imp <- addLines4MCA(Fj23.imp, 
                    col4Var = col4Levels$color4Variables[
                               which(importantVar23)],
                    axis_h = axis3,
                    axis_v = axis2,
                    size = .9, linetype = 1, alpha = .4)
 b0031.BaseMap.Fj23 <-  b0030.BaseMap.Fj23 + lines4J23.imp
 print( b0031.BaseMap.Fj23)

## Warning: ggrepel: 47 unlabeled data points (too many overlaps). Consider
## increasing max.overlaps

Dimension 1 = dimension 3, dimension 2 = dimension 2. This is really messy, but we can see that different levels of drugs are related to different levels of personality. For example, we can see that the different levels of personalities are related to heroin. Particularly, which levels of each personlity is related to which level of heroin intake.

4.2.7 Bootstrap columns (different levels per category)

4.2.7.1 Bootstrap dimension 1

c0001.Levels.BR  <- PrettyBarPlot2(
  resMCA.inf$Inference.Data$fj.boots$tests$boot.ratios[,1], # BR
  main = 'Bootstrap Ratios for Columns : Dimension 1',
                             threshold = 2,
  color4bar = gplots::col2hex(col4Labels),
  horizontal = F,
  signifOnly = T
)
print(c0001.Levels.BR)

This shows that the different levels of each personality related to different levels of each drug intake. We do have to note that this is a reliability test to see if it is similar to the biplots.

4.2.7.2 Bootstrap dimension 2

c0001.Levels.BR_2  <- PrettyBarPlot2(
  resMCA.inf$Inference.Data$fj.boots$tests$boot.ratios[,2], # BR
  main = 'Bootstrap Ratios for Columns : Dimension 2',
                             threshold = 2,
  color4bar = gplots::col2hex(col4Labels),
  horizontal = F,
  signifOnly = T
)
print(c0001.Levels.BR_2)

4.2.7.3 Bootstrap dimension 3

c0001.Levels.BR_3  <- PrettyBarPlot2(
  resMCA.inf$Inference.Data$fj.boots$tests$boot.ratios[,3], # BR
  main = 'Bootstrap Ratios for Columns : Dimension 3',
                             threshold = 2,
  color4bar = gplots::col2hex(col4Labels),
  horizontal = F,
  signifOnly = T
)
print(c0001.Levels.BR_3)

4.2.8 Biplot Based on country

Fi <- resMCA$ExPosition.Data$fi
colCity <- c('indianred4', #UK
             'gold', #CA
             'lightpink2', #US
             'orange', #OT
             'Blue', #AU
             'green', #IE
             'purple' #NZ
             )
#We can find the order of the countries using unique(qualitative_df[,'COUNTRY'])

nI <- nrow(Fi)
col4I.City <- rep("",nI)

for (i in 1:length(colCity) ){
  lindex <- qualitative_df[,'COUNTRY'] %in% unique(qualitative_df[,'COUNTRY'])[i]
  col4I.City[lindex] <- colCity[i]
}
# generate the set of maps
BaseMap.Fi <- createFactorMap(X = Fi , # resMCA$ExPosition.Data$fj,
                        axis1 = axis1, axis2 = axis2,
                        title = 'MCA. Observations (by country)', 
                        col.points = col4I.City,
                        alpha.points = .08, 
                        cex = .9,
                        col.labels = col4I.City,
                        text.cex = 2.5, 
                        force = 2)
# make the maps
d0001.BaseMapNoLabels.Fi  <- BaseMap.Fi$zeMap_background +
                                 BaseMap.Fi$zeMap_dots + 
                                 labels4MCA 

## ----plotaMapi, fig.width= 8---------------------------------------------
# print(d0001.BaseMapNoLabels.Fi)

4.2.8.1 Biplot with country means

BootCube.Gr <- PTCA4CATA::Boot4Mean(resMCA$ExPosition.Data$fi, 
                                 design = qualitative_df$COUNTRY,
                                 niter = 100,
                                 suppressProgressBar = TRUE)
countryMeans <- PTCA4CATA::getMeans(resMCA$ExPosition.Data$fi, qualitative_df$COUNTRY)

col4Mean_recode <- dplyr::recode(rownames(countryMeans),
                     AU = 'Blue',
                     CA = 'gold',
                     IE = 'green',
                     NZ = 'purple',
                     OT = 'orange',
                     UK = 'indianred4',
                     US = 'lightpink2'
                     )
names(col4Mean_recode) <- rownames(countryMeans)

# colCity <- c('darkblue', 'red4')
MapGroup <- PTCA4CATA::createFactorMap(countryMeans,
                            # use the constraint from the main map
                            constraints = BaseMap.Fi$constraints,
                            col.points = col4Mean_recode,
                            cex = 7,  # size of the dot (bigger)
                            col.labels = col4Mean_recode,
                            text.cex = 6)
d002.Map.I.withMeans <- d0001.BaseMapNoLabels.Fi  +
                          MapGroup$zeMap_dots + MapGroup$zeMap_text
print(d002.Map.I.withMeans)

It seems like country is based mostly on dimension 1. UK (left side gray colored) is very different compared to all countries. UK and US/NZ contribute most to Dimension 1. Canada and Ireland are very similar as that they overlap.

4.2.8.2 Biplot with country confidence interval

GraphElli <- PTCA4CATA::MakeCIEllipses(BootCube.Gr$BootCube[,1:2,],
                            names.of.factors = c("Dimension 1","Dimension 2"),
                            col = col4Mean_recode,
                            p.level = .95)
d003.Map.I.withCI <-  d0001.BaseMapNoLabels.Fi + 
                          MapGroup$zeMap_text +  GraphElli
print(d003.Map.I.withCI)

It seems like the confidence interval for NZ is very large compared to UK and US. This may suggest that NZ and UK are not that different. However, UK compared to the other countries are different because the confidence intervals do not overlap.

4.2.8.3 Tolerance Intervall Biplot

GraphTI.Hull <- PTCA4CATA::MakeToleranceIntervals(resMCA$ExPosition.Data$fi,
                            design = as.factor(qualitative_df$COUNTRY),
                            # line below is needed
                            names.of.factors =  c("Dim1","Dim2"), # needed 
                            col = col4Mean_recode,
                            line.size = .50, 
                            line.type = 3,
                            alpha.ellipse = .2,
                            alpha.line    = .4,
                            p.level       = .75)

d005.Map.I.withTIHull <- d002.Map.I.withMeans  +
                           GraphTI.Hull + MapGroup$zeMap_dots +
                           MapGroup$zeMap_text + MapGroup$zeMap_dots

# plot it
# dev.new()
print(d005.Map.I.withTIHull)

Tolerance interval: a certain proportion of population falls in.It seems like they all kind of overlap in a sense.

4.3 Conclusion

Dimension 1 and 2 are contributing to the most variance. Therefore, it makes more sense to analyze these two.

We can see that MCA is similar to PCA.

Dimension 1: Tells us which personality is more likely to participate in drugs.

Dimension 2: Differentiate which personality is more likely to participate in which drugs.

Chapter 5 Discriminant Correspondence Analysis (DiCA) on Drug Consumption

DiCA performs MCA on the sum of the group means based on nominal/qualitative dataset. It is similar to supervised machine learning algorithm that classifies observations based on a-priori groups. DiCA first creates a new space using Sum of Observation Groups (rows) based on group, then projects the observations and variables into that same space to see which classification would fit best.

Similar to MCA, the data needs to be binned and disjunticvely coded before the analysis.

More info: DiCA and BADA are similar, the only difference is the type of data that is being analyzed.

The math behind DiCA is similar to MCA. The only difference is that DiCA uses the summation of counts from the disjunctively coded data (similar to dummy coding).

Overview: - Perform MCA - Perform DiCA - check under/overfitting

DiCa is very similar to BADA. We will be using the Drug Consumption data for this analysis. Our design will be based on different countries (we will have a group mean for each country).

5.1 Load the data

function to use later

load data

load("DrugConsumption.RData")
qualitative <- data[,1:5]
personality <- data[,6:12]
drugs <- data[,13:31]
country <- data [,"COUNTRY"]
qualitative_without_country <- qualitative[,-4]
personalityXdrug <- data[,6:31]
personalityXdrug_no_country <- data[,-4] 

# a <- knitr::kable(data, format = "html")
# kableExtra::scroll_box(a, width = "500px", height = "500px", fixed_thead = T)

a <- data
dplyr::glimpse(a)

## Rows: 1,885
## Columns: 31
## $ AGE         <dbl> 3, 2, 3, 1, 3, 6, 4, 3, 3, 5, 2, 4, 5, 5, 5, 5, 3, 4, 5, 3…
## $ GENDER      <fct> F, M, M, F, F, F, M, M, F, M, F, M, F, F, F, M, F, M, M, M…
## $ EDUCATION   <dbl> 6, 9, 6, 8, 9, 4, 8, 2, 6, 8, 7, 5, 7, 6, 6, 7, 5, 2, 7, 6…
## $ COUNTRY     <fct> UK, UK, UK, UK, UK, CA, US, UK, CA, UK, UK, OT, UK, CA, UK…
## $ ETHNICITY   <fct> WA, WH, WH, WH, WH, WH, WH, WH, WH, WH, WH, WH, WH, WH, WH…
## $ NSCORE      <dbl> 28, 18, 20, 23, 32, 18, 20, 13, 31, 22, 15, 13, 45, 17, 16…
## $ ESCORE      <dbl> 20, 36, 29, 18, 12, 22, 16, 36, 39, 24, 29, 24, 25, 29, 33…
## $ OSCORE      <dbl> 17, 30, 15, 21, 18, 10, 18, 15, 14, 11, 13, 22, 24, 21, 24…
## $ ASCORE      <dbl> 18, 29, 13, 28, 22, 36, 22, 22, 29, 28, 19, 11, 13, 30, 20…
## $ CSCORE      <dbl> 25, 24, 17, 29, 33, 35, 31, 35, 32, 26, 36, 21, 19, 41, 35…
## $ IMPULSIVITY <dbl> 4, 3, 2, 2, 4, 2, 4, 5, 2, 2, 5, 6, 8, 3, 8, 3, 4, 2, 3, 2…
## $ SS          <dbl> 3, 6, 8, 3, 6, 2, 7, 5, 2, 4, 7, 10, 7, 4, 9, 5, 1, 4, 6, …
## $ ALCOHOL     <dbl> 6, 6, 7, 5, 5, 3, 7, 6, 5, 7, 6, 6, 6, 2, 7, 6, 7, 7, 7, 5…
## $ AMPHET      <dbl> 3, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 3, 1, 2, 3, 2…
## $ AMYL        <dbl> 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1…
## $ BENZOS      <dbl> 3, 1, 1, 4, 1, 1, 1, 1, 1, 2, 1, 1, 5, 1, 1, 1, 2, 1, 3, 1…
## $ CAFF        <dbl> 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7…
## $ CANNABIS    <dbl> 1, 5, 4, 3, 4, 1, 2, 1, 1, 2, 3, 5, 4, 1, 1, 2, 4, 7, 4, 2…
## $ CHOC        <dbl> 6, 7, 5, 5, 7, 5, 6, 5, 7, 7, 6, 6, 6, 1, 7, 6, 6, 5, 7, 7…
## $ COKE        <dbl> 1, 4, 1, 3, 1, 1, 1, 1, 1, 1, 3, 3, 2, 1, 1, 3, 1, 2, 3, 1…
## $ CRACK       <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
## $ ECSTASY     <dbl> 1, 5, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 2, 3, 2…
## $ HEROIN      <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
## $ KETAMINE    <dbl> 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1…
## $ LEGALH      <dbl> 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1…
## $ LSD         <dbl> 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1…
## $ METH        <dbl> 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 7…
## $ MUSHROOMS   <dbl> 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2, 2, 1…
## $ NICOTINE    <dbl> 3, 5, 1, 3, 3, 7, 7, 1, 7, 7, 3, 7, 7, 2, 7, 1, 7, 7, 1, 2…
## $ SEMER       <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
## $ VSA         <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1…

5.1.1 histogram binning

#histogram to know how much to bin
library(ggplot2)
library(reshape2)
ggplot(melt(personalityXdrug),aes(x=value)) + geom_histogram() + facet_wrap(~variable)

## Warning: attributes are not identical across measure variables; they will be
## dropped

We bin the data based on the histograms.

#recode/bin quantitative data
## cluser from column 1 to 5 and bin at 4 based on histogram
personalityXdrug_no_country[, c(5:9)] <- data.frame(
    lapply(data[, c(6:10)], 
    function(x) Ckmedian.1d.dp(x, k = 5)$cluster), 
    row.names = rownames(data)) 

personalityXdrug_no_country[, c(10:11)] <- data.frame(
    lapply(data[, c(11:12)], 
    function(x) Ckmedian.1d.dp(x, k = 3)$cluster), 
    row.names = rownames(data))

personalityXdrug_no_country[, c(12:30)] <- data.frame(
    lapply(data[, c(13:31)], 
    function(x) Ckmedian.1d.dp(x, k = 2)$cluster), 
    row.names = rownames(data))

5.1.2 Computation of Eigen Values

# Computations ----
## Run DiCA  ----
resDiCA <- tepDICA(personalityXdrug, 
                   make_data_nominal = TRUE, 
                   DESIGN = qualitative$COUNTRY,
                   graphs = FALSE)

##  Inferences ----
set.seed(25) # set the seed
# for random so that we all have the same results. 
nIter <- 25
resDiCA.inf <- tepDICA.inference.battery(personalityXdrug,
                 DESIGN = qualitative$COUNTRY,
                 test.iters = nIter,
                 graphs = FALSE)

## [1] "It is estimated that your iterations will take 0.38 minutes."
## [1] "R is not in interactive() mode. Resample-based tests will be conducted. Please take note of the progress bar."
## ================================================================================

##error pops up if I include data that is not numerical

5.1.3 Group Means

Group Means based on country

Drugconsumption_means<- PTCA4CATA::getMeans(
               resDiCA$TExPosition.Data$fii, 
               qualitative$COUNTRY)

Drugconsumption_means

##            V1          V2          V3           V4           V5           V6
## AU  0.2920642 -0.01691787 -0.49575312  0.209105036 -0.115031592  0.108775408
## CA  0.1471334 -0.46937351  0.03120892 -0.017529340  0.043098939  0.032045063
## IE  0.1155778  0.06896327 -0.22691557  0.136134580  0.748520210 -0.298192069
## NZ  0.3108478  0.30248654  0.07511820 -0.374553033  0.683405924  1.405775535
## OT  0.1835446  0.02418580 -0.14922774 -0.305051243 -0.023620502 -0.065349320
## UK -0.2867310  0.01413764  0.01201397  0.009714978 -0.006279727  0.001737127
## US  0.4403071  0.03813958  0.05975673  0.027355591 -0.011817022 -0.006874631

Getting group means with row and column names

scale_drug_means <- scale(personalityXdrug, center = T, scale = T)

Drugconsumption_means_2 <- PTCA4CATA::getMeans(
               scale_drug_means, 
               qualitative$COUNTRY)

Drugconsumption_means_2

##         NSCORE     ESCORE      OSCORE      ASCORE      CSCORE   IMPULSIVITY
## AU -0.08667547  0.1065528  0.18782943  0.01368664 -0.15053440  0.2947968529
## CA  0.10924574 -0.1853636 -0.09398204 -0.28212579  0.02966078  0.0777709098
## IE -0.10086456  0.3214777  0.02793205  0.05851728 -0.23510234 -0.2359384354
## NZ  0.38075354 -0.6764856  0.21828607  0.14441857 -0.12021759 -0.0002500672
## OT -0.06561516 -0.1114116  0.41238264 -0.29951508 -0.34998709  0.0820411257
## UK -0.11963189  0.1025404 -0.28958720  0.14573507  0.19211156 -0.1981018736
## US  0.22967308 -0.1554398  0.44892542 -0.17036123 -0.26645329  0.3216740091
##             SS     ALCOHOL      AMPHET        AMYL      BENZOS         CAFF
## AU  0.43761125  0.00986656  0.68119597  0.61300193  0.39547022  0.031129913
## CA  0.03928271 -0.35612598  0.29882466 -0.20305389  0.12017682 -0.093759914
## IE  0.25561631 -0.36432843  0.42578053  0.41636848  0.04538592  0.328516654
## NZ  0.38541647 -0.02630099 -0.30308792  1.12111669  0.71481056  0.283659436
## OT  0.32633069  0.01316738  0.03711404  0.16234060  0.02768588 -0.190762665
## UK -0.30386715  0.08205775 -0.35582509  0.05796397 -0.34406019  0.008500793
## US  0.43921413 -0.08860646  0.53378713 -0.19576291  0.57385882  0.021764153
##      CANNABIS        CHOC        COKE       CRACK     ECSTASY      HEROIN
## AU  0.4580003 -0.01288765  0.15043686 -0.02369617  0.60741456 -0.03930498
## CA  0.2106612 -0.43554663  0.35681761  0.50955577  0.15820553  0.20506947
## IE  0.4199503  0.17751407  0.28996544 -0.11661475  0.50737932  0.02512101
## NZ  0.6166769  0.63651820  0.02559550 -0.35554824 -0.06922834  0.41167697
## OT  0.4306943 -0.02787084  0.06144227 -0.03157063  0.22087578 -0.08298362
## UK -0.4857126  0.12106183 -0.25156245 -0.21021992 -0.30398753 -0.25498395
## US  0.7212211 -0.16381360  0.37753502  0.33079571  0.42178444  0.46268570
##       KETAMINE     LEGALH          LSD        METH  MUSHROOMS   NICOTINE
## AU  0.26201159  0.3495177  0.890180426  0.03787348  0.4153388  0.1470733
## CA  0.27771393  0.1671862  0.166860651  0.16817542  0.3818065  0.1550094
## IE -0.05674583  0.4157422 -0.007738207 -0.07683877 -0.2300064  0.7040253
## NZ  0.35308514  1.2538962  0.897632033  0.22681129  0.6906525 -0.3316292
## OT  0.20026681  0.1988639  0.515705604 -0.06448690  0.3750971  0.3347004
## UK -0.11327424 -0.3733857 -0.444042640 -0.34779700 -0.4340747 -0.2525068
## US  0.09997589  0.5715355  0.602885528  0.63633001  0.6362908  0.3416020
##          SEMER        VSA
## AU  0.52120019  0.3000916
## CA -0.05993216 -0.1278888
## IE -0.05993216  0.7965488
## NZ -0.05993216  1.6278259
## OT -0.05993216  0.1220166
## UK -0.05993216 -0.2304050
## US  0.08655075  0.3536735

5.1.4 Heat map based on group means

corrplot3 <- corrplot::corrplot((as.matrix(Drugconsumption_means_2)), method = 'color',
                            tl.pos = "lt", 
                            tl.col = "black",
                            tl.cex = 0.7,
                            is.corr = FALSE,
                            addCoefasPercent = TRUE,
                            number.cex=0.5,
                            col=colorRampPalette(c("white","brown"))(200))

heat_map <- recordPlot()

This is the group mean based on country and correlation of the variables/columns. For example, NZ has greater AMYL, VSA, and legal H than other countries.

5.1.5 Scree plot

# The ScreePlot. Fixed Effects. ----
# Get the ScreePlot
# scree for ev ----
PlotScree(ev = resDiCA$TExPosition.Data$eigs,
          p.ev = resDiCA.inf$Inference.Data$components$p.vals,
   title = 'BADA Drug Consumption: Scree Plot with P-Values',
   plotKaiser = T, 
   color4Kaiser = ggplot2::alpha('darkorchid4', .5),
   lwd4Kaiser  = 2)

# Save the plot for pptx
a0002.Scree.sv <- recordPlot()

It seems like all are significant/important dimensions? However, I’m skeptical and will only look at the first 2 dimensions.

5.2 DiCA Analysis

5.2.1 change color of the countries

countryColors <- recode(resDiCA$Plotting.Data$fii.col, 
                     "#305ABF"= 'indianred4', 
                     "#84BF30"= 'gold', 
                     "#BF30AD" = 'lightpink2',
                     "#30BFA7" = "orange",
                     "#BF7D30" = "Blue",
                     "#5430BF" = "green",
                     "#36BF30" =  "purple")

5.2.2 Maps of loadings and factors

# I-set map ----
# a graph of the observations
Imap <- PTCA4CATA::createFactorMap(
  resDiCA$TExPosition.Data$fii,
  col.points = countryColors,
  col.labels = countryColors,
  alpha.points = .02,
  display.labels = F
)
# make labels ----
label4Map <- createxyLabels.gen(1,2,
           lambda = resDiCA$TExPosition.Data$eigs,
           tau = resDiCA$TExPosition.Data$t)

#get new variable to use for row/country color
col4Mean_recode <- recode(rownames(Drugconsumption_means),
                     AU = "Blue",
                     CA = 'gold',
                     IE = "green",
                     NZ =  "purple",
                      OT = "orange",
                     UK = 'indianred4',
                     US = 'lightpink2'
                     )
names(col4Mean_recode) <- rownames(Drugconsumption_means)

# the map
MapGroup <- PTCA4CATA::createFactorMap(Drugconsumption_means,
           # use the constraints from the main map
              constraints = Imap$constraints,
              col.points = col4Mean_recode,
              cex = 7,  # size of the dot (bigger)
              col.labels = col4Mean_recode,
              text.cex = 6)
# The map with observations and group means
a003.DICA <- Imap$zeMap + 
               label4Map +
               MapGroup$zeMap_dots + 
               MapGroup$zeMap_text
print(a003.DICA)

The group means for the country seems like almost all countries overlap except for UK. We can see if this is true based on confidence interval

Confidence Intervals

# Confidence intervals
### Bootstrapped CI ----
#_________________________________________________
# Create Confidence Interval Plots
# use function MakeCIEllipses 
# from package PTCA4CATA
# First get the order of colors for the ellipses
truc <- unique(rownames(
   resDiCA.inf$Inference.Data$boot.data$fi.boot.data$boots))
col4Means.ordered <- col4Mean_recode[order(truc)]
#
GraphElli <- PTCA4CATA::MakeCIEllipses(
  resDiCA.inf$Inference.Data$boot.data$fi.boot.data$boots,
  col = col4Means.ordered, 
  centers = resDiCA$TExPosition.Data$fi,
  p.level = .95
)
#_________________________________________________
### Pretty Imap ----
# create the I-map with Observations, 
# means and confidence intervals
#
a004.DICA.withCI <- Imap$zeMap_background + 
                         Imap$zeMap_dots + 
                         MapGroup$zeMap_dots + 
                         MapGroup$zeMap_text +
                         GraphElli + label4Map +
  ggtitle('DICA: Group Centers with CI and Observations')
#_________________________________________________
# plot it!
# dev.new()
print(a004.DICA.withCI)

The confidence intervals are all over the place, really difficult to know what is really going on. But it seems like the confidence intervals are large and overlap, suggesting that the countries do not differ at all.

Hull Graph (Tolerance Level)

###  with Hull ----
Fii <- resDiCA$TExPosition.Data$fii
# use function MakeToleranceIntervals 
#     from package PTCA4CATA
colnames(Fii) <- paste0('Dimension', 1:ncol(Fii))
GraphHull <- PTCA4CATA::MakeToleranceIntervals(
                   Fii,
                   design = qualitative$COUNTRY,
                   col = col4Mean_recode,
                   type = 'hull',
                   p.level = 1.0,
                   alpha.ellipse = .02)
#(
a006.DICA.withHull <-  Imap$zeMap_background + 
                       Imap$zeMap_dots       + 
                       MapGroup$zeMap_dots   + 
                       MapGroup$zeMap_text   +
                       GraphHull + label4Map +
 ggtitle('DICA: Group Centers with Hulls and Observations')
#)
# To print the Hulls
print(a006.DICA.withHull)

The spread of the data is really wide for Canada. However, a lot of the countries overlap.

Loadings aka factor/column Scores

# variables fs----------
var <- colnames(personalityXdrug)
var.color <- prettyGraphsColorSelection(n.colors = ncol(personalityXdrug)) 
col4Levels <- coloringLevels(
              rownames(resDiCA$TExPosition.Data$fj), var.color)
#too many var to assign color manually so we use prettyGraphsColorSelection() to automatically assign colors

#Column Factor Scores----
Fj <- resDiCA$TExPosition.Data$fj
baseMap.j <- PTCA4CATA::createFactorMap(Fj,
                                        col.points   = col4Levels$color4Levels,
                                        alpha.points =  .3,
                                        col.labels   = col4Levels$color4Levels,
                                        alpha.labels = .05)
# arrows
zeArrows <- addLines4MCA(Fj, col4Var = col4Levels$color4Levels)

# A graph for the J-set
b001.aggMap.j <- baseMap.j$zeMap_background + # background
  baseMap.j$zeMap_dots +  # dots
  baseMap.j$zeMap_text +  # names
  label4Map # lables for the axes
b002.aggMap.j <- b001.aggMap.j + zeArrows
b003.aggMap.j <- baseMap.j$zeMap_background + # background
  baseMap.j$zeMap_text +  # names
  zeArrows + label4Map +
  ggtitle('DICA: Col Factor Scores - All Levels')

b003.aggMap.j 

This is not pretty. I have 26 quantitative columns that was spread out, meaning my 26 original columns is more than 50 columns due to binning. We can check this.

#the number of loadings after binning
nrow(Fj)

## [1] 360

Wow, that’s actually a lot. No wonder it’s such a mess.

5.2.3 Contribution Map

Putting the loading/column scores onto the factor map

### Contribution Maps ----
CtrJ12 <- data4PCCAR::ctr4Variables(
                     resDiCA$TExPosition.Data$cj)
baseMap.ctrj <- PTCA4CATA::createFactorMap(CtrJ12,
                     col.points   = var.color,
                     alpha.points =  .3,
                     col.labels   = var.color)
#_________________________________________________
b001a.BaseMap.Ctrj <- baseMap.ctrj$zeMap + 
                       label4Map +
  ggtitle('Variables Contributions Map')
b001aa.BaseMapNoDot.Ctrj  <- 
  baseMap.ctrj$zeMap_background +
  baseMap.ctrj$zeMap_text + label4Map 
print(b001a.BaseMap.Ctrj)

5.2.4 Contribution barplots

We will only look at the significant ones because it will get messy if we look at all 360 loadings.

Dimension 1

### Ctr J-set ----
# 
ctrj <- resDiCA$TExPosition.Data$cj
signed.ctrj <- ctrj * sign(Fj)
####  CtrJ 1 ====
c001.plotCtrj.1 <- PrettyBarPlot2(
  bootratio = round(100*signed.ctrj[,1]), 
  threshold = 100 / nrow(signed.ctrj), 
  ylim = NULL, 
  color4bar = gplots::col2hex(col4Levels$color4Levels),
  color4ns = "gray75", 
  plotnames = TRUE, 
  main = 'Important Contributions Variables. Dim 1.', 
  ylab = "Signed Contributions",
  signifOnly = T)
# print(c001.plotCtrj.1)

Dimension 2

c002.plotCtrj.2 <- PrettyBarPlot2(
  bootratio = round(100*signed.ctrj[,2]), 
  threshold = 100 / nrow(signed.ctrj), 
  ylim = NULL, 
  color4bar = gplots::col2hex(col4Levels$color4Level),
  color4ns = "gray75", 
  plotnames = TRUE, 
  main = 'Important Contributions Variables. Dim 2.', 
  ylab = "Signed Contributions",
  signifOnly = T)
# print(c002.plotCtrj.2)

Bootstrap ratios This bootstrap ratio dno not look at the different levels of each loading/columns.

BRj <- resDiCA.inf$Inference.Data$boot.data$fj.boot.data$tests$boot.ratios
# BR1
d001.plotBRj.1 <- PrettyBarPlot2(
  bootratio = BRj[,1], 
  threshold = 2, 
  ylim = NULL, 
  color4bar = gplots::col2hex(var.color),
  color4ns = "gray75", 
  plotnames = TRUE, 
  main = 'Bootstrap Ratios Variable Levels. Dim 1.', 
  ylab = "Bootstrap Ratios")
# print(d001.plotBRj.1)

d003.plotBRj.2 <- PrettyBarPlot2(
  bootratio = BRj[,2], 
  threshold = 2, 
  ylim = NULL, 
  color4bar = gplots::col2hex(var.color),
  color4ns = "gray75", 
  plotnames = TRUE, 
  main = 'Bootstrap Ratios Variable Levels. Dim 2.', 
  ylab = "Bootstrap Ratios")
# print(d003.plotBRj.2)

 grid.arrange(
    as.grob(c001.plotCtrj.1),
    as.grob(c002.plotCtrj.2),
    as.grob(d001.plotBRj.1),
    as.grob(d003.plotBRj.2),
    ncol = 2,nrow = 2,
    top = textGrob("Barplots for variables", gp = gpar(fontsize = 25, font = 3))
  )

BothCtrJ <- recordPlot() # you need this line to be able to save them in the end

Barplots show the different levels of each variable. It not only tells us how it is related within the variables but also how different levels of each variable is related to other different levels of variables.

Dimension 1: Which personality is more likely to partake in drugs. For example, Impulsive & sensation seeking is related to drugs. Degrees of implusivity and sensation seeking is related to degrees of drugs.

Dimension 2 is based more on personality and which personality are more likely to participate in these different types of drugs. Furthermore, this tells us which levels of personality is related to which levels of drugs

The bootstraps just made everything more significant, however it is still difficult to interpret due to everything being significant and there’s a lot of loadings due to binning. The main goal of bootstrap is to see whether variables are stable.

5.3 Accuracy (Prediction & Confusion Matrix)

Fixed Effect We will now look at how good our prediction on new observations. We will first look at the fixed effects first

Fixed Effects: train data and test to see how well it performs

Fixed Accuracy Table/ Data confusion matrix

fixed <- as.data.frame(resDiCA.inf$Inference.Data$loo.data$fixed.confuse)
a <- knitr::kable(fixed, format = 'html')
kableExtra::scroll_box(a, width = "400px", height = "400px")

	.UK	.CA	.US	.OT	.AU	.IE	.NZ
.UK	725	30	61	30	11	5	1
.CA	45	11	48	4	1	0	0
.US	36	17	231	22	8	0	0
.OT	67	13	83	31	8	1	0
.AU	62	7	67	12	19	3	0
.IE	86	3	25	11	2	9	0
.NZ	23	6	42	8	5	2	4

Fixed Accuracy Percentage

resDiCA.inf$Inference.Data$loo.data$fixed.acc

## [1] 0.5464191

Random Effect After training with sample data, input new data to see how well it performs

#random cm
random <- as.data.frame(resDiCA.inf$Inference.Data$loo.data$loo.confuse)
a <- knitr::kable(random, format = 'html')
kableExtra::scroll_box(a, width = "400px", height = "500px")

	.UK.actual	.CA.actual	.US.actual	.OT.actual	.AU.actual	.IE.actual	.NZ.actual
.UK.predicted	723	32	61	31	11	5	1
.CA.predicted	45	4	49	4	1	0	0
.US.predicted	36	20	227	22	11	2	2
.OT.predicted	68	13	84	27	9	1	2
.AU.predicted	62	8	68	13	13	4	0
.IE.predicted	87	3	25	13	3	6	0
.NZ.predicted	23	7	43	8	6	2	0

Random Accuracy Percentage

resDiCA.inf$Inference.Data$loo.data$loo.acc

## [1] 0.530504

The random accuracy is very similar to the fixed accuracy. We can assume that DICA was reliable.

5.4 Conclusion

Dimension 1: Which personality is more likely to partake in drugs. rows = Participants Cols = impulsivity & Sensation seeking are more likely to participate in different drugs with different degrees of intensity.

Dimension 2: Which personality is more likely to partake in which drugs at what degree. Row = participants Col = personality and different drugs

Performance: Both seems stable and are around the same percentage. We should use this model because Fixed effect= around 55%, not great, but not bad either Random effect = around 53%, not great, but not bad either

Conclusion: DiCA vs BADA are the same in terms of the accuracy of prediction. however, I would go with BADA due to the visualization of the graphs are easier to interpret.

Chapter 6 Partial Least Squares Correlation (PLSC)

Partial Least Squares Correlation/Covariance: A technique that takes the basis of PCA and used to analyze the relationship of two sets of variables (data sets) of the same observations (two data tables that are similar based on observation). Compared to PCA where we try to find the maximum inertia (largest variance of observation of each component based on the columns/loadings & rows/factor scores), PLSC assess the maximum covariance(variability/direction) between 2 sets of variables (2 latent variables aka hidden variables from the two data data sets). The main goal is to find what is common between the two data tables of the same observations when they are not orthogonal (multicollinearity) and when there are more variables/columns than observations (p >> n).

Step by step process:

1. Perform SVD to decompose the correlation matrix between the data tables into weights for each data table(Find the weights of P in data table 1 & find the weights of Q on data table 2). The first step is to find commonality between the two data sets after centering and scaling the data for each data sets.

2. Find the latent variables (component) by multiplying the weights with their respective data sets. This is to maximize the covariance by looking at the diagonal matrix of the SVD. Through this, we get Lx=XP for latent 1 and Ly=YP for latent 2.

More info:

• Lx1 compared to Ly2 (Sugar and fruity from I dataset vs. Sweetness from J dataset). This means that the latent variables are similar due to these specific columns from dataset I and J. We can assume this means that they share highly similarity of the latent variables due to these specific variables (sugar & fruity vs sweetness). We will have to check based on the graphs and barplots.

• Lx2 compared to Ly2 (Sodium from I dataset vs salt J dataset).

• Lx1,Ly1 vs Lx2,Ly2 are different and are different latent variables.

Other notes:

-Partial least squares regression (PLSR) is very similar to PLSC but it creates latent variables from one data table to predict values from the other data table. Furthermore, instead of finding the maximum variance between response and independent variables, it finds a linear regression model by projecting the predicted variables and the observable variables to a new space. The dependent variable are numeric.

-Canonical Correlation Analysis: a method that focuses on the correlation between two data sets. This is done by reducing the dimensions of the two data sets and finding the pairs of component that have the maximum correlation.

More info: Partial least squares methods: partial least squares correlation and partial least square regression Herve Abdi & Lynne Williams 2013

Our data for analysis: For our analysis, we will be looking at a survey sent to participants from different countries asking their personality and how often they partake in an array of drugs.

6.1 Loading data

load("DrugConsumption.RData")
qualitative <- data[,1:5]
personality <- data[,6:12]
drugs <- data[,13:31]

View the data

# a <- knitr::kable(data, format = "html")
# kableExtra::scroll_box(a, width = "500px", height = "500px", fixed_thead = T)

a <- data
dplyr::glimpse(a)

## Rows: 1,885
## Columns: 31
## $ AGE         <dbl> 3, 2, 3, 1, 3, 6, 4, 3, 3, 5, 2, 4, 5, 5, 5, 5, 3, 4, 5, 3…
## $ GENDER      <fct> F, M, M, F, F, F, M, M, F, M, F, M, F, F, F, M, F, M, M, M…
## $ EDUCATION   <dbl> 6, 9, 6, 8, 9, 4, 8, 2, 6, 8, 7, 5, 7, 6, 6, 7, 5, 2, 7, 6…
## $ COUNTRY     <fct> UK, UK, UK, UK, UK, CA, US, UK, CA, UK, UK, OT, UK, CA, UK…
## $ ETHNICITY   <fct> WA, WH, WH, WH, WH, WH, WH, WH, WH, WH, WH, WH, WH, WH, WH…
## $ NSCORE      <dbl> 28, 18, 20, 23, 32, 18, 20, 13, 31, 22, 15, 13, 45, 17, 16…
## $ ESCORE      <dbl> 20, 36, 29, 18, 12, 22, 16, 36, 39, 24, 29, 24, 25, 29, 33…
## $ OSCORE      <dbl> 17, 30, 15, 21, 18, 10, 18, 15, 14, 11, 13, 22, 24, 21, 24…
## $ ASCORE      <dbl> 18, 29, 13, 28, 22, 36, 22, 22, 29, 28, 19, 11, 13, 30, 20…
## $ CSCORE      <dbl> 25, 24, 17, 29, 33, 35, 31, 35, 32, 26, 36, 21, 19, 41, 35…
## $ IMPULSIVITY <dbl> 4, 3, 2, 2, 4, 2, 4, 5, 2, 2, 5, 6, 8, 3, 8, 3, 4, 2, 3, 2…
## $ SS          <dbl> 3, 6, 8, 3, 6, 2, 7, 5, 2, 4, 7, 10, 7, 4, 9, 5, 1, 4, 6, …
## $ ALCOHOL     <dbl> 6, 6, 7, 5, 5, 3, 7, 6, 5, 7, 6, 6, 6, 2, 7, 6, 7, 7, 7, 5…
## $ AMPHET      <dbl> 3, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 3, 1, 2, 3, 2…
## $ AMYL        <dbl> 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1…
## $ BENZOS      <dbl> 3, 1, 1, 4, 1, 1, 1, 1, 1, 2, 1, 1, 5, 1, 1, 1, 2, 1, 3, 1…
## $ CAFF        <dbl> 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7…
## $ CANNABIS    <dbl> 1, 5, 4, 3, 4, 1, 2, 1, 1, 2, 3, 5, 4, 1, 1, 2, 4, 7, 4, 2…
## $ CHOC        <dbl> 6, 7, 5, 5, 7, 5, 6, 5, 7, 7, 6, 6, 6, 1, 7, 6, 6, 5, 7, 7…
## $ COKE        <dbl> 1, 4, 1, 3, 1, 1, 1, 1, 1, 1, 3, 3, 2, 1, 1, 3, 1, 2, 3, 1…
## $ CRACK       <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
## $ ECSTASY     <dbl> 1, 5, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 2, 3, 2…
## $ HEROIN      <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
## $ KETAMINE    <dbl> 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1…
## $ LEGALH      <dbl> 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1…
## $ LSD         <dbl> 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1…
## $ METH        <dbl> 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 7…
## $ MUSHROOMS   <dbl> 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2, 2, 1…
## $ NICOTINE    <dbl> 3, 5, 1, 3, 3, 7, 7, 1, 7, 7, 3, 7, 7, 2, 7, 1, 7, 7, 1, 2…
## $ SEMER       <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
## $ VSA         <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1…

6.1.1 Heat Plot

XY.cor <- cor(personality,drugs)
# Plot it with corrplot
corrplot(XY.cor, method = "color")

#record
heatmap <- recordPlot()

From our correlation plot, we see that agreeableness,contentiousness, extrovert is negatively correlated to the different drugs. Impulsivity and sensation seeking is highly correlated to the drugs compared to neuroticism and openness.

6.2 PLSC Anaylsis Begins

PLSC function

#PLSC
resPLSC <- tepPLS(personality, 
                   drugs, 
                   DESIGN = qualitative$COUNTRY,
                   graphs = FALSE)

This function is needed to perform the permutation inference test. This will help us find the p-values for each eigen value for our screen plot and eigen value tests.

#inference
##nIter ----
nIter = 100
### 1. Permutation test ----
resPerm4PLSC <- perm4PLSC(
                personality, # First Data matrix 
                drugs, # Second Data matrix
                nIter = 100 # How many iterations
)

#to find the correct p-values needed for the eigen values for scree plot
## instead of permutating by labels of observations, we will permutate based on cols of each data matrix
resPErm4PLSC_col <- perm4PLSC(
                personality, # First Data matrix 
                drugs, # Second Data matrix
                permType = 'byColumns',
                nIter = 100 # How many iterations
)

Bootstrap for barplots and stability test

#bootstrap
resBoot4PLSC <- Boot4PLSC(
           personality, # First Data matrix 
           drugs, # Second Data matrix
           nIter = 100, # How many iterations
           Fi = resPLSC$TExPosition.Data$fi,
           Fj = resPLSC$TExPosition.Data$fj,
           nf2keep = 3,
           critical.value = 2,
           eig = TRUE,
           alphaLevel = .05)

6.2.1 Scree plot

The Scree plot for eigen values

# scree for eigen values
PlotScree(ev = resPLSC$TExPosition.Data$eigs,
title = 'Drugs & Personality: Inertia Scree Plot',
          p.ev = resPErm4PLSC_col$pEigenvalues,
          plotKaiser = TRUE, 
  color4Kaiser = ggplot2::alpha('darkorchid4', .5),
          lwd4Kaiser  = 2)

a0001.Scree.ev <- recordPlot()

It seems like the first 3 dimensions are significant and explains the most variance. However, I will only look at the first as they explain the most compared to all. Keep in mind that only the first dimension is reliable because it is above kaiser line.

We can also check the p-values for the eigen values

resPErm4PLSC_col$pEigenvalues

## [1] 0.01 0.01 0.01 0.17 0.22 0.87 0.65

PLSC uses singular values which is squared of the eigen values. This means that values that are large, becomes larger because we square it.

Scree plot for Singular Values

# scree for sv ----
PlotScree(ev = resPLSC$TExPosition.Data$eigs^(1/2),
    title = 'Personality x Drug:  Singular Values Scree Plot',
          plotKaiser = T, 
    color4Kaiser = ggplot2::alpha('darkorchid4', .5),
          lwd4Kaiser  = 2)

# Save the plot
a0002.Scree.sv <- recordPlot()

It seems like dimension 1 explains the most variance and is the significant one (above kaiser line).

6.2.2 Salience Map

This is a barplot just to give a sense of what to expect from our two data sets.

#### Contributions ----
# get the color schemes for I (first data table)
Fi   <- resPLSC$TExPosition.Data$fi
col4I <- prettyGraphsColorSelection(nrow(Fi), 
                              starting.color = 1)
ctri <- resPLSC$TExPosition.Data$ci
signed.ctri <- ctri * sign(Fi)

# get the color schemes for J (second data table)
Fj   <- resPLSC$TExPosition.Data$fj
col4J <- prettyGraphsColorSelection(nrow(Fj), 
                          starting.color = 42)
ctrj <- resPLSC$TExPosition.Data$cj

signed.ctrj <- ctrj * sign(Fj)

# I = first data table
#  Personality dimension 1/latent 1 
barX1<- PrettyBarPlot2(
        bootratio = round(100*signed.ctri[,1]), 
        threshold = 0, 
        ylim = NULL, 
        color4bar = gplots::col2hex(col4I),
        color4ns = "gray75", 
        plotnames = TRUE, 
    main = 'Personality Viarables; LV1; Dim 1', 
    ylab = "Signed Contributions",
    horizontal = F)

# J (second data table)
# Drug Consumption dimension 1/latent 1
barY1 <- PrettyBarPlot2(
          bootratio = round(100*signed.ctrj[,1]), 
          threshold = 0, 
                       ylim = NULL, 
          color4bar = gplots::col2hex(col4J),
                       color4ns = "gray75", 
                       plotnames = TRUE, 
    main = 'Drug Consumption; LV1 ; Dim 1', 
    ylab = "Signed Contributions",
    horizontal = F)

# I = first data table
# personality dimension 2/latent 2
barX2 <- PrettyBarPlot2(
        bootratio = round(100*signed.ctri[,2]), 
        threshold = 0, 
        ylim = NULL, 
        color4bar = gplots::col2hex(col4I),
        color4ns = "gray75", 
        plotnames = TRUE, 
    main = 'Personality Viarables; LV2; Dim 2', 
    ylab = "Signed Contributions",
    horizontal = F)

#J = second data table
# Drug Consumption dimension 2/latent 2
barY2 <- PrettyBarPlot2(
          bootratio = round(100*signed.ctrj[,2]), 
          threshold = 0, 
                       ylim = NULL, 
          color4bar = gplots::col2hex(col4J),
                       color4ns = "gray75", 
                       plotnames = TRUE, 
    main = 'Drug Consumption; LV2 ; Dim 2', 
    ylab = "Signed Contributions",
    horizontal = F)

For latent variable 1

#combine all graphs
 grid.arrange(
    as.grob(barX1),
    as.grob(barY1),
    ncol = 1,nrow = 2,
    top = textGrob("Barplots Latent Variable 1 for variables", gp = gpar(fontsize = 25, font = 3))
  )

barplot_XY<- recordPlot() # you need this line to be able to save them in the end

It seems agreeableness and conscientiousness are going opposite directions to all of the drugs while the rest of the other personalities are going same directions of the drugs This means that SS, impulsivity, and openness personalities more likely to partake in these drugs.

For latent variable 2

#combine all graphs
 grid.arrange(
    as.grob(barX2),
    as.grob(barY2),
    ncol = 1,nrow = 2,
    top = textGrob("Barplots Latent Variable 2 for variables", gp = gpar(fontsize = 25, font = 3))
  )

barplot_XY<- recordPlot() # you need this line to be able to save them in the end

For latent variable 2, it seems like this barplot shows which personalities are likely to partake on which drugs.

6.3 PLSC Analysis

6.3.1 Latent Maps

Change colors

# check what are the different colors 
unique(resPLSC$Plotting.Data$fii.col)

##     [,1]     
## 1   "#84BF30"
## 6   "#BF30AD"
## 7   "#30BFA7"
## 12  "#BF7D30"
## 19  "#5430BF"
## 133 "#36BF30"
## 163 "#BF3060"

unique(qualitative$COUNTRY)

## [1] UK CA US OT AU IE NZ
## Levels: AU CA IE NZ OT UK US

countryColors <- recode(resPLSC$Plotting.Data$fii.col, 
                     "#BF3060"= 'goldenrod2', #NZ 
                    "#84BF30"=  "indianred4", #UK
                     "#BF30AD" = 'gold', #CA
                     "#30BFA7" = "lightpink2", #US
                     "#BF7D30" = "orange", #OT
                     "#5430BF" = "Blue", #AU
                     "#36BF30" =  "green") #IE

6.3.1.1 Latent Variable 1

laDim = 1
lv1.xy <- cbind(
    resPLSC$TExPosition.Data$lx[,laDim, drop = FALSE],
    resPLSC$TExPosition.Data$ly[,laDim, drop = FALSE])
colnames(lv1.xy) <- 
      c(paste0('LX',laDim),paste0('LY',laDim))
lv1 <- createFactorMap(lv1.xy,
   title = 'PLSC: First Pair of Latent Variables',
                       col.points = countryColors,
                       alpha.points = .04,
                       col.labels = countryColors,
                       display.label = F,
                       alpha.labels = .05)
a001.LV1 <- lv1$zeMap + 
          xlab(paste0("X Latent Variable ", laDim)) +
          ylab(paste0("Y Latent Variable ", laDim))

# print(a001.LV1)

Country group means

# Add groups etc.
# Groups in LV space

Drugconsumption_means_PLSC<- PTCA4CATA::getMeans(
               lv1.xy, 
               qualitative$COUNTRY)

The latent variables means based on country

Recode the colors based on country/design

#  "#BF3060"= 'goldenrod2', #NZ 
# "#84BF30"=  "indianred4", #UK
#  "#BF30AD" = 'gold', #CA
#  "#30BFA7" = "lightpink2", #US
#  "#BF7D30" = "orange", #OT
#  "#5430BF" = "Blue", #AU
#  "#36BF30" =  "green") #IE
col4Mean_recode <- recode(rownames(Drugconsumption_means_PLSC),
                     AU = "Blue",
                     CA = 'gold',
                     IE = "green",
                     NZ =  "goldenrod2",
                     OT = "orange",
                     UK = 'indianred4',
                     US = 'lightpink2'
                     )

names(col4Mean_recode) <- rownames(Drugconsumption_means_PLSC)

#### the map ----
MapGroup <- PTCA4CATA::createFactorMap(Drugconsumption_means_PLSC,
           # use the constraint from the main map
           constraints = lv1$constraints,
           col.points = col4Mean_recode,
           cex = 7,  # size of the dot (bigger)
           col.labels = col4Mean_recode,
           text.cex = 6)

a003.lv1.withMeans <- a001.LV1 +
    MapGroup$zeMap_dots + MapGroup$zeMap_text
# print(a003.lv1.withMeans)

Confidence Interval

# Bootstrap for CI:
BootCube.Gr <- PTCA4CATA::Boot4Mean(lv1.xy, 
    design = qualitative$COUNTRY,
    niter = 100,
    suppressProgressBar = TRUE)
# Create the ellipses
#### Bootstrapped CI ----
##_________________________________________________
# Create Confidence Interval Plots
# use function MakeCIEllipses from package PTCA4CATA
dimnames(BootCube.Gr$BootCube)[[2]] <- c("LX1","LY1")
GraphElli <- PTCA4CATA::MakeCIEllipses(
                BootCube.Gr$BootCube[,1:2,],
                names.of.factors = c("LX1","LY1"),
                col = col4Mean_recode,
                p.level = .95
)

a004.lv1.withCI <-  a001.LV1 + 
                     MapGroup$zeMap_text + 
                     MapGroup$zeMap_dots +
                     GraphElli
# plot it!
# print(a004.lv1.withCI)

6.3.1.2 Latent variable 2

laDim = 2
lv2.xy <- cbind(
    resPLSC$TExPosition.Data$lx[,laDim, drop = FALSE],
    resPLSC$TExPosition.Data$ly[,laDim, drop = FALSE])
colnames(lv2.xy) <- 
      c(paste0('LX',laDim),paste0('LY',laDim))
lv2 <- createFactorMap(lv2.xy,
   title = 'PLSC: Second Pair of Latent Variables',
                       col.points = countryColors,
                       alpha.points = .08,
                       col.labels = countryColors,
                       display.label = F,
                       alpha.labels = .05)
a001.LV2 <- lv2$zeMap + 
          xlab(paste0("X Latent Variable ", laDim)) +
          ylab(paste0("Y Latent Variable ", laDim))

# print(a001.LV2)

# Add groups etc.
# Groups in LV space

Drugconsumption_means_PLSC_2<- PTCA4CATA::getMeans(
               lv2.xy, 
               qualitative$COUNTRY)

Recode the colors based on country/design

col4Mean_recode_2 <- recode(rownames(Drugconsumption_means_PLSC_2),
                     AU = "Blue",
                     CA = 'gold',
                     IE = "green",
                     NZ =  "goldenrod2",
                     OT = "orange",
                     UK = 'indianred4',
                     US = 'lightpink2'
                     )

names(col4Mean_recode_2) <- rownames(Drugconsumption_means_PLSC_2)

Group mean for latent 2

#### the mean map ----
MapGroup_2 <- PTCA4CATA::createFactorMap(Drugconsumption_means_PLSC_2,
           # use the constraint from the main map
           constraints = lv2$constraints,
           col.points = col4Mean_recode_2,
           cex = 7,  # size of the dot (bigger)
           col.labels = col4Mean_recode_2,
           text.cex = 6)

a003.lv2.withMeans <- a001.LV2 +
    MapGroup_2$zeMap_dots + MapGroup_2$zeMap_text
# print(a003.lv2.withMeans)

Confidence Interval for latent variable 2

# Bootstrap for CI:
BootCube.Gr_2 <- PTCA4CATA::Boot4Mean(lv2.xy, 
    design = qualitative$COUNTRY,
    niter = 100,
    suppressProgressBar = TRUE)
# Create the ellipses
#### Bootstrapped CI ----
##_________________________________________________
# Create Confidence Interval Plots
# use function MakeCIEllipses from package PTCA4CATA
dimnames(BootCube.Gr_2$BootCube)[[2]] <- c("LX2","LY2")
GraphElli_2 <- PTCA4CATA::MakeCIEllipses(
                BootCube.Gr_2$BootCube[,1:2,],
                names.of.factors = c("LX2","LY2"),
                col = col4Mean_recode_2,
                p.level = .95
)

a004.lv2.withCI <-  a001.LV2 + 
                     MapGroup_2$zeMap_text + 
                     MapGroup_2$zeMap_dots +
                     GraphElli_2
# plot it!
# print(a004.lv2.withCI)

6.3.1.3 Combine the two latent variable mapes

grid.arrange(
    as.grob(a003.lv1.withMeans),
    as.grob(a004.lv1.withCI),
    as.grob(a003.lv2.withMeans),
    as.grob(a004.lv2.withCI),
    ncol = 2,nrow = 2,
    top = textGrob("Latent 1 and 2 pairings", gp = gpar(fontsize = 25, font = 3))
  )

Latent_1_2_map <- recordPlot() # you need this line to be able to save them in the end

LV1: It seems like UK is further away from the other countries, meaning UK is different compared to the other countries. However, lets see if they are really different based on confidence intervals. Looking at the confidence interval , UK does not overlaps with NZ while NZ overlapps with all other countries. We can suggest that UK is different compared to all other countries.

LV2: All the countries overlap for latent variable 2, suggesting they are not different at all.The confidence interval for latent variable 2 all overlap for the countries, they are very similar/no clear distinction.

Combining the two different pairs of latent variables as one graph, we can clearly see that for latent variable 1, UK is very different compared to the other countries.

6.3.2 Contribution barplots

6.3.2.1 Latent Variable 1 & 2

The barplots shows the loadings that make up the pairs of latent variables.

#### Contributions ----
# get the color schemes for I (first data table)
Fi   <- resPLSC$TExPosition.Data$fi
col4I <- prettyGraphsColorSelection(nrow(Fi), 
                              starting.color = 1)
ctri <- resPLSC$TExPosition.Data$ci
signed.ctri <- ctri * sign(Fi)

# get the color schemes for J (second data table)
Fj   <- resPLSC$TExPosition.Data$fj
col4J <- prettyGraphsColorSelection(nrow(Fj), 
                          starting.color = 42)
ctrj <- resPLSC$TExPosition.Data$cj

signed.ctrj <- ctrj * sign(Fj)

# I = first data table
#  Personality dimension 1/latent 1 
a020.plotCtri.1 <- PrettyBarPlot2(
        bootratio = round(100*signed.ctri[,1]), 
        threshold = 100/ nrow(signed.ctri), 
        ylim = NULL, 
        color4bar = gplots::col2hex(col4I),
        color4ns = "gray75", 
        plotnames = TRUE, 
    main = 'Personality Viarables; LV1; Dim 1', 
    ylab = "Signed Contributions",
    signifOnly = T,
    font.size = 2,
    horizontal = F)

# J (second data table)
# Drug Consumption dimension 1/latent 1
a021.plotCtrj.1 <- PrettyBarPlot2(
          bootratio = round(100*signed.ctrj[,1]), 
          threshold = 100 / nrow(signed.ctrj), 
                       ylim = NULL, 
          color4bar = gplots::col2hex(col4J),
                       color4ns = "gray75", 
                       plotnames = TRUE, 
    main = 'Drug Consumption; LV1 ; Dim 1', 
    ylab = "Signed Contributions",
    signifOnly = T,
    horizontal = F)

# I = first data table
# personality dimension 2/latent 2
a020.plotCtri.2 <- PrettyBarPlot2(
        bootratio = round(100*signed.ctri[,2]), 
        threshold = 100/ nrow(signed.ctri), 
        ylim = NULL, 
        color4bar = gplots::col2hex(col4I),
                    color4ns = "gray75", 
                    plotnames = TRUE, 
    main = 'Personality Viarables; LV2; Dim 2', 
    ylab = "Signed Contributions",
    signifOnly = T,
    horizontal = F)

#J = second data table
# Drug Consumption dimension 2/latent 2
a021.plotCtrj.2 <- PrettyBarPlot2(
          bootratio = round(100*signed.ctrj[,2]), 
          threshold = 100 / nrow(signed.ctrj), 
                       ylim = NULL, 
          color4bar = gplots::col2hex(col4J),
                       color4ns = "gray75", 
                       plotnames = TRUE, 
    main = 'Drug Consumption; LV2 ; Dim 2', 
    ylab = "Signed Contributions",
    signifOnly = T,
    horizontal = F)

#combine all graphs
 grid.arrange(
    as.grob(a020.plotCtri.1),
    as.grob(a021.plotCtrj.1),
    as.grob(a020.plotCtri.2),
    as.grob(a021.plotCtrj.2),
    ncol = 1,nrow = 4,
    top = textGrob("Barplots for variables", gp = gpar(fontsize = 25, font = 3))
  )

barplot_latent_1<- recordPlot() # you need this line to be able to save them in the end

Based on this graph, we can assume that for latent variable 1 and 1 in dimension 1, sensation seeking, impulsivity and openess are related to multiple drugs.

For latent variable 2 and 2 for dimension 2, neuroticism is related to meth, heroine and benzos while openness and estravert is related to mushrooms, LSD, and ecstasy.

This barplot for the latent variables suggest that the difference between UK and the other countries are due to these specific loadings.

I would interpret this as:

For Dimension 1/LV1: UK have more individuals with personalities(SS,openness, & impulsivity) that likely to partake in drugs. We can see this by looking at how many participants are in the UK

qualitative %>% count(COUNTRY)

##   COUNTRY    n
## 1      AU   54
## 2      CA   87
## 3      IE   20
## 4      NZ    5
## 5      OT  118
## 6      UK 1044
## 7      US  557

For Dimension 2/LV2: There is no difference amongst the different countries. This is basically suggesting that the different personalities within all the countries from this data set is related to different types of drugs.

6.3.2.2 Bootstrap ratios for latent 1 and 2

This is to test if the barplots variables are stable

Combine I-J

# LV1 Personality
a030.plotBRi.1 <- PrettyBarPlot2(
    bootratio = resBoot4PLSC$bootRatios.i[,1], 
    threshold = 2, 
    ylim = NULL, 
    color4bar = gplots::col2hex(col4I),
    color4ns = "gray75", 
    plotnames = TRUE, 
    main = 'Bootstrap Ratios. Personality: LV1/Dim 1', 
    ylab = "Bootstrap Ratios",
    signifOnly = T,
    horizontal = F)

#LV 2 drugs
a031.plotBRj.1 <- PrettyBarPlot2(
    bootratio = resBoot4PLSC$bootRatios.j[,1],
    threshold = 2, 
    ylim = NULL, 
    color4bar = gplots::col2hex(col4J),
    color4ns = "gray75", 
    plotnames = TRUE, 
    main = 'Bootstrap Ratios. Drugs: LV1/Dim 1', 
    ylab = "Bootstrap Ratios",
    signifOnly = T,
    font.size = 3,
    horizontal = F)

# LV1 Personality
a030.plotBRi.2 <- PrettyBarPlot2(
    bootratio = resBoot4PLSC$bootRatios.i[,2], 
    threshold = 2, 
    ylim = NULL, 
    color4bar = gplots::col2hex(col4I),
    color4ns = "gray75", 
    plotnames = TRUE, 
    main = 'Bootstrap Ratios. Personality: LV2/Dim 2', 
    ylab = "Bootstrap Ratios",
    signifOnly = T,
    horizontal = F)

#LV 2 drugs
a031.plotBRj.2 <- PrettyBarPlot2(
    bootratio = resBoot4PLSC$bootRatios.j[,2],
    threshold = 2, 
    ylim = NULL, 
    color4bar = gplots::col2hex(col4J),
    color4ns = "gray75", 
    plotnames = TRUE, 
    main = 'Bootstrap Ratios. Drugs: LV2/Dim 2', 
    ylab = "Bootstrap Ratios",
    signifOnly = T,
    horizontal = F)

 grid.arrange(
    as.grob(a030.plotBRi.1),
    as.grob(a031.plotBRj.1),
    as.grob(a030.plotBRi.2),
    as.grob(a031.plotBRj.2),
    ncol = 1,nrow = 4,
    top = textGrob("BootStrap Barplots for variables", gp = gpar(fontsize = 25, font = 3))
  )

barplot_latent_2<- recordPlot() # you need this line to be able to save them in the end

From the bootstrap ratio barplot:

– For dimension 1 : consentioucness and agreeablness are negatively related to the drugs.

– For dimension 2: Neuroticism is highly related to Meth, heroin, crack, and benzos. Sensation seeking, contentiousness, agreeableness, openess, and extravert are related to mushroom, lsd, ecstasy, cannabis, and alcohol.

We can conclude from our bootstrap, our original variables are stable/consistent.

6.4 Conclusion

Interpreting the loadings (personality and drug consumption) with the factor scores (countries):

Latent variables

Component 1: Lxy1

• We find that for the latent variable pairs, Lxy 1 separates UK from the different countries.

Component 2: Lxy2

• Lxy 2 doesn’t separate the countries.

Barplots: shows why the countries are different based on the loadings.

• Dimension 1: Particularly, it shows which personality is more likely to participate in drugs. We can see that sensation seeking, impulsivity and openness is related to drug consumption. Bootstrap ratio further shows that neuroticism is associated with drug consumptions. In relation to the difference of countries (UK vs other countries), I would interpret in that UK have more individuals with personalities that are likely to partake in drugs.

• Dimension 2: Shows which personality is likely to participate in which drugs. In relation to the difference of countries. Within this dataset (all the countries are not much different), the different personalities are related to different drugs.

Chapter 7 DiSTATIS

DiSTATIS is a 3-way multi-dimensional scaling technique. DiSTATIS is the general technique for multi-dimensional scaling while MFA is somewhat the advanced version of it. DiSTATIS technique is mostly rooted in distance analysis, particularly a set of distance of matrices. DiSTATIS can look at both rows and columns where we can tell the difference between rows and columns based on pre-assigned group. Furthermore, this technique is similar to k-clustering or hcluster where data is grouped together in how similar or different they are.

The main point is that with DiSTATIS we can project all of this onto the same space using the euclidean distance to see how similar or different the data is based on pre-assigned group.

Techniques to project onto the map:

• find the distance for variable/row.

• convert the distance into covariance through PCA by double centering the distance.(Also known as a compromise)

• Project it onto the map.

More info can be found: DISTATIS: The Analysis of Multiple Distance Matrices 2005

7.1 Data set: African Wine

This data set is a wine data set where judges sort the different kind of wines into groups. The wines are either French or from South Africa. Judges were were 1 in 4 conditions: Judges from France Vs. South Africa and judges with vs without information (labels of the wine).

data <- read.csv("French and South African Wines C.csv")
data <- as.data.frame(data)


#remove the first column
ratings <- data[,-1]

#add the first column values as row names
rownames(ratings) <- data[,1]

ratings_2 <- ratings[-15,]

ratings_3 <- sapply( ratings_2, as.numeric )

ratings_3 <- as.data.frame(ratings_3)

# Convert judge info into a seperate data table
judge <- ratings[15,]
judge_t <- as.data.frame(t(judge))

judge_list <- judge_t[,1]

#colors for the judges
color4Judges   <-  prettyGraphsColorSelection(n.colors = ncol(ratings_3))

# a <- knitr::kable(data, format = "html")
# kableExtra::scroll_box(a, width = "500px", height = "200px", fixed_thead = T)

a <- data
dplyr::glimpse(a)

## Rows: 15
## Columns: 57
## $ X   <chr> "FCAR", "SRUD", "FBAU", "FROC", "SFED", "SREY", "SKLE", "FCLL", "F…
## $ J1  <chr> "2", "1", "1", "1", "1", "3", "1", "2", "2", "3", "3", "2", "3", "…
## $ J2  <chr> "2", "1", "1", "1", "2", "1", "1", "2", "1", "1", "2", "2", "2", "…
## $ J3  <chr> "2", "5", "3", "2", "4", "2", "5", "4", "5", "1", "3", "4", "1", "…
## $ J4  <chr> "3", "3", "2", "2", "2", "2", "3", "3", "1", "1", "3", "3", "3", "…
## $ J5  <chr> "4", "5", "3", "2", "1", "5", "5", "4", "2", "2", "3", "2", "3", "…
## $ J6  <chr> "1", "4", "3", "2", "3", "3", "2", "2", "2", "1", "4", "3", "4", "…
## $ J7  <chr> "3", "1", "3", "2", "2", "1", "3", "2", "1", "2", "1", "3", "3", "…
## $ J8  <chr> "1", "3", "2", "2", "3", "1", "1", "1", "1", "1", "3", "2", "2", "…
## $ J9  <chr> "2", "3", "1", "3", "1", "1", "1", "2", "3", "1", "1", "2", "2", "…
## $ J10 <chr> "1", "1", "1", "2", "2", "2", "3", "3", "3", "4", "4", "4", "4", "…
## $ J11 <chr> "4", "3", "2", "4", "2", "4", "3", "1", "1", "2", "2", "4", "4", "…
## $ J12 <chr> "4", "1", "3", "2", "4", "1", "1", "3", "3", "3", "3", "2", "2", "…
## $ J13 <chr> "2", "4", "2", "1", "3", "3", "1", "1", "4", "4", "3", "3", "2", "…
## $ J14 <chr> "5", "4", "3", "5", "6", "6", "6", "4", "2", "4", "4", "3", "5", "…
## $ J15 <chr> "3", "2", "2", "4", "3", "4", "4", "1", "1", "2", "2", "4", "3", "…
## $ J16 <chr> "1", "2", "2", "2", "1", "3", "3", "3", "3", "3", "2", "1", "3", "…
## $ J17 <chr> "4", "4", "2", "3", "3", "2", "2", "3", "4", "1", "4", "1", "1", "…
## $ J18 <chr> "1", "2", "3", "2", "3", "3", "1", "2", "3", "2", "3", "1", "3", "…
## $ J19 <chr> "4", "1", "5", "4", "1", "4", "2", "2", "3", "2", "2", "4", "3", "…
## $ J20 <chr> "2", "3", "2", "2", "1", "1", "1", "3", "3", "3", "1", "3", "3", "…
## $ J21 <chr> "1", "2", "4", "2", "1", "4", "3", "2", "1", "1", "4", "3", "3", "…
## $ J22 <chr> "1", "1", "2", "2", "2", "3", "3", "4", "3", "2", "3", "1", "4", "…
## $ J23 <chr> "3", "3", "3", "1", "2", "2", "1", "3", "1", "3", "3", "2", "1", "…
## $ J24 <chr> "3", "2", "3", "1", "3", "1", "2", "3", "4", "2", "4", "1", "1", "…
## $ J25 <chr> "4", "4", "4", "3", "2", "3", "2", "4", "1", "3", "2", "2", "1", "…
## $ J26 <chr> "2", "1", "3", "2", "1", "3", "4", "4", "1", "2", "3", "1", "1", "…
## $ J27 <chr> "2", "3", "2", "3", "2", "2", "3", "1", "2", "1", "1", "3", "2", "…
## $ J28 <chr> "2", "1", "1", "2", "1", "1", "2", "4", "4", "3", "3", "1", "3", "…
## $ J29 <chr> "2", "8", "7", "4", "7", "1", "2", "2", "9", "5", "6", "6", "3", "…
## $ J30 <chr> "1", "1", "2", "3", "2", "1", "1", "4", "4", "2", "1", "3", "4", "…
## $ J31 <chr> "2", "1", "1", "2", "2", "2", "2", "2", "1", "1", "1", "3", "4", "…
## $ J32 <chr> "3", "2", "3", "1", "1", "1", "1", "3", "3", "2", "2", "3", "3", "…
## $ J33 <chr> "2", "2", "3", "1", "5", "4", "5", "5", "6", "2", "3", "2", "1", "…
## $ J34 <chr> "2", "1", "1", "3", "4", "5", "4", "3", "5", "4", "2", "3", "5", "…
## $ J35 <chr> "4", "3", "2", "4", "1", "5", "5", "6", "6", "3", "7", "8", "4", "…
## $ J36 <chr> "3", "2", "3", "1", "4", "2", "2", "3", "1", "2", "2", "3", "1", "…
## $ J37 <chr> "1", "4", "3", "1", "1", "2", "1", "2", "3", "4", "4", "1", "2", "…
## $ J38 <chr> "1", "2", "1", "2", "1", "3", "4", "3", "2", "3", "3", "3", "1", "…
## $ J39 <chr> "5", "5", "5", "7", "2", "6", "3", "5", "4", "4", "6", "1", "5", "…
## $ J40 <chr> "4", "1", "4", "3", "4", "1", "1", "2", "3", "1", "4", "2", "2", "…
## $ J41 <chr> "5", "4", "3", "5", "2", "1", "1", "5", "5", "2", "4", "5", "5", "…
## $ J42 <chr> "8", "3", "8", "8", "5", "8", "4", "6", "6", "1", "1", "2", "8", "…
## $ J43 <chr> "1", "5", "4", "2", "4", "3", "2", "1", "5", "5", "1", "3", "1", "…
## $ J44 <chr> "1", "2", "1", "3", "1", "5", "5", "4", "4", "2", "2", "3", "4", "…
## $ J45 <chr> "4", "4", "3", "4", "5", "2", "5", "3", "1", "4", "5", "4", "3", "…
## $ J46 <chr> "1", "5", "6", "4", "1", "2", "1", "7", "2", "1", "2", "3", "2", "…
## $ J47 <chr> "5", "4", "3", "1", "4", "5", "3", "1", "2", "4", "4", "3", "3", "…
## $ J48 <chr> "1", "1", "3", "1", "2", "3", "3", "2", "1", "2", "2", "1", "1", "…
## $ J49 <chr> "2", "4", "2", "1", "1", "4", "3", "3", "3", "4", "4", "2", "1", "…
## $ J50 <chr> "4", "1", "1", "4", "7", "1", "2", "3", "2", "4", "1", "6", "2", "…
## $ J51 <chr> "3", "5", "5", "2", "1", "5", "5", "3", "5", "5", "2", "4", "4", "…
## $ J52 <chr> "1", "3", "3", "3", "2", "1", "3", "1", "1", "2", "3", "2", "1", "…
## $ J53 <chr> "3", "4", "1", "1", "2", "4", "1", "1", "3", "4", "2", "2", "3", "…
## $ J54 <chr> "4", "3", "2", "3", "5", "5", "4", "2", "3", "1", "1", "4", "5", "…
## $ J55 <chr> "1", "3", "5", "2", "1", "5", "5", "2", "3", "4", "4", "4", "3", "…
## $ J56 <chr> "2", "1", "1", "1", "1", "2", "1", "1", "1", "2", "3", "1", "2", "…

#turn judge description as nominal
nominal.Judges <- makeNominalData(
                       as.data.frame(judge_t))

# get the colors
color4Judges.list <- prettyGraphs::createColorVectorsByDesign(
                                  nominal.Judges)

Getting the color of the different judges

unique(color4Judges.list$gc)

##                       [,1]     
## judge_info.Fr Info    "#305ABF"
## judge_info.Fr No Info "#84BF30"
## judge_info.SA Info    "#BF30AD"
## judge_info.SA No Info "#30BFA7"

# "#305ABF" = blue ,"#84BF30" = green , "#BF30AD" = purple, "#30BFA7" = teal

#view of the judge descriptions
b <- knitr::kable(nominal.Judges, format = "html")
kableExtra::scroll_box(b, width = "500px", height = "500px")

	judge_info.Fr Info	judge_info.Fr No Info	judge_info.SA Info	judge_info.SA No Info
J1	1	0	0	0
J2	1	0	0	0
J3	1	0	0	0
J4	1	0	0	0
J5	1	0	0	0
J6	1	0	0	0
J7	1	0	0	0
J8	1	0	0	0
J9	1	0	0	0
J10	1	0	0	0
J11	0	1	0	0
J12	0	1	0	0
J13	0	1	0	0
J14	0	1	0	0
J15	0	1	0	0
J16	0	1	0	0
J17	0	1	0	0
J18	0	1	0	0
J19	0	1	0	0
J20	0	1	0	0
J21	0	1	0	0
J22	0	1	0	0
J23	0	1	0	0
J24	0	1	0	0
J25	0	1	0	0
J26	0	1	0	0
J27	0	1	0	0
J28	0	0	1	0
J29	0	0	1	0
J30	0	0	1	0
J31	0	0	1	0
J32	0	0	1	0
J33	0	0	1	0
J34	0	0	1	0
J35	0	0	1	0
J36	0	0	1	0
J37	0	0	1	0
J38	0	0	1	0
J39	0	0	1	0
J40	0	0	1	0
J41	0	0	1	0
J42	0	0	0	1
J43	0	0	0	1
J44	0	0	0	1
J45	0	0	0	1
J46	0	0	0	1
J47	0	0	0	1
J48	0	0	0	1
J49	0	0	0	1
J50	0	0	0	1
J51	0	0	0	1
J52	0	0	0	1
J53	0	0	0	1
J54	0	0	0	1
J55	0	0	0	1
J56	0	0	0	1

7.1.1 Correlation plot

#compute covariance
rating_covariance <- cor(ratings_3)

corrplot(rating_covariance, method = "color",
         tl.cex = .5)

This is to see how similar or different judge’s grouping of the different wines.

7.2 Distatis Analysis

Function to compute distance of data set.

DistanceCube <- DistanceFromSort(ratings_3)
resDistatis <- distatis(DistanceCube)

Function to get group means based on judges

# Get the factors from the Cmat analysis
G <- resDistatis$res4Cmat$G 

# Compute the mean by groups of Judges
JudgesMeans.tmp <- aggregate(G, 
                      list(judge_list), mean) 
JudgesMeans <- JudgesMeans.tmp[,2:ncol(
                             JudgesMeans.tmp )] 
rownames(JudgesMeans) <- JudgesMeans.tmp[,1]

Bootstrap for the group means

BootCube <- PTCA4CATA::Boot4Mean(G, 
                       design = judge_list,
                       niter = 100,
                       suppressProgressBar = TRUE)

Compute SK

groupK <- computePartial4Groups(
   resDistatis = resDistatis,
   DESIGN = judge_list
)
alpha_k <- groupK $groupAlpha
F_k     <- groupK $groupFS

7.2.1 Analysis on the judges

Scree plot of judges: We plot the scree so we can see how much variance is explained per dimension. We use the between-judges to look at the scree.

scree <- PlotScree(ev = resDistatis$res4Cmat$eigValues,
title = "RV-map: Explained Variance per Dimension",
plotKaiser = TRUE)

scree <- recordPlot()

It seems like the first couple of dimensions explain the most variance. We will only look at the first two.

7.2.1.1 Biplot

RV Plot: Biplot to see all the judges on the same plane

gg.rv.graph.out <- createFactorMap(
             X = resDistatis$res4Cmat$G, 
             axis1 = 1, axis2 = 2, 
             title = "Judges: RVMap", 
             col.points = color4Judges.list$oc, 
             col.labels = color4Judges.list$oc)

labels4RV <- createxyLabels.gen(
        lambda = resDistatis$res4Cmat$eigValues, 
        tau    = resDistatis$res4Cmat$tau,
        axisName = "Dimension ")

a2a.gg.RVmap <- gg.rv.graph.out$zeMap + labels4RV

a2b.gg.RVmap <- gg.rv.graph.out$zeMap_background +
                gg.rv.graph.out$zeMap_dots + 
                labels4RV

print(a2a.gg.RVmap )

The 56 judges are colored by how they grouped by the design (FR vs SA & Info vs no Info)

Confidence Interval with group means

# ----RVwithCI-----------------------------------
# First the means
# A tweak for colors
in.tmp <- sort(rownames(color4Judges.list$gc), 
                        index.return = TRUE)$ix
col4Group <- color4Judges.list$gc[in.tmp]
#
gg.rv.means <- PTCA4CATA::createFactorMap(
        JudgesMeans,
        axis1 = 1, axis2 = 2, 
        constraints = gg.rv.graph.out$constraints,
        col.points =  col4Group,
        alpha.points = 1, # no transparency
        col.labels = col4Group)
#
 dimnames(BootCube$BootCube)[[2]] <- 
     paste0('dim ',1: dim(BootCube$BootCube)[[2]])
  #c('Dim1','Dim2') 
GraphElli.rv <- MakeCIEllipses(
        BootCube$BootCube[,1:2,],
        names.of.factors = c("dim 1","dim 2"), 
        col = col4Group, 
        p.level = .95)
a2d.gg.RVMap.CI <- a2b.gg.RVmap +
                  gg.rv.means$zeMap_dots + 
                  GraphElli.rv 
# dev.new()
print(a2d.gg.RVMap.CI)

color4Judges.list$gc

##                       [,1]     
## judge_info.Fr Info    "#305ABF"
## judge_info.Fr No Info "#84BF30"
## judge_info.SA Info    "#BF30AD"
## judge_info.SA No Info "#30BFA7"

#FR info = blue
#FR no info = green
#SA Info = pink
#SA no info = teal

It seems like almost all of them are overlapping, meaning the judge descriptions are not much different from each other. However, the french judges with info vs SA with info are different.

7.2.1.2 HCA on judges

Let’s look at the dimensions based on judge description

knitr::kable(JudgesMeans[,1:3])

	dim 1	dim 2	dim 3
Fr Info	0.4342582	0.0418181	0.0889197
Fr No Info	0.4810248	-0.0610898	-0.0117954
SA Info	0.5645092	0.0644610	-0.0355182
SA No Info	0.5247804	-0.0429363	0.0322169

Hierarchical Clustering Analysis (HCA)

 D <- dist(resDistatis$res4Cmat$G,
               method = "euclidean")
 fit <- hclust(D, method = "ward.D2")
 a05.tree4participants <- fviz_dend(fit,  
     k = 1, 
     k_colors = 'burlywood4', 
     label_cols = color4Judges.list$oc[fit$order],
     cex = .7, xlab = 'Participants',
     main = 'Cluster Analysis: Participants') 
 
print(a05.tree4participants)

How we look at this dendogram is which vertical line is the longest. Then, looking at the longest vertical line we then draw a horizontal line across it to count the groups. Based on this, we have 5 groups.

7.2.2 Analysis on Products

7.2.2.1 Cluster Correlation plot: Products

S corPlot: This plot looks at how similar or different the products are based on how they were grouped by the judges (Cluster correlation map)

color4Products <- #  
   prettyGraphsColorSelection(n.colors = 
                nrow(resDistatis$res4Splus$F))

rownames(color4Products) <- rownames(ratings_2)
rownames(resDistatis$res4Splus$Splus) <- rownames(ratings_2)

Reorder cluster correlation map (HCA)

plot <- corrplot::corrplot(
   resDistatis$res4Splus$Splus, 
   # title = "The S Map", title is cut off when used 
   order = 'hclust', # order from a HCA
   method = "color",
   col = brewer.pal(n = 8, name = "PRGn"),
   number.cex = 0.8, 
   tl.col = color4Products,
   mar = c(0,0,0,0),
   addgrid.col = "grey", 
   tl.srt = 50)

b002.ScorrMap.OrderedByHCA <- recordPlot() 

From this graph, we can see which product is related.

7.2.2.2 Scree plot on products

Scree plot for compromise: This is a scree plot after the HCA/clustering. Basically, this looks at the dimensions that explains the most variance after clustering (on how the products were grouped by the judges).

 scree.S.out <- PlotScree(
   ev = resDistatis$res4Splus$eigValues, 
   title = "Compromise: Explained Variance per Dimension",
   plotKaiser = T)

 b1.Scree.S <- recordPlot()

This tells us that the first 7 dimensions explains the most variance. However we will only look at the first two.

7.2.2.3 Biplot for Products

Looking at products based on how they were grouped by the judges

#adding row names to the map
rownames(resDistatis$res4Splus$F) <- rownames(ratings_2)

# 4.1 Get the bootstrap factor scores 
 #    (with default 1000 iterations)
BootF <- BootFactorScores(
                resDistatis$res4Splus$PartialF)

## [1] Bootstrap On Factor Scores. Iterations #: 
## [2] 1000

# 5.2 a compromise plot
# General title for the compromise factor plots:
genTitle4Compromise = 'Compromise.'
# To get graphs with axes 1 and 2:
h_axis = 1
v_axis = 2
# To get graphs with say 2 and 3 
# change the values of v_axis and h_axis
gg.compromise.graph.out <- createFactorMap(
                resDistatis$res4Splus$F,
                axis1 = h_axis, 
                axis2 = v_axis,
                title = genTitle4Compromise,
                col.points = color4Products,
                col.labels = color4Products) 
# NB for the lines below 
#  You need DISTATIS version > 1.0.0
#  to get the eigen values 
#   and tau for the compromise
label4S <- createxyLabels.gen(
      x_axis   = h_axis, y_axis = v_axis,
      lambda   = resDistatis$res4Splus$eigValues, 
      tau      = resDistatis$res4Splus$tau,
      axisName = "Dimension ")
b2.gg.Smap <-  gg.compromise.graph.out$zeMap + 
                   label4S 

# print(b2.gg.Smap)

Bootstrap of the compromise plot

#5.4 a bootstrap confidence interval plot 
# 5.3  create the ellipses
## With Ellipses -----
gg.boot.graph.out.elli <- MakeCIEllipses(
           data = BootF[,c(h_axis,v_axis),],
           names.of.factors = 
                c(paste0('Factor ',h_axis),
                  paste0('Factor ',v_axis)),
           col = color4Products,
)  
# Add ellipses to compromise graph
b3.gg.map.elli <- gg.compromise.graph.out$zeMap +
                  gg.boot.graph.out.elli + label4S 
#
## ----plot with ellipse -----------
print(b3.gg.map.elli)

Some of the confidence intervals for the products are overlapping, suggesting that they are very similar based on the judges grouping. For example, SRUD SRAD and SBEA are very similar based on how the judges grouped them.

7.2.2.4 Dendogram Analysis of Products

HCA of products

##  HCA products ------
nFac4Prod = 3 # there's only 3 factors
D4Prod <- dist(
           resDistatis$res4Splus$F[,1:nFac4Prod],
           method = "euclidean")
 fit4Prod <- hclust(D4Prod, method = "ward.D2")
 b3.tree4Product <- fviz_dend(fit4Prod,  k = 1, 
       k_colors = 'burlywood4', 
      label_cols = color4Products[fit4Prod$order],
       cex = .7, xlab = 'Products',
       main = 'Cluster Analysis: Products') 

## Warning: `guides(<scale> = FALSE)` is deprecated. Please use `guides(<scale> =
## "none")` instead.

 print(b3.tree4Product)

Here, we can see that there are 4 groups in total.

7.2.3 Partial Map on Products

Partial Map: The main goal for this technique is to look at how the products are partitioned based on the judges description.

# get the partial map
map4PFS <- createPartialFactorScoresMap(
         factorScores = resDistatis$res4Splus$F,      
         partialFactorScores = F_k,  
         axis1 = 1, axis2 = 2,
         colors4Blocks = color4Judges.list$gc, 
         colors4Items = as.vector(color4Products), 
         names4Partial = dimnames(F_k)[[3]], # 
         font.labels = 'bold')
# partial maps
d1.partialFS.map.byProducts <- 
          gg.compromise.graph.out$zeMap + 
          map4PFS$mapColByItems + label4S 

d2.partialFS.map.byCategories  <- 
          gg.compromise.graph.out$zeMap + 
          map4PFS$mapColByBlocks + label4S 

## ----S with Categories, based on product names
print(d1.partialFS.map.byProducts )

## ----S with Categories.2, based on judge descriptions
print(d2.partialFS.map.byCategories)

How to interpret the map: each product has the same color of the description of the judges that influneces the grouping. The length of the line from the product determines how much it influences the product. The further away, the less influence on the grouping of the products. What is also interesting is that if we look at judges that were french and were provided wine labels (fr info) for SKLE & FHUE we can see that these two points are very close, suggesting that they are very similar.

7.3 Conclusion

For the judges descriptions:

• Dimension 1: Tells us what type of judges (FR vs. SA & Info vs. No Info) and how similar they are based on how they grouped the products

For the products: how similar the products are based on how they were grouped by the judges

• Dimension 1 explains how different the wines were based on how they were grouped by the judges

• Dimension 2 explains the description of the judges (SA vs. F & info vs no info)

Conclusion:

• Some of the judges were similar based on how they grouped the products.

• Some of the products are similar based on how they were grouped by the judges.

Overall, some of the wines are very similar in the way they are grouped based on how the judges grouped them. For example, we can see that SBEA, SRUD & SRAD are very similar because they are very close. In the other hand, FCLP is very different compared to the others.

Chapter 8 Multiple Factor Analysis

Multiple Factor Analysis (MFA) is a technique that based in the roots of PCA (Also MCA, if using qualitative data). MFA is also similar to DiSTATIS in that it can be used to analyze rows/observations and columns/variables to see the difference between rows/observations and columns/variables based on pre-assigned group. However, the difference lies in how the data is pre-processed before being fed into the function. MFA can take more than 3 data tables of the same rows/observations.

Step by step process:

1. Perform a PCA without scale on the individual tables of the same observation to find the weights by taking inverse of the first eigen values of each group of variables.

2. Normalize each cell data table by applying weights to them.

3. Combine all cell data tables as grand table.

4. Perform GPCA (grand PCA) on the grand table (all individual data tables of each candidate’s rating on the same set of products).

8.1 Data set: Wine

This data set is a wine data set where judges sort the different kind of wines into groups. The wines are either French or from South Africa. Judges were were 1 in 4 conditions: Judges from France Vs. South Africa and judges with vs without information (labels of the wine).

data <- read.csv("French and South African Wines C.csv")
data <- as.data.frame(data)


#remove the first column
ratings <- data[,-1]

#add the first column values as row names
rownames(ratings) <- data[,1]

ratings_2 <- ratings[-15,]

ratings_3 <- sapply(ratings_2, as.numeric )

ratings_3 <- as.data.frame(ratings_3)

rownames(ratings_3) <- rownames(ratings_2)

ratings_matrix <- as.matrix(ratings_3)

# Convert judge info into a separate data table
judge <- ratings[15,]
judge_t <- as.data.frame(t(judge))

judge_list <- judge_t[,1]

View the data

# a <- knitr::kable(data, format = "html")
# kableExtra::scroll_box(a, width = "500px", height = "500px", fixed_thead = T)

a <- data
dplyr::glimpse(a)

## Rows: 15
## Columns: 57
## $ X   <chr> "FCAR", "SRUD", "FBAU", "FROC", "SFED", "SREY", "SKLE", "FCLL", "F…
## $ J1  <chr> "2", "1", "1", "1", "1", "3", "1", "2", "2", "3", "3", "2", "3", "…
## $ J2  <chr> "2", "1", "1", "1", "2", "1", "1", "2", "1", "1", "2", "2", "2", "…
## $ J3  <chr> "2", "5", "3", "2", "4", "2", "5", "4", "5", "1", "3", "4", "1", "…
## $ J4  <chr> "3", "3", "2", "2", "2", "2", "3", "3", "1", "1", "3", "3", "3", "…
## $ J5  <chr> "4", "5", "3", "2", "1", "5", "5", "4", "2", "2", "3", "2", "3", "…
## $ J6  <chr> "1", "4", "3", "2", "3", "3", "2", "2", "2", "1", "4", "3", "4", "…
## $ J7  <chr> "3", "1", "3", "2", "2", "1", "3", "2", "1", "2", "1", "3", "3", "…
## $ J8  <chr> "1", "3", "2", "2", "3", "1", "1", "1", "1", "1", "3", "2", "2", "…
## $ J9  <chr> "2", "3", "1", "3", "1", "1", "1", "2", "3", "1", "1", "2", "2", "…
## $ J10 <chr> "1", "1", "1", "2", "2", "2", "3", "3", "3", "4", "4", "4", "4", "…
## $ J11 <chr> "4", "3", "2", "4", "2", "4", "3", "1", "1", "2", "2", "4", "4", "…
## $ J12 <chr> "4", "1", "3", "2", "4", "1", "1", "3", "3", "3", "3", "2", "2", "…
## $ J13 <chr> "2", "4", "2", "1", "3", "3", "1", "1", "4", "4", "3", "3", "2", "…
## $ J14 <chr> "5", "4", "3", "5", "6", "6", "6", "4", "2", "4", "4", "3", "5", "…
## $ J15 <chr> "3", "2", "2", "4", "3", "4", "4", "1", "1", "2", "2", "4", "3", "…
## $ J16 <chr> "1", "2", "2", "2", "1", "3", "3", "3", "3", "3", "2", "1", "3", "…
## $ J17 <chr> "4", "4", "2", "3", "3", "2", "2", "3", "4", "1", "4", "1", "1", "…
## $ J18 <chr> "1", "2", "3", "2", "3", "3", "1", "2", "3", "2", "3", "1", "3", "…
## $ J19 <chr> "4", "1", "5", "4", "1", "4", "2", "2", "3", "2", "2", "4", "3", "…
## $ J20 <chr> "2", "3", "2", "2", "1", "1", "1", "3", "3", "3", "1", "3", "3", "…
## $ J21 <chr> "1", "2", "4", "2", "1", "4", "3", "2", "1", "1", "4", "3", "3", "…
## $ J22 <chr> "1", "1", "2", "2", "2", "3", "3", "4", "3", "2", "3", "1", "4", "…
## $ J23 <chr> "3", "3", "3", "1", "2", "2", "1", "3", "1", "3", "3", "2", "1", "…
## $ J24 <chr> "3", "2", "3", "1", "3", "1", "2", "3", "4", "2", "4", "1", "1", "…
## $ J25 <chr> "4", "4", "4", "3", "2", "3", "2", "4", "1", "3", "2", "2", "1", "…
## $ J26 <chr> "2", "1", "3", "2", "1", "3", "4", "4", "1", "2", "3", "1", "1", "…
## $ J27 <chr> "2", "3", "2", "3", "2", "2", "3", "1", "2", "1", "1", "3", "2", "…
## $ J28 <chr> "2", "1", "1", "2", "1", "1", "2", "4", "4", "3", "3", "1", "3", "…
## $ J29 <chr> "2", "8", "7", "4", "7", "1", "2", "2", "9", "5", "6", "6", "3", "…
## $ J30 <chr> "1", "1", "2", "3", "2", "1", "1", "4", "4", "2", "1", "3", "4", "…
## $ J31 <chr> "2", "1", "1", "2", "2", "2", "2", "2", "1", "1", "1", "3", "4", "…
## $ J32 <chr> "3", "2", "3", "1", "1", "1", "1", "3", "3", "2", "2", "3", "3", "…
## $ J33 <chr> "2", "2", "3", "1", "5", "4", "5", "5", "6", "2", "3", "2", "1", "…
## $ J34 <chr> "2", "1", "1", "3", "4", "5", "4", "3", "5", "4", "2", "3", "5", "…
## $ J35 <chr> "4", "3", "2", "4", "1", "5", "5", "6", "6", "3", "7", "8", "4", "…
## $ J36 <chr> "3", "2", "3", "1", "4", "2", "2", "3", "1", "2", "2", "3", "1", "…
## $ J37 <chr> "1", "4", "3", "1", "1", "2", "1", "2", "3", "4", "4", "1", "2", "…
## $ J38 <chr> "1", "2", "1", "2", "1", "3", "4", "3", "2", "3", "3", "3", "1", "…
## $ J39 <chr> "5", "5", "5", "7", "2", "6", "3", "5", "4", "4", "6", "1", "5", "…
## $ J40 <chr> "4", "1", "4", "3", "4", "1", "1", "2", "3", "1", "4", "2", "2", "…
## $ J41 <chr> "5", "4", "3", "5", "2", "1", "1", "5", "5", "2", "4", "5", "5", "…
## $ J42 <chr> "8", "3", "8", "8", "5", "8", "4", "6", "6", "1", "1", "2", "8", "…
## $ J43 <chr> "1", "5", "4", "2", "4", "3", "2", "1", "5", "5", "1", "3", "1", "…
## $ J44 <chr> "1", "2", "1", "3", "1", "5", "5", "4", "4", "2", "2", "3", "4", "…
## $ J45 <chr> "4", "4", "3", "4", "5", "2", "5", "3", "1", "4", "5", "4", "3", "…
## $ J46 <chr> "1", "5", "6", "4", "1", "2", "1", "7", "2", "1", "2", "3", "2", "…
## $ J47 <chr> "5", "4", "3", "1", "4", "5", "3", "1", "2", "4", "4", "3", "3", "…
## $ J48 <chr> "1", "1", "3", "1", "2", "3", "3", "2", "1", "2", "2", "1", "1", "…
## $ J49 <chr> "2", "4", "2", "1", "1", "4", "3", "3", "3", "4", "4", "2", "1", "…
## $ J50 <chr> "4", "1", "1", "4", "7", "1", "2", "3", "2", "4", "1", "6", "2", "…
## $ J51 <chr> "3", "5", "5", "2", "1", "5", "5", "3", "5", "5", "2", "4", "4", "…
## $ J52 <chr> "1", "3", "3", "3", "2", "1", "3", "1", "1", "2", "3", "2", "1", "…
## $ J53 <chr> "3", "4", "1", "1", "2", "4", "1", "1", "3", "4", "2", "2", "3", "…
## $ J54 <chr> "4", "3", "2", "3", "5", "5", "4", "2", "3", "1", "1", "4", "5", "…
## $ J55 <chr> "1", "3", "5", "2", "1", "5", "5", "2", "3", "4", "4", "4", "3", "…
## $ J56 <chr> "2", "1", "1", "1", "1", "2", "1", "1", "1", "2", "3", "1", "2", "…

variables for later use

# get the parameters
nJudges     <- length(ratings_3)
nProducts   <- nrow(ratings_3)
type <- rep("c", nJudges)
vector_judge <- rep(1, each = 56)
names(vector_judge) <- colnames(ratings_3)

nVar4Judges <- vector_judge
ratings     <- ratings_3
namesOfJudges <- colnames(ratings_3)

8.2 MFA

8.2.1 run MFA

nVar4Judges

##  J1  J2  J3  J4  J5  J6  J7  J8  J9 J10 J11 J12 J13 J14 J15 J16 J17 J18 J19 J20 
##   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1 
## J21 J22 J23 J24 J25 J26 J27 J28 J29 J30 J31 J32 J33 J34 J35 J36 J37 J38 J39 J40 
##   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1 
## J41 J42 J43 J44 J45 J46 J47 J48 J49 J50 J51 J52 J53 J54 J55 J56 
##   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1

resMFA <- FactoMineR::MFA(ratings_matrix,
     group = nVar4Judges, #how you would like to group your columns (each column is 1 group)
     type = rep("s", nJudges), #type of data for your grouping of columns
     name.group = namesOfJudges,
     graph = TRUE  # TRUE first pass only
)

Partial Axes: We can see which judges are more related compared to others. Judges that are closer together are similar. While judges that are opposite directions, are not similar. We can also see which dimension contributes the most from the judges based on how long the lines are.

Individual Factor Scores: This map tells us how the products are grouped based on the judges’ grouping.

Correlation Circle: This map tells us how similar or different the judges are. Judges that are very close are similar for example the top left group of judegs are similar. Judges that are different are opposite directions.

Individual factor map: This map shows us how similar or different the products based on how the judges grouped them. We can see that SBEA & SRAA is very similar. SREY and FCLL is different because they are further away from each other.

Group Representation: This map shows how similar or different on how the judges’ grouped the products. Most of the judges were very similar on how they grouped the products. But some were very different as well. We can see that J44 grouped things different compared to the rest. J16 is a bit further away from the rest of the judges, so it’s a little different.This map is suppose to group the columns the ways you want it to look like. For example, a judge can have multiple columns with each column having different groupings (salty, sweet, aroma) on the product. However, in our case each individual judge only gives one grouping for each product.

8.2.2 Distatis

Now we are going to fit the MFA values into diSTATIS

# Run Distatis ----
## Distances
# Get the cube of distances for distatis
cubeOfDistances <- array(data = NA,
      dim = c(nProducts, nProducts, nJudges),
      dimnames = list(rownames(ratings), 
      rownames(ratings), namesOfJudges )
)

lindex <- cumsum(c(0,nVar4Judges))
for (k in 1: nJudges){
  mat_k <- ratings[,(lindex[k] + 1) : lindex[k + 1]]
  norm_mat_k <- scale0(mat_k) 
  # NB Squared Euclidean
  cubeOfDistances[,,k] <- as.matrix(dist(norm_mat_k)^2)
}

# distatis ----
resDistatis <- distatis(cubeOfDistances)
# **** Graphs ----
color4Judges   <-  prettyGraphsColorSelection(n.colors = ncol(ratings_3))
color4Products <- prettyGraphsColorSelection(n.colors = nrow(ratings_3))
# ... MFA ----
val.p   <- resMFA$eig[,1]
val.tau <- resMFA$eig[,2] 
ctr.Judges.mfa <- resMFA$group$coord

8.2.3 Scree plot

scree.mfa <- PlotScree(
  ev = val.p, 
  title = "MFA: Explained Variance per Dimension",
  plotKaiser = T)

a.a0001.Scree.mfa <- recordPlot() # Save the plot

It seems like the first 7 dimensions are significant based on the kaiser line. However we will look at the first two only as they explain the most variance.

8.2.4 Judge Factor Plot

## ----RVGplot------------------------------------
# get the eigenvalues for RV
RV.eig <- eigen(resMFA$group$RV[1:nJudges, 1:nJudges], 
                symmetric = TRUE)
G.mfa  <- firstpos(RV.eig$vectors) %*% 
                        diag(RV.eig$values^(1/2))
rownames(G.mfa) <- namesOfJudges
colnames(G.mfa) <- paste0('Dimension ', 1:ncol(G.mfa))
mfa.rv.eig <- RV.eig$values
mfa.rv.tau <-  round(100 * mfa.rv.eig / sum(mfa.rv.eig))

# Create the layers of the map
gg.rv.graph.out.mfa <- createFactorMap(
  X = as.data.frame(G.mfa), 
  axis1 = 1, axis2 = 2, 
  title = "MFA. Judges: RVMap", 
  col.points = color4Judges, 
  col.labels = color4Judges)
# create the labels for the dimensions of the RV map
labels4RV.mfa <- createxyLabels.gen(
  lambda =  mfa.rv.eig, 
  tau    =  mfa.rv.tau,
  axisName = "Dimension ")
# # Create the map from the layers
# Here with labels and dots
a.a2a.gg.RVmap.mfa <- gg.rv.graph.out.mfa$zeMap + 
                        labels4RV.mfa
print(a.a2a.gg.RVmap.mfa)

As you can see here, the RV map for the different judges is difficult to interpret because we have so many (56!). The general idea is that all judges are on the right side of the graph (same dimensions), suggesting that they are somewhat related at least. If they were on the oppose side of the graph (different dimensions), then they are not similar. However, we can compare each individual judge if needed.

What is also interesting is that dimension 1 and 2 are almost equal in terms of how much variance is being explained. This is possibly due to that we have a lot of plots in this map.

8.2.5 Product Factor Scores

# Global Factor Scores ----
constraints.mfa <- minmaxHelper(resMFA$ind$coord.partiel)
F.mfa <- resMFA$ind$coord
# To get graphs with axes 1 and 2:
h_axis = 1
v_axis = 2
genTitle4Compromise = 'Compromise / Global Map. mfa'
gg.compromise.graph.out.mfa <- createFactorMap(
  F.mfa,
  axis1 = h_axis, 
  axis2 = v_axis,
  title = genTitle4Compromise,
  col.points = color4Products ,
  col.labels = color4Products,
  constraints = constraints.mfa)
label4S.mfa <- createxyLabels.gen(
  x_axis   = h_axis, y_axis = v_axis,
  lambda   = resMFA$eig[,1] , 
  tau      = round(resMFA$eig[,2]),
  axisName = "Dimension ")
b2.gg.Smap.mfa <-  
  gg.compromise.graph.out.mfa$zeMap + label4S.mfa 
print(b2.gg.Smap.mfa)

From this map, we can that the products are very similar based on the judges’ grouping. However, we can possible say that SFED is different compared to SREY while FHUE is different compared to FCLP.

8.2.6 Partial Factor Scores

# Format MFA results to match distatis
# Partial coordinates in MFA
F_long <- resMFA$ind$coord.partiel 
F_k.mfa <-  array(data = NA,
                  dim = c(nProducts, ncol(F_long), nJudges),
                  dimnames = list(rownames(ratings_matrix), 
                                  colnames(F_long), namesOfJudges)
)
for (k in 1 : nJudges){
  row2keep <- seq(k, nProducts*nJudges, nJudges)
  F_k.mfa[,,k] <- F_long[row2keep,]
}

map4PFS.mfa <- createPartialFactorScoresMap(
  factorScores = F.mfa,      
  partialFactorScores = F_k.mfa,  
  axis1 = 1, axis2 = 2,
  colors4Items = as.vector(color4Products), 
  names4Partial = dimnames(F_k.mfa)[[3]], # 
  font.labels = 'bold'
)
d1.partialFS.map.mfa.byProducts <- 
  gg.compromise.graph.out.mfa$zeMap + 
  map4PFS.mfa$mapColByItems + label4S.mfa 
d2.partialFS.map.mfa.byCategories  <- 
  gg.compromise.graph.out.mfa$zeMap + 
  map4PFS.mfa$mapColByBlocks + label4S.mfa 

#print the maps
print(d1.partialFS.map.mfa.byProducts)

print(d2.partialFS.map.mfa.byCategories)

This is so messy. I cannot interpret it. This is most likely because we have 56 judges per product and we have 14 products. Multiple those two variables and we have a huge number. However, what we can maybe see from this mess is that J31 (top left) has a different opinion of grouping the products based on how far it is away from everyone.

Circle of Correlation on how similar the judges grouped the products

# Compute correlation between variables & factors
cor.ratings <- cor(ratings, F.mfa)
col4J  <- rep(color4Judges, times = nVar4Judges)
jolie.ggplot.J <- PTCA4CATA::createFactorMap(
  cor.ratings,
  col.points = col4J, col.labels = col4J, 
  constraints = list(minx = -1, miny = -1,
                     maxx = 1 , maxy = 1),
  alpha.points = .05)
# draw the circle
e1.jolieggMap.J <- jolie.ggplot.J$zeMap + 
  addCircleOfCor() + label4S.mfa


print(e1.jolieggMap.J)

Here on the correlation circle, we can see which judges are similar vs not similar. We can see that J15 is similar to the other judges on the same plane compared to J13 and the other judges near that area.

#  Add some arrows
arrows <- addArrows(cor.ratings, color = col4J)  
e2.jolieggMap.J <- e1.jolieggMap.J + 
  arrows 
print(e2.jolieggMap.J)

This arrow of correlation plot didn’t really help much but it does help show which judges are further away from the others. For example while J15 is similar to J42 , there is a small difference as it is a bit further away.

Show only the significant values

#make no dots appear
e3.jolieggMap.J <- jolie.ggplot.J$zeMap_background +
  jolie.ggplot.J$zeMap_text + arrows +
  addCircleOfCor() + label4S.mfa 

# Gray the small values ----
corLevels <- rowSums(cor.ratings[,1:2]^2) 
threshold <-  .75
col4J.gray <- col4J
col4J.gray[corLevels < threshold] <- 'gray85'
jolie.ggplot.J.gray <- PTCA4CATA::createFactorMap(
  cor.ratings,
  col.points = col4J.gray, col.labels = col4J.gray, 
  constraints = list(minx = -1, miny = -1,
                     maxx = 1 , maxy = 1)   )
arrows.gray <- addArrows(cor.ratings, 
                         color = col4J.gray)  
e8.jolieggMap.J.gray <- 
  jolie.ggplot.J$zeMap_background +
  jolie.ggplot.J.gray$zeMap_text + 
  arrows.gray +
  addCircleOfCor() + label4S.mfa 
print(e8.jolieggMap.J.gray)

This graph shows that J15 and J16 contributes the most values, meaning they are very different compared to all the other judges and have the longest lines (furthest distance).

8.3 Distatis Comparison to MFA

8.3.1 Scree plot

scree.distatis <- PlotScree(
  ev = resDistatis$res4Cmat$eigValues, 
  title = "distatis: RV Explained Variance per Dimension",
  plotKaiser = T)

f.a0001.Scree.distatis <- recordPlot() # Save the plot

scree.distatis.S <- PlotScree(
  ev = resDistatis$res4Splus$eigValues, 
  title = "distatis: Compromise Explained Variance per Dimension",
  plotKaiser = T)

f.a0003.Scree.distatis <- recordPlot() # Save the plot

8.3.2 RV Plot

# get the eigenvalues for RV
G.distatis  <- firstpos(resDistatis$res4Cmat$G)
colnames(G.distatis) <- paste0('Dimension ',
                          1:ncol(G.distatis))
distatis.rv.eig <- resDistatis$res4Cmat$eigValues
distatis.rv.tau <-  round(100 *distatis.rv.eig / 
                          sum(distatis.rv.eig))

# Create the layers of the map
gg.rv.graph.out.distatis <- createFactorMap(
  X = as.data.frame(G.distatis), 
  axis1 = 1, axis2 = 2, 
  title = "DISTATIS. Judges: RVMap", 
  col.points = color4Judges, 
  col.labels = color4Judges)
# create the labels for the dimensions of the RV map
labels4RV.distatis <- createxyLabels.gen(
  lambda =  distatis.rv.eig, 
  tau    =  distatis.rv.tau,
  axisName = "Dimension ")
# # Create the map from the layers
# Here with labels and dots
f.a0002.a2a.gg.RVmap.distatis <- 
      gg.rv.graph.out.distatis$zeMap + 
      labels4RV.distatis

print(f.a0002.a2a.gg.RVmap.distatis)

The RV map for the judges is very similar to MFA. There is no difference.

8.3.3 Global Factor Scores

# Global Factor Scores ----
constraints.distatis <- minmaxHelper(
  matrix(resDistatis$res4Splus$PartialF[,1:2,], 
         nrow = nProducts * nJudges, ncol = 2) )
F.distatis <- resDistatis$res4Splus$F

# To get graphs with axes 1 and 2:
h_axis = 1
v_axis = 2
genTitle4Compromise.distatis = 'Compromise / Global Map. DISTATIS'
gg.compromise.graph.out.distatis <- createFactorMap(
  F.distatis,
  axis1 = h_axis, 
  axis2 = v_axis,
  title = genTitle4Compromise.distatis,
  col.points = color4Products ,
  col.labels = color4Products,
  constraints = constraints.distatis)
label4S.distatis <- createxyLabels.gen(
  x_axis   = h_axis, y_axis = v_axis,
  lambda   = resMFA$eig[,1] , 
  tau      = round(resMFA$eig[,2]),
  axisName = "Dimension ")
f2.gg.Smap.distatis <-  
     gg.compromise.graph.out.distatis$zeMap + 
     label4S.distatis 

print(f2.gg.Smap.distatis)

Here, we can see whether the products are similar or different based on how the judges grouped them. We can see FCAR and SFED are similar. This graph is not much different than MFA except it is flipped upside down.

8.3.4 Partial Factor Sctores

#Format MFA results to match distatis
# Partial coordinates in MFA

F_k.distatis <- resDistatis$res4Splus$PartialF
map4PFS.distatis <- createPartialFactorScoresMap(
  factorScores = F.distatis,      
  partialFactorScores = F_k.distatis,  
  axis1 = 1, axis2 = 2,
  colors4Items = as.vector(color4Products), 
  names4Partial = dimnames(F_k.distatis)[[3]], # 
  font.labels = 'bold'
)
h1.partialFS.map.distatis.byProducts <- 
  gg.compromise.graph.out.distatis$zeMap + 
  map4PFS.distatis$mapColByItems + 
  label4S.distatis 
h2.partialFS.map.distatis.byCategories  <- 
  gg.compromise.graph.out.distatis$zeMap + 
  map4PFS.distatis$mapColByBlocks + 
  label4S.distatis 

print(h1.partialFS.map.distatis.byProducts)

print(h2.partialFS.map.distatis.byCategories)

Again, we can’t really see what is going on here because we have 59 judges and 14 products. This map is similar to MFA.

8.3.5 Circle of Correlation Judges

# Compute correlation between variables & factors
cor.ratings.distatis <- cor(ratings, F.distatis)
col4J  <- rep(color4Judges, times = nVar4Judges)
jolie.ggplot.J.distatis <- PTCA4CATA::createFactorMap(
  cor.ratings.distatis,
  col.points = col4J, col.labels = col4J, 
  constraints = list(minx = -1, miny = -1,
                     maxx = 1 , maxy = 1))
# draw the circle
i1.jolieggMap.J.distatis <- jolie.ggplot.J.distatis$zeMap + 
  addCircleOfCor() + label4S.distatis
print(i1.jolieggMap.J.distatis) 

We can see from this graph which judges are similar based on their grouping. This map is similar to MFA but flipped upside down.

#  Add some arrows
arrows.distatis <- addArrows(cor.ratings.distatis, color = col4J)  
i2.jolieggMap.J.distatis <- i1.jolieggMap.J.distatis + 
  arrows.distatis 

#no dots
i3.jolieggMap.J.distatis <- jolie.ggplot.J.distatis$zeMap_background +
  jolie.ggplot.J.distatis$zeMap_text + arrows.distatis +
  addCircleOfCor() + label4S.distatis 

print(i3.jolieggMap.J.distatis)

Correlation graph for Distatis is similar to MFA. We can interpret that J16 is similar to J55 but a small difference due to the length of the lines.

Significant correlation plot

# Gray the small values ----
corLevels <- rowSums(cor.ratings.distatis[,1:2]^2) 
threshold <-  .75
col4J.gray <- col4J
col4J.gray[corLevels < threshold] <- 'gray85'
jolie.ggplot.J.gray.distatis <- PTCA4CATA::createFactorMap(
  cor.ratings.distatis,
  col.points = col4J.gray, col.labels = col4J.gray, 
  constraints = list(minx = -1, miny = -1,
                     maxx = 1 , maxy = 1)   )
arrows.gray.distatis <- addArrows(cor.ratings.distatis, 
                                  color = col4J.gray)  
i8.jolieggMap.J.gray.distatis <- 
  jolie.ggplot.J.distatis$zeMap_background +
  jolie.ggplot.J.gray.distatis$zeMap_text + 
  arrows.gray.distatis +
  addCircleOfCor() + label4S.distatis 
print(i8.jolieggMap.J.gray.distatis)

Similar to MFA, only J15 and J16 contribute the most variance (the most different) compared to the other judges. The only difference is the map is switched upside down compared to the MFA

8.4 Conclusion

From our results, we can see that certain judges provide more variance to the grouping of the products compared to others (J16 & J15).

We can see that MFA and DiSTATIS shows similar analysis. The only difference between the two is that MFA is pre-procseesed before putting it into DiSTATIS function. Furthermore, the maps are showing the same results. Only difference is the maps are flipped upside down.