# Chapter 4 Analyzing Survey Data

This is a tutorial for using the survey package to analyze complex survey data. “Complex” surveys are those with stratification and/or clustering. The package handles weights, and adjusts statistical tests for the survey design.

## 4.1 Defining the Survey Design

Below are some examples defining the most common survey designs. The survey package includes the Student performance in California schools data set (api), a record of the Academic Performance Index based on standardized testing. api contains sub-data sets that illustrate the design types.

• apisrs is a simple random sample of (n = 200) schools,
• apistrat is stratified sample of 3 school types (elementary, middle, high) with simple random sampling of different sizes in each stratum,
• apiclus2 is a two-stage cluster sample of schools within districts.

You create a survey design object with the svydesign(data, ...) function. There are parameter settings for each design type.

### 4.1.1 Simple Random Sample

A simple random sample has no clusters, so indicate this with ids = ~1. The response weights will always be the same, equaling the population size divided by the sample size. Typically, the response weight is identified in a column. There is another parameter called the finite population correction (fpc) that is used to reduce the variance when a substantial fraction of the total population has been sampled. Set fpc to the stratum population size. A simple random sample has no strata, so it will always be the same, equaling the population size.

For apisrs the population size is 6,194 (the number of schools in California). The sample size is 200, so the response weights all equal 6,194 / 200 = 30.97.

nrow(apisrs)
## [1] 200
apisrs %>% count(pw, fpc)
##      pw  fpc   n
## 1 30.97 6194 200

Here is the design object.

apisrs_design <- svydesign(
data = apisrs,
weights = ~pw,
fpc = ~fpc,
ids = ~1
)
summary(apisrs_design)
## Independent Sampling design
## svydesign(data = apisrs, weights = ~pw, fpc = ~fpc, ids = ~1)
## Probabilities:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
## 0.03229 0.03229 0.03229 0.03229 0.03229 0.03229
## Population size (PSUs): 6194
## Data variables:
##  [1] "cds"      "stype"    "name"     "sname"    "snum"     "dname"
##  [7] "dnum"     "cname"    "cnum"     "flag"     "pcttest"  "api00"
## [13] "api99"    "target"   "growth"   "sch.wide" "comp.imp" "both"
## [19] "awards"   "meals"    "ell"      "yr.rnd"   "mobility" "acs.k3"
## [25] "acs.46"   "acs.core" "pct.resp" "not.hsg"  "hsg"      "some.col"
## [37] "api.stu"  "pw"       "fpc"

### 4.1.2 Stratified Sample

Define a stratified sample by specifying with the strata parameter. The schools in apistrat are stratified based on the school type E = Elementary, M = Middle, and H = High School. For each school type, a simple random sample of schools was taken: $$n_E$$ = 100, $$n_M$$ = 50, and $$n_H$$ = 50. The 100 elementary schools represent 100 / 4,421 of the state’s elementary schools, so their weights = 44.21. Similarly, the weights are 50 / 1,018 = 20.36 for the middle schools, and 50 / 755 = 15.10 for the high schools.

apistrat %>%
count(stype, pw, fpc) %>%
mutate(pw*n = pw * n) %>%
flextable() %>% colformat_num(j = 2, digits = 2) %>% colformat_int(j = c(3:5))

stype

pw

fpc

n

pw*n

E

44.2099990844727

4,421

100

4,421

H

15.1000003814697

755

50

755

M

20.3600006103516

1,018

50

1,018

Total

-

6,194

200

6,194

Here is the design object.

apistrat_design <- svydesign(
data = apistrat,
weights = ~pw,
fpc = ~fpc,
ids = ~1,
strata = ~stype
)
summary(apistrat_design)
## Stratified Independent Sampling design
## svydesign(data = apistrat, weights = ~pw, fpc = ~fpc, ids = ~1,
##     strata = ~stype)
## Probabilities:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
## 0.02262 0.02262 0.03587 0.04014 0.05339 0.06623
## Stratum Sizes:
##              E  H  M
## obs        100 50 50
## design.PSU 100 50 50
## actual.PSU 100 50 50
## Population stratum sizes (PSUs):
##    E    H    M
## 4421  755 1018
## Data variables:
##  [1] "cds"      "stype"    "name"     "sname"    "snum"     "dname"
##  [7] "dnum"     "cname"    "cnum"     "flag"     "pcttest"  "api00"
## [13] "api99"    "target"   "growth"   "sch.wide" "comp.imp" "both"
## [19] "awards"   "meals"    "ell"      "yr.rnd"   "mobility" "acs.k3"
## [25] "acs.46"   "acs.core" "pct.resp" "not.hsg"  "hsg"      "some.col"
## [37] "api.stu"  "pw"       "fpc"

### 4.1.3 Clustered

Define a clustered sample by specifying the the cluster ids from largest to smallest level. The schools in apiclus2 are clustered in two stages, first by the (fpc1 = 757) school districts and a random sample of (n = 40) school districts (dnum) were selected. Then a random sample of (n <= 5) schools (snum) were selected from the fpc2 schools in the selected school districts.

apiclus_design <- svydesign(
id = ~dnum + snum,
data = apiclus2,
weights = ~pw,
fpc = ~fpc1 + fpc2
)
summary(apiclus_design)
## 2 - level Cluster Sampling design
## With (40, 126) clusters.
## svydesign(id = ~dnum + snum, data = apiclus2, weights = ~pw,
##     fpc = ~fpc1 + fpc2)
## Probabilities:
##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max.
## 0.003669 0.037743 0.052840 0.042390 0.052840 0.052840
## Population size (PSUs): 757
## Data variables:
##  [1] "cds"      "stype"    "name"     "sname"    "snum"     "dname"
##  [7] "dnum"     "cname"    "cnum"     "flag"     "pcttest"  "api00"
## [13] "api99"    "target"   "growth"   "sch.wide" "comp.imp" "both"
## [19] "awards"   "meals"    "ell"      "yr.rnd"   "mobility" "acs.k3"
## [25] "acs.46"   "acs.core" "pct.resp" "not.hsg"  "hsg"      "some.col"
## [37] "api.stu"  "pw"       "fpc1"     "fpc2"

### 4.1.4 Example: NHANES

Let’s create a complex survey design for the National Health and Nutrition Examination Survey (NHANES). The survey collected 78 attributes of (n = 20,293) persons.

data(NHANESraw, package = "NHANES")
NHANESraw <- NHANESraw %>%
mutate(WTMEC4YR = WTMEC2YR / 2) # correction to weights

The survey used a 4-stage design: stage 0 stratified the US by geography and proportion of minority populations; stage 1 randomly selected counties within strata; stage 2 randomly seleted city blocks within counties; stage 3 randomly selected households within city blocks; and stage 4 randomly selected persons within households. When there are multiple levels of clusters like this, the convention is to assign the first cluster to ids. Set nest = TRUE because the cluster ids are nested within the strata (i.e., they are not unique).

NHANES_design <- svydesign(
data = NHANESraw,
strata = ~SDMVSTRA,
ids = ~SDMVPSU,
nest = TRUE,
weights = ~WTMEC4YR
)
summary(NHANES_design)
## Stratified 1 - level Cluster Sampling design (with replacement)
## With (62) clusters.
## svydesign(data = NHANESraw, strata = ~SDMVSTRA, ids = ~SDMVPSU,
##     nest = TRUE, weights = ~WTMEC4YR)
## Probabilities:
##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max.
## 8.986e-06 5.664e-05 1.054e-04       Inf 1.721e-04       Inf
## Stratum Sizes:
##             75  76  77  78  79  80  81  82  83  84  85  86  87  88  89  90  91
## obs        803 785 823 829 696 751 696 724 713 683 592 946 598 647 251 862 998
## design.PSU   2   2   2   2   2   2   2   2   2   2   2   3   2   2   2   3   3
## actual.PSU   2   2   2   2   2   2   2   2   2   2   2   3   2   2   2   3   3
##             92  93  94  95  96  97  98  99 100 101 102 103
## obs        875 602 688 722 676 608 708 682 700 715 624 296
## design.PSU   3   2   2   2   2   2   2   2   2   2   2   2
## actual.PSU   3   2   2   2   2   2   2   2   2   2   2   2
## Data variables:
##  [1] "ID"               "SurveyYr"         "Gender"           "Age"
##  [5] "AgeMonths"        "Race1"            "Race3"            "Education"
##  [9] "MaritalStatus"    "HHIncome"         "HHIncomeMid"      "Poverty"
## [13] "HomeRooms"        "HomeOwn"          "Work"             "Weight"
## [17] "Length"           "HeadCirc"         "Height"           "BMI"
## [21] "BMICatUnder20yrs" "BMI_WHO"          "Pulse"            "BPSysAve"
## [25] "BPDiaAve"         "BPSys1"           "BPDia1"           "BPSys2"
## [29] "BPDia2"           "BPSys3"           "BPDia3"           "Testosterone"
## [33] "DirectChol"       "TotChol"          "UrineVol1"        "UrineFlow1"
## [37] "UrineVol2"        "UrineFlow2"       "Diabetes"         "DiabetesAge"
## [45] "Depressed"        "nPregnancies"     "nBabies"          "Age1stBaby"
## [49] "SleepHrsNight"    "SleepTrouble"     "PhysActive"       "PhysActiveDays"
## [53] "TVHrsDay"         "CompHrsDay"       "TVHrsDayChild"    "CompHrsDayChild"
## [57] "Alcohol12PlusYr"  "AlcoholDay"       "AlcoholYear"      "SmokeNow"
## [61] "Smoke100"         "SmokeAge"         "Marijuana"        "AgeFirstMarij"
## [65] "RegularMarij"     "AgeRegMarij"      "HardDrugs"        "SexEver"
## [69] "SexAge"           "SexNumPartnLife"  "SexNumPartYear"   "SameSex"
## [73] "SexOrientation"   "WTINT2YR"         "WTMEC2YR"         "SDMVPSU"
## [77] "SDMVSTRA"         "PregnantNow"      "WTMEC4YR"

Survey weights for minorities are typically lower to account for their large sample sizes relative to population representation. You can see how the weights sum to the sub-populations and the total population.

NHANESraw %>%
group_by(Race1) %>%
summarize(.groups = "drop",
Sum(WTMEC4YR) = sum(WTMEC4YR),
Avg(WTMEC4YR) = mean(WTMEC4YR),
n = n()) %>%
mutate(Avg * n = Avg(WTMEC4YR) * n) %>%
flextable::flextable() %>%
flextable::colformat_int(j = c(2:5))

Race1

Sum(WTMEC4YR)

Avg(WTMEC4YR)

n

Avg * n

Black

37,241,616

8,026.210

4,640

37,241,616

Hispanic

18,951,150

8,579.063

2,209

18,951,150

Mexican

30,719,158

8,215.875

3,739

30,719,158

White

193,966,274

26,236.477

7,393

193,966,274

Other

23,389,002

10,116.350

2,312

23,389,002

Total

304,267,200

61,173.976

20,293

304,267,200

The survey package functions handle the survey designs and weights. The population figures from the table above could have been built with svytable().

svytable(~Race1, design = NHANES_design) %>%
as.data.frame() %>%
mutate(prop = Freq / sum(Freq) * 100) %>%
arrange(desc(prop)) %>%
flextable() %>%
colformat_int(j = 2) %>%
colformat_num(j = 3, suffix = "%", digits = 0)

Race1

Freq

prop

White

193,966,274

63.748664%

Black

37,241,616

12.239773%

Mexican

30,719,158

10.096112%

Other

23,389,002

7.686994%

Hispanic

18,951,150

6.228456%

Total

304,267,200

100.000000%

## 4.2 Exploring Categorical Items

Create a contingency table by including two variables in svytable(). Here is contingency table for self-reported health by depression expressed as a 100% stacked bar chart.

svytable(~Depressed + HealthGen, design = NHANES_design) %>%
data.frame() %>%
group_by(HealthGen) %>%
mutate(n_HealthGen = sum(Freq), Prop_Depressed = Freq / sum(Freq)) %>%
ggplot(aes(x = HealthGen, y = Prop_Depressed, fill = Depressed)) +
geom_col() +
coord_flip() +
theme_minimal() +
scale_fill_brewer()

Perform a chi-square test of independence on contingency tables using the svychisq() function. Here is a test ofthe null hypothesis that depression is independent of general health.

svychisq(~Depressed + HealthGen, design = NHANES_design, statistic = "Chisq")
##
##  Pearson's X^2: Rao & Scott adjustment
##
## data:  svychisq(~Depressed + HealthGen, design = NHANES_design, statistic = "Chisq")
## X-squared = 1592.7, df = 8, p-value < 2.2e-16

The chi-square test with Rao & Scott adjustment is evidently not a standard chi-square test. Maybe in how it factors in survey design? The test statistic is usually $$X^2 = \sum (O - E)^2 / E.$$

O <- svytable(~Depressed + HealthGen, design = NHANES_design) %>% as.matrix()
E <- sum(O) * prop.table(O, 1) * prop.table(O, 2)
(X2 <- sum((O - E)^2 / E))
## [1] 16025254
pchisq(X2, df = (nrow(O)-1) * (ncol(O) - 1), lower.tail = FALSE)
## [1] 0

which is what chisq.test() does.

svytable(~Depressed + HealthGen, design = NHANES_design) %>%
as.matrix() %>%
chisq.test()
##
##  Pearson's Chi-squared test
##
## data:  .
## X-squared = 16025254, df = 8, p-value < 2.2e-16

## 4.3 Exploring Quantitative Data

The svymean(), svytotal(), and svyquantile() functions summarize quantitative variables. To group by a factor variable, use svyby().

svyquantile(x = ~SleepHrsNight,
design = NHANES_design,
na.rm = TRUE,
quantiles = c(.01, .25, .50, .75, .99))
## $SleepHrsNight ## quantile ci.2.5 ci.97.5 se ## 0.01 4 4 5 0.2457588 ## 0.25 6 6 7 0.2457588 ## 0.5 7 7 8 0.2457588 ## 0.75 8 8 9 0.2457588 ## 0.99 10 10 11 0.2457588 ## ## attr(,"hasci") ## [1] TRUE ## attr(,"class") ## [1] "newsvyquantile" svymean(x = ~SleepHrsNight, design = NHANES_design, na.rm = TRUE) ## mean SE ## SleepHrsNight 6.9292 0.0166 svyby(formula = ~SleepHrsNight, by = ~Depressed, FUN = svymean, design = NHANES_design, na.rm = TRUE, keep.names = FALSE) %>% ggplot(aes(x = Depressed, y = SleepHrsNight, ymin = SleepHrsNight - 2*se, ymax = SleepHrsNight + 2*se)) + geom_col(fill = "lightblue") + geom_errorbar(width = 0.5) You need raw data for the distribution plots, so be sure to weight the variables. NHANESraw %>% ggplot(aes(x = SleepHrsNight, weight = WTMEC4YR)) + geom_histogram(binwidth = 1, fill = "lightblue", color = "#FFFFFF", na.rm = TRUE) NHANESraw %>% filter(!is.na(SleepHrsNight) & !is.na(Gender)) %>% group_by(Gender) %>% mutate(WTMEC4YR_std = WTMEC4YR / sum(WTMEC4YR)) %>% ggplot(aes(x = SleepHrsNight, Weight = WTMEC4YR_std)) + geom_density(bw = 0.6, fill = "lightblue") + labs(x = "Sleep Hours per Night") + facet_wrap(~Gender, labeller = "label_both") ## Warning: The following aesthetics were dropped during ## statistical transformation: Weight ## ℹ This can happen when ggplot fails to infer the ## correct grouping structure in the data. ## ℹ Did you forget to specify a group aesthetic or ## to convert a numerical variable into a factor? ## The following aesthetics were dropped during ## statistical transformation: Weight ## ℹ This can happen when ggplot fails to infer the ## correct grouping structure in the data. ## ℹ Did you forget to specify a group aesthetic or ## to convert a numerical variable into a factor? Test whether the population averages differ with a two-sample survey-weighted t-test. Use the svytest() function to incorporate the survey design. svyttest(formula = SleepHrsNight ~ Gender, design = NHANES_design) ## Warning in summary.glm(g): observations with zero weight not used for ## calculating dispersion ## Warning in summary.glm(glm.object): observations with zero weight not used for ## calculating dispersion ## ## Design-based t-test ## ## data: SleepHrsNight ~ Gender ## t = -3.4077, df = 32, p-value = 0.001785 ## alternative hypothesis: true difference in mean is not equal to 0 ## 95 percent confidence interval: ## -0.15506427 -0.03904047 ## sample estimates: ## difference in mean ## -0.09705237 ## 4.4 Modeling Quantitative Data Scatterplots need to adjust for the sampling weights. You can do this with the size or alpha aesthetics. p1 <- NHANESraw %>% filter(Age == 20) %>% ggplot(aes(x = Height, y = Weight, color = Gender, size = WTMEC4YR)) + geom_jitter(width = 0.3, height = 0, alpha = 0.3) + guides(size = FALSE) + theme(legend.position = "top") + labs(color = "") ## Warning: The <scale> argument of guides() cannot be ## FALSE. Use "none" instead as of ggplot2 3.3.4. p2 <- NHANESraw %>% filter(Age == 20) %>% ggplot(aes(x = Height, y = Weight, color = Gender, alpha = WTMEC4YR)) + geom_jitter(width = 0.3, height = 0) + guides(alpha = FALSE) + theme(legend.position = "top") + labs(color = "") gridExtra::grid.arrange(p1, p2, nrow = 1) ## Warning: Removed 4 rows containing missing values ## (geom_point()). ## Warning: Removed 4 rows containing missing values ## (geom_point()). Fit a regression line with geom_smooth(). NHANESraw %>% filter(!is.na(Weight) & !is.na(Height)) %>% ggplot(aes(x = Height, y = Weight, size = WTMEC4YR)) + geom_point(alpha = 0.1) + geom_smooth(method = "lm", se = FALSE, mapping = aes(weight = WTMEC4YR), formula = y ~ x, color = "blue") + geom_smooth(method = "lm", se = FALSE, mapping = aes(weight = WTMEC4YR), formula = y ~ poly(x, 2), color = "orange") + geom_smooth(method = "lm", se = FALSE, mapping = aes(weight = WTMEC4YR), formula = y ~ poly(x, 3), color = "red") + guides(size = FALSE)  ## Warning: Using size aesthetic for lines was deprecated in ## ggplot2 3.4.0. ## ℹ Please use linewidth instead. ## Warning: The following aesthetics were dropped during ## statistical transformation: weight, size ## ℹ This can happen when ggplot fails to infer the ## correct grouping structure in the data. ## ℹ Did you forget to specify a group aesthetic or ## to convert a numerical variable into a factor? ## The following aesthetics were dropped during ## statistical transformation: weight, size ## ℹ This can happen when ggplot fails to infer the ## correct grouping structure in the data. ## ℹ Did you forget to specify a group aesthetic or ## to convert a numerical variable into a factor? ## The following aesthetics were dropped during ## statistical transformation: weight, size ## ℹ This can happen when ggplot fails to infer the ## correct grouping structure in the data. ## ℹ Did you forget to specify a group aesthetic or ## to convert a numerical variable into a factor? Model a regression line with svyglm(). Let’s build a model to predict, BPSysAve, a person’s systolic blood pressure reading, using BPDiaAve, a person’s diastolic blood pressure reading and Diabetes, whether or not they were diagnosed with diabetes. drop_na(NHANESraw, Diabetes, BPDiaAve, BPSysAve) %>% ggplot(mapping = aes(x = BPDiaAve, y = BPSysAve, size = WTMEC4YR, color = Diabetes)) + geom_point(alpha = 0.2) + guides(size = FALSE) + geom_smooth(method = "lm", formula = y ~ x, se = FALSE, mapping = aes(weight = WTMEC4YR)) ## Warning: The following aesthetics were dropped during ## statistical transformation: weight, size ## ℹ This can happen when ggplot fails to infer the ## correct grouping structure in the data. ## ℹ Did you forget to specify a group aesthetic or ## to convert a numerical variable into a factor? mod <- svyglm(BPSysAve ~ BPDiaAve*Diabetes, design = NHANES_design) summary(mod) ## ## Call: ## svyglm(formula = BPSysAve ~ BPDiaAve * Diabetes, design = NHANES_design) ## ## Survey design: ## svydesign(data = NHANESraw, strata = ~SDMVSTRA, ids = ~SDMVPSU, ## nest = TRUE, weights = ~WTMEC4YR) ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 83.58652 2.05537 40.667 < 2e-16 *** ## BPDiaAve 0.49964 0.02623 19.047 < 2e-16 *** ## DiabetesYes 25.36616 3.56587 7.114 6.53e-08 *** ## BPDiaAve:DiabetesYes -0.22132 0.05120 -4.323 0.000156 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## (Dispersion parameter for gaussian family taken to be 279.1637) ## ## Number of Fisher Scoring iterations: 2 ## 4.5 Survey Administration ### 4.5.1 Instrument: Validity and Reliability ### 4.5.2 Data Collection The second phase of a survey analysis is to collect the responses and perform an exploratory data analysis to familiarize yourself with the data. ### 4.5.3 Frequencies brand_rep is a brand reputation survey of n = 599 respondents answering nine 5-point Likert-scale items. The responses come in as numeric, and you will want to leave them that way for most analyses. brand_rep <- read_csv(url("https://assets.datacamp.com/production/repositories/4494/datasets/59b5f2d717ddd647415d8c88aa40af6f89ed24df/brandrep-cleansurvey-extraitem.csv")) psych::response.frequencies(brand_rep) ## 1 2 3 4 5 miss ## well_made 0.6493739 0.2665474 0.04830054 0.014311270 0.02146691 0 ## consistent 0.6779964 0.2343470 0.04651163 0.019677996 0.02146691 0 ## poor_workman_r 0.7584973 0.1699463 0.04472272 0.007155635 0.01967800 0 ## higher_price 0.3130590 0.2593918 0.18425760 0.119856887 0.12343470 0 ## lot_more 0.1484794 0.1735242 0.15563506 0.162790698 0.35957066 0 ## go_up 0.1842576 0.1824687 0.19677996 0.177101968 0.25939177 0 ## stands_out 0.4275492 0.3041145 0.13953488 0.067978533 0.06082290 0 ## unique 0.4579606 0.2933810 0.11985689 0.069767442 0.05903399 0 ## one_of_a_kind 0.6225403 0.1949911 0.08944544 0.023255814 0.06976744 0 Summarize Likert response with the likert::likert() function. This is the one place where you will need the items to be treated as factors. brand_rep %>% data.frame() %>% # read_csv() returns a tibble mutate(across(everything(), as.factor)) %>% # likert() uses factors likert::likert() %>% plot() + labs(title = "Brand Reputation Survey") + theme(legend.position = "top") Missing values may mean respondents did not understand the question or did not want to reveal their answer. If <5% of survey responses have no missing values, you can just drop those responses. If missing values are a problem, try the Hmisc::naclus() to see which items tend to be missing in the same record. This survey is clean. nrow(brand_rep) - nrow(na.omit(brand_rep)) # num cases ## [1] 0 colSums(is.na(brand_rep)) # num cases by col ## well_made consistent poor_workman_r higher_price lot_more ## 0 0 0 0 0 ## go_up stands_out unique one_of_a_kind ## 0 0 0 0 ### 4.5.4 Correlations You will want to identify items that correlate highly with each other, but not highly outside their group. These patterns are the basis of mapping factors to the latent variables. Factors are the concrete survey items; latent variables are the abstract concepts they are intended to supply, like brand loyalty or customer satisfaction. The correlation plot below appears to have 3 groups, plus a stand-alone variable (one_of_a_kind). #psych::corr.test(brand_rep) corrplot::corrplot(cor(brand_rep), method = "circle") ## 4.6 Item Reduction The third phase, explores the mapping of the factors (aka “manifest variables”) to the latent variable’s “dimensions” and refines the survey to exclude factors that do not map to a dimension. A latent variable may have several dimensions. E.g., “brand loyalty” may consist of “brand identification”, “perceived value”, and “brand trust”. Exploratory factor analysis (EFA), identifies the dimensions in the data, and whether any items do not reveal information about the latent variable. EFA establishes the internal reliability, whether similar items produce similar scores. Start with a parallel analysis and scree plot. This will suggest the number of factors in the data. Use this number as the input to an exploratory factor analysis. ### 4.6.1 Parallel Analysis A scree plot is a line plot of the eigenvalues. An eigenvalue is the proportion of variance explained by each factor. Only factors with eigenvalues greater than those from uncorrelated data are useful. You want to find a sharp reduction in the size of the eigenvalues (like a cliff), with the rest of the smaller eigenvalues constituting rubble (scree!). After the eigenvalues drop dramatically in size, additional factors add relatively little to the information already extracted. Parallel analysis helps to make the interpretation of scree plots more objective. The eigenvalues are plotted along with eigenvalues of simulated variables with population correlations of 0. The number of eigenvalues above the point where the two lines intersect is the suggested number of factors. The rationale for parallel analysis is that useful factors account for more variance than could be expected by chance. psych::fa.parallel() compares a scree of your data set to a random data set to identify the number of factors. The elbow below here is at 3 factors. psych::fa.parallel(brand_rep) ## Parallel analysis suggests that the number of factors = 3 and the number of components = 2 ### 4.6.2 Exporatory Factor Analysis Use psych::fa() to perform the factor analysis with your chosen number of factors. The number of factors may be the result of your parallel analysis, or the opinion of the SMEs. In this case, we’ll go with the 3 factors identified by the parallel analysis. brand_rep_efa <- psych::fa(brand_rep, nfactors = 3) ## Loading required namespace: GPArotation # psych::scree(brand_rep) # psych makes scree plot's too. psych::fa.diagram(brand_rep_efa) Using EFA, you may tweak the number of factors or drop poorly-loading items. Each item should load highly to one and only one dimension. This one dimension is the item’s primary loading. Generally, a primary loading > .7 is excellent, >.6 is very good, >.5 is good, >.4 is fair, and <.4 is poor. Here are the factor loadings from the 3 factor model. brand_rep_efa$loadings
##
##                MR2    MR1    MR3
## consistent      0.947
## poor_workman_r  0.739
## higher_price    0.127  0.632  0.149
## lot_more               0.850
## go_up                  0.896
## stands_out                    1.008
## unique                        0.896
## one_of_a_kind   0.309  0.295  0.115
##
##                  MR2   MR1   MR3
## Proportion Var 0.262 0.224 0.206
## Cumulative Var 0.262 0.486 0.692

The brand-rep survey items load to 3 factors well except for the one_of_a_kind item. Its primary factor loading (0.309) is poor. The others are either very good (.6-.7) and excellent (>.7) range.

Look at the model eigenvalues. There should be one eigenvalue per dimension. Eigenvalues a little less than one may be contaminating the model.

brand_rep_efa$e.value ## [1] 4.35629549 1.75381015 0.98701607 0.67377072 0.39901205 0.35865598 0.23915591 ## [8] 0.14238807 0.08989556 Look at the factor score correlations. They should all be around 0.6. Much smaller means they are not describing the same latent variable. Much larger means they are describing the same dimension of the latent variable. brand_rep_efa$r.scores
##           [,1]      [,2]      [,3]
## [1,] 1.0000000 0.3147485 0.4778759
## [2,] 0.3147485 1.0000000 0.5428218
## [3,] 0.4778759 0.5428218 1.0000000

If you have a poorly loaded dimension, drop factors one at a time from the scale. one_of_a_kind loads across all three factors, but does not load strongly onto any one factor. one_of_a_kind is not clearly measuring any dimension of the latent variable. Drop it and try again.

brand_rep_efa <- psych::fa(brand_rep %>% select(-one_of_a_kind), nfactors = 3)
brand_rep_efa$loadings ## ## Loadings: ## MR2 MR3 MR1 ## well_made 0.887 ## consistent 0.958 ## poor_workman_r 0.735 ## higher_price 0.120 0.596 0.170 ## lot_more 0.845 ## go_up 0.918 ## stands_out 0.990 ## unique 0.916 ## ## MR2 MR3 MR1 ## SS loadings 2.261 1.915 1.850 ## Proportion Var 0.283 0.239 0.231 ## Cumulative Var 0.283 0.522 0.753 brand_rep_efa$e.value
## [1] 4.00884429 1.75380537 0.97785883 0.42232527 0.36214534 0.24086772 0.14404264
## [8] 0.09011054
brand_rep_efa$r.scores ## [,1] [,2] [,3] ## [1,] 1.0000000 0.2978009 0.4802138 ## [2,] 0.2978009 1.0000000 0.5351654 ## [3,] 0.4802138 0.5351654 1.0000000 This is better. We have three dimensions of brand reputation: • items well_made, consistent, and poor_workman_r describe Product Quality, • items higher_price, lot_more, and go_up describe Willingness to Pay, and • items stands_out and unique describe Product Differentiation Even if the data and your theory suggest otherwise, explore what happens when you include more or fewer factors in your EFA. psych::fa(brand_rep, nfactors = 2)$loadings
##
##                MR1    MR2
## consistent             0.932
## poor_workman_r         0.728
## higher_price    0.745
## lot_more        0.784 -0.131
## go_up           0.855 -0.103
## stands_out      0.591  0.286
## unique          0.540  0.307
## one_of_a_kind   0.378  0.305
##
##                  MR1   MR2
## Proportion Var 0.299 0.276
## Cumulative Var 0.299 0.575
psych::fa(brand_rep, nfactors = 4)$loadings ## ## Loadings: ## MR2 MR1 MR3 MR4 ## well_made 0.879 ## consistent 0.964 ## poor_workman_r 0.732 ## higher_price 0.552 0.150 0.168 ## lot_more 0.835 ## go_up 0.929 ## stands_out 0.976 ## unique 0.932 ## one_of_a_kind 0.999 ## ## MR2 MR1 MR3 MR4 ## SS loadings 2.244 1.866 1.846 1.029 ## Proportion Var 0.249 0.207 0.205 0.114 ## Cumulative Var 0.249 0.457 0.662 0.776 The two-factor loading worked okay. The 4 factor loading only loaded one variable to the fourth factor. In this example the SME expected a three-factor model and the data did not contradict the theory, so stick with three. Whereas the item generation phase tested for item equivalence, the EFA phase tests for internal reliability (consistency) of items. Internal reliability means the survey produces consistent results. The more common statistics for assessing internal reliability are Cronbach’s Alpha, and split-half. ### 4.6.3 Cronbach’s Alpha In general, an alpha <.6 is unacceptable, <.65 is undesirable, <.7 is minimally acceptable, <.8 is respectable, <.9 is very good, and >=.9 suggests items are too alike. A very low alpha means items may not be measuring the same construct, so you should drop items. A very high alpha means items are multicollinear, and you should drop items. Here is Cronbach’s alpha for the brand reputation survey, after removing the poorly-loading one_of_a_kind variable. psych::alpha(brand_rep[, 1:8])$total$std.alpha ## [1] 0.8557356 This value is in the “very good” range. Cronbach’s alpha is often used to measure the reliability of a single dimension. Here are the values for the 3 dimensions. psych::alpha(brand_rep[, 1:3])$total$std # Product Quality ## [1] 0.8918025 psych::alpha(brand_rep[, 4:6])$total$std # Willingness to Pay ## [1] 0.8517566 psych::alpha(brand_rep[, 7:8])$total$std # Product Differentiation ## [1] 0.951472 Alpha is >0.7 for each dimension. Sometimes the alpha for our survey as a whole is greater than that of the dimensions. This can happen because Cronbach’s alpha is sensitive to the number of items. Over-inflation of the alpha statistic can be a concern when working with surveys containing a large number of items. ### 4.6.4 Split-Half Use psych::splitHalf() to split the survey in half and test whether all parts of the survey contribute equally to measurement. This method is much less popular than Cronbach’s alpha. psych::splitHalf(brand_rep[, 1:8]) ## Split half reliabilities ## Call: psych::splitHalf(r = brand_rep[, 1:8]) ## ## Maximum split half reliability (lambda 4) = 0.93 ## Guttman lambda 6 = 0.92 ## Average split half reliability = 0.86 ## Guttman lambda 3 (alpha) = 0.86 ## Guttman lambda 2 = 0.87 ## Minimum split half reliability (beta) = 0.66 ## Average interitem r = 0.43 with median = 0.4 ## 4.7 Confirmatory Factor Analysis Whereas EFA is used to develop a theory of the number of factors needed to explain the relationships among the survey items, confirmatory factor analysis (CFA) is a formal hypothesis test of the EFA theory. CFA measures construct validity, that is, whether you are really measuring what you claim to measure. These notes explain how to use CFA, but do not explain the theory. For that you need to learn about dimensionality reduction, and structural equation modeling. Use the lavaan package (latent variable analysis package), passing in the model definition. Here is the model for the three dimensions in the brand reputation survey. Lavaan’s default estimator is maximum likelihood, which assumes normality. You can change it to MLR which uses robust standard errors to mitigate non-normality. The summary prints a ton of output. Concentrate on the lambda - the factor loadings. brand_rep_mdl <- paste( "PrdQl =~ well_made + consistent + poor_workman_r", "WillPay =~ higher_price + lot_more + go_up", "PrdDff =~ stands_out + unique", sep = "\n" ) brand_rep_cfa <- lavaan::cfa(model = brand_rep_mdl, data = brand_rep[, 1:8], estimator = "MLR") # lavaan::summary(brand_rep_cfa, fit.measures = TRUE, standardized = TRUE) semPlot::semPaths(brand_rep_cfa, rotation = 4) ## Registered S3 method overwritten by 'kutils': ## method from ## print.likert likert lavaan::inspect(brand_rep_cfa, "std")$lambda
##                PrdQl WillPy PrdDff
## consistent     0.932  0.000  0.000
## poor_workman_r 0.730  0.000  0.000
## higher_price   0.000  0.733  0.000
## lot_more       0.000  0.812  0.000
## go_up          0.000  0.902  0.000
## stands_out     0.000  0.000  0.976
## unique         0.000  0.000  0.930

The CFA hypothesis test is a chi-square test, so is sensitive to normality assumptions and n-size. Other fit measure are reported too: * Comparative Fit Index (CFI) (look for value >.9) * Tucker-Lewis Index (TLI) (look for value >.9) * Root mean squared Error of Approximation (RMSEA) (look for value <.05)

There are actually 78 fit measures to choose from! Focus on CFI and TLI.

lavaan::fitMeasures(brand_rep_cfa, fit.measures = c("cfi", "tli"))
##   cfi   tli
## 0.980 0.967

This output indicates a good model because both measures are >.9. Check the standardized estimates for each item. The standardized factor loadings are the basis of establishing construct validity. While we call these measures ‘loadings,’ they are better described as correlations of each manifest item with the dimensions. As you calculated, the difference between a perfect correlation and the observed is considered ‘error.’ This relationship between the so-called ‘true’ and ‘observed’ scores is the basis of classical test theory.

lavaan::standardizedSolution(brand_rep_cfa) %>%
filter(op == "=~") %>%
select(lhs, rhs, est.std, pvalue)
##       lhs            rhs est.std pvalue
## 1   PrdQl      well_made   0.914      0
## 2   PrdQl     consistent   0.932      0
## 3   PrdQl poor_workman_r   0.730      0
## 4 WillPay   higher_price   0.733      0
## 5 WillPay       lot_more   0.812      0
## 6 WillPay          go_up   0.902      0
## 7  PrdDff     stands_out   0.976      0
## 8  PrdDff         unique   0.930      0

If you have a survey that meets your assumptions, performs well under EFA, but fails under CFA, return to your survey and revisit your scale, examine the CFA modification indices, factor variances, etc.

## 4.8 Convergent/Discriminant Validity

Construct validity means the survey measures what it intends to measure. It is composed of convergent validity and discriminant validity. Convergent validity means factors address the same concept. Discriminant validity means factors address different aspects of the concept.

Test for construct validity after assessing CFA model strength (with CFI, TFI, and RMSEA) – a poor-fitting model may have greater construct validity than a better-fitting model. Use the semTools::reliability() function. The average variance extracted (AVE) measures convergent validity (avevar) and should be > .5. The composite reliability (CR) measures discriminant validity (omega) and should be > .7.

semTools::reliability(brand_rep_cfa)
##            PrdQl   WillPay    PrdDff
## alpha  0.8926349 0.8521873 0.9514719
## omega  0.9017514 0.8605751 0.9520117
## omega2 0.9017514 0.8605751 0.9520117
## omega3 0.9019287 0.8646975 0.9520114
## avevar 0.7573092 0.6756174 0.9084654

These values look good for all three dimensions. As an aside, alpha is Cronbach’s alpha. Do not be tempted to test reliability and validity in the same step. Start with reliability because it is a necessary but insufficient condition for validity. By checking for internal consistency first, as measured by alpha, then construct validity, as measured by AVE and CR, you establish the necessary reliability of the scale as a whole was met, then took it to the next level by checking for construct validity among the unique dimensions.

At this point you have established that the latent and manifest variables are related as hypothesized, and that the survey measures what you intended to measure, in this case, brand reputation.

## 4.9 Replication

The replication phase establishes criterion validity and stability (reliability). Criterion validity is a measure of the relationship between the construct and some external measure of interest. Measure criterion validity with concurrent validity, how well items correlate with an external metric measured at the same time, and with predictive validity, how well an item predicts an external metric. Stability means the survey produces similar results over repeated test-retest administrations.

### 4.9.1 Criterion Validity

#### 4.9.1.1 Concurrent Validity

Concurrent validity is a measure of whether our latent construct is significantly correlated to some outcome measured at the same time.

Suppose you have an additional data set of consumer spending on the brand. The consumer’s perception of the brand should correlate with their spending. Before checking for concurrent validity, standardize the data so that likert and other variable types are on the same scale.

set.seed(20201004)
brand_rep <- brand_rep %>%
mutate(spend = ((well_made + consistent + poor_workman_r)/3 * 5 +
(higher_price + lot_more + go_up)/3 * 3 +
(stands_out + unique)/2 * 2) / 10)
brand_rep$spend <- brand_rep$spend + rnorm(559, 5, 4) # add randomness
brand_rep_scaled <- scale(brand_rep)

Do respondents with higher scores on our the brand reputation scale also tend to spend more at the store? Build model, and latentize spend as Spndng and model with the ~~ operator. Fit the model with the semTools::sem() function.

brand_rep_cv_mdl <- paste(
"PrdQl =~ well_made + consistent + poor_workman_r",
"WillPay =~ higher_price + lot_more + go_up",
"PrdDff =~ stands_out + unique",
"Spndng =~ spend",
"Spndng ~~ PrdQl + WillPay + PrdDff",
sep = "\n"
)
brand_rep_cv <- lavaan::sem(data = brand_rep_scaled, model = brand_rep_cv_mdl)

Here are the standardized covariances. Because the data is standardized, interpret these as correlations. The p-vales are not significant because the spending data was random.

lavaan::standardizedSolution(brand_rep_cv) %>%
filter(rhs == "Spndng") %>%
select(-op, -rhs)
##       lhs est.std    se     z pvalue ci.lower ci.upper
## 1   PrdQl   0.174 0.043 4.092      0    0.091    0.258
## 2 WillPay   0.241 0.043 5.654      0    0.157    0.324
## 3  PrdDff   0.198 0.041 4.779      0    0.117    0.279
## 4  Spndng   1.000 0.000    NA     NA    1.000    1.000
semPlot::semPaths(brand_rep_cv, whatLabels = "est.std", edge.label.cex = .8, rotation = 2)

Each dimension of brand reputation is positively correlated to spending history and the relationships are all significant.

#### 4.9.1.2 Predictive Validity

Predictive validity is established by regressing some future outcome on your established construct. Assess predictive validity just as you would with any linear regression – regression estimates and p-values (starndardizedSolution()), and the r-squared coefficient of determination inspect().

Build a regression model with the single ~ operator. Then fit the model to the data as before.

brand_rep_pv_mdl <- paste(
"PrdQl =~ well_made + consistent + poor_workman_r",
"WillPay =~ higher_price + lot_more + go_up",
"PrdDff =~ stands_out + unique",
"spend ~ PrdQl + WillPay + PrdDff",
sep = "\n"
)
brand_rep_pv <- lavaan::sem(data = brand_rep_scaled, model = brand_rep_pv_mdl)
#lavaan::summary(brand_rep_pv, standardized = T, fit.measures = T, rsquare = T)
semPlot::semPaths(brand_rep_pv, whatLabels = "est.std", edge.label.cex = .8, rotation = 2)

lavaan::standardizedSolution(brand_rep_pv) %>%
filter(op == "~") %>%
mutate_if(is.numeric, round, digits = 3)
##     lhs op     rhs est.std    se     z pvalue ci.lower ci.upper
## 1 spend  ~   PrdQl   0.094 0.048 1.949  0.051   -0.001    0.189
## 2 spend  ~ WillPay   0.182 0.052 3.486  0.000    0.080    0.284
## 3 spend  ~  PrdDff   0.060 0.055 1.092  0.275   -0.047    0.167
lavaan::inspect(brand_rep_pv, "r2")
##      well_made     consistent poor_workman_r   higher_price       lot_more
##          0.837          0.867          0.533          0.537          0.658
##          go_up     stands_out         unique          spend
##          0.816          0.951          0.866          0.072

There is a statistically significant relationship between one dimension of brand quality (Willingness to Pay) and spending. At this point you may want to drop the other two dimensions. However, the R^2 is not good - only 7% of the variability in Spending can be explained by the three dimension of our construct.

Factor scores represent individual respondents’ standing on a latent factor. While not used for scale validation per se, factor scores can be used for customer segmentation via clustering, network analysis and other statistical techniques.

brand_rep_cfa <- lavaan::cfa(brand_rep_pv_mdl, data = brand_rep_scaled)

brand_rep_cfa_scores <- lavaan::predict(brand_rep_cfa) %>% as.data.frame()
psych::describe(brand_rep_cfa_scores)
##         vars   n mean   sd median trimmed  mad   min  max range  skew kurtosis
## PrdQl      1 559    0 0.88  -0.50   -0.19 0.09 -0.57 4.02  4.59  2.31     6.18
## WillPay    2 559    0 0.69   0.00    0.01 0.86 -1.15 1.16  2.31 -0.05    -1.18
## PrdDff     3 559    0 0.96  -0.04   -0.15 1.17 -0.88 2.54  3.41  1.09     0.35
##           se
## PrdQl   0.04
## WillPay 0.03
## PrdDff  0.04
psych::multi.hist(brand_rep_cfa_scores)

map(brand_rep_cfa_scores, shapiro.test)
## $PrdQl ## ## Shapiro-Wilk normality test ## ## data: .x[[i]] ## W = 0.66986, p-value < 2.2e-16 ## ## ##$WillPay
##
##  Shapiro-Wilk normality test
##
## data:  .x[[i]]
## W = 0.94986, p-value = 7.811e-13
##
##
## $PrdDff ## ## Shapiro-Wilk normality test ## ## data: .x[[i]] ## W = 0.82818, p-value < 2.2e-16 These scores are not normally distributed, which makes clustering a great choice for modeling factor scores. Clustering does not mean distance-based clustering, such as K-means, in this context. Mixture models consider data as coming from a distribution which itself is a mixture of clusters. To learn more about model-based clustering in the Hierarchical and Mixed Effects Models DataCamp course. Factor scores can be extracted from a structural equation model and used as inputs in other models. For example, you can use the factor scores from the brand reputation dimensions as regressors for a regrssion on spending. brand_rep_fs_reg_dat <- bind_cols(brand_rep_cfa_scores, spend = brand_rep$spend)
brand_rep_fs_reg <- lm(spend ~ PrdQl + WillPay + PrdDff, data = brand_rep_fs_reg_dat)
summary(brand_rep_fs_reg)$coef ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 7.1555354 0.1591195 44.9695738 3.363705e-187 ## PrdQl 0.4260875 0.2062002 2.0663774 3.925620e-02 ## WillPay 1.1365087 0.2805960 4.0503388 5.842799e-05 ## PrdDff 0.1714031 0.2181813 0.7855993 4.324375e-01 The coefficients and r-squared of the lm() and sem() models closely resemble each other, but keeping the regression inside the lavaan framework provides more information (as witnessed in the higher estimates and r-squared). A construct, once validated, can be combined with a wide range of outcomes and models to produce valuable information about consumer behavior and habits. ### 4.9.2 Test-Retest Reliability Test-retest reliability is the ability to achieve the same result from a respondent at two closely-spaced points in time (repeated measures). Suppose you had two surveys, identified by an id field. # svy_1 <- brand_rep[sample(1:559, 300),] %>% as.data.frame() # svy_2 <- brand_rep[sample(1:559, 300),] %>% as.data.frame() # survey_test_retest <- psych::testRetest(t1 = svy_1, t2 = svy_2, id = "id") # survey_test_retest$r12

An r^2 <.7 is unacceptable, <.9 good, and >.9 very good. This one is unacceptable.

One way to check for replication is by splitting the data in half.

# svy <- bind_rows(svy_1, svy_2, .id = "time")
#
# psych::describeBy(svy, "time")
#
# brand_rep_test_retest <- psych::testRetest(
#   t1 = filter(svy, time == 1),
#   t2 = filter(svy, time == 2),
#   id = "id")
#
# brand_rep_test_retest$r12 If the correlation of scaled scores across time 1 and time 2 is greater than .9, that indicates very strong test-retest reliability. This measure can be difficult to collect because it requires the same respondents to answer the survey at two points in time. However, it’s a good technique to have in your survey development toolkit. When validating a scale, it’s a good idea to split the survey results into two samples, using one for EFA and one for CFA. This works as a sort of cross-validation such that the overall fit of the model is less likely due to chance of any one sample’s makeup. # brand_rep_efa_data <- brand_rep[1:280,] # brand_rep_cfa_data <- brand_rep[281:559,] # # efa <- psych::fa(brand_rep_efa_data, nfactors = 3) # efa$loadings
#
# brand_rep_cfa <- lavaan::cfa(brand_rep_mdl, data = brand_rep_cfa_data)
# lavaan::inspect(brand_rep_cfa, what = "call")
#
# lavaan::fitmeasures(brand_rep_cfa)[c("cfi","tli","rmsea")]

## 4.10 Reporting

There are seven key areas to report:

• Explain the study objective, explicitly identifying the research question.
• Motivate the research in the context of previous work.
• Explain the method and rationale, including the instrument and its psychometric properties, it development/testing, sample selection, and data collection. Explain and justify the analytical methods.
• Present the results in a concise and factual manner.
• Interpret and discuss the findings.
• Draw conclusions.

### References

———. 2021. Survey: Analysis of Complex Survey Samples. http://r-survey.r-forge.r-project.org/survey/.