10 Exploratory Data Analysis at Scale

Wk4_Data <- 'Week_4/DATA'

library('kableExtra')
library('tidyverse')
#> ── Attaching packages ─────────────────────────────────────── tidyverse 1.3.2 ──
#> ✔ ggplot2 3.4.0      ✔ purrr   1.0.0 
#> ✔ tibble  3.1.8      ✔ dplyr   1.0.10
#> ✔ tidyr   1.2.1      ✔ stringr 1.5.0 
#> ✔ readr   2.1.3      ✔ forcats 0.5.2 
#> ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
#> ✖ dplyr::filter()     masks stats::filter()
#> ✖ dplyr::group_rows() masks kableExtra::group_rows()
#> ✖ dplyr::lag()        masks stats::lag()

A_DATA_2 <- readRDS(file.path(Wk4_Data,'A_DATA_2.RDS'))

A_DATA_TBL_2.t_ks_result.furrr <- readRDS(file.path(Wk4_Data,'A_DATA_TBL_2.t_ks_result.furrr.RDS'))

FEATURE_TYPE <- readRDS(file.path(Wk4_Data,'FEATURE_TYPE.RDS'))

numeric_features <- FEATURE_TYPE$numeric_features
categorical_features <- FEATURE_TYPE$categorical_features

\(~\)

10.1 Investigate Results

We can make make our results scroll-able if we use kableExtra:

A_DATA_TBL_2.t_ks_result.furrr %>%
  select(Feature, mean_diff_est , ttest.pvalue, kstest.pvalue, N_Target, mean_Target, sd_Target, N_Control, mean_Control, sd_Control) %>%
  mutate(across(where(is.numeric), round, 3))  %>%
  kableExtra::kbl() %>% # kableExtra from here down
  kable_paper() %>%
  scroll_box(width = "100%", height = "200px")

Feature	mean_diff_est	ttest.pvalue	kstest.pvalue	N_Target	mean_Target	sd_Target	N_Control	mean_Control	sd_Control
PHAFSTHR	0.403	0.000	0.000	2002	11.331	3.756	17227	10.928	3.542
BPXML1	15.089	0.000	0.000	6212	160.122	26.533	66292	145.032	22.621
PHAFSTMN	0.210	0.607	0.953	2002	29.980	17.260	17227	29.770	17.302
BPXDI4	0.966	0.223	0.039	665	67.738	19.806	7687	66.772	16.984
BPXPLS	-0.084	0.625	0.286	6246	74.368	13.046	66470	74.452	12.639
BPXDI1	2.029	0.000	0.000	5749	68.184	14.977	61757	66.156	14.895
BPXDI3	1.496	0.000	0.000	5740	67.521	15.235	60517	66.024	15.184
BPXDI2	1.964	0.000	0.000	5810	68.058	14.962	61041	66.094	14.996
BPXSY4	16.500	0.000	0.000	665	135.919	23.407	7687	119.419	20.331
BPXCHR	-3.384	0.519	0.817	15	96.933	19.783	15152	100.317	17.076
BPXSY3	13.999	0.000	0.000	5740	131.256	20.739	60518	117.258	18.055
Age	31.486	0.000	0.000	6807	61.527	14.780	88740	30.041	23.635
BPXSY2	14.419	0.000	0.000	5810	132.280	21.146	61041	117.860	18.455
BPXSY1	14.842	0.000	0.000	5749	133.321	21.342	61757	118.478	18.858
BPXDAR	0.835	0.069	0.016	1356	66.953	16.481	20711	66.118	14.444
BMXHEAD	NaN	NA	NA	0	NaN	NA	0	NaN	NA
LBDHDD	-5.268	0.000	0.000	4704	48.553	14.488	45568	53.821	15.288
LBDHDDSI	-0.136	0.000	0.000	4704	1.256	0.375	45568	1.392	0.395
BPXSAR	16.998	0.000	0.000	1356	134.541	22.248	20711	117.542	19.343
LBXAPB	7.345	0.000	0.001	228	101.175	28.270	2837	93.830	28.515
LBDAPBSI	0.073	0.000	0.001	228	1.012	0.283	2837	0.938	0.285
LBDLDLM	-8.538	0.000	0.000	381	100.360	38.142	2167	108.898	35.153
LBDLDMSI	-0.221	0.000	0.000	381	2.595	0.986	2167	2.816	0.909
LBDLDLN	-10.763	0.000	0.000	385	99.764	38.508	2182	110.527	35.851
LBDLDNSI	-0.278	0.000	0.000	385	2.580	0.996	2182	2.858	0.927
BMXSAD3	3.322	0.000	0.000	126	25.225	4.049	960	21.904	5.050
BMXSAD4	3.312	0.000	0.000	126	25.214	3.995	960	21.903	5.053
LBDLDL	-9.006	0.000	0.000	2180	100.390	36.569	18145	109.396	35.225
LBDLDLSI	-0.233	0.000	0.000	2180	2.596	0.946	18145	2.829	0.911
BMXSAD1	5.005	0.000	0.000	1923	25.616	4.551	18661	20.610	4.732
BMXSAD2	5.018	0.000	0.000	1923	25.612	4.570	18661	20.594	4.741
BMDAVSAD	5.005	0.000	0.000	1923	25.624	4.558	18661	20.619	4.734
BMXLEG	-1.397	0.000	0.000	5857	37.535	4.280	65870	38.932	4.238
LBXTC	-1.827	0.009	0.000	4704	181.063	46.312	45567	182.890	41.105
LBDTCSI	-0.047	0.009	0.000	4704	4.682	1.198	45567	4.730	1.063
LBXGLU	60.692	0.000	0.000	2352	160.710	68.044	18897	100.018	18.194
BMXCALF	1.379	0.000	0.000	1811	38.211	4.791	27783	36.832	5.016
BMXARML	4.123	0.000	0.000	6057	37.580	2.993	81107	33.457	6.720
BMXSUB	7.916	0.000	0.000	2175	23.325	7.689	44885	15.409	8.668
URXUCR2	-42.900	0.000	0.000	636	94.338	56.189	6600	137.238	73.324
URDUCR2S	-3792.372	0.000	0.000	636	8339.484	4967.065	6600	12131.855	6481.851
BMXTRI	4.759	0.000	0.000	2843	21.026	8.490	48206	16.267	8.204
BMXARMC	6.334	0.000	0.000	6054	34.672	5.408	81091	28.338	7.442
Poverty_Income_Ratio	-0.072	0.000	0.000	6059	2.225	1.502	80794	2.296	1.610
BMAEXLEN	22.610	0.000	0.000	450	286.258	93.019	8321	263.647	79.125
URXUCR	-11.888	0.000	0.000	4840	113.749	71.374	49920	125.637	81.455
BMXTHICR	1.186	0.000	0.000	1742	52.439	7.866	27456	51.253	8.012
LBDGLUSI	3.369	0.000	0.000	2352	8.921	3.777	18897	5.552	1.010
LBXTR	43.659	0.000	0.000	2260	157.202	159.082	18367	113.543	93.182
LBDTRSI	0.493	0.000	0.000	2260	1.775	1.796	18367	1.282	1.052
BMXRECUM	-0.479	0.912	0.944	5	89.640	9.141	7205	90.119	8.600
URXUMA2	39.630	0.000	0.000	636	55.181	209.726	6600	15.550	77.535
URDUMA2S	39.630	0.000	0.000	636	55.181	209.726	6600	15.550	77.535
BMXHIP	6.579	0.000	0.000	773	110.892	15.731	5098	104.313	14.460
URXCRS	-1050.899	0.000	0.000	4840	10055.402	6309.499	49920	11106.301	7200.588
BMXHT	10.458	0.000	0.000	6322	165.672	10.725	80726	155.214	23.879
BMXWAIST	23.277	0.000	0.000	5913	108.049	16.341	77653	84.773	21.414
PEASCTM1	149.680	0.000	0.000	4674	711.582	200.877	67355	561.902	267.205
URDACT2	88.039	0.000	0.000	636	102.720	580.014	6600	14.680	93.315
BMXWT	25.929	0.000	0.000	6306	88.116	23.925	83464	62.187	29.774
URXUMA	141.019	0.000	0.000	4840	172.300	793.228	49919	31.281	200.764
URXUMS	141.019	0.000	0.000	4840	172.300	793.228	49919	31.281	200.764
LBXIN	7.997	0.000	0.000	2284	21.110	35.946	18490	13.113	12.496
LBDINSI	47.984	0.000	0.000	2284	126.662	215.673	18490	78.678	74.976
BMXBMI	7.043	0.000	0.000	6268	31.948	7.566	80343	24.905	7.355
URDACT	141.427	0.000	0.000	3660	171.371	719.616	35584	29.944	532.255
WTSAF2YR	-19537.782	0.000	0.000	2474	61002.970	68801.008	20100	80540.753	82941.640

We can make a corresponding plot to go along with our table:

plot <- A_DATA_TBL_2.t_ks_result.furrr %>%
  mutate(Feature_Prevalence_Pct = round(N_Target/(N_Target+N_Control)*100,2)) %>%
  ggplot(aes(x=round(ttest.pvalue,4) , y= round(kstest.pvalue,4), color = Feature, size = Feature_Prevalence_Pct)) +
  geom_point() + 
  theme(legend.position='none') 

plot
#> Warning: Removed 1 rows containing missing values (`geom_point()`).

We can also make our plot interactive with plotly::ggplotly

plotly::ggplotly(plot)

\(~\)

10.2 tableby

The arsenal package also contains other helpful functions in terms of Exploratory Data Analysis:

library('arsenal')
library('knitr')


A_DATA_2 %>%
  head()
#> # A tibble: 6 × 133
#>    SEQN DIABE…¹ AGE_A…²   Age Gender Race  USAF  Birth…³ Grade…⁴ Grade…⁵ Marit…⁶
#>   <dbl>   <dbl>   <dbl> <dbl> <chr>  <chr> <chr> <chr>   <chr>   <chr>   <chr>  
#> 1     1       0      NA     2 Female Black <NA>  USA     <NA>    <NA>    <NA>   
#> 2     2       0      NA    77 Male   White Yes   USA     <NA>    Colleg… <NA>   
#> 3     3       0      NA    10 Female White <NA>  <NA>    3rd gr… <NA>    <NA>   
#> 4     4       0      NA     1 Male   Black <NA>  USA     <NA>    <NA>    <NA>   
#> 5     5       0      NA    49 Male   White Yes   USA     <NA>    Colleg… Married
#> 6     6       0      NA    19 Female Other No    USA     More t… <NA>    Never …
#> # … with 122 more variables: Pregnant <chr>, Household_Icome <chr>,
#> #   Family_Income <chr>, Poverty_Income_Ratio <dbl>, yr_range <chr>,
#> #   PEASCST1 <dbl>, PEASCTM1 <dbl>, PEASCCT1 <dbl>, BPXCHR <dbl>,
#> #   BPQ150A <dbl>, BPQ150B <dbl>, BPQ150C <dbl>, BPQ150D <dbl>, BPAARM <dbl>,
#> #   BPACSZ <dbl>, BPXPLS <dbl>, BPXDB <dbl>, BPXPULS <dbl>, BPXPTY <dbl>,
#> #   BPXML1 <dbl>, BPXSY1 <dbl>, BPXDI1 <dbl>, BPAEN1 <dbl>, BPXSY2 <dbl>,
#> #   BPXDI2 <dbl>, BPAEN2 <dbl>, BPXSY3 <dbl>, BPXDI3 <dbl>, BPAEN3 <dbl>, …

We can perform many of the analyses we did in the last part easily with the tableby function

tableby(DIABETES_char ~ Age + Gender + Race + Marital_Status + Grade_Level + LBXGLU,
        data = A_DATA_2 %>%
          mutate(DIABETES_char = case_when(DIABETES == 1 ~ "Diabetics", 
                                   DIABETES == 0 ~ "Non-Diabetics",
                                   is.na(DIABETES) ~ "Unknown Diabetic Status"))) %>%
  summary(pfootnote=TRUE)

	Diabetics (N=6807)	Non-Diabetics (N=88740)	Unknown Diabetic Status (N=5769)	Total (N=101316)	p value
Age					< 0.001¹
Mean (SD)	61.527 (14.780)	30.041 (23.635)	11.989 (24.525)	31.128 (24.943)
Range	1.000 - 85.000	1.000 - 85.000	0.000 - 85.000	0.000 - 85.000
Gender					0.019²
Female	3371 (49.5%)	45188 (50.9%)	2864 (49.6%)	51423 (50.8%)
Male	3436 (50.5%)	43552 (49.1%)	2905 (50.4%)	49893 (49.2%)
Race					< 0.001²
Black	1823 (26.8%)	20692 (23.3%)	1129 (19.6%)	23644 (23.3%)
Mexican American	1373 (20.2%)	19426 (21.9%)	1650 (28.6%)	22449 (22.2%)
Other	611 (9.0%)	8376 (9.4%)	510 (8.8%)	9497 (9.4%)
Other Hispanic	592 (8.7%)	7201 (8.1%)	501 (8.7%)	8294 (8.2%)
White	2408 (35.4%)	33045 (37.2%)	1979 (34.3%)	37432 (36.9%)
Marital_Status					< 0.001²
N-Miss	157	35025	4620	39802
Divorced	838 (12.6%)	4599 (8.6%)	162 (14.1%)	5599 (9.1%)
Living with partner	224 (3.4%)	3892 (7.2%)	70 (6.1%)	4186 (6.8%)
Married	3634 (54.6%)	24349 (45.3%)	595 (51.8%)	28578 (46.5%)
Never married	604 (9.1%)	15531 (28.9%)	137 (11.9%)	16272 (26.5%)
Separated	253 (3.8%)	1541 (2.9%)	39 (3.4%)	1833 (3.0%)
Widowed	1097 (16.5%)	3803 (7.1%)	146 (12.7%)	5046 (8.2%)
Grade_Level					0.040²
N-Miss	6681	59497	5663	71841
10th grade	11 (8.7%)	2177 (7.4%)	9 (8.5%)	2197 (7.5%)
11th grade	12 (9.5%)	2146 (7.3%)	8 (7.5%)	2166 (7.3%)
12th grade, no diploma	0 (0.0%)	438 (1.5%)	2 (1.9%)	440 (1.5%)
1st grade	6 (4.8%)	2126 (7.3%)	2 (1.9%)	2134 (7.2%)
2nd grade	4 (3.2%)	2102 (7.2%)	1 (0.9%)	2107 (7.1%)
3rd grade	6 (4.8%)	2031 (6.9%)	6 (5.7%)	2043 (6.9%)
4th grade	6 (4.8%)	2036 (7.0%)	12 (11.3%)	2054 (7.0%)
5th grade	9 (7.1%)	2110 (7.2%)	10 (9.4%)	2129 (7.2%)
6th grade	14 (11.1%)	2219 (7.6%)	5 (4.7%)	2238 (7.6%)
7th grade	8 (6.3%)	2178 (7.4%)	11 (10.4%)	2197 (7.5%)
8th grade	11 (8.7%)	2292 (7.8%)	12 (11.3%)	2315 (7.9%)
9th grade	18 (14.3%)	2195 (7.5%)	15 (14.2%)	2228 (7.6%)
Don’t Know	0 (0.0%)	9 (0.0%)	0 (0.0%)	9 (0.0%)
GED or equivalent	0 (0.0%)	115 (0.4%)	0 (0.0%)	115 (0.4%)
High school graduate	9 (7.1%)	1422 (4.9%)	7 (6.6%)	1438 (4.9%)
Less than 5th grade	0 (0.0%)	26 (0.1%)	0 (0.0%)	26 (0.1%)
Less than 9th grade	1 (0.8%)	277 (0.9%)	1 (0.9%)	279 (0.9%)
More than high school	6 (4.8%)	980 (3.4%)	3 (2.8%)	989 (3.4%)
Never attended / kindergarten only	5 (4.0%)	2362 (8.1%)	2 (1.9%)	2369 (8.0%)
Refused	0 (0.0%)	2 (0.0%)	0 (0.0%)	2 (0.0%)
LBXGLU					< 0.001¹
N-Miss	4455	69843	5326	79624
Mean (SD)	160.710 (68.044)	100.018 (18.194)	118.363 (36.091)	106.974 (34.273)
Range	21.000 - 584.000	38.000 - 422.000	77.000 - 451.000	21.000 - 584.000

Linear Model ANOVA
Pearson’s Chi-squared test

\(~\)

10.3 Assessing Cohort Balance

Let’s assume there was a cohort assigned to the patient, we can take our algorithm from Section 7.2

set.seed(85763099)

A_DATA_2$rand_class <- sample(c('Rand_A','Rand_B'), 
                            size = nrow(A_DATA_2),  
                            replace = TRUE)

A_DATA_2$rand_sort <- runif(nrow(A_DATA_2))

A_DATA_2 <- A_DATA_2 %>%
  arrange(rand_sort) %>%
  mutate(rn = row_number()) %>%
  mutate(rn_mod_5 = rn %% 5 ) %>% 
  mutate(rand_class = if_else( rn_mod_5 == 0, 
                              "Rand_C", 
                              rand_class)) %>%
  select(-rn, -rn_mod_5, -rand_sort) %>%
  mutate(rand_class = as.factor(rand_class))

We can check for balance in the cohorts among some of the features perhaps, Age, Race, and Gender with:

tableby(rand_class ~ Age + Gender + Race, 
        data = A_DATA_2) %>%
  summary(pfootnote=TRUE)

	Rand_A (N=40552)	Rand_B (N=40501)	Rand_C (N=20263)	Total (N=101316)	p value
Age					0.017¹
Mean (SD)	31.059 (24.930)	30.977 (24.899)	31.570 (25.054)	31.128 (24.943)
Range	0.000 - 85.000	0.000 - 85.000	0.000 - 85.000	0.000 - 85.000
Gender					0.209²
Female	20557 (50.7%)	20472 (50.5%)	10394 (51.3%)	51423 (50.8%)
Male	19995 (49.3%)	20029 (49.5%)	9869 (48.7%)	49893 (49.2%)
Race					0.122²
Black	9357 (23.1%)	9654 (23.8%)	4633 (22.9%)	23644 (23.3%)
Mexican American	9047 (22.3%)	8935 (22.1%)	4467 (22.0%)	22449 (22.2%)
Other	3856 (9.5%)	3761 (9.3%)	1880 (9.3%)	9497 (9.4%)
Other Hispanic	3333 (8.2%)	3292 (8.1%)	1669 (8.2%)	8294 (8.2%)
White	14959 (36.9%)	14859 (36.7%)	7614 (37.6%)	37432 (36.9%)

Linear Model ANOVA
Pearson’s Chi-squared test

Above we see that both p-values of Race, and Gender are above .05 meaning the distributions of Race, and Gender appear to be random among the cohorts of rand_class; so here, the cohorts rand_class are well-balanced on Race, and Gender.

We see that the p-value of Age appears to be significant, however, the distributions appear to be similar:

A_DATA_2 %>%
  ggplot(aes(x=Age, color=rand_class)) +
  geom_density()

\(~\)

10.4 Missing Data

A_DATA_2.Num <- A_DATA_2 %>%
  select(SEQN, DIABETES, Gender, Race, Family_Income, all_of(FEATURE_TYPE$numeric_features))

The first function is the Amelia::missmap function which can be used as follows.

tic <- Sys.time()

Amelia::missmap(as.data.frame(A_DATA_2.Num))

toc <- Sys.time()

time.Amelia <- difftime(toc , tic , units = "secs")

Next we will review some of the functionality within the mice package:

library(mice)
#> 
#> Attaching package: 'mice'
#> The following object is masked from 'package:stats':
#> 
#>     filter
#> The following objects are masked from 'package:base':
#> 
#>     cbind, rbind
library(VIM)
#> Loading required package: colorspace
#> Loading required package: grid
#> VIM is ready to use.
#> Suggestions and bug-reports can be submitted at: https://github.com/statistikat/VIM/issues
#> 
#> Attaching package: 'VIM'
#> The following object is masked from 'package:datasets':
#> 
#>     sleep
library(lattice)

tic <- Sys.time()

#plot the missing values
plot.miss <- aggr(A_DATA_2.Num,  
     numbers=TRUE, 
     sortVars=TRUE, 
     labels=colnames(A_DATA_2.Num), 
     cex.axis=.7, 
     gap=3, 
     ylab=c("Proportion of missingness","Missingness Pattern"))
#> Warning in plot.aggr(res, ...): not enough vertical space to display frequencies
#> (too many combinations)
#> 
#>  Variables sorted by number of missings: 
#>              Variable      Count
#>               BMXSAD3 0.98905405
#>               BMXSAD4 0.98905405
#>               BMXHEAD 0.97536421
#>               LBDLDLM 0.97403174
#>              LBDLDMSI 0.97403174
#>               LBDLDLN 0.97384421
#>              LBDLDNSI 0.97384421
#>                LBXAPB 0.96934344
#>              LBDAPBSI 0.96934344
#>                BMXHIP 0.94039441
#>               URXUCR2 0.92734612
#>              URDUCR2S 0.92734612
#>               URXUMA2 0.92734612
#>              URDUMA2S 0.92734612
#>               URDACT2 0.92734612
#>                BPXDI4 0.91643965
#>                BPXSY4 0.91643965
#>              BMAEXLEN 0.90838564
#>              BMXRECUM 0.88619764
#>                BPXCHR 0.80783884
#>              PHAFSTHR 0.80652612
#>              PHAFSTMN 0.80652612
#>                LBDLDL 0.79523471
#>              LBDLDLSI 0.79523471
#>               BMXSAD1 0.79276718
#>               BMXSAD2 0.79276718
#>              BMDAVSAD 0.79276718
#>                 LBXTR 0.79212563
#>               LBDTRSI 0.79212563
#>                 LBXIN 0.79067472
#>               LBDINSI 0.79067472
#>                LBXGLU 0.78589759
#>              LBDGLUSI 0.78589759
#>                BPXDAR 0.78019266
#>                BPXSAR 0.78019266
#>              WTSAF2YR 0.77256307
#>              BMXTHICR 0.70892060
#>               BMXCALF 0.70493308
#>                URDACT 0.60521537
#>                BMXSUB 0.50978128
#>                 LBXTC 0.49466027
#>               LBDTCSI 0.49466027
#>                LBDHDD 0.49465040
#>              LBDHDDSI 0.49465040
#>                BMXTRI 0.46913617
#>                URXUMA 0.45004738
#>                URXUMS 0.45004738
#>                URXUCR 0.45003751
#>                URXCRS 0.45003751
#>         Family_Income 0.44252635
#>                BPXDI3 0.33547515
#>                BPXSY3 0.33546528
#>                BPXDI2 0.32948399
#>                BPXSY2 0.32948399
#>                BPXDI1 0.32314738
#>                BPXSY1 0.32314738
#>                BMXLEG 0.28113032
#>                BPXML1 0.27300722
#>                BPXPLS 0.27083580
#>              PEASCTM1 0.24495637
#>              BMXWAIST 0.16401161
#>                BMXBMI 0.13341427
#>                 BMXHT 0.12907142
#>               BMXARMC 0.09402266
#>               BMXARML 0.09381539
#>  Poverty_Income_Ratio 0.09076553
#>                 BMXWT 0.06054325
#>              DIABETES 0.05694066
#>                  SEQN 0.00000000
#>                Gender 0.00000000
#>                  Race 0.00000000
#>                   Age 0.00000000

toc <- Sys.time()

time.mice <- difftime(toc , tic , units = "secs")

#Drawing margin plot
marginplot(A_DATA_2.Num[, c("Age", "BMXARML")], 
           cex.numbers = 1.2, 
           pch = 19)

#Drawing margin plot
marginplot(A_DATA_2.Num[, c("Age", "BMXSAD3")], 
           cex.numbers = 1.2, 
           pch = 19)

Here’s a function to give to return the a tibble of features with percentage of non-missing values:

Features_Percent_Complete <- function(data, Percent_Complete = 0){
  
  SumNa <- function(col){sum(is.na(col))}
  
  na_sums <- data %>% 
    summarise_all(SumNa) %>%
    tidyr::pivot_longer(everything() ,names_to = 'feature', values_to = 'SumNa') %>%
    arrange(-SumNa) %>%
    mutate(PctNa = SumNa/nrow(data)) %>%
    mutate(PctComp = (1 - PctNa)*100)
  
  data_out <- na_sums %>%
    filter(PctComp >= Percent_Complete)
  
return(data_out)
}

Let’s first define the features that have at least 70% data:

features_all_percent_compete <- Features_Percent_Complete(A_DATA_2.Num, 0)

features_all_percent_compete
#> # A tibble: 72 × 4
#>   feature   SumNa PctNa PctComp
#>   <chr>     <int> <dbl>   <dbl>
#> 1 BMXSAD3  100207 0.989    1.09
#> 2 BMXSAD4  100207 0.989    1.09
#> 3 BMXHEAD   98820 0.975    2.46
#> 4 LBDLDLM   98685 0.974    2.60
#> 5 LBDLDMSI  98685 0.974    2.60
#> 6 LBDLDLN   98666 0.974    2.62
#> # … with 66 more rows

Then we can graph this as:

features_all_percent_compete %>%
  ggplot(aes(x=reorder(feature, PctComp), y =PctComp, fill=feature)) +
  geom_bar(stat = "identity") +
  coord_flip() +
  theme(legend.position = "none")

For the remainder of the majority of this discussion we’ll limit ourselves to features that have at least 65% of data:

features_65_num <- Features_Percent_Complete(A_DATA_2.Num, 65) %>%
  filter(feature %in% c(FEATURE_TYPE$numeric_features) )

features_65_num
#> # A tibble: 18 × 4
#>   feature SumNa PctNa PctComp
#>   <chr>   <int> <dbl>   <dbl>
#> 1 BPXDI3  33989 0.335    66.5
#> 2 BPXSY3  33988 0.335    66.5
#> 3 BPXDI2  33382 0.329    67.1
#> 4 BPXSY2  33382 0.329    67.1
#> 5 BPXDI1  32740 0.323    67.7
#> 6 BPXSY1  32740 0.323    67.7
#> # … with 12 more rows

Note again we can make a function for this graph above:

Feature_Percent_Complete_Graph <- function(data, Percent_Complete = 0 ){
  table <- Features_Percent_Complete(data, Percent_Complete)
  
  plot1 <- table %>%
    ggplot(aes(x=reorder(feature, PctComp), y =PctComp, fill=feature)) +
    geom_bar(stat = "identity") +
    coord_flip() +
    theme(legend.position = "none")
  return(plot1)
}

Let’s time it:

tic <- Sys.time()

Feature_Percent_Complete_Graph(A_DATA_2.Num, 0)

toc <- Sys.time()

FPC.time <- difftime(toc , tic , units = 'secs')

10.4.0.1 Speed-Ups

Amelia is an older package, we can see that our function is

as.numeric(time.Amelia) / as.numeric(FPC.time)
#> [1] 187.8361

times faster than Amelia::missmap

It also outperforms the mice results by:

as.numeric(time.mice) / as.numeric(FPC.time)
#> [1] 167.5308

10.4.1 Missing Value Imputation

Note in the below summarise_at we are passing in a number of functions including a function to count the number of missing values n_miss, n, min, max, mean, median, and sd, these summary statistics are computed _at each of the vars we pass in:

summary_table <- A_DATA_2.Num %>%
  filter(!is.na(DIABETES)) %>%
  group_by(Gender, Race, Family_Income) %>%
  summarise_at(vars(all_of(features_65_num$feature)), 
               list( 
                 n = ~n(),
                 n_miss = ~sum(is.na(.x)),
                 min = ~min(.x , na.rm = TRUE),
                 max = ~max(.x , na.rm = TRUE),
                 mean = ~mean(.x , na.rm = TRUE),
                 median = ~median(.x , na.rm = TRUE),
                 sd = ~sd(.x , na.rm = TRUE)),
               .groups='keep'
               ) 
#> Warning in min(.x, na.rm = TRUE): no non-missing arguments to min; returning Inf

#> Warning in min(.x, na.rm = TRUE): no non-missing arguments to min; returning Inf

#> Warning in min(.x, na.rm = TRUE): no non-missing arguments to min; returning Inf

#> Warning in min(.x, na.rm = TRUE): no non-missing arguments to min; returning Inf

#> Warning in min(.x, na.rm = TRUE): no non-missing arguments to min; returning Inf

#> Warning in min(.x, na.rm = TRUE): no non-missing arguments to min; returning Inf

#> Warning in min(.x, na.rm = TRUE): no non-missing arguments to min; returning Inf

#> Warning in min(.x, na.rm = TRUE): no non-missing arguments to min; returning Inf

#> Warning in min(.x, na.rm = TRUE): no non-missing arguments to min; returning Inf

#> Warning in min(.x, na.rm = TRUE): no non-missing arguments to min; returning Inf

#> Warning in min(.x, na.rm = TRUE): no non-missing arguments to min; returning Inf

#> Warning in min(.x, na.rm = TRUE): no non-missing arguments to min; returning Inf

#> Warning in min(.x, na.rm = TRUE): no non-missing arguments to min; returning Inf

#> Warning in min(.x, na.rm = TRUE): no non-missing arguments to min; returning Inf

#> Warning in min(.x, na.rm = TRUE): no non-missing arguments to min; returning Inf

#> Warning in min(.x, na.rm = TRUE): no non-missing arguments to min; returning Inf

#> Warning in min(.x, na.rm = TRUE): no non-missing arguments to min; returning Inf

#> Warning in min(.x, na.rm = TRUE): no non-missing arguments to min; returning Inf

#> Warning in min(.x, na.rm = TRUE): no non-missing arguments to min; returning Inf

#> Warning in min(.x, na.rm = TRUE): no non-missing arguments to min; returning Inf
#> Warning in max(.x, na.rm = TRUE): no non-missing arguments to max; returning
#> -Inf

#> Warning in max(.x, na.rm = TRUE): no non-missing arguments to max; returning
#> -Inf

#> Warning in max(.x, na.rm = TRUE): no non-missing arguments to max; returning
#> -Inf

#> Warning in max(.x, na.rm = TRUE): no non-missing arguments to max; returning
#> -Inf

#> Warning in max(.x, na.rm = TRUE): no non-missing arguments to max; returning
#> -Inf

#> Warning in max(.x, na.rm = TRUE): no non-missing arguments to max; returning
#> -Inf

#> Warning in max(.x, na.rm = TRUE): no non-missing arguments to max; returning
#> -Inf

#> Warning in max(.x, na.rm = TRUE): no non-missing arguments to max; returning
#> -Inf

#> Warning in max(.x, na.rm = TRUE): no non-missing arguments to max; returning
#> -Inf

#> Warning in max(.x, na.rm = TRUE): no non-missing arguments to max; returning
#> -Inf

#> Warning in max(.x, na.rm = TRUE): no non-missing arguments to max; returning
#> -Inf

#> Warning in max(.x, na.rm = TRUE): no non-missing arguments to max; returning
#> -Inf

#> Warning in max(.x, na.rm = TRUE): no non-missing arguments to max; returning
#> -Inf

#> Warning in max(.x, na.rm = TRUE): no non-missing arguments to max; returning
#> -Inf

#> Warning in max(.x, na.rm = TRUE): no non-missing arguments to max; returning
#> -Inf

#> Warning in max(.x, na.rm = TRUE): no non-missing arguments to max; returning
#> -Inf

#> Warning in max(.x, na.rm = TRUE): no non-missing arguments to max; returning
#> -Inf

#> Warning in max(.x, na.rm = TRUE): no non-missing arguments to max; returning
#> -Inf

#> Warning in max(.x, na.rm = TRUE): no non-missing arguments to max; returning
#> -Inf

#> Warning in max(.x, na.rm = TRUE): no non-missing arguments to max; returning
#> -Inf

summary_table %>%
  glimpse()
#> Rows: 150
#> Columns: 129
#> Groups: Gender, Race [10]
#> $ Gender                      <chr> "Female", "Female", "Female", "Female", "F…
#> $ Race                        <chr> "Black", "Black", "Black", "Black", "Black…
#> $ Family_Income               <chr> "$ 0 to $ 4,999", "$ 5,000 to $ 9,999", "$…
#> $ BPXDI3_n                    <int> 364, 459, 522, 541, 456, 171, 544, 827, 57…
#> $ BPXSY3_n                    <int> 364, 459, 522, 541, 456, 171, 544, 827, 57…
#> $ BPXDI2_n                    <int> 364, 459, 522, 541, 456, 171, 544, 827, 57…
#> $ BPXSY2_n                    <int> 364, 459, 522, 541, 456, 171, 544, 827, 57…
#> $ BPXDI1_n                    <int> 364, 459, 522, 541, 456, 171, 544, 827, 57…
#> $ BPXSY1_n                    <int> 364, 459, 522, 541, 456, 171, 544, 827, 57…
#> $ BMXLEG_n                    <int> 364, 459, 522, 541, 456, 171, 544, 827, 57…
#> $ BPXML1_n                    <int> 364, 459, 522, 541, 456, 171, 544, 827, 57…
#> $ BPXPLS_n                    <int> 364, 459, 522, 541, 456, 171, 544, 827, 57…
#> $ PEASCTM1_n                  <int> 364, 459, 522, 541, 456, 171, 544, 827, 57…
#> $ BMXWAIST_n                  <int> 364, 459, 522, 541, 456, 171, 544, 827, 57…
#> $ BMXBMI_n                    <int> 364, 459, 522, 541, 456, 171, 544, 827, 57…
#> $ BMXHT_n                     <int> 364, 459, 522, 541, 456, 171, 544, 827, 57…
#> $ BMXARMC_n                   <int> 364, 459, 522, 541, 456, 171, 544, 827, 57…
#> $ BMXARML_n                   <int> 364, 459, 522, 541, 456, 171, 544, 827, 57…
#> $ Poverty_Income_Ratio_n      <int> 364, 459, 522, 541, 456, 171, 544, 827, 57…
#> $ BMXWT_n                     <int> 364, 459, 522, 541, 456, 171, 544, 827, 57…
#> $ Age_n                       <int> 364, 459, 522, 541, 456, 171, 544, 827, 57…
#> $ BPXDI3_n_miss               <int> 110, 142, 155, 117, 136, 36, 134, 206, 138…
#> $ BPXSY3_n_miss               <int> 110, 142, 155, 117, 136, 36, 134, 206, 138…
#> $ BPXDI2_n_miss               <int> 114, 145, 160, 112, 129, 34, 126, 208, 138…
#> $ BPXSY2_n_miss               <int> 114, 145, 160, 112, 129, 34, 126, 208, 138…
#> $ BPXDI1_n_miss               <int> 113, 148, 163, 129, 139, 38, 138, 219, 145…
#> $ BPXSY1_n_miss               <int> 113, 148, 163, 129, 139, 38, 138, 219, 145…
#> $ BMXLEG_n_miss               <int> 116, 139, 165, 111, 134, 35, 126, 199, 129…
#> $ BPXML1_n_miss               <int> 101, 130, 143, 104, 119, 31, 113, 184, 126…
#> $ BPXPLS_n_miss               <int> 101, 131, 142, 104, 119, 31, 113, 183, 126…
#> $ PEASCTM1_n_miss             <int> 131, 145, 170, 229, 155, 48, 187, 294, 229…
#> $ BMXWAIST_n_miss             <int> 59, 66, 79, 66, 55, 25, 73, 91, 55, 57, 26…
#> $ BMXBMI_n_miss               <int> 37, 36, 45, 42, 32, 15, 39, 59, 36, 32, 11…
#> $ BMXHT_n_miss                <int> 36, 36, 44, 42, 32, 14, 39, 59, 34, 32, 11…
#> $ BMXARMC_n_miss              <int> 41, 40, 58, 44, 41, 20, 50, 65, 39, 42, 20…
#> $ BMXARML_n_miss              <int> 43, 42, 57, 45, 40, 22, 52, 66, 39, 42, 21…
#> $ Poverty_Income_Ratio_n_miss <int> 0, 0, 0, 0, 0, 171, 0, 0, 0, 0, 0, 0, 0, 1…
#> $ BMXWT_n_miss                <int> 24, 17, 30, 27, 22, 9, 23, 36, 21, 19, 6, …
#> $ Age_n_miss                  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, …
#> $ BPXDI3_min                  <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 26,…
#> $ BPXSY3_min                  <dbl> 84, 82, 74, 82, 86, 80, 82, 80, 78, 82, 86…
#> $ BPXDI2_min                  <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 36,…
#> $ BPXSY2_min                  <dbl> 86, 86, 72, 80, 86, 84, 84, 84, 76, 82, 86…
#> $ BPXDI1_min                  <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 42,…
#> $ BPXSY1_min                  <dbl> 88, 80, 74, 82, 84, 84, 86, 84, 78, 84, 86…
#> $ BMXLEG_min                  <dbl> 27.6, 27.7, 24.9, 27.1, 27.2, 29.8, 28.0, …
#> $ BPXML1_min                  <dbl> 110, 0, 110, 100, 110, 120, 110, 110, 110,…
#> $ BPXPLS_min                  <dbl> 50, 46, 46, 44, 46, 50, 46, 46, 50, 48, 36…
#> $ PEASCTM1_min                <dbl> 6, 9, 6, 37, 7, 46, 8, 3, 7, 6, 5, 7, 6, 1…
#> $ BMXWAIST_min                <dbl> 38.7, 43.3, 40.5, 46.2, 42.7, 43.9, 44.2, …
#> $ BMXBMI_min                  <dbl> 13.40, 12.50, 13.41, 13.40, 12.70, 13.30, …
#> $ BMXHT_min                   <dbl> 78.5, 81.6, 82.8, 86.0, 82.5, 92.3, 86.9, …
#> $ BMXARMC_min                 <dbl> 13.8, 13.8, 13.9, 14.6, 13.7, 14.0, 13.1, …
#> $ BMXARML_min                 <dbl> 13.9, 15.0, 14.4, 16.0, 14.0, 15.6, 16.0, …
#> $ Poverty_Income_Ratio_min    <dbl> 0.00, 0.12, 0.27, 2.25, 0.30, Inf, 0.48, 0…
#> $ BMXWT_min                   <dbl> 7.7, 8.9, 8.5, 10.0, 8.9, 10.0, 8.9, 8.2, …
#> $ Age_min                     <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …
#> $ BPXDI3_max                  <dbl> 116, 126, 112, 110, 108, 102, 112, 118, 11…
#> $ BPXSY3_max                  <dbl> 224, 212, 218, 210, 196, 222, 212, 232, 20…
#> $ BPXDI2_max                  <dbl> 126, 124, 110, 114, 110, 104, 108, 122, 11…
#> $ BPXSY2_max                  <dbl> 190, 228, 212, 208, 194, 216, 210, 234, 20…
#> $ BPXDI1_max                  <dbl> 124, 124, 106, 114, 104, 112, 116, 116, 11…
#> $ BPXSY1_max                  <dbl> 208, 230, 220, 208, 198, 216, 222, 238, 20…
#> $ BMXLEG_max                  <dbl> 47.2, 46.8, 48.2, 48.5, 50.0, 45.0, 46.7, …
#> $ BPXML1_max                  <dbl> 240, 250, 888, 220, 220, 250, 888, 888, 24…
#> $ BPXPLS_max                  <dbl> 120, 116, 130, 120, 112, 110, 140, 118, 12…
#> $ PEASCTM1_max                <dbl> 1131, 1536, 1243, 1439, 1274, 1086, 1142, …
#> $ BMXWAIST_max                <dbl> 165.5, 171.6, 156.8, 163.5, 158.8, 151.7, …
#> $ BMXBMI_max                  <dbl> 57.80, 82.10, 77.50, 68.70, 84.40, 64.70, …
#> $ BMXHT_max                   <dbl> 184.8, 186.4, 180.9, 187.8, 184.1, 177.4, …
#> $ BMXARMC_max                 <dbl> 54.0, 57.3, 51.7, 58.1, 48.5, 51.5, 58.3, …
#> $ BMXARML_max                 <dbl> 42.0, 43.9, 43.2, 43.0, 43.9, 41.9, 43.0, …
#> $ Poverty_Income_Ratio_max    <dbl> 0.43, 0.95, 1.35, 5.00, 1.85, -Inf, 2.31, …
#> $ BMXWT_max                   <dbl> 157.5, 187.7, 191.6, 193.7, 219.6, 173.4, …
#> $ Age_max                     <dbl> 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80…
#> $ BPXDI3_mean                 <dbl> 65.09449, 67.35016, 64.74659, 65.93396, 66…
#> $ BPXSY3_mean                 <dbl> 117.7087, 123.9558, 122.2616, 116.7594, 12…
#> $ BPXDI2_mean                 <dbl> 65.25600, 67.51592, 65.16575, 66.54079, 66…
#> $ BPXSY2_mean                 <dbl> 117.6400, 125.1401, 123.3702, 117.6364, 12…
#> $ BPXDI1_mean                 <dbl> 66.28685, 68.47588, 65.14206, 66.51942, 66…
#> $ BPXSY1_mean                 <dbl> 117.8725, 125.1447, 122.8969, 117.5485, 12…
#> $ BMXLEG_mean                 <dbl> 38.63629, 38.14375, 37.86022, 38.96349, 37…
#> $ BPXML1_mean                 <dbl> 143.3840, 149.3617, 155.3879, 143.5515, 14…
#> $ BPXPLS_mean                 <dbl> 77.87072, 76.92073, 75.43684, 74.52174, 75…
#> $ PEASCTM1_mean               <dbl> 543.7597, 563.0000, 554.6705, 637.6538, 55…
#> $ BMXWAIST_mean               <dbl> 83.58656, 89.62290, 88.33115, 89.26379, 84…
#> $ BMXBMI_mean                 <dbl> 26.12003, 27.81638, 27.77199, 27.98603, 26…
#> $ BMXHT_mean                  <dbl> 148.9064, 150.9000, 150.5358, 156.7822, 15…
#> $ BMXARMC_mean                <dbl> 28.14458, 29.31217, 29.28405, 30.31288, 28…
#> $ BMXARML_mean                <dbl> 32.38505, 32.86547, 33.19505, 34.53145, 32…
#> $ Poverty_Income_Ratio_mean   <dbl> 0.10730769, 0.46331155, 0.70402299, 4.6876…
#> $ BMXWT_mean                  <dbl> 60.84735, 65.91312, 66.12337, 70.02490, 63…
#> $ Age_mean                    <dbl> 24.66484, 33.91285, 35.53831, 33.82994, 31…
#> $ BPXDI3_median               <dbl> 66, 68, 66, 68, 68, 70, 66, 68, 66, 68, 68…
#> $ BPXSY3_median               <dbl> 114, 118, 116, 112, 116, 122, 114, 116, 11…
#> $ BPXDI2_median               <dbl> 66, 66, 66, 68, 68, 68, 66, 66, 66, 66, 66…
#> $ BPXSY2_median               <dbl> 114, 120, 116, 114, 116, 122, 114, 116, 11…
#> $ BPXDI1_median               <dbl> 66, 68, 66, 68, 68, 68, 66, 66, 66, 68, 68…
#> $ BPXSY1_median               <dbl> 114, 120, 118, 114, 116, 122, 114, 116, 11…
#> $ BMXLEG_median               <dbl> 38.95, 38.20, 38.00, 39.05, 38.00, 38.30, …
#> $ BPXML1_median               <dbl> 140, 140, 140, 140, 140, 140, 140, 140, 14…
#> $ BPXPLS_median               <dbl> 78, 76, 74, 74, 74, 74, 76, 76, 74, 74, 74…
#> $ PEASCTM1_median             <dbl> 567.0, 576.5, 603.5, 657.5, 590.0, 667.0, …
#> $ BMXWAIST_median             <dbl> 80.80, 90.20, 91.60, 89.40, 86.70, 99.60, …
#> $ BMXBMI_median               <dbl> 23.700, 26.660, 27.230, 26.900, 25.500, 30…
#> $ BMXHT_median                <dbl> 158.45, 159.00, 158.35, 161.40, 158.50, 16…
#> $ BMXARMC_median              <dbl> 28.00, 29.80, 30.10, 30.70, 29.30, 32.50, …
#> $ BMXARML_median              <dbl> 34.60, 35.40, 35.70, 36.00, 35.15, 36.00, …
#> $ Poverty_Income_Ratio_median <dbl> 0.085, 0.410, 0.680, 5.000, 0.860, NA, 1.0…
#> $ BMXWT_median                <dbl> 59.65, 67.10, 70.05, 70.55, 64.95, 76.75, …
#> $ Age_median                  <dbl> 19.0, 28.0, 31.0, 35.0, 24.0, 42.0, 26.0, …
#> $ BPXDI3_sd                   <dbl> 16.55506, 17.14552, 16.63621, 15.14111, 16…
#> $ BPXSY3_sd                   <dbl> 19.76127, 23.39178, 23.76564, 18.70560, 20…
#> $ BPXDI2_sd                   <dbl> 15.90050, 16.45426, 16.46419, 14.67869, 16…
#> $ BPXSY2_sd                   <dbl> 19.00957, 23.64783, 23.73587, 18.99184, 21…
#> $ BPXDI1_sd                   <dbl> 14.07457, 15.88685, 15.49407, 13.23757, 15…
#> $ BPXSY1_sd                   <dbl> 19.40844, 24.38247, 23.34555, 18.58550, 21…
#> $ BMXLEG_sd                   <dbl> 3.544325, 3.598819, 3.773591, 3.155392, 3.…
#> $ BPXML1_sd                   <dbl> 20.31134, 24.59216, 69.76214, 18.14822, 21…
#> $ BPXPLS_sd                   <dbl> 12.51924, 12.55665, 12.28718, 11.47804, 12…
#> $ PEASCTM1_sd                 <dbl> 248.5154, 272.9288, 271.9278, 245.9812, 25…
#> $ BMXWAIST_sd                 <dbl> 25.32800, 26.69575, 25.81373, 21.50595, 24…
#> $ BMXBMI_sd                   <dbl> 9.509575, 10.418529, 9.889805, 8.842373, 9…
#> $ BMXHT_sd                    <dbl> 23.88131, 22.45114, 21.96616, 16.70671, 22…
#> $ BMXARMC_sd                  <dbl> 8.507530, 9.030110, 8.539857, 7.509531, 8.…
#> $ BMXARML_sd                  <dbl> 6.895394, 6.810559, 6.819470, 5.132052, 6.…
#> $ Poverty_Income_Ratio_sd     <dbl> 0.09773131, 0.20340318, 0.26553590, 0.5764…
#> $ BMXWT_sd                    <dbl> 33.33993, 35.45615, 33.77805, 29.73478, 33…
#> $ Age_sd                      <dbl> 20.24840, 25.68241, 26.35764, 22.24478, 25…

We can actually restructure this table if we use some dplyr:

summary_table %>%
  pivot_longer(cols = contains(features_65_num$feature))
#> # A tibble: 18,900 × 5
#> # Groups:   Gender, Race [10]
#>   Gender Race  Family_Income  name          value
#>   <chr>  <chr> <chr>          <chr>         <dbl>
#> 1 Female Black $ 0 to $ 4,999 BPXDI3_n      364  
#> 2 Female Black $ 0 to $ 4,999 BPXDI3_n_miss 110  
#> 3 Female Black $ 0 to $ 4,999 BPXDI3_min      0  
#> 4 Female Black $ 0 to $ 4,999 BPXDI3_max    116  
#> 5 Female Black $ 0 to $ 4,999 BPXDI3_mean    65.1
#> 6 Female Black $ 0 to $ 4,999 BPXDI3_median  66  
#> # … with 18,894 more rows

We can also just focus on the counts of missing values:

summary_table %>%
  pivot_longer(cols = contains(features_65_num$feature)) %>%
  filter(str_detect(name, "n_miss")) %>%
  arrange(-value)
#> # A tibble: 2,700 × 5
#> # Groups:   Gender, Race [10]
#>   Gender Race  Family_Income name          value
#>   <chr>  <chr> <chr>         <chr>         <dbl>
#> 1 Female White <NA>          BPXDI3_n_miss  2701
#> 2 Female White <NA>          BPXSY3_n_miss  2700
#> 3 Female White <NA>          BPXDI2_n_miss  2619
#> 4 Female White <NA>          BPXSY2_n_miss  2619
#> 5 Male   White <NA>          BPXDI3_n_miss  2516
#> 6 Male   White <NA>          BPXSY3_n_miss  2516
#> # … with 2,694 more rows

We can impute missing values with the mean for the column, perhaps by Gender, Race, Family_Income

A_DATA_2.Num.impute <- A_DATA_2.Num %>%
  filter(!is.na(DIABETES)) %>%
  mutate_at(vars(Gender, Race), ~if_else(is.na(.x), "Missing", .x)) %>%
  group_by(Gender, Race) %>%
  mutate_at(vars(features_65_num$feature),  ~if_else(is.na(.x), mean(.x, na.rm = TRUE), .x)) %>%
  ungroup() %>%
  select(SEQN, DIABETES, Gender, Race, Family_Income, all_of(features_65_num$feature))

We can recompute our summary table - and if we were so inclined see if any of these values changed significantly:

A_DATA_2.Num.impute %>%
  group_by(Gender, Race, Family_Income) %>%
  summarise_at(vars(all_of(features_65_num$feature)), 
               list( 
                 n = ~n(),
                 n_miss = ~sum(is.na(.x)),
                 min = ~min(.x , na.rm = TRUE),
                 max = ~max(.x , na.rm = TRUE),
                 mean = ~mean(.x , na.rm = TRUE),
                 median = ~median(.x , na.rm = TRUE),
                 sd = ~sd(.x , na.rm = TRUE)),
               .groups='keep'
               ) %>%
  ungroup() %>%
  pivot_longer(contains(features_65_num$feature)) %>%
  filter(str_detect(name,"n_miss")) %>%
  arrange(-value)
#> # A tibble: 2,700 × 5
#>   Gender Race  Family_Income  name          value
#>   <chr>  <chr> <chr>          <chr>         <dbl>
#> 1 Female Black $ 0 to $ 4,999 BPXDI3_n_miss     0
#> 2 Female Black $ 0 to $ 4,999 BPXSY3_n_miss     0
#> 3 Female Black $ 0 to $ 4,999 BPXDI2_n_miss     0
#> 4 Female Black $ 0 to $ 4,999 BPXSY2_n_miss     0
#> 5 Female Black $ 0 to $ 4,999 BPXDI1_n_miss     0
#> 6 Female Black $ 0 to $ 4,999 BPXSY1_n_miss     0
#> # … with 2,694 more rows

But for now let’s just look at the number of missing values:

Feature_Percent_Complete_Graph(A_DATA_2.Num.impute,0)

\(~\)

10.5 Correlated Features

Often we will want to search for highly correlated features in order to reduce the number of them in a model.

In the extreme case, consider some variables that are perfectly correlated, i.e. linearly dependent.

Let’s say you’re using logistic regression to predict the probability of having Diabetes, where one of the predictor variables will be Age, but for some reason you read in two input variables \(x_1\) = Age in months and \(x_2\) = Age in years. Clearly, \(x_1\) and \(x_2\) are linearly dependent as \[x_1 \cdot 12 = x_2 \].

In your model, you will estimate coefficients \(\beta_i\) such that

\[\log \left( \frac{P(DIABETES == 1)}{P(DIABETES == 0)} \right)=\beta_0+\beta_1x_1+\beta_2x_2+(\cdots)\]

Due to the linear dependence, we can rewrite this:

\[\log \left( \frac{P(DIABETES == 1)}{P(DIABETES == 0)} \right)=\beta_0 + \beta_1x_1 + \beta_2\cdot ( x_1 \cdot 12) + (\cdots)\]

\[\log \left( \frac{P(DIABETES == 1)}{P(DIABETES == 0)} \right)=\beta_0 + x_1 ( \beta_1 + 12 \cdot \beta_2 ) + (\cdots)\]

SUPPOSE (for some reason) that the Age in years has no influence on Diabetic status.

Then you would expect your model to find \[\beta_1 = \beta_2=0\].

However, due to linear dependence, the model cannot distinguish between \(\beta_1=12\), \(\beta_2 = -1\) or any other combination where \(\beta_1 + 12 \cdot \beta_2 = 0\)

Thus on the one hand, your model itself might become numerically unstable. On the other hand, you can no longer interpret the results correctly.

The same is true when variables are no longer linearly dependent but highly correlated.

GGally::ggcorr(A_DATA_2.Num.impute %>% select(all_of(features_65_num$feature)),
               method = c("pairwise", "pearson"),
               nbreaks = 10,
               label = TRUE)
#> Registered S3 method overwritten by 'GGally':
#>   method from   
#>   +.gg   ggplot2

In the graph below we highlighted all the correlations with absolute value greater than \(.6\):

library('GGally')

ggcorr(A_DATA_2.Num.impute %>% select(all_of(features_65_num$feature)),
       geom = "blank", label = TRUE, hjust = 0.75) +
  geom_point(size = 10, aes(color = coefficient > 0, alpha = abs(coefficient) > 0.6)) +
  scale_alpha_manual(values = c("TRUE" = 0.25, "FALSE" = 0)) +
  guides(color = FALSE, alpha = FALSE)
#> Warning: The `<scale>` argument of `guides()` cannot be `FALSE`. Use "none" instead as
#> of ggplot2 3.3.4.

library('corrr')
cor.corr_data <- correlate(A_DATA_2.Num.impute %>% select(all_of(features_65_num$feature)))
#> Correlation computed with
#> • Method: 'pearson'
#> • Missing treated using: 'pairwise.complete.obs'

cor.corr_data %>%
  glimpse()
#> Rows: 18
#> Columns: 19
#> $ term                 <chr> "BPXDI3", "BPXSY3", "BPXDI2", "BPXSY2", "BPXDI1",…
#> $ BPXDI3               <dbl> NA, 0.38920867, 0.83967525, 0.36443328, 0.7753937…
#> $ BPXSY3               <dbl> 0.38920867, NA, 0.38726807, 0.90335402, 0.3855849…
#> $ BPXDI2               <dbl> 0.83967525, 0.38726807, NA, 0.40498199, 0.7965398…
#> $ BPXSY2               <dbl> 0.36443328, 0.90335402, 0.40498199, NA, 0.3885915…
#> $ BPXDI1               <dbl> 0.77539372, 0.38558496, 0.79653986, 0.38859151, N…
#> $ BPXSY1               <dbl> 0.33796063, 0.85488694, 0.36272962, 0.87340339, 0…
#> $ BMXLEG               <dbl> 0.17755182, 0.08128816, 0.17371484, 0.07458496, 0…
#> $ BPXML1               <dbl> 0.29189725, 0.71441600, 0.31253493, 0.72838772, 0…
#> $ BPXPLS               <dbl> -0.04218378, -0.13904054, -0.04813751, -0.1459656…
#> $ PEASCTM1             <dbl> 0.04786246, 0.09740853, 0.05094561, 0.09772117, 0…
#> $ BMXWAIST             <dbl> 0.21484511, 0.28490823, 0.22377722, 0.28689614, 0…
#> $ BMXBMI               <dbl> 0.19847274, 0.25144750, 0.20672813, 0.25229207, 0…
#> $ BMXHT                <dbl> 0.1497063, 0.1099074, 0.1519246, 0.1080122, 0.157…
#> $ BMXARMC              <dbl> 0.20485012, 0.21875540, 0.21110377, 0.21854444, 0…
#> $ BMXARML              <dbl> 0.1432836, 0.1448157, 0.1464210, 0.1440410, 0.152…
#> $ Poverty_Income_Ratio <dbl> 0.08229314, 0.02244386, 0.08448418, 0.02402425, 0…
#> $ BMXWT                <dbl> 0.21497276, 0.22357424, 0.22145776, 0.22239343, 0…
#> $ Age                  <dbl> 0.20551574, 0.43888769, 0.22400775, 0.45189382, 0…

Let’s find all the rows where Age has a correlation of .5 or more with another variable.

cor.corr_data %>%
   filter(Age > .5)
#> # A tibble: 4 × 19
#>   term   BPXDI3 BPXSY3 BPXDI2 BPXSY2 BPXDI1 BPXSY1 BMXLEG BPXML1  BPXPLS PEASC…¹
#>   <chr>   <dbl>  <dbl>  <dbl>  <dbl>  <dbl>  <dbl>  <dbl>  <dbl>   <dbl>   <dbl>
#> 1 BMXWA…  0.215  0.285  0.224  0.287  0.233  0.287  0.157  0.238 -0.0462   0.432
#> 2 BMXAR…  0.205  0.219  0.211  0.219  0.219  0.217  0.231  0.173 -0.0769   0.510
#> 3 BMXAR…  0.143  0.145  0.146  0.144  0.153  0.148  0.339  0.131 -0.104    0.599
#> 4 BMXWT   0.215  0.224  0.221  0.222  0.229  0.220  0.301  0.187 -0.0701   0.521
#> # … with 8 more variables: BMXWAIST <dbl>, BMXBMI <dbl>, BMXHT <dbl>,
#> #   BMXARMC <dbl>, BMXARML <dbl>, Poverty_Income_Ratio <dbl>, BMXWT <dbl>,
#> #   Age <dbl>, and abbreviated variable name ¹PEASCTM1

Let’s find all the rows where any feature has a correlation of .7 or more with another variable:

cor.corr_data %>%
   filter_at(vars(all_of(features_65_num$feature)), any_vars( . > .7 ))
#> # A tibble: 13 × 19
#>   term   BPXDI3 BPXSY3 BPXDI2 BPXSY2 BPXDI1 BPXSY1 BMXLEG BPXML1  BPXPLS PEASC…¹
#>   <chr>   <dbl>  <dbl>  <dbl>  <dbl>  <dbl>  <dbl>  <dbl>  <dbl>   <dbl>   <dbl>
#> 1 BPXDI3 NA      0.389  0.840  0.364  0.775  0.338 0.178   0.292 -0.0422  0.0479
#> 2 BPXSY3  0.389 NA      0.387  0.903  0.386  0.855 0.0813  0.714 -0.139   0.0974
#> 3 BPXDI2  0.840  0.387 NA      0.405  0.797  0.363 0.174   0.313 -0.0481  0.0509
#> 4 BPXSY2  0.364  0.903  0.405 NA      0.389  0.873 0.0746  0.728 -0.146   0.0977
#> 5 BPXDI1  0.775  0.386  0.797  0.389 NA      0.404 0.178   0.320 -0.0498  0.0569
#> 6 BPXSY1  0.338  0.855  0.363  0.873  0.404 NA     0.0773  0.739 -0.156   0.103 
#> # … with 7 more rows, 8 more variables: BMXWAIST <dbl>, BMXBMI <dbl>,
#> #   BMXHT <dbl>, BMXARMC <dbl>, BMXARML <dbl>, Poverty_Income_Ratio <dbl>,
#> #   BMXWT <dbl>, Age <dbl>, and abbreviated variable name ¹PEASCTM1

Above we used an _at version of a dplyr function, in this case it will apply the function any_vars( . > .7 ) to the columns corr_features.

cor.corr_data %>% 
  focus(BMXWT, BMXARMC, BMXARML, BMXWAIST)
#> # A tibble: 14 × 5
#>   term   BMXWT BMXARMC BMXARML BMXWAIST
#>   <chr>  <dbl>   <dbl>   <dbl>    <dbl>
#> 1 BPXDI3 0.215   0.205   0.143    0.215
#> 2 BPXSY3 0.224   0.219   0.145    0.285
#> 3 BPXDI2 0.221   0.211   0.146    0.224
#> 4 BPXSY2 0.222   0.219   0.144    0.287
#> 5 BPXDI1 0.229   0.219   0.153    0.233
#> 6 BPXSY1 0.220   0.217   0.148    0.287
#> # … with 8 more rows

cor.corr_data %>%
  focus(BMXWT, BMXARMC, BMXARML, BMXWAIST, mirror = TRUE) %>%  # Focus only on these
  shave() %>% # Remove the upper triangle
  fashion()   # Print in nice format
#>       term BMXWT BMXARMC BMXARML BMXWAIST
#> 1 BMXWAIST   .87                         
#> 2  BMXARMC   .94                         
#> 3  BMXARML   .83     .81                 
#> 4    BMXWT           .94     .83      .87

cor.corr_data %>%
  focus(BMXWT, BMXARMC, BMXARML, BMXWAIST, mirror = TRUE) %>%
  shave(upper = FALSE) %>% 
  rplot()

The findCorrelation function in the caret library is useful to find correlated features as well:

correlated_features <- caret::findCorrelation( cor(A_DATA_2.Num.impute %>% select(all_of(features_65_num$feature)) ),
                        cutoff = .8,
                        names = TRUE )

correlated_features
#> [1] "BMXWT"    "BMXARMC"  "BMXWAIST" "BMXARML"  "BPXSY2"   "BPXSY3"   "BPXDI2"

The DataExplorer package also has a nice plot_correlate function:


DataExplorer::plot_correlation(A_DATA_2.Num.impute %>% select(all_of(features_65_num$feature)), maxcat = 5L)

\(~\)

10.6 Principal Component Analysis

Principal Component Analysis describes an orthogonal (preserves inner product) linear transformation of the data; where the data are mapped into a new coordinate system for which the first dimension (the first principal component) contains the greatest variance of the data; the second dimension contains the second greatest variance; and so on.

We will showcase how Principal Component Analysis (PCA) can yield the Principal Components (PCs) can be utilized as effectively as features in a predictive model.

First will split the data:

set.seed(4321)

A_DATA_2.Num.impute <- A_DATA_2.Num.impute %>%
  mutate(DIABETES_factor = as.factor(DIABETES))


PCA_train.sample <- sample(A_DATA_2.Num.impute$SEQN,
                    nrow(A_DATA_2.Num.impute)*.65,
                    replace = FALSE)

PCA.train <- A_DATA_2.Num.impute %>%
  filter(SEQN %in% PCA_train.sample) 

PCA.test <- A_DATA_2.Num.impute %>%
  filter(!(SEQN %in% PCA_train.sample))

10.6.1 Fit PCA Model

Below, we set center and scale to TRUE so R will center (subtract the mean) and scale (divide by the standard deviation) by each numeric column (i.e., z-score, normalize. or standardize the data). Now we can fit a PCA model on the training data:

A_DATA_2.pca.model <- prcomp(PCA.train %>% select(all_of(features_65_num$feature)), 
                              center=TRUE,
                              scale=TRUE)

A_DATA_2.pca.sum <- summary(A_DATA_2.pca.model)
A_DATA_2.pca.sum
#> Importance of components:
#>                           PC1    PC2    PC3     PC4     PC5     PC6     PC7
#> Standard deviation     2.5932 1.8469 1.3480 1.11308 0.99830 0.92194 0.90134
#> Proportion of Variance 0.3736 0.1895 0.1010 0.06883 0.05537 0.04722 0.04513
#> Cumulative Proportion  0.3736 0.5631 0.6641 0.73290 0.78826 0.83548 0.88062
#>                            PC8     PC9    PC10    PC11   PC12    PC13    PC14
#> Standard deviation     0.67495 0.62261 0.51934 0.49271 0.4799 0.40570 0.37000
#> Proportion of Variance 0.02531 0.02154 0.01498 0.01349 0.0128 0.00914 0.00761
#> Cumulative Proportion  0.90593 0.92746 0.94245 0.95593 0.9687 0.97787 0.98548
#>                           PC15    PC16    PC17    PC18
#> Standard deviation     0.30437 0.29754 0.22877 0.16703
#> Proportion of Variance 0.00515 0.00492 0.00291 0.00155
#> Cumulative Proportion  0.99062 0.99554 0.99845 1.00000

You can see there are 18 principal components, with each explaining a proportion of the variability in the data. For example, PC1 explains 37.36% of the total variance; the first 5 principal components account for over 78.826% of the variance; the first 10 principal components account for over 94.245% of the variance.

10.6.2 Plot Principal Components

The biplot is used to visualize principal components. This plots the first and second principal components. The closer the variable is the the center, the less contribution that variable has to either principal component. The configuration of arrows reflects the relations of the variables. The cosine of the angle (the dot product) between the arrows reflects the correlation between the variables they represent, and the principal component.

AMR::ggplot_pca(A_DATA_2.pca.model, arrows_colour = 'red') # note here that the assumption is to plot PC1 V PC2

(#fig:PCA-plot-1-2 )biplot PC1 Vs PC2

This is the first versus the third principal component, again notice the magnitude and direction of the vector with relation to the first principal component:

AMR::ggplot_pca(A_DATA_2.pca.model,
                choices = c(1,3), # here we specify PC1 V PC3 
                arrows_colour = 'red')

(#fig:PCA-plot-1-3 )biplot PC1 Vs PC3

And now here’s a look at the second versus the third, we see everything is concentrated near the origin (Remark when we zoomed in with coord_cartesian some points were excluded from the Figure below)

AMR::ggplot_pca(A_DATA_2.pca.model,
                choices = c(2,3),
                arrows_colour = 'red',
                arrows_size = 1,
                arrows_textsize = 4) +
  coord_cartesian(xlim = c(-5,5), ylim=c(-5,5))

(#fig:PCA-plot-2-3 )biplot PC2 Vs PC3

10.6.3 Scree Plot

We aim to find the components with the maximum variance so we can retain as much information about the original dataset as possible.

To determine the number of principal components to retain in our analysis we need to compute the proportion of variance explained.

We can plot the cumulative proportion of variance explained in a scree plot to determine the number of principal components to retain:

var_exp <- A_DATA_2.pca.model$sdev^2

# Proportion of variance explained
pct_var_exp <- var_exp/sum(var_exp)

Prop_Var_Explained_df <- as_tibble(cbind(var_exp, pct_var_exp))

Prop_Var_Explained_df$PC <- 1:nrow(Prop_Var_Explained_df)


Prop_Var_Explained_df <- Prop_Var_Explained_df %>%
  mutate(cum_pct = cumsum(pct_var_exp)) 

Prop_Var_Explained_df %>%
  ggplot(aes(x=PC, y=cum_pct)) +
  geom_point() +
  geom_smooth()
#> `geom_smooth()` using method = 'loess' and formula = 'y ~ x'

min_cum_pct <- min(Prop_Var_Explained_df$cum_pct)

min_cum_pct
#> [1] 0.3735985

pc_var2 <- Prop_Var_Explained_df %>%
  filter(cum_pct <= max(0.8, min_cum_pct))

pc_var2 %>%
  head()
#> # A tibble: 5 × 4
#>   var_exp pct_var_exp    PC cum_pct
#>     <dbl>       <dbl> <int>   <dbl>
#> 1   6.72       0.374      1   0.374
#> 2   3.41       0.190      2   0.563
#> 3   1.82       0.101      3   0.664
#> 4   1.24       0.0688     4   0.733
#> 5   0.997      0.0554     5   0.788

ggplot(pc_var2, aes(x = reorder(PC, pct_var_exp), y = pct_var_exp)) +
    geom_bar(stat = "identity") +
    scale_y_continuous(labels = scales::percent) +
    coord_flip() +
    labs(x = "Principal Components", y = "% Variance Explained")

Next let’s get the eigenvectors

rotation_tibble <- as_tibble(A_DATA_2.pca.model$rotation , rownames='Feature')

rotation_tibble %>%
  head()
#> # A tibble: 6 × 19
#>   Feature   PC1   PC2    PC3      PC4     PC5    PC6     PC7      PC8      PC9
#>   <chr>   <dbl> <dbl>  <dbl>    <dbl>   <dbl>  <dbl>   <dbl>    <dbl>    <dbl>
#> 1 BPXDI3  0.183 0.259 -0.469 -0.0709  -0.0571  0.119 -0.0323  0.0113   0.00534
#> 2 BPXSY3  0.226 0.348  0.240  0.00761  0.0530 -0.125 -0.0362  0.148   -0.300  
#> 3 BPXDI2  0.189 0.267 -0.459 -0.0724  -0.0623  0.121 -0.0284 -0.00765  0.0146 
#> 4 BPXSY2  0.228 0.352  0.248  0.0106   0.0458 -0.116 -0.0335  0.129   -0.279  
#> 5 BPXDI1  0.192 0.262 -0.432 -0.0645  -0.0549  0.109 -0.0266 -0.0253   0.0381 
#> 6 BPXSY1  0.226 0.343  0.258  0.0241   0.0408 -0.104 -0.0363  0.0732  -0.188  
#> # … with 9 more variables: PC10 <dbl>, PC11 <dbl>, PC12 <dbl>, PC13 <dbl>,
#> #   PC14 <dbl>, PC15 <dbl>, PC16 <dbl>, PC17 <dbl>, PC18 <dbl>

Get the values per the feature

rotation_tibble_T <- rotation_tibble %>%
  pivot_longer(-Feature, names_to = 'variable', values_to = 'value') 

rotation_tibble_T %>%
  head()
#> # A tibble: 6 × 3
#>   Feature variable   value
#>   <chr>   <chr>      <dbl>
#> 1 BPXDI3  PC1       0.183 
#> 2 BPXDI3  PC2       0.259 
#> 3 BPXDI3  PC3      -0.469 
#> 4 BPXDI3  PC4      -0.0709
#> 5 BPXDI3  PC5      -0.0571
#> 6 BPXDI3  PC6       0.119

Plot the results of the feature’s contribution to the PC - lets do the 1st below:

rotation_tibble_T %>%
  filter(variable == paste0("PC", 1)) %>%
  ggplot(aes(x = Feature, y = value)) +
  geom_bar(stat = "identity") +
  coord_flip() +
  ylab("Relative Importance")

10.6.3.1 proc.pca

We can functionalize the entire above process and enhance the proportion of variance explained graph with a ggplot:

proc.pca <- function(data ){
  
  # fit PCA model 
  data.pca <- prcomp(data, 
                      center=TRUE,
                      scale=TRUE)



 biplot_function <- function(choices = c(1,2) , # these are the things I probably want to pass into biplot
                              arrows_colour = "red",
                              arrows_size = 1,
                              arrows_textsize = 4, # the ... should pass anything else 
                              ...){
    AMR::ggplot_pca(data.pca,
                    choices = choices,
                    arrows = TRUE,
                    arrows_colour = arrows_colour, # I think Red is easier to see on Black
                    arrows_size = arrows_size, # bigger arrows
                    arrows_textsize = 4, #bigger text size 
                    ...)
   }

 
  # Proportion of variance explained
  var_exp <- data.pca$sdev^2
  
  pct_var_exp <- var_exp/sum(var_exp)

  Prop_Var_Explained_df <- as_tibble(cbind(var_exp ,pct_var_exp))

  Prop_Var_Explained_df$PC <- 1:nrow(Prop_Var_Explained_df)

  Prop_Var_Explained_df <- Prop_Var_Explained_df %>%
    mutate(cum_pct = cumsum(pct_var_exp)) %>%
    select(PC, var_exp, pct_var_exp, cum_pct)

  # scree plot 
  scree_plot <- Prop_Var_Explained_df %>%
    ggplot(aes(x=PC, y=cum_pct)) +
    geom_point() +
    geom_smooth()

  # PC_var_Explained_bar
  min_cum_pct <- min(Prop_Var_Explained_df$cum_pct)

  PC_var_Explained_bar <- function(variance_cap = 0.8){
  
    pc_var2 <- Prop_Var_Explained_df %>%
      filter(cum_pct <= max(variance_cap, min_cum_pct))

    PC_var_Explained_bar <- ggplot(pc_var2, aes(x = reorder(PC, pct_var_exp), y = pct_var_exp)) +
      geom_bar(stat = "identity") +
      scale_y_continuous(labels = scales::percent) +
      coord_flip() +
      labs(x = "Principal Components", y = "% Variance Explained")

    return(PC_var_Explained_bar)
    }

  #Feature_Imp_PC
   
   # eigenvectors
  rotation_tibble <- as_tibble(A_DATA_2.pca.model$rotation , rownames='Feature')
    
   # get the values per the feature  
  rotation_tibble_T <- rotation_tibble %>%
    pivot_longer(-Feature, names_to = 'variable', values_to = 'value')

    # here's a nice plot of the feature's contribution to the PC 
  Feature_Imp_PC <- function(PC_Num = 1){
    rotation_tibble_T %>%
      filter(variable == paste0("PC", PC_Num)) %>%
      mutate(Feature = reorder(Feature, value)) %>%
      ggplot(aes(x = Feature, y = value)) +
      geom_bar(stat = "identity") +
      coord_flip() +
      ylab("Relative Importance") +
      labs(title = paste0("PC",PC_Num))
    }

#QED

return(list(PCA_Sum = summary(data.pca) ,
            Prop_Var_Explained_df = Prop_Var_Explained_df,
            scree_plot = scree_plot,
            biplot = biplot_function, 
            PC_var_Explained_bar = PC_var_Explained_bar,
            Feature_Imp_PC = Feature_Imp_PC)
       ) 
}

10.6.3.1.1 test proc.pca

Now we can test our function:

OUTPUT.proc.pca <- proc.pca(PCA.train %>% select(all_of(features_65_num$feature)))

Let’s check out the Relative Feature Importance in PC8, for instance:

OUTPUT.proc.pca$Feature_Imp_PC(8)

Again note how most of the variance is explained by the PC1:

OUTPUT.proc.pca$PC_var_Explained_bar(1)

Also note the high Feature Relative Importance of features like BMXWT , Age and BMXBMI in this graph in Figure below:

OUTPUT.proc.pca$Feature_Imp_PC(1)

(#fig:out-proc-pca-FE1 )Feature Relative Importance - PC1

and the corresponding magnitude and direction of the vector in the plot in Figure below:

OUTPUT.proc.pca$biplot(c(1,2))  +
  coord_cartesian(ylim=c(-2.5,2.5))

Figure 10.1: biplot PC1 V PC2

At the same time, note the above magnitude and direction of BPXPLS corresponding to the negative feature relative importance in PC2 in Figure below:

OUTPUT.proc.pca$Feature_Imp_PC(2)

Getting back on track, we recall

OUTPUT.proc.pca$Prop_Var_Explained_df %>%
  filter(3 <= PC & PC <= 7 ) %>%
  kbl() %>%
  kable_paper("striped", full_width = F) %>%
  row_spec(3, bold = T, color = "white", background = "#D7261E")

PC	var_exp	pct_var_exp	cum_pct
3	1.8172379	0.1009577	0.6640650
4	1.2389422	0.0688301	0.7328952
5	0.9966022	0.0553668	0.7882619
6	0.8499791	0.0472211	0.8354830
7	0.8124173	0.0451343	0.8806173

So knowing 5 Principal Components accounts for about 84.3% of the variance of the data observed in the PCA.train dataset.

10.6.4 Modeling with PCs

We will quickly compare models using:

all of the numeric features
the first 5 PCs
5 random features

First, we’ll use the predict to predict the PCAs onto PCA.train, we then attach those predictions to PCA.train with the cbind:

PCA.PCA.train <- cbind(PCA.train, predict(A_DATA_2.pca.model, PCA.train))

PCA.PCA.train %>%
  glimpse()
#> Rows: 62,105
#> Columns: 42
#> $ SEQN                 <dbl> 46160, 7875, 29649, 30336, 78095, 91585, 49315, 4…
#> $ DIABETES             <dbl> 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
#> $ Gender               <chr> "Female", "Female", "Female", "Female", "Female",…
#> $ Race                 <chr> "White", "White", "Black", "White", "Other Hispan…
#> $ Family_Income        <chr> "$ 0 to $ 4,999", NA, NA, NA, "$ 0 to $ 4,999", "…
#> $ BPXDI3               <dbl> 64.00000, 76.00000, 66.11492, 68.00000, 90.00000,…
#> $ BPXSY3               <dbl> 100.0000, 90.0000, 118.8599, 116.0000, 212.0000, …
#> $ BPXDI2               <dbl> 70.00000, 72.00000, 66.05193, 68.00000, 98.00000,…
#> $ BPXSY2               <dbl> 102.0000, 94.0000, 119.4753, 118.0000, 200.0000, …
#> $ BPXDI1               <dbl> 70.00000, 80.00000, 65.88837, 70.00000, 92.00000,…
#> $ BPXSY1               <dbl> 104.0000, 96.0000, 119.5301, 118.0000, 204.0000, …
#> $ BMXLEG               <dbl> 42.50000, 38.50000, 38.95650, 37.90000, 36.40000,…
#> $ BPXML1               <dbl> 130.0000, 120.0000, 146.0224, 150.0000, 220.0000,…
#> $ BPXPLS               <dbl> 82.00000, 94.00000, 76.42917, 64.00000, 84.00000,…
#> $ PEASCTM1             <dbl> 484.0000, 690.0000, 126.0000, 651.0000, 821.0000,…
#> $ BMXWAIST             <dbl> 82.60000, 92.30000, 51.70000, 79.60000, 77.70000,…
#> $ BMXBMI               <dbl> 24.58000, 26.23000, 15.37000, 24.01000, 21.50000,…
#> $ BMXHT                <dbl> 163.5000, 163.7000, 116.6000, 147.6000, 156.0000,…
#> $ BMXARMC              <dbl> 29.20000, 31.90000, 17.70000, 23.80000, 23.10000,…
#> $ BMXARML              <dbl> 36.30000, 35.70000, 23.00000, 31.20000, 33.40000,…
#> $ Poverty_Income_Ratio <dbl> 0.29, 4.51, 2.15, 5.00, 0.02, 3.31, 0.57, 1.36, 1…
#> $ BMXWT                <dbl> 65.70000, 70.30000, 20.90000, 52.30000, 52.40000,…
#> $ Age                  <dbl> 19, 14, 5, 48, 51, 42, 23, 10, 38, 14, 80, 81, 44…
#> $ DIABETES_factor      <fct> 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
#> $ PC1                  <dbl> -0.8752034, -0.5059119, -3.6213223, -0.2741463, 5…
#> $ PC2                  <dbl> -1.357014885, -2.061359525, 2.470355725, 0.563217…
#> $ PC3                  <dbl> -1.5343935, -2.9989493, -0.2988726, -0.1222763, 2…
#> $ PC4                  <dbl> -0.27355317, -0.87481013, 0.22573202, 1.12254192,…
#> $ PC5                  <dbl> 1.41675221, -1.48805756, 0.38779207, -1.78643532,…
#> $ PC6                  <dbl> 0.02359763, -0.95201204, -0.54183187, 0.29288089,…
#> $ PC7                  <dbl> -0.68676342, -0.21594779, 0.51125002, 0.67567141,…
#> $ PC8                  <dbl> -0.56022967, 0.13797055, -0.24147595, 0.10008451,…
#> $ PC9                  <dbl> 0.12384792, 0.35119581, -0.06303190, 0.29760064, …
#> $ PC10                 <dbl> 0.17544669, 0.09482743, 0.07976258, 0.50663257, -…
#> $ PC11                 <dbl> -0.253751180, -0.475424150, 0.062030753, 0.028768…
#> $ PC12                 <dbl> 0.1393515881, 0.0479600873, 0.2915428526, -0.2226…
#> $ PC13                 <dbl> -0.316442862, 0.218737608, 0.003815722, 0.0271017…
#> $ PC14                 <dbl> -0.02763135, -0.10887048, 0.01630495, 0.09848579,…
#> $ PC15                 <dbl> 0.02440551, -0.20997949, 0.14573544, 0.11248075, …
#> $ PC16                 <dbl> -0.0213528612, 0.2328745767, -0.0009075528, 0.058…
#> $ PC17                 <dbl> 0.008782960, 0.097489083, 0.011334128, -0.2087617…
#> $ PC18                 <dbl> -0.122097162, 0.007710023, 0.132307401, -0.144167…

Recall our features were

features_65_num$feature
#>  [1] "BPXDI3"               "BPXSY3"               "BPXDI2"              
#>  [4] "BPXSY2"               "BPXDI1"               "BPXSY1"              
#>  [7] "BMXLEG"               "BPXML1"               "BPXPLS"              
#> [10] "PEASCTM1"             "BMXWAIST"             "BMXBMI"              
#> [13] "BMXHT"                "BMXARMC"              "BMXARML"             
#> [16] "Poverty_Income_Ratio" "BMXWT"                "Age"

We can make a formula:

features_plus <- paste0(features_65_num$feature, collapse = " + ")
features_plus
#> [1] "BPXDI3 + BPXSY3 + BPXDI2 + BPXSY2 + BPXDI1 + BPXSY1 + BMXLEG + BPXML1 + BPXPLS + PEASCTM1 + BMXWAIST + BMXBMI + BMXHT + BMXARMC + BMXARML + Poverty_Income_Ratio + BMXWT + Age"

feature_fomula <- paste0("DIABETES_factor ~ ", features_plus)
feature_fomula
#> [1] "DIABETES_factor ~ BPXDI3 + BPXSY3 + BPXDI2 + BPXSY2 + BPXDI1 + BPXSY1 + BMXLEG + BPXML1 + BPXPLS + PEASCTM1 + BMXWAIST + BMXBMI + BMXHT + BMXARMC + BMXARML + Poverty_Income_Ratio + BMXWT + Age"

Let’s also choose 5 random features for good measure

set.seed(86753094)
feature_rand_5_formula <- paste0("DIABETES_factor ~ ", 
                                 paste0(sample(features_65_num$feature, 5, replace = FALSE ), 
                                        collapse = " + "))
feature_rand_5_formula
#> [1] "DIABETES_factor ~ BMXLEG + BPXSY3 + Poverty_Income_Ratio + BMXHT + PEASCTM1"

And now we can train our three models:

logit_PCA <- glm(DIABETES_factor ~ PC1 + PC2 + PC3 + PC4 + PC5,
                 data = PCA.PCA.train,
                 family = binomial(link = 'logit'))

logit_features <- glm(as.formula(feature_fomula),
                      data = PCA.PCA.train,
                      family = binomial(link = 'logit'))

logit_rand_5 <- glm(as.formula(feature_rand_5_formula),
                     data = PCA.PCA.train,
                     family = binomial(link = 'logit'))

There is an issue: the test set does not know what it’s PCAs are at this moment, the PCA model was trained on the training set. Therefore, we will need to apply the PCAs model to the test set to get the predicted PCAs:

PCA.PCA.test <- cbind(PCA.test, predict(A_DATA_2.pca.model, PCA.test))

We can utilize our helper from the last chapter,

NOTE THE CHANGES MADE BELOW

we added additional parameters for the target and the level, now this helper-function can be utilized with other data-frames that do not contain the column DIABETES_factor, this will still add on three additional columns to the data frame to be scored probs, pred and pred_factor where pred is set to 1 if over the mean probability score of the target at the level in the training dataset.

logit_model_scorer <- function(my_model, my_data, target , level){ 
  # extracts models name 
  my_model_name <- deparse(substitute(my_model))
  
  enquo_target <- enquo(target)
  
  # store model name into a new column called model
  data.s <- my_data %>%
    mutate(model = my_model_name)
  
  # store the training data someplace
  train_data.s <- my_model$data
  
  # score the training data 
  train_data.s$probs <- predict(my_model, 
                                train_data.s, 
                                'response')
  
  # threshold query
  threshold_value_query <- train_data.s %>% 
                        group_by(!!enquo_target) %>%
                        summarise(mean_prob = mean(probs, na.rm=TRUE)) %>%
                        ungroup() %>%
                        filter(!!enquo_target == level)
  # threshold value                      
  threshold_value <- threshold_value_query$mean_prob
  
  # score test data
  data.s$probs <- predict(my_model, 
                          data.s, 
                          'response')
  
  # use threshold to make prediction  
  data.s <- data.s %>%
    mutate(pred = if_else(probs > threshold_value, 1,0)) %>%
    mutate(pred_factor = as.factor(pred))             
  
  # return scored data 
  return(data.s)
}

And we can get a quick comparison of the ROC Curves

10.6.5 Effectiveness of PCs as features

library('yardstick')
#> For binary classification, the first factor level is assumed to be the event.
#> Use the argument `event_level = "second"` to alter this as needed.
#> 
#> Attaching package: 'yardstick'
#> The following object is masked from 'package:readr':
#> 
#>     spec

compare_models <- bind_rows(
  logit_model_scorer(logit_PCA, PCA.PCA.test, DIABETES_factor, 1),
  logit_model_scorer(logit_features, PCA.PCA.test, DIABETES_factor, 1),
  logit_model_scorer(logit_rand_5, PCA.PCA.test, DIABETES_factor, 1),
) 


model_AUCS <- compare_models %>%
  group_by(model) %>%
  roc_auc(truth= DIABETES_factor, probs, event_level = "second") %>%
  mutate(model_AUC = paste(model , " AUC : ", round(.estimate,4)))

compare_models %>%
  left_join(model_AUCS) %>%
  group_by(model_AUC) %>%
  roc_curve(truth= DIABETES_factor, probs, event_level = "second") %>%
  autoplot()
#> Joining, by = "model"

In this instance the model with the first 5 PCAs performed better than a model with 5 features chosen at random.


Percent_Diff <- function(x,y){
   return(abs(x-y)/mean(x,y)*100) 
}


percent_diff_auc_Table <- model_AUCS %>%
  select(-model_AUC) %>%
  pivot_wider(names_from = model, values_from = .estimate) %>%
  mutate(percent_diff_auc.all_PCA = Percent_Diff(logit_features, logit_PCA)) %>%
  mutate(percent_diff_auc.PCA_rand_5 = Percent_Diff(logit_PCA, logit_rand_5)) 
  

percent_diff_auc_Table %>%
  glimpse()
#> Rows: 1
#> Columns: 7
#> $ .metric                     <chr> "roc_auc"
#> $ .estimator                  <chr> "binary"
#> $ logit_features              <dbl> 0.8827921
#> $ logit_PCA                   <dbl> 0.8496852
#> $ logit_rand_5                <dbl> 0.7806718
#> $ percent_diff_auc.all_PCA    <dbl> 3.750239
#> $ percent_diff_auc.PCA_rand_5 <dbl> 8.122242

So there’s only a 3.75% percent difference in Area between the logistic regression model with all of the features and the model with the first 5 PCs, and there is a 8.12% percent difference between the model with the first 5 PCs and a model with 5 features chosen at random.

\(~\)

10.7 k-means Clustering

k-means clustering that aims to partition \(n\) observations into \(k\) clusters in which each observation belonging to the cluster with the nearest mean (cluster centers or cluster centroid), serving as a prototype of the cluster.

set.seed(657)
Sample_Id <- sample(A_DATA_2.Num.impute$SEQN, 
                    nrow(A_DATA_2.Num.impute)*.4, 
                    replace = FALSE)

DATA_65.impute.sample <- A_DATA_2.Num.impute %>%
  filter(SEQN %in% Sample_Id) %>%
  mutate_at(all_of(features_65_num$feature), scale)
#> Warning: Using `all_of()` outside of a selecting function was deprecated in tidyselect
#> 1.2.0.
#> ℹ See details at
#>   <https://tidyselect.r-lib.org/reference/faq-selection-context.html>

10.7.1 How do we estimate the number of clusters?

proc.kmeans <- function(data, k = 1:9){

kclusts <-
  tibble(k = k) %>%
  mutate(
    kclust = map(k, ~kmeans(data, .x)),
    glanced = map(kclust, glance),
    augmented = map(kclust, augment, data)
  )

kmeans_model <- function(final_k = 5){
   (kclusts %>%
     filter(k == final_k))$kclust[[1]]
}

id_vars <- colnames(kclusts)

cluster_assignments <- kclusts %>%
  unnest(cols = augmented)

vars <- setdiff(colnames(cluster_assignments), c(id_vars,'.cluster'))

within_cluster_variation_plot <- kclusts %>%
  unnest(cols = c(glanced)) %>%
  ggplot(aes(k, tot.withinss))  +
  geom_line() +
  geom_point() +
  labs(title = "Within Cluster Variation versus number of Clusters")

kmeans.pca <- cluster_assignments %>%
  select(all_of(vars)) %>%
  prcomp()

cluster_assignments_plot <- kmeans.pca %>%
  augment(cluster_assignments) %>%
  ggplot( aes(x = .fittedPC1, y = .fittedPC2, color = .cluster )) +
  geom_point(alpha = 0.8) +
  facet_wrap(~ k) +
  labs(title = "k-means clusterings")

return(list(cluster_assignments_plot= cluster_assignments_plot,
            within_cluster_variation_plot=within_cluster_variation_plot,
            kmeans_model = kmeans_model))
}

library('broom')

tic <- Sys.time()

proc.kmeans.results <- proc.kmeans(DATA_65.impute.sample %>%
                                     select(all_of(features_65_num$feature)), 
                                   k = 3:11)

toc <- Sys.time()

toc - tic 
#> Time difference of 1.97179 secs

proc.kmeans.results$cluster_assignments_plot

cluster_wgss <- function(data, nc=15){ 

clusters <- 1:nc 

wss <- purrr::map_dfr(clusters, 
                      function(N_Clusters){ 
                        wgss = sum(kmeans(data, centers = N_Clusters)$withinss) 
                        
                        tibble(N_Clusters, wgss) 
                        } ) 
}

DATA_65.impute.sample %>%  
  select(all_of(features_65_num$feature)) %>% 
  cluster_wgss(nc=10) %>%        
  ggplot(aes(x= N_Clusters, y= wgss)) + 
  geom_point() + 
  geom_line() +  
  labs(title = "Within group Sum of Squares Plot",
        x="Number of Clusters",
        y="Within groups sum of squares") 
#> Warning: did not converge in 10 iterations

Above, we see a bend in the curve at around 5 so below we will run a kmeans experiment with (\(k\)) centers = 5.

kmeans has an additional options, here I chose 15 for the number of random starting positions:

set.seed(12345)

km <- kmeans(DATA_65.impute.sample %>% select(all_of(features_65_num$feature)) , 
             centers = 5,
             iter.max= 15,
             nstart = 15)

km
#> K-means clustering with 5 clusters of sizes 5298, 14412, 7873, 4354, 6281
#> 
#> Cluster means:
#>        BPXDI3      BPXSY3      BPXDI2      BPXSY2      BPXDI1      BPXSY1
#> 1 -1.18757431 -0.97753315 -1.21899580 -0.99079920 -1.21528212 -0.99530237
#> 2 -0.03717229 -0.31811843 -0.04569126 -0.31710079 -0.05774348 -0.30791737
#> 3  0.46523940  0.20793980  0.47323149  0.20193256  0.48459640  0.19459877
#> 4  0.74498644  1.89536565  0.79672958  1.91549832  0.82192371  1.91668911
#> 5 -0.01257841 -0.02003481 -0.01241362 -0.01760612 -0.01960191 -0.02651225
#>        BMXLEG      BPXML1      BPXPLS   PEASCTM1    BMXWAIST      BMXBMI
#> 1 -0.83034756 -0.91876984  0.57067272  0.1965214 -0.69573116 -0.69358384
#> 2  0.17445304 -0.26608589 -0.11958988  0.1863414 -0.01865886 -0.08802613
#> 3  0.39760560  0.14150239  0.02309796  0.4195544  1.17269198  1.21648161
#> 4 -0.19257897  1.78317204 -0.36077635  0.5614782  0.55100870  0.36744076
#> 5 -0.06478202 -0.02794252  0.01418170 -1.5084462 -1.22222587 -0.99251027
#>        BMXHT    BMXARMC    BMXARML Poverty_Income_Ratio       BMXWT        Age
#> 1 -0.2898939 -0.6003371 -0.2688240          -0.18203388 -0.59570920 -0.6876412
#> 2  0.3712827  0.1157080  0.3367366           0.05027904  0.09559683  0.0637058
#> 3  0.6681734  1.1599990  0.7658356           0.20253611  1.21095460  0.5397692
#> 4  0.3846618  0.4127052  0.4967846           0.06072111  0.41992904  1.3187380
#> 5 -1.7115770 -1.4992181 -1.8502216          -0.25778578 -1.52585430 -1.1568848
#> 
#> Clustering vector:
#>     [1] 2 2 2 2 3 5 3 4 2 2 4 2 2 2 5 2 4 1 2 2 3 2 3 2 2 4 2 1 4 4 4 2 5 2 5 2
#>    [37] 2 3 5 1 4 5 3 3 4 1 2 2 1 5 2 5 5 2 4 4 3 3 2 3 1 5 2 5 3 3 2 4 1 2 1 3
#>    [73] 4 2 4 2 4 1 3 2 5 1 1 1 3 2 2 5 2 4 4 2 5 2 2 5 4 2 1 1 2 2 2 3 2 1 2 2
#>   [109] 2 3 5 1 5 2 2 2 2 1 2 5 2 2 2 2 4 3 1 1 3 4 2 3 2 1 1 1 2 1 1 4 5 3 2 1
#>   [145] 3 4 2 4 5 3 4 4 2 5 1 1 2 5 5 3 1 4 2 2 1 3 5 2 2 2 5 4 2 3 4 5 5 3 5 2
#>   [181] 2 5 5 2 2 5 3 4 3 1 5 5 2 5 2 3 2 5 2 1 1 2 2 3 5 3 3 1 4 2 2 3 3 2 3 3
#>   [217] 2 2 3 5 2 4 2 2 4 1 1 4 1 5 2 4 3 3 5 2 2 3 4 5 5 3 5 2 1 5 4 5 2 5 2 1
#>   [253] 3 5 2 3 2 4 4 2 2 2 3 5 2 1 3 2 1 3 4 2 2 1 4 2 2 2 4 2 5 2 5 1 3 1 5 5
#>   [289] 2 5 2 2 3 1 1 3 5 2 3 4 2 1 1 2 5 4 2 4 3 1 2 2 2 2 5 2 2 3 5 2 3 4 5 1
#>   [325] 4 5 5 5 3 3 2 1 2 3 1 2 2 3 5 5 5 2 2 4 2 2 4 3 5 3 4 3 2 5 3 2 4 2 5 2
#>   [361] 2 3 1 4 5 2 2 1 2 4 5 2 2 2 3 3 3 3 5 4 2 3 3 3 2 2 4 1 2 2 2 3 4 2 5 2
#>   [397] 4 4 1 1 2 3 4 3 3 2 5 4 4 4 2 3 2 3 2 5 1 4 2 5 5 5 5 4 5 5 5 5 1 2 2 5
#>   [433] 1 2 2 4 2 4 5 5 2 5 2 3 4 2 4 2 3 3 3 5 1 1 5 5 3 3 1 2 3 2 5 4 2 5 1 1
#>   [469] 5 2 2 1 2 4 4 4 3 5 4 2 2 4 2 2 5 2 1 5 1 5 1 2 4 3 3 2 1 3 5 3 5 2 5 2
#>   [505] 5 5 3 2 2 3 2 4 3 3 4 2 2 1 2 2 2 5 3 1 5 1 5 3 1 2 2 5 2 3 2 5 2 2 3 4
#>   [541] 4 2 2 3 5 2 3 3 5 2 3 2 2 3 2 2 5 5 5 2 2 2 2 3 3 5 3 2 3 2 5 3 5 3 4 2
#>   [577] 2 2 2 2 3 2 1 1 2 2 2 5 2 2 1 3 3 5 3 2 4 2 3 3 2 2 2 3 2 4 1 4 5 5 2 2
#>   [613] 3 3 5 5 1 3 1 2 5 2 2 2 3 1 4 2 5 2 4 5 5 5 2 4 2 2 4 2 2 2 2 3 1 4 2 3
#>   [649] 2 4 5 3 2 3 2 3 2 1 2 3 1 2 4 1 3 1 2 1 2 1 4 1 3 2 3 2 2 2 3 2 2 1 2 1
#>   [685] 3 1 2 3 3 2 3 3 3 5 1 2 4 3 2 5 5 2 3 5 3 2 5 5 2 5 5 2 5 5 2 5 2 1 2 3
#>   [721] 5 5 1 3 5 3 5 2 2 2 2 2 1 1 3 1 2 4 5 5 3 1 4 2 3 1 2 2 1 2 2 2 1 1 3 2
#>   [757] 3 2 2 1 3 2 2 2 2 3 1 4 3 2 3 2 5 2 4 1 2 2 5 4 2 5 1 3 5 3 1 2 2 5 3 3
#>   [793] 1 5 2 2 4 3 1 3 3 3 2 4 2 2 3 2 3 2 3 2 3 2 1 5 3 1 3 2 2 2 5 2 3 5 4 1
#>   [829] 1 2 1 2 2 4 3 4 3 2 2 3 2 3 2 2 2 3 5 2 2 1 3 2 5 5 2 5 1 3 4 2 2 2 2 2
#>   [865] 2 3 5 2 5 5 2 5 2 3 2 5 5 1 3 3 4 3 2 5 2 2 1 4 1 2 4 2 1 2 1 3 1 3 5 3
#>   [901] 3 2 4 4 3 2 2 3 2 2 4 2 2 1 3 2 5 4 5 1 2 3 4 2 3 2 4 2 3 3 4 2 1 2 2 2
#>   [937] 2 5 2 2 5 2 5 5 5 3 3 5 1 2 2 1 1 2 3 2 3 2 2 2 3 1 2 2 2 5 5 2 4 5 2 2
#>   [973] 2 1 2 2 5 2 2 3 2 2 2 2 2 4 2 4 1 3 5 2 5 3 5 2 3 5 4 2 2 2 2 3 3 1 2 2
#>  [1009] 2 1 4 2 3 2 3 2 5 3 2 2 2 1 2 2 4 2 2 2 3 3 4 2 5 5 2 2 5 2 3 2 2 2 5 2
#>  [1045] 2 2 5 4 3 2 2 1 1 2 3 2 5 4 5 1 2 2 3 2 3 3 3 2 3 5 4 3 2 4 5 1 5 2 3 3
#>  [1081] 2 2 3 2 3 4 2 2 2 2 1 3 5 4 1 3 5 5 5 1 1 4 2 2 3 2 1 3 2 4 2 1 3 2 2 4
#>  [1117] 5 4 3 5 4 4 3 1 3 2 2 5 2 2 5 3 2 2 1 5 2 5 4 3 2 2 1 3 2 5 2 2 5 2 2 3
#>  [1153] 3 2 4 3 2 5 1 1 2 1 4 3 3 4 2 3 3 2 2 5 3 2 1 3 4 4 1 3 2 3 2 4 5 2 3 2
#>  [1189] 3 5 4 2 2 2 2 4 2 3 2 5 5 5 2 2 5 2 3 2 3 4 2 3 2 2 5 4 2 4 3 2 5 1 3 4
#>  [1225] 3 1 1 3 1 5 1 4 2 1 3 3 5 1 2 3 2 2 4 4 3 2 2 1 3 1 2 5 2 1 1 2 2 2 2 2
#>  [1261] 1 2 4 2 4 2 1 1 2 2 5 2 2 5 2 1 5 2 2 2 2 2 3 2 2 2 3 2 2 4 1 2 3 4 3 5
#>  [1297] 1 1 3 5 1 2 2 5 2 5 2 4 3 4 2 3 1 1 1 1 5 2 3 5 2 1 2 2 2 4 2 3 4 2 4 2
#>  [1333] 3 2 5 2 3 5 2 2 2 2 2 2 2 2 3 3 2 4 1 2 2 5 2 5 2 5 4 3 5 5 2 5 2 5 3 2
#>  [1369] 4 3 1 3 4 2 3 5 2 3 2 2 4 2 5 3 2 3 5 5 2 3 1 4 1 2 3 2 1 4 2 5 3 4 3 2
#>  [1405] 2 3 2 1 2 5 3 2 3 2 1 1 2 4 1 4 2 2 2 4 2 3 2 1 5 2 3 3 1 3 2 4 4 4 1 5
#>  [1441] 3 2 3 2 2 4 3 5 2 2 2 2 3 1 1 3 2 1 5 3 3 5 3 3 1 3 5 1 2 3 2 4 5 2 2 3
#>  [1477] 2 3 5 2 5 2 2 3 4 4 1 3 5 3 3 4 2 4 5 4 3 4 3 4 3 3 5 2 3 5 3 2 4 1 2 2
#>  [1513] 4 2 3 3 2 1 4 5 2 3 2 2 3 5 3 5 3 1 1 5 2 4 3 2 1 5 3 3 2 3 1 2 4 4 5 2
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#> [34381] 3 3 5 1 3 2 2 2 2 2 1 2 1 1 4 2 3 3 1 5 3 1 5 5 4 2 3 1 3 1 2 4 1 1 3 3
#> [34417] 3 2 2 3 4 2 2 1 2 4 1 2 4 2 2 2 2 2 2 5 1 2 3 1 2 3 1 5 2 4 2 3 2 4 2 3
#> [34453] 2 2 4 2 2 3 2 2 2 4 3 4 5 2 3 2 1 3 4 3 1 2 5 3 4 3 3 5 2 3 3 2 3 4 5 3
#> [34489] 5 2 1 1 5 2 3 2 4 1 5 2 5 1 2 5 3 4 1 4 3 3 3 5 2 1 2 1 2 2 2 1 2 2 5 5
#> [34525] 1 3 1 5 2 1 3 5 4 4 3 2 2 4 4 2 4 2 2 2 5 2 2 2 1 2 2 2 3 2 5 2 2 2 1 5
#> [34561] 4 2 3 4 5 4 2 1 3 2 3 2 2 1 2 5 2 4 2 1 3 5 3 1 2 1 2 5 2 1 1 3 2 1 4 1
#> [34597] 2 2 5 1 3 5 3 2 5 2 1 2 5 2 2 4 5 3 2 3 1 5 4 2 3 2 4 2 2 2 2 2 2 2 1 4
#> [34633] 1 5 5 2 5 2 1 4 4 4 2 2 1 1 1 5 2 3 1 5 5 2 3 2 5 2 3 2 3 5 5 2 4 3 2 2
#> [34669] 1 2 3 1 1 5 2 2 3 2 4 1 2 2 5 2 3 4 2 4 5 2 2 2 2 1 2 3 1 3 2 2 1 2 4 3
#> [34705] 4 3 1 5 4 1 3 1 2 2 3 1 3 5 1 3 2 2 1 5 1 5 4 1 1 3 2 2 5 5 2 4 5 4 2 2
#> [34741] 2 3 2 2 1 2 2 5 5 3 1 2 4 5 1 3 3 4 2 5 2 5 4 1 5 2 2 5 2 4 2 4 5 2 2 3
#> [34777] 2 5 5 1 2 1 2 1 4 1 2 2 1 4 2 3 5 4 3 3 1 2 3 2 4 2 3 2 5 5 5 3 2 5 3 2
#> [34813] 2 3 3 5 5 4 2 1 3 1 4 3 3 2 3 1 2 2 1 2 2 4 1 3 5 3 3 3 5 2 4 4 2 4 2 2
#> [34849] 4 2 5 2 3 2 5 2 2 3 3 2 2 5 4 4 5 2 5 1 5 5 4 2 2 1 5 1 5 3 2 1 1 3 2 2
#> [34885] 4 4 3 4 2 3 3 1 2 1 2 4 5 5 2 3 2 2 3 4 2 2 2 1 1 3 3 5 2 4 3 2 2 3 2 2
#> [34921] 3 5 4 3 1 5 2 2 2 2 4 5 1 2 5 3 2 2 2 4 2 3 2 5 2 3 2 1 4 2 3 4 3 2 3 4
#> [34957] 4 2 3 1 5 2 2 1 1 2 3 2 3 3 3 3 3 2 3 1 3 3 2 2 1 5 5 5 3 4 2 4 3 3 5 2
#> [34993] 2 2 2 3 3 4 2 2 1 2 5 5 3 5 2 2 4 4 1 5 4 3 3 3 3 2 2 2 2 2 4 4 2 1 2 3
#> [35029] 2 1 2 2 2 5 4 2 5 5 3 4 5 2 5 3 1 2 5 2 3 3 2 4 4 2 3 3 3 2 2 5 3 4 2 4
#> [35065] 2 2 1 2 1 2 5 1 2 5 5 2 2 5 3 5 2 5 2 5 3 5 3 1 1 2 2 1 2 3 3 3 2 5 3 4
#> [35101] 2 2 3 4 2 3 3 2 1 2 4 5 4 3 2 3 3 2 5 2 3 1 3 2 4 1 4 2 2 2 3 2 3 2 1 3
#> [35137] 5 2 3 1 3 2 2 2 3 3 4 3 1 3 2 2 2 1 3 3 2 2 3 3 3 3 2 5 1 3 1 5 5 2 5 3
#> [35173] 4 4 4 2 2 2 1 5 5 3 1 2 5 3 5 3 5 3 5 5 2 2 4 4 5 1 1 4 4 5 5 3 3 3 3 2
#> [35209] 2 2 4 3 5 2 3 2 3 4 2 1 3 3 2 5 2 2 2 2 3 3 5 2 2 2 4 5 3 3 5 2 3 2 3 2
#> [35245] 2 2 3 3 2 5 2 3 2 4 1 3 2 2 3 2 3 2 5 2 4 2 2 4 5 2 2 2 2 4 3 3 2 2 2 3
#> [35281] 5 2 4 5 2 2 2 1 2 3 3 5 1 2 1 2 3 1 2 1 3 4 5 1 4 2 2 3 3 5 2 3 1 2 2 2
#> [35317] 2 2 3 1 2 3 1 1 5 3 2 4 1 2 2 1 3 5 2 2 3 4 2 3 4 5 2 5 3 5 4 3 4 4 5 5
#> [35353] 5 3 2 2 1 2 4 3 3 2 5 5 1 2 2 4 5 5 1 4 5 4 2 2 5 1 3 2 3 3 5 3 1 5 5 5
#> [35389] 2 3 1 3 2 2 1 5 1 1 5 2 2 2 4 4 5 3 2 2 4 2 2 3 2 2 2 5 3 2 2 3 3 4 1 2
#> [35425] 5 2 2 4 2 5 2 1 2 5 3 1 2 2 5 4 5 2 3 1 3 3 3 2 1 3 2 5 5 5 1 2 2 2 2 3
#> [35461] 2 2 2 2 2 5 3 2 3 2 3 2 1 3 1 4 2 2 2 3 2 3 2 2 2 2 2 2 2 2 3 5 5 1 5 2
#> [35497] 4 2 5 2 4 2 2 5 4 3 3 3 2 3 1 1 2 2 5 3 2 4 1 4 2 2 3 5 2 2 2 2 2 2 2 2
#> [35533] 2 1 5 2 3 2 2 5 3 1 2 2 1 4 5 1 3 4 2 1 2 2 2 3 3 3 2 2 3 2 4 1 2 4 2 1
#> [35569] 1 2 3 2 4 2 2 3 2 3 3 2 2 2 1 5 3 5 4 5 5 1 2 2 1 5 4 2 4 3 4 5 2 2 2 2
#> [35605] 4 2 2 3 2 5 5 5 2 2 2 1 5 5 2 2 3 3 2 5 4 2 5 1 1 2 2 2 2 2 3 1 4 5 5 2
#> [35641] 2 2 2 3 5 5 1 1 3 4 5 3 3 3 2 3 2 3 3 2 1 1 1 2 2 5 3 2 3 2 4 1 1 1 2 2
#> [35677] 3 2 2 2 2 5 3 2 3 4 2 5 5 4 4 2 1 3 2 3 3 5 1 3 3 2 2 3 5 3 3 5 2 3 2 2
#> [35713] 4 3 2 5 3 5 2 3 2 3 5 4 5 3 3 2 2 2 2 1 2 5 1 5 5 5 2 2 3 1 2 2 2 4 2 1
#> [35749] 1 2 2 2 5 5 5 4 5 1 2 1 3 1 2 1 5 3 2 1 3 2 2 3 2 2 2 2 1 5 1 4 2 3 1 5
#> [35785] 5 4 2 2 2 5 2 1 2 5 3 3 4 4 2 3 2 4 2 2 4 2 1 1 3 2 2 4 2 1 1 5 3 3 2 4
#> [35821] 5 2 5 1 2 4 2 3 5 2 2 1 3 2 2 5 3 2 5 3 3 2 3 5 2 1 3 4 2 4 3 3 2 3 2 4
#> [35857] 1 5 2 2 2 2 1 3 4 2 1 2 2 2 5 4 1 2 5 5 2 4 2 5 4 2 3 4 2 2 2 5 3 5 1 1
#> [35893] 2 2 3 4 2 1 1 5 1 4 2 3 3 3 2 3 5 1 3 3 1 3 2 2 2 4 1 3 4 4 4 5 5 3 5 2
#> [35929] 3 3 1 3 5 2 2 5 3 2 3 2 2 2 3 5 1 2 4 1 1 2 2 5 2 2 3 2 3 5 1 2 3 2 1 4
#> [35965] 5 4 3 1 4 2 5 2 2 2 3 5 1 2 4 2 1 1 2 5 2 2 3 1 2 2 4 3 2 3 2 2 2 2 2 2
#> [36001] 3 2 2 2 1 3 3 5 2 3 4 4 1 2 3 1 3 2 3 2 1 1 5 3 2 5 2 1 3 5 5 5 5 2 2 3
#> [36037] 3 2 2 1 4 5 2 3 5 3 1 1 1 1 2 5 4 4 4 2 5 2 1 3 2 2 2 3 2 3 3 3 2 2 1 2
#> [36073] 2 2 3 2 2 2 2 1 4 2 1 2 2 3 1 2 5 2 1 2 2 2 3 2 2 5 2 2 3 1 5 5 2 3 2 2
#> [36109] 1 2 1 2 2 2 2 2 4 3 5 5 1 1 1 3 2 1 1 2 2 2 2 4 5 2 3 3 2 4 4 3 4 5 3 5
#> [36145] 3 2 2 2 2 3 2 2 2 5 1 3 5 2 3 3 2 3 2 2 3 2 3 2 2 4 5 2 5 2 2 2 3 2 3 3
#> [36181] 2 2 2 4 2 2 2 2 2 5 2 5 5 2 4 1 1 2 2 2 1 1 2 1 3 2 4 3 2 2 3 2 2 1 2 3
#> [36217] 3 3 5 1 2 2 4 1 2 2 3 2 2 4 5 5 2 3 2 3 2 2 2 3 4 5 2 4 2 2 1 5 2 2 2 2
#> [36253] 5 1 5 2 1 3 5 4 4 5 3 5 5 1 2 4 5 2 5 3 4 2 2 2 2 4 1 2 2 2 4 5 5 1 3 4
#> [36289] 2 2 2 3 2 2 5 1 3 2 1 1 1 4 2 5 2 1 1 2 2 2 2 2 5 3 2 5 5 4 3 1 2 2 5 2
#> [36325] 5 2 1 4 4 3 1 2 4 2 3 5 4 4 2 5 2 2 3 3 1 2 3 2 2 2 2 4 2 3 4 4 2 1 2 1
#> [36361] 3 5 5 2 2 5 5 2 1 1 2 2 2 5 4 5 5 2 2 3 2 1 4 5 2 2 4 3 4 2 3 2 2 4 2 2
#> [36397] 1 5 5 2 2 3 2 3 1 2 2 2 5 2 2 2 4 2 2 2 2 2 3 2 3 1 3 5 5 1 1 1 2 2 4 5
#> [36433] 2 1 3 3 1 4 3 5 2 2 2 2 1 1 3 2 3 2 2 2 5 2 3 5 2 1 2 4 3 2 5 2 3 4 2 3
#> [36469] 1 1 2 4 4 2 5 3 2 1 5 4 1 2 4 1 3 3 1 2 2 3 2 2 1 5 2 4 2 5 2 2 1 2 2 2
#> [36505] 3 3 5 1 1 1 1 5 1 3 2 5 3 2 5 3 1 4 2 2 3 2 3 4 5 5 1 2 5 3 4 2 2 3 2 1
#> [36541] 1 1 3 1 4 3 5 2 3 3 2 2 5 2 3 2 2 3 2 3 2 1 2 3 2 2 1 3 3 5 4 1 4 2 1 4
#> [36577] 1 5 5 5 1 2 2 5 2 2 2 2 3 5 3 2 2 2 2 2 3 3 3 1 3 2 3 5 1 3 5 2 3 5 2 1
#> [36613] 3 3 2 2 3 5 2 2 5 3 3 3 2 2 3 3 5 3 2 1 2 2 4 3 5 2 5 2 2 1 3 4 1 5 1 3
#> [36649] 1 2 4 4 1 3 2 2 4 5 2 5 2 2 2 3 4 4 2 3 2 2 2 5 2 2 3 1 3 2 1 5 4 5 2 1
#> [36685] 4 2 3 2 4 1 2 2 5 4 1 3 2 4 5 1 2 3 2 1 2 3 2 1 3 5 2 2 3 5 4 3 1 5 2 5
#> [36721] 2 2 3 1 3 2 3 1 2 4 2 2 5 3 1 1 5 4 4 5 5 3 3 5 3 5 3 3 3 2 2 5 2 2 2 2
#> [36757] 2 2 5 4 5 2 2 4 2 1 5 1 2 2 2 4 2 2 5 2 2 3 2 5 3 5 3 2 4 1 3 2 2 5 1 3
#> [36793] 1 4 2 5 3 2 1 4 2 3 4 4 1 5 2 1 4 1 4 2 2 2 5 5 2 4 2 1 1 2 4 3 4 2 1 2
#> [36829] 2 4 2 5 3 4 2 2 2 1 2 2 2 2 3 1 2 5 1 5 4 3 2 3 5 2 2 2 4 3 2 1 5 4 4 1
#> [36865] 3 2 3 5 2 3 2 2 2 5 5 2 3 1 2 2 2 3 2 2 3 2 2 2 2 1 2 2 2 5 1 3 4 3 2 3
#> [36901] 2 2 1 2 2 2 2 2 2 3 5 2 2 1 2 5 3 2 4 3 2 3 1 2 3 2 4 4 5 2 1 1 3 3 3 3
#> [36937] 4 5 1 2 2 5 3 2 1 5 2 4 1 1 1 4 3 2 5 2 4 4 5 3 2 5 3 2 1 3 4 2 2 5 2 3
#> [36973] 1 2 1 3 5 1 2 3 2 4 4 2 2 3 3 5 1 1 2 2 3 1 2 2 3 5 5 4 1 4 5 1 5 2 3 1
#> [37009] 4 1 3 5 4 1 1 4 2 3 4 2 4 4 4 4 2 2 1 2 1 1 4 1 3 4 2 5 2 2 5 3 2 3 5 4
#> [37045] 2 2 4 3 2 5 3 1 3 1 3 3 2 2 4 5 3 3 2 5 2 2 2 5 2 2 2 4 3 3 5 1 1 2 1 2
#> [37081] 1 2 1 5 1 2 3 3 1 4 5 4 3 3 1 1 3 4 4 3 4 3 5 2 5 3 2 2 2 3 3 5 2 5 2 2
#> [37117] 2 2 1 4 2 2 2 3 4 2 5 4 2 2 5 2 2 3 5 2 3 2 3 2 5 2 3 2 3 2 3 3 2 3 1 3
#> [37153] 3 5 1 3 4 2 5 4 4 4 1 4 3 2 1 2 1 3 2 3 5 1 4 4 1 3 2 1 2 3 2 3 5 1 5 3
#> [37189] 2 4 2 4 3 4 2 2 2 2 2 3 4 3 5 3 3 1 2 2 3 5 1 2 1 4 2 3 2 2 3 5 4 2 5 3
#> [37225] 2 5 4 2 5 2 4 3 5 2 2 2 2 1 3 5 5 2 1 2 2 2 5 3 4 3 2 2 1 5 4 2 2 2 5 2
#> [37261] 2 4 3 2 3 4 2 4 5 2 4 2 3 4 1 2 2 3 1 1 1 5 4 3 2 3 2 4 4 1 1 5 5 4 5 1
#> [37297] 4 2 1 2 2 3 1 5 3 3 5 4 3 2 3 3 2 4 3 5 4 4 4 1 3 2 2 1 5 1 5 3 1 2 2 2
#> [37333] 2 1 3 2 5 2 3 4 3 2 4 2 2 2 5 3 5 2 1 2 3 2 1 3 3 4 2 2 2 2 2 2 3 2 5 5
#> [37369] 4 2 2 2 3 3 3 2 2 4 3 2 3 2 1 1 1 2 3 3 2 2 2 5 2 2 4 5 3 4 2 2 2 1 2 3
#> [37405] 3 2 4 1 3 1 2 5 2 2 4 2 2 4 4 3 5 2 3 5 2 2 2 5 1 4 3 5 2 1 1 2 1 2 2 2
#> [37441] 2 2 5 4 3 1 4 4 2 1 3 1 3 2 3 1 2 3 1 4 5 2 1 3 2 4 3 2 2 4 3 5 3 2 4 5
#> [37477] 2 3 2 3 2 3 3 2 2 2 2 5 3 2 1 1 2 3 4 2 5 3 2 4 2 2 5 3 2 3 2 5 1 2 2 2
#> [37513] 1 3 4 4 2 3 4 2 2 3 5 5 5 2 2 2 2 5 3 2 4 1 2 3 2 3 3 3 2 2 1 2 5 2 2 2
#> [37549] 4 2 5 1 3 3 2 2 4 2 2 2 5 2 2 4 3 2 3 2 2 2 5 3 4 2 3 2 5 4 5 2 2 2 5 3
#> [37585] 3 3 2 3 2 2 2 2 5 2 2 3 4 3 4 2 2 1 3 1 1 4 4 1 2 2 3 2 4 2 2 2 5 4 2 1
#> [37621] 1 2 2 3 2 2 1 3 2 3 5 1 1 3 1 2 5 2 5 3 3 2 4 2 2 3 5 2 2 5 2 5 2 1 4 2
#> [37657] 5 2 4 2 1 4 5 2 3 1 2 2 3 3 2 1 1 3 3 5 2 2 2 5 1 2 3 2 1 1 2 2 4 1 2 3
#> [37693] 2 1 2 5 3 2 5 2 4 3 1 1 4 5 2 2 5 1 2 4 4 2 2 2 3 5 3 2 2 2 1 3 2 2 5 2
#> [37729] 3 2 5 4 5 5 3 2 1 2 2 3 4 3 5 5 4 2 2 2 2 2 4 3 5 4 4 5 4 4 2 3 5 5 2 2
#> [37765] 1 3 2 1 2 2 3 2 1 5 3 2 3 2 2 5 2 4 2 1 1 4 2 4 5 3 4 5 2 2 1 3 3 3 4 3
#> [37801] 2 3 5 4 4 4 4 5 5 2 1 4 1 1 3 2 2 2 4 3 2 2 3 5 5 5 2 2 5 2 5 4 2 2 5 2
#> [37837] 3 3 3 2 1 1 5 5 3 4 2 2 1 5 2 4 2 4 2 5 3 3 2 5 5 4 1 2 2 2 2 5 3 3 5 2
#> [37873] 4 2 4 2 1 2 1 2 5 5 2 1 3 4 1 2 3 4 2 5 2 4 1 1 2 2 5 3 3 2 5 3 2 3 1 3
#> [37909] 2 1 4 4 3 3 3 2 2 2 2 3 2 1 2 2 2 4 1 3 3 3 2 3 1 2 2 2 1 1 2 5 2 4 5 2
#> [37945] 2 4 3 5 2 3 2 3 3 2 2 2 2 4 2 2 3 3 2 1 1 3 2 4 3 5 3 3 5 2 1 2 2 2 3 2
#> [37981] 1 2 3 5 5 2 2 3 5 2 3 1 1 3 2 2 2 3 1 2 2 1 2 5 5 2 2 5 1 5 2 2 1 2 2 1
#> [38017] 4 4 1 2 3 3 2 5 1 1 3 2 1 5 2 3 1 2 3 4 2 1 3 2 1 1 5 3 5 3 3 5 4 2 3 5
#> [38053] 4 4 1 3 2 5 4 3 3 3 3 2 2 2 5 2 3 1 3 1 3 2 3 2 3 1 2 2 2 2 5 2 2 5 5 2
#> [38089] 3 5 4 4 4 4 5 2 1 2 3 2 2 4 3 2 1 1 1 1 2 3 3 2 3 3 5 5 2 2 1 5 3 2 2 5
#> [38125] 5 3 2 1 1 2 2 1 5 5 5 3 5 2 2 2 2 2 2 2 5 4 2 5 2 5 2 4 1 3 1 3 4 2 2 5
#> [38161] 1 1 4 5 2 5 5 2 3 3 5 2 5 5 3 2 5 4 2 3 3 2 1 4 1 1 5 5 3 5 2 3 2 2 1 2
#> [38197] 4 3 3 1 1 3 4 4 2 2 5 2 4 2 3 2 2 2 4 2 2 5
#> 
#> Within cluster sum of squares by cluster:
#> [1]  61973.94 110615.23  82522.73  74106.14  30632.03
#>  (between_SS / total_SS =  47.7 %)
#> 
#> Available components:
#> 
#> [1] "cluster"      "centers"      "totss"        "withinss"     "tot.withinss"
#> [6] "betweenss"    "size"         "iter"         "ifault"

proc.kmeans.results$kmeans_model(5)
#> K-means clustering with 5 clusters of sizes 7926, 5220, 14430, 4366, 6276
#> 
#> Cluster means:
#>        BPXDI3      BPXSY3      BPXDI2      BPXSY2      BPXDI1      BPXSY1
#> 1  0.46188807  0.20536309  0.46921717  0.20002435  0.48104800  0.19216582
#> 2 -1.19959080 -0.97977434 -1.23154632 -0.99348269 -1.22795122 -0.99800508
#> 3 -0.03958390 -0.32250547 -0.04782192 -0.32172411 -0.06026541 -0.31243106
#> 4  0.74504593  1.89305052  0.79686121  1.91325640  0.82269579  1.91464700
#> 5 -0.01286421 -0.01985189 -0.01264647 -0.01756086 -0.01993961 -0.02620911
#>        BMXLEG      BPXML1      BPXPLS   PEASCTM1    BMXWAIST      BMXBMI
#> 1  0.39808341  0.14019410  0.01764375  0.4189000  1.16793701  1.20993770
#> 2 -0.83984222 -0.92116389  0.56677239  0.1972831 -0.70049693 -0.69730563
#> 3  0.17158591 -0.27066267 -0.11199121  0.1851560 -0.02345797 -0.09234715
#> 4 -0.19299637  1.78090194 -0.36014814  0.5604144  0.55219747  0.36891860
#> 5 -0.06446706 -0.02747904  0.01434730 -1.5086982 -1.22257336 -0.99237732
#>        BMXHT    BMXARMC    BMXARML Poverty_Income_Ratio      BMXWT         Age
#> 1  0.6680187  1.1557629  0.7646280           0.20186905  1.2061885  0.53995774
#> 2 -0.2960422 -0.6052468 -0.2739096          -0.18484863 -0.6002403 -0.68944986
#> 3  0.3686741  0.1110423  0.3339697           0.04915597  0.0910803  0.05724003
#> 4  0.3848402  0.4143226  0.4972752           0.06229332  0.4211034  1.31828346
#> 5 -1.7128036 -1.4997549 -1.8516444          -0.25755218 -1.5264217 -1.15716635
#> 
#> Clustering vector:
#>     [1] 3 3 3 1 1 5 1 4 3 3 4 1 3 3 5 1 4 2 3 3 1 3 1 3 3 4 3 2 4 4 4 3 5 3 5 3
#>    [37] 3 1 5 2 4 5 1 1 4 2 3 3 2 5 3 5 5 3 4 4 1 1 3 1 2 5 3 5 1 1 3 4 2 3 2 1
#>    [73] 4 3 4 3 4 2 1 3 5 2 2 2 1 3 3 5 3 4 4 3 5 3 3 5 4 3 2 2 3 3 3 1 3 2 3 3
#>   [109] 1 1 5 2 5 3 3 3 3 2 3 5 3 3 3 3 4 1 2 2 1 4 3 1 3 2 2 2 3 2 2 4 5 1 3 2
#>   [145] 1 4 3 4 5 1 4 4 1 5 2 2 3 5 5 1 2 4 3 3 2 1 5 3 3 3 5 4 3 1 4 5 5 1 5 3
#>   [181] 3 5 5 3 3 5 1 4 1 2 5 5 3 5 3 1 3 5 3 2 2 3 3 1 5 1 1 2 4 3 3 1 1 3 1 1
#>   [217] 3 3 1 5 3 4 3 3 4 2 2 4 2 5 3 4 1 1 5 3 3 1 4 5 5 1 5 3 2 5 4 5 3 5 3 2
#>   [253] 1 5 3 1 3 4 4 3 3 3 1 5 3 2 1 3 2 1 4 3 3 2 4 3 3 3 4 3 5 3 5 2 1 2 5 5
#>   [289] 3 5 3 3 1 2 2 1 5 3 1 4 3 2 2 3 5 4 3 4 1 2 3 3 3 3 5 3 3 1 5 3 1 4 5 2
#>   [325] 4 5 5 5 1 1 3 2 3 1 2 3 3 1 5 5 5 3 3 4 3 3 4 1 5 1 4 1 3 5 1 3 4 3 5 3
#>   [361] 3 1 2 4 5 3 3 2 3 4 5 3 3 3 1 1 1 1 5 4 3 1 1 1 3 3 4 2 3 3 3 1 4 3 5 3
#>   [397] 4 4 2 2 3 1 4 1 1 3 5 4 4 4 3 1 3 1 3 5 2 4 3 5 5 5 5 4 5 5 5 5 2 3 3 5
#>   [433] 2 3 3 4 3 4 5 5 3 5 3 1 4 3 4 3 1 1 1 5 2 2 5 5 1 1 2 3 1 3 5 4 3 5 2 2
#>   [469] 5 3 3 2 3 4 4 4 1 5 4 3 3 4 3 3 5 3 2 5 2 5 2 3 4 1 1 3 2 1 5 1 5 3 5 3
#>   [505] 5 5 1 3 3 1 3 4 1 1 4 3 3 2 3 3 3 5 1 2 5 2 5 1 2 3 3 5 3 1 3 5 3 3 1 4
#>   [541] 4 3 3 1 5 3 1 1 5 3 1 3 3 1 3 3 5 5 5 3 3 3 3 1 1 5 1 3 1 3 5 1 5 1 4 3
#>   [577] 3 3 3 3 1 3 2 2 3 3 3 5 3 3 3 1 1 5 1 3 4 3 1 1 3 3 3 1 3 4 2 4 5 5 3 3
#>   [613] 1 1 5 5 2 1 2 3 5 3 3 3 1 2 4 3 5 3 4 5 5 5 3 4 3 3 4 3 3 3 3 1 2 4 3 1
#>   [649] 3 4 5 1 3 1 3 1 3 2 3 1 2 3 4 2 1 2 3 2 3 2 4 2 1 3 1 3 3 1 1 3 3 2 3 2
#>   [685] 1 2 3 1 1 3 1 1 1 5 2 3 4 1 3 5 5 3 1 5 1 3 5 5 3 5 5 3 5 5 3 5 3 2 3 1
#>   [721] 5 5 2 1 5 1 5 3 3 3 3 3 2 2 1 2 3 4 5 5 1 2 4 3 1 2 3 3 2 3 3 3 2 2 1 3
#>   [757] 1 3 3 2 1 3 3 3 3 1 2 4 1 3 1 3 5 3 4 2 3 3 5 4 3 5 2 1 5 1 2 3 3 5 1 1
#>   [793] 2 5 3 3 4 1 2 1 1 1 3 4 3 3 1 3 1 3 1 3 1 3 2 5 1 2 1 3 3 3 5 3 1 5 4 2
#>   [829] 2 3 2 3 3 4 1 4 1 3 3 1 3 1 3 3 3 1 5 3 3 2 1 3 5 5 3 5 2 1 4 3 3 3 3 3
#>   [865] 3 1 5 3 5 5 3 5 3 1 3 5 5 2 1 1 4 1 3 5 3 3 2 4 2 3 4 3 2 3 2 1 2 1 5 1
#>   [901] 1 3 4 4 1 3 3 1 3 3 4 3 3 2 1 3 5 4 5 2 3 1 4 3 1 3 4 3 1 1 4 3 2 3 3 3
#>   [937] 3 5 3 3 5 3 5 5 5 1 1 5 2 3 3 2 2 3 1 3 1 3 3 3 1 2 3 3 3 5 5 3 4 5 3 3
#>   [973] 3 2 3 3 5 3 3 1 3 3 3 3 3 4 3 4 2 1 5 3 5 1 5 3 1 5 4 3 3 3 3 1 1 2 3 3
#>  [1009] 3 2 4 3 1 3 1 3 5 1 3 3 3 2 3 3 4 3 3 3 1 1 4 3 5 5 3 3 5 3 1 3 3 3 5 3
#>  [1045] 3 3 5 4 1 3 3 2 2 3 1 3 5 4 5 2 3 3 1 3 1 1 1 3 1 5 4 1 3 4 5 2 5 3 1 1
#>  [1081] 3 3 1 3 1 4 3 3 3 3 2 1 5 4 2 1 5 5 5 2 2 4 3 3 1 3 2 1 3 4 3 2 1 3 3 4
#>  [1117] 5 4 1 5 4 4 1 2 1 3 3 5 3 3 5 1 3 3 2 5 3 5 4 1 3 3 2 1 3 5 3 3 5 3 3 1
#>  [1153] 1 3 4 1 3 5 2 2 3 2 4 1 1 4 3 1 1 3 3 5 1 3 2 1 4 4 2 1 3 1 3 4 5 3 1 3
#>  [1189] 1 5 4 3 3 3 3 4 3 1 3 5 5 5 3 3 5 3 1 3 1 4 3 1 3 3 5 4 3 4 1 3 5 2 1 4
#>  [1225] 1 2 2 1 2 5 2 4 3 2 1 1 5 2 3 1 3 3 4 4 1 3 3 2 1 2 3 5 3 2 2 3 3 3 3 3
#>  [1261] 2 3 4 3 4 3 2 2 3 3 5 3 3 5 3 2 5 3 3 3 3 3 1 3 3 3 1 3 3 4 2 3 1 4 1 5
#>  [1297] 2 2 1 5 2 3 3 5 3 5 3 4 1 4 3 1 2 2 2 3 5 3 1 5 3 2 3 3 3 4 3 1 4 3 4 3
#>  [1333] 1 3 5 3 1 5 3 3 3 3 3 3 3 3 1 1 3 4 2 3 3 5 3 5 3 5 4 1 5 5 3 5 3 5 1 3
#>  [1369] 4 1 2 1 4 3 1 5 3 1 3 3 4 3 5 1 3 1 5 5 3 1 2 4 2 3 1 3 2 4 3 5 1 4 1 3
#>  [1405] 3 1 3 2 3 5 1 3 1 3 2 2 3 4 2 4 3 3 3 4 3 1 3 2 5 3 1 1 2 1 3 4 4 4 2 5
#>  [1441] 1 3 1 3 3 4 1 5 3 3 3 3 1 2 2 1 3 2 5 1 1 5 1 1 2 1 5 2 3 1 3 4 5 3 3 1
#>  [1477] 3 1 5 3 5 3 3 1 4 4 2 1 5 1 1 4 3 4 5 4 1 4 1 4 1 1 5 3 1 5 1 3 4 2 3 3
#>  [1513] 4 3 1 1 3 2 4 5 3 1 3 3 1 5 1 5 1 2 2 5 3 4 1 3 2 5 1 1 3 1 2 3 4 4 5 3
#>  [1549] 3 1 3 4 1 3 2 5 5 5 5 3 5 3 1 3 1 5 2 5 1 3 3 4 4 1 1 4 3 3 3 3 4 4 5 1
#>  [1585] 3 5 1 3 4 5 3 3 3 1 3 1 4 3 1 1 3 5 5 3 3 3 3 3 1 5 5 1 1 2 3 2 3 4 2 3
#>  [1621] 3 4 3 3 4 3 4 3 1 5 1 5 1 3 3 3 3 3 1 4 3 2 3 1 1 3 4 2 2 4 3 5 1 2 2 5
#>  [1657] 3 3 1 3 5 2 3 3 4 3 1 3 3 5 5 3 5 2 3 5 4 4 1 3 3 2 3 3 3 2 3 5 4 4 5 3
#>  [1693] 5 1 1 2 4 3 2 2 3 1 3 1 5 1 5 4 4 5 3 5 5 1 1 1 3 4 5 1 3 5 3 3 5 5 1 3
#>  [1729] 3 3 3 1 5 5 3 4 3 4 5 3 5 3 3 3 1 3 3 3 4 3 1 3 3 3 1 1 2 3 2 5 5 5 3 3
#>  [1765] 3 1 1 5 5 3 4 3 4 4 5 3 3 3 3 4 5 1 4 3 4 4 3 3 3 3 1 5 3 1 3 3 5 4 3 4
#>  [1801] 2 5 4 1 3 3 3 4 3 3 3 3 4 3 2 3 4 2 5 4 5 2 3 3 1 3 3 1 3 3 3 1 1 5 3 3
#>  [1837] 5 2 1 3 5 2 5 3 3 4 3 1 2 2 1 2 1 4 5 2 4 1 1 3 3 5 3 3 2 2 2 5 3 3 5 5
#>  [1873] 5 5 2 3 2 1 3 1 3 4 3 3 3 5 5 2 1 2 2 3 2 1 3 1 3 5 1 3 5 3 2 2 5 1 3 5
#>  [1909] 5 2 3 1 5 5 3 5 3 3 1 2 3 5 5 3 3 5 2 3 3 2 4 3 4 5 1 2 3 3 1 3 3 3 3 3
#>  [1945] 5 4 1 1 5 4 2 5 2 3 1 5 4 5 1 5 3 3 2 4 1 5 5 3 1 5 1 3 3 3 3 3 3 5 3 4
#>  [1981] 5 3 1 5 1 3 3 3 3 5 3 1 4 1 3 2 5 3 4 3 1 3 2 3 1 1 1 2 3 3 3 3 4 4 3 3
#>  [2017] 3 4 3 5 1 3 4 2 4 3 1 5 4 3 3 3 1 1 1 4 3 3 3 4 1 5 3 3 3 3 3 3 3 2 1 4
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#> [34921] 1 5 4 1 2 5 3 3 3 3 4 5 2 3 5 1 3 3 3 4 3 1 3 5 3 1 3 2 4 3 1 4 1 3 1 4
#> [34957] 4 1 1 2 5 3 3 2 2 3 1 3 1 1 1 1 1 3 1 2 1 1 3 3 2 5 5 5 1 4 3 4 1 1 5 3
#> [34993] 3 3 3 1 1 4 3 3 2 3 5 5 1 5 3 3 4 4 2 5 4 1 1 1 1 3 3 3 3 3 4 4 3 2 3 1
#> [35029] 3 2 3 3 3 5 4 3 5 5 1 4 5 3 5 1 3 3 5 3 1 1 3 4 4 3 1 1 1 3 3 5 1 4 3 4
#> [35065] 3 3 2 3 2 3 5 2 3 5 5 3 3 5 1 5 3 5 3 5 1 5 1 2 2 3 3 2 3 1 1 1 3 5 1 4
#> [35101] 3 3 1 4 3 1 1 3 2 3 4 5 4 1 3 1 1 3 5 3 1 2 1 3 4 2 4 3 3 3 1 3 1 3 2 1
#> [35137] 5 3 1 2 1 3 3 3 1 1 4 1 2 1 3 3 3 2 1 1 3 3 1 1 1 1 3 5 2 1 2 5 5 3 5 1
#> [35173] 4 4 4 3 3 3 2 5 5 1 2 3 5 1 5 1 5 1 5 5 3 3 4 4 5 2 2 4 4 5 5 1 1 1 1 3
#> [35209] 3 3 4 1 5 3 1 3 1 4 3 2 1 1 3 5 3 3 3 3 1 1 5 3 3 3 4 5 1 1 5 3 1 3 1 3
#> [35245] 3 3 1 1 3 5 3 1 3 4 2 1 3 3 1 3 1 3 5 3 4 3 3 4 5 3 3 3 3 4 1 1 3 3 3 1
#> [35281] 5 3 4 5 3 3 3 2 3 1 1 5 2 3 2 3 1 2 3 2 1 4 5 2 4 3 3 1 1 5 3 1 2 3 3 3
#> [35317] 3 3 1 2 3 1 2 2 5 1 3 4 2 1 3 2 1 5 3 3 1 4 3 1 4 5 3 5 1 5 4 1 4 4 5 5
#> [35353] 5 1 3 3 2 3 4 1 1 3 5 5 2 3 3 4 5 5 2 4 5 4 3 3 5 2 1 3 1 1 5 1 2 5 5 5
#> [35389] 3 1 2 1 3 3 2 5 2 2 5 3 3 3 4 4 5 1 3 3 4 3 3 1 3 3 3 5 1 3 3 1 1 4 2 3
#> [35425] 5 3 3 4 3 5 3 2 3 5 1 2 3 3 5 4 5 3 1 2 1 1 1 3 2 1 3 5 5 5 2 3 3 3 3 1
#> [35461] 3 3 3 3 3 5 1 3 1 3 1 3 2 1 2 4 3 3 3 1 3 1 3 3 3 3 3 3 3 3 1 5 5 2 5 3
#> [35497] 4 3 5 3 4 3 3 5 4 1 1 1 3 1 2 2 3 3 5 1 3 4 2 4 3 3 1 5 3 3 3 3 3 3 3 3
#> [35533] 3 2 5 3 1 3 3 5 1 2 3 3 2 4 5 2 1 4 3 2 3 3 3 1 1 1 3 3 1 3 4 2 3 4 3 2
#> [35569] 2 3 1 3 4 3 3 1 3 1 1 3 3 3 2 5 1 5 4 5 5 2 3 3 2 5 4 3 4 1 4 5 3 3 3 3
#> [35605] 4 3 3 1 3 5 5 5 3 3 3 2 5 5 3 3 1 1 3 5 4 3 5 2 2 3 3 3 3 3 1 2 4 5 5 3
#> [35641] 3 3 3 1 5 5 2 2 1 4 5 1 1 1 3 1 3 1 1 3 2 2 2 3 3 5 1 3 1 3 4 2 3 2 3 3
#> [35677] 1 3 3 3 3 5 1 3 1 4 3 5 5 4 4 3 2 1 3 1 1 5 2 1 1 3 3 1 5 1 1 5 3 1 3 3
#> [35713] 4 1 3 5 1 5 3 1 3 1 5 4 5 1 1 3 3 3 3 2 3 5 2 5 5 5 3 3 1 2 3 3 3 4 3 2
#> [35749] 2 1 3 3 5 5 5 4 5 2 3 2 1 2 3 2 5 1 3 2 1 3 3 1 3 3 3 3 2 5 2 4 3 1 2 5
#> [35785] 5 4 3 3 3 5 3 2 3 5 1 1 4 4 3 1 3 4 3 3 4 3 2 2 1 3 3 4 3 2 2 5 1 1 3 4
#> [35821] 5 3 5 2 3 4 3 1 5 3 3 2 1 3 3 5 1 3 5 1 1 3 1 5 3 2 1 4 3 4 1 1 3 1 3 4
#> [35857] 2 5 3 3 3 3 2 1 4 3 2 3 3 3 5 4 2 3 5 5 3 4 3 5 4 3 1 4 3 3 3 5 1 5 2 2
#> [35893] 3 3 1 4 3 2 2 5 2 4 3 1 1 1 3 1 5 2 1 1 2 1 3 3 3 4 2 1 4 4 4 5 5 1 5 3
#> [35929] 1 1 2 1 5 3 3 5 1 3 1 3 3 3 1 5 2 3 4 2 2 3 3 5 3 3 1 3 1 5 2 3 1 3 2 4
#> [35965] 5 4 1 2 4 3 5 3 3 3 1 5 2 3 4 3 2 2 3 5 3 3 1 2 3 3 4 1 3 1 3 3 3 3 3 3
#> [36001] 1 3 3 3 2 1 1 5 3 1 4 4 2 3 1 2 1 3 1 3 2 2 5 1 3 5 3 2 1 5 5 5 5 3 3 1
#> [36037] 1 3 3 2 4 5 3 1 5 1 2 2 2 2 3 5 4 4 4 3 5 3 2 1 3 3 3 1 3 1 1 1 3 3 2 3
#> [36073] 3 3 1 3 3 3 3 2 4 3 2 3 3 1 2 3 5 3 2 3 3 3 1 3 3 5 3 3 1 2 5 5 3 1 3 3
#> [36109] 2 3 2 3 3 3 3 3 4 1 5 5 2 2 2 1 3 2 2 3 3 3 3 4 5 3 1 1 3 4 4 1 4 5 1 5
#> [36145] 1 3 3 3 3 1 3 3 3 5 2 1 5 3 1 1 3 1 3 3 1 3 1 3 3 4 5 3 5 3 3 3 1 3 1 1
#> [36181] 3 3 3 4 3 3 3 3 3 5 3 5 5 3 4 2 2 3 3 3 2 2 3 2 1 3 4 1 3 3 1 3 3 2 3 1
#> [36217] 1 1 5 2 3 3 4 2 3 3 1 3 3 4 5 5 3 1 3 1 3 3 3 1 4 5 3 4 3 3 2 5 3 3 3 3
#> [36253] 5 2 5 3 2 1 5 4 4 5 1 5 5 3 3 4 5 3 5 1 4 3 3 3 3 4 2 3 3 3 4 5 5 2 1 4
#> [36289] 3 3 3 1 3 3 5 2 1 3 2 2 2 4 3 5 3 2 2 3 3 3 3 3 5 1 3 5 5 4 1 2 3 3 5 3
#> [36325] 5 3 2 4 4 1 2 3 4 3 1 5 4 4 3 5 3 3 1 1 2 3 1 3 3 3 3 4 3 1 4 4 3 2 3 2
#> [36361] 1 5 5 3 3 5 5 3 2 2 3 3 3 5 4 5 5 3 3 1 3 2 4 5 3 3 4 1 4 3 1 3 3 4 3 3
#> [36397] 2 5 5 3 3 1 3 1 2 3 3 3 5 3 3 3 4 3 3 3 3 3 1 3 1 2 1 5 5 2 2 2 3 3 4 5
#> [36433] 3 2 1 1 2 4 1 5 3 3 3 3 2 2 1 3 1 3 3 3 5 3 1 5 3 2 3 4 1 3 5 3 1 4 3 1
#> [36469] 2 2 3 4 4 3 5 1 3 2 5 4 2 3 4 2 1 1 2 3 3 1 3 3 2 5 3 4 3 5 3 3 2 3 3 3
#> [36505] 1 1 5 2 2 2 2 5 2 1 3 5 1 3 5 1 2 4 3 3 1 3 1 4 5 5 2 3 5 1 4 3 3 1 3 2
#> [36541] 2 2 1 2 4 1 5 3 1 1 3 3 5 3 1 3 3 1 3 1 3 2 3 1 3 3 2 1 1 5 4 2 4 3 2 4
#> [36577] 2 5 5 5 2 3 3 5 3 3 3 3 1 5 1 3 3 3 3 3 1 1 1 2 1 3 1 5 2 1 5 3 1 5 3 2
#> [36613] 1 1 3 3 1 5 3 3 5 1 1 1 3 3 1 1 5 1 3 2 3 3 4 1 5 3 5 3 3 2 1 4 2 5 2 1
#> [36649] 2 3 4 4 2 1 3 3 4 5 3 5 3 3 3 1 4 4 3 1 3 3 3 5 3 3 1 2 1 3 2 5 4 5 3 2
#> [36685] 4 3 1 3 4 2 3 3 5 4 2 1 3 4 5 2 3 1 3 2 3 1 3 2 1 5 3 3 1 5 4 1 2 5 3 5
#> [36721] 3 3 1 2 1 3 1 2 3 4 3 3 5 1 2 2 5 4 4 5 5 1 1 5 1 5 1 1 1 3 3 5 3 3 3 3
#> [36757] 3 3 5 4 5 3 3 4 3 2 5 2 3 3 3 4 3 3 5 3 3 1 3 5 1 5 1 3 4 2 1 3 3 5 2 1
#> [36793] 2 4 3 5 1 3 2 4 3 1 4 4 2 5 3 2 4 2 4 3 3 3 5 5 3 4 3 2 2 3 4 1 4 3 2 3
#> [36829] 3 4 3 5 1 4 3 3 3 2 3 3 3 3 1 2 3 5 2 5 4 1 3 1 5 3 3 3 4 1 3 2 5 4 4 2
#> [36865] 1 3 1 5 3 1 3 3 3 5 5 3 1 2 3 3 3 1 3 3 1 3 3 3 3 2 3 3 3 5 2 1 4 1 3 1
#> [36901] 3 3 2 3 3 3 3 3 3 1 5 3 3 2 3 5 1 3 4 1 3 1 2 3 1 3 4 4 5 3 2 2 1 1 1 1
#> [36937] 4 5 2 3 3 5 1 3 2 5 3 4 2 3 2 4 1 3 5 3 4 4 5 1 3 5 1 3 2 1 4 3 3 5 3 1
#> [36973] 2 3 2 1 5 2 3 1 3 4 4 3 3 1 1 5 2 2 3 3 1 2 3 3 1 5 5 4 2 4 5 2 5 3 1 2
#> [37009] 4 2 1 5 4 2 2 4 3 1 4 3 4 4 4 4 3 3 2 3 2 2 4 2 1 4 3 5 3 3 5 1 3 1 5 4
#> [37045] 3 3 4 1 3 5 1 2 1 2 1 1 3 3 4 5 1 1 4 5 3 3 3 5 3 3 3 4 1 1 5 2 2 3 2 3
#> [37081] 2 3 2 5 2 3 1 1 2 4 3 4 1 1 2 2 1 4 4 1 4 1 5 3 5 1 3 3 3 1 1 5 3 5 3 3
#> [37117] 3 3 2 4 3 3 3 1 4 3 5 4 3 3 5 3 3 1 5 3 1 3 1 3 5 3 1 3 1 3 1 1 3 1 2 1
#> [37153] 1 5 2 1 4 3 5 4 4 4 2 4 1 3 2 3 3 1 3 1 5 2 4 4 2 1 3 2 3 1 3 1 5 2 5 1
#> [37189] 3 4 3 4 1 4 3 3 3 3 3 1 4 1 5 1 1 2 3 3 1 5 2 3 2 4 3 1 3 3 1 5 4 3 5 1
#> [37225] 3 5 4 3 5 3 4 1 5 3 3 3 3 2 1 5 5 3 2 3 3 3 5 1 4 1 3 3 2 5 4 3 3 3 5 3
#> [37261] 3 4 1 3 1 4 3 4 5 3 4 3 1 4 2 3 3 1 2 2 2 5 4 1 3 1 3 4 4 2 2 5 5 4 5 2
#> [37297] 4 3 2 3 3 1 2 5 1 1 5 4 1 3 1 1 3 4 1 5 4 4 4 2 1 3 3 2 5 2 5 1 2 3 3 3
#> [37333] 3 2 1 3 5 3 1 4 1 3 4 3 3 3 5 1 5 3 2 3 1 3 2 1 1 4 3 3 3 3 3 3 1 3 5 5
#> [37369] 4 3 3 3 1 1 1 3 3 4 1 3 1 3 2 2 2 3 1 1 3 3 3 5 3 3 4 5 1 4 3 3 3 2 3 1
#> [37405] 1 3 4 2 1 2 3 5 1 3 4 3 3 4 4 1 5 3 1 5 3 3 3 5 2 4 1 5 3 2 2 3 2 3 3 3
#> [37441] 3 3 5 4 1 2 4 4 3 2 1 2 1 3 1 2 3 1 2 4 5 3 2 1 3 4 1 3 3 4 1 5 1 3 4 5
#> [37477] 3 1 3 1 3 1 1 3 3 3 3 5 1 3 2 2 3 1 4 3 5 1 3 4 3 3 5 1 1 1 3 5 2 3 3 3
#> [37513] 2 1 4 4 3 1 4 3 3 1 5 5 5 3 3 3 3 5 1 3 4 2 3 1 3 1 1 1 3 3 2 3 5 3 3 3
#> [37549] 4 3 5 2 1 1 3 3 4 3 3 3 5 3 3 4 1 3 1 3 3 3 5 1 4 3 1 3 5 4 5 3 3 3 5 1
#> [37585] 1 1 3 1 3 3 3 3 5 3 3 1 4 1 4 3 3 2 1 2 2 4 4 2 3 3 1 3 4 3 3 3 5 4 3 2
#> [37621] 2 3 3 1 3 3 2 1 3 1 5 2 2 1 2 3 5 3 5 1 1 3 4 3 3 1 5 3 3 5 3 5 3 2 4 3
#> [37657] 5 3 4 3 2 4 5 3 1 2 3 3 1 1 3 2 2 1 1 5 3 3 3 5 2 3 1 3 2 2 3 3 4 2 3 1
#> [37693] 3 2 3 5 1 3 5 3 4 1 2 2 4 5 3 1 5 2 3 4 4 3 3 3 1 5 1 3 3 3 2 1 3 3 5 3
#> [37729] 1 3 5 4 5 5 1 3 2 3 3 1 4 1 5 5 4 3 3 3 3 3 4 1 5 4 4 5 4 4 3 1 5 5 3 3
#> [37765] 2 1 3 2 3 3 1 3 2 5 1 3 1 3 3 5 3 4 3 2 2 4 3 4 5 1 4 5 3 3 2 1 1 1 4 1
#> [37801] 3 1 5 4 4 4 4 5 5 3 2 4 2 2 1 3 3 3 4 1 3 3 1 5 5 5 3 3 5 3 5 4 3 3 5 3
#> [37837] 1 1 1 3 2 2 5 5 1 4 3 3 2 5 3 4 3 4 3 5 1 1 3 5 5 4 2 3 3 3 3 5 1 1 5 3
#> [37873] 4 3 4 3 2 3 2 3 5 5 3 2 1 4 2 3 1 4 3 5 3 4 2 2 3 3 5 1 1 3 5 1 3 1 2 1
#> [37909] 3 2 4 4 1 1 1 3 3 3 3 1 3 2 3 3 3 4 2 1 1 1 3 1 2 3 3 3 2 2 3 5 3 4 5 3
#> [37945] 3 4 1 5 3 1 3 1 1 3 3 3 3 4 3 3 1 1 3 2 2 1 3 4 1 5 1 1 5 3 2 3 3 3 1 3
#> [37981] 2 3 1 5 5 3 3 1 5 3 1 2 2 1 3 3 3 1 2 1 3 2 3 5 5 3 3 5 2 5 3 3 2 3 3 2
#> [38017] 4 4 2 3 1 1 1 5 2 2 1 3 2 5 3 1 2 3 1 4 3 2 1 3 2 2 5 1 5 1 1 5 4 3 1 5
#> [38053] 4 4 2 1 3 5 4 1 1 1 1 3 3 3 5 3 1 2 1 2 1 3 1 3 1 2 3 3 3 3 5 3 3 5 5 3
#> [38089] 1 5 4 4 4 4 5 3 2 3 1 3 1 4 1 3 2 2 2 2 3 1 1 3 1 1 5 5 3 3 2 5 1 3 3 5
#> [38125] 5 1 3 2 2 3 3 2 5 5 5 1 5 3 3 3 3 3 3 3 5 4 3 5 3 5 3 4 2 1 2 1 4 3 3 5
#> [38161] 2 2 4 5 3 5 5 3 1 1 5 3 5 5 1 3 5 4 3 1 1 3 2 4 2 2 5 5 1 5 3 1 3 3 2 3
#> [38197] 4 1 1 2 2 1 4 4 3 3 5 3 4 3 1 3 3 3 4 3 3 5
#> 
#> Within cluster sum of squares by cluster:
#> [1]  82994.12  61206.77 110798.28  74278.60  30572.70
#>  (between_SS / total_SS =  47.7 %)
#> 
#> Available components:
#> 
#> [1] "cluster"      "centers"      "totss"        "withinss"     "tot.withinss"
#> [6] "betweenss"    "size"         "iter"         "ifault"

We can now attach the clusters back to the data-frame:

df.cluster <- cbind(DATA_65.impute.sample, cluster=km$cluster)

df.cluster %>%
  select(cluster, all_of(colnames(df.cluster)))  %>%
  head()
#>   cluster   SEQN DIABETES Gender  Race      Family_Income     BPXDI3     BPXSY3
#> 1       2  49315        0 Female Black $10,000 to $14,999 0.93103091 -1.0509978
#> 2       2  12656        0   Male Other               <NA> 0.61498539 -1.1784710
#> 3       2  75508        0   Male White $35,000 to $44,999 0.93103091 -0.9235245
#> 4       2  39286        0 Female Black               <NA> 0.45696263  0.2237348
#> 5       3  96820        0   Male White $20,000 to $24,999 0.14091710 -0.2861583
#> 6       5 101514        0 Female Other $25,000 to $34,999 0.02758816 -0.2632931
#>       BPXDI2     BPXSY2        BPXDI1     BPXSY1     BMXLEG     BPXML1
#> 1 0.45560151 -1.3159807  0.2908770907 -1.5604544  0.8364784 -0.8312833
#> 2 0.77614628 -1.0666961  0.7728788919 -1.0748524 -0.3969271 -0.3202789
#> 3 0.61587389 -0.8174115  0.9335461590 -1.0748524  1.6500011  0.1907255
#> 4 0.29532912  0.4290114 -0.1911247104  0.5033540  0.3903530  0.7017299
#> 5 0.45560151 -0.3188424  0.7728788919 -0.4678500  1.1513904 -0.3202789
#> 6 0.01842516 -0.2704832  0.0009582346 -0.2705987 -0.6899635 -0.2279253
#>       BPXPLS     PEASCTM1   BMXWAIST     BMXBMI      BMXHT     BMXARMC
#> 1  0.6742685  0.721939570  0.7788639  0.5919049  0.4208552  0.56225066
#> 2 -0.2218206 -0.178901222 -0.1850887 -0.1390703  0.2104422  0.24176592
#> 3  1.3911398  1.351235051 -0.5338554 -0.5820856  1.1237243  0.10242472
#> 4 -2.1932166  0.549529849  0.5415088  0.3510154  0.4477165  0.06062237
#> 5  1.7495754  0.140934143  1.0937228  0.4008546  1.3878598  1.21715426
#> 6  0.1725892  0.008049367 -1.9289225 -1.3850507 -2.7398171 -1.80654961
#>      BMXARML Poverty_Income_Ratio       BMXWT        Age DIABETES_factor
#> 1  0.9094677           -1.1137083  0.58471445 -0.3785749               0
#> 2  0.2779486           -0.6380018 -0.03573447  0.2342063               0
#> 3  0.6726480           -0.7487827  0.18244537 -0.7462436               0
#> 4  0.7673759            1.7731137  0.44153393  0.4793187               0
#> 5  0.9884076           -0.6836175  1.19493620  0.6018750               0
#> 6 -2.5165231           -0.8204646 -1.71299201 -1.1956165               0

10.7.1.1 So what?

We can modify cat_feature_explore from Section 9.2.2.2 to create:

proc_chi_square <- function(data, factor, feature){

  enquo_feature <- enquo(feature)
  enquo_factor <- enquo(factor)
  
  tmp <- data %>%
    select(!!enquo_factor, !!enquo_feature) %>%
    collect()
  
   table1 <- table(tmp) 
   
   table2 <- table(tmp %>% select(!!enquo_feature, !!enquo_factor) )
   
   plot_chi_square_residuals <-  vcd::mosaic(table2, gp = vcd::shading_max)
  
   plot_balloon <- gplots::balloonplot(table2, 
                                       main ="Balloon Plot for Gender by Diabetes \n Area is proportional to Freq.")
   
   return( list(Frequency_Table = table2,
                plot_chi_square_residuals = plot_chi_square_residuals,
                plot_balloon = plot_balloon))
   
}

Now let’s run Chi-Square on it:

proc_chi_square(df.cluster, DIABETES, cluster)
#> $Frequency_Table
#>        DIABETES
#> cluster     0     1
#>       1  5241    57
#>       2 13694   718
#>       3  6747  1126
#>       4  3523   831
#>       5  6275     6
#> 
#> $plot_chi_square_residuals
#>         DIABETES     0     1
#> cluster                     
#> 1                 5241    57
#> 2                13694   718
#> 3                 6747  1126
#> 4                 3523   831
#> 5                 6275     6
#> 
#> $plot_balloon
#> NULL

10.7.2 Effectiveness of k-means clusters as features

If you’re still skeptical because df.cluster is a sample of A_DATA_2.Num.impute then we can project the clusters onto all of A_DATA_2.Num.impute:

A_DATA_2.Num.impute_scale <- A_DATA_2.Num.impute %>%
  mutate_at(all_of(features_65_num$feature), scale)

library(clue)

cluster <- clue::cl_predict(km,
                             A_DATA_2.Num.impute_scale %>% select(all_of(features_65_num$feature))
                            )

A_DATA_2.Num.impute_scale$.cluster <- factor(as.numeric(as.character(cluster)))

Now run proc_chi_square on it:

proc_chi_square(A_DATA_2.Num.impute_scale, DIABETES, .cluster)
#> $Frequency_Table
#>         DIABETES
#> .cluster     0     1
#>        1 13005   186
#>        2 34482  1779
#>        3 16694  2776
#>        4  8875  2052
#>        5 15684    14
#> 
#> $plot_chi_square_residuals
#>          DIABETES     0     1
#> .cluster                     
#> 1                 13005   186
#> 2                 34482  1779
#> 3                 16694  2776
#> 4                  8875  2052
#> 5                 15684    14
#> 
#> $plot_balloon
#> NULL

Above, we can see Diabetics are much more concentrated in Clusters 2, 3, & 4 than the others.

A_DATA_2.Num.impute_scale %>%
  group_by(.cluster) %>%
  summarise(N_Diabetic = sum(DIABETES)) 
#> # A tibble: 5 × 2
#>   .cluster N_Diabetic
#>   <fct>         <dbl>
#> 1 1               186
#> 2 2              1779
#> 3 3              2776
#> 4 4              2052
#> 5 5                14

\(~\)

10.8 Disscussion

The functions in this chapter compared to the others.

9 Functional dbplyr, purrr, and furrr

11 Packages for Automated Exploratory Data Analysis