Principle Component Analysis (R)

What is Principle Component Analysis (PCA)?

Questionnaires and surveys often have items that reflect multiple sub-scales or multiple related constructs, in a single measure. For example, the empathy quotient (Lawrence et al. 2004) has multiple types of empathy within it:

  • cognitive empathy

  • emotional reactivity

  • social skills

Let’s use one publicly available data set to check if we can replicate these subscales:

https://github.com/bhismalab/EyeTracking_PlosOne_2017/blob/master/EQ_Data.csv

eq <- read.csv("EQ_Data.csv")
# focusing on q1.raw to q40.raw
eq_processed <- eq[,c(paste("Q",1:40,".processed",sep=""))]
rmarkdown::paged_table(eq_processed)
colnames(eq_processed) <- paste("eq",1:40, sep="")
rmarkdown::paged_table(data.frame(cor(eq_processed)))

Confirmatory factor analysis

We already have three subscales for the empathy quotient, so let’s confirm whether our current data is consistent with them. The subscales are:

Some of the items below are reverse scored

The items that would go against the component, e.g. “I find it hard to know what to do in a social situation” if the participant agreed are reverse scored, meaning that strongly disagreeing increases their score for the relevant subscale (in this case “social skills”).

Cognitive:

  • 14: I am good at predicting how someone will feel.

  • 15: I am quick to spot when someone in a group is feeling awkward or uncomfortable.

  • 29: I can sense if I am intruding, even if the other person doesn’t tell me.

  • 34: I can tune into how someone else feels rapidly and intuitively.

  • 35: I can easily work out what another person might want to talk about.

Social Skills:

  • 2: I find it difficult to explain to others things that I understand easily, when they don’t understand it first time.

  • 4: I find it hard to know what to do in a social situation.

  • 7: Friendships and relationships are just too difficult, so I tend not to bother with them.

  • 8: I often find it difficult to judge if something is rude or polite.

  • 21: I don’t tend to find social situations confusing.

Emotion:

  • 3: I really enjoy caring for other people.

  • 16: If I say something that someone else is offended by, I think that that’s their problem, not mine.

  • 19: Seeing people cry doesn’t really upset me.

  • 33: I usually stay emotionally detached when watching a film.

  • 39: I tend to get emotionally involved with a friend’s problems.

Let’s start by looking at correlation matrices to see if the items tend to correlate with each other in the current data set:

Cognitive empathy

cor(eq_processed[,c("eq14","eq15","eq29","eq34","eq35")])
          eq14       eq15      eq29      eq34       eq35
eq14 1.0000000 0.34501695 0.2905797 0.4416185 0.27504384
eq15 0.3450170 1.00000000 0.3721252 0.2532783 0.07192026
eq29 0.2905797 0.37212520 1.0000000 0.2468024 0.28854264
eq34 0.4416185 0.25327834 0.2468024 1.0000000 0.23509011
eq35 0.2750438 0.07192026 0.2885426 0.2350901 1.00000000

So far looking good, as everything positively correlates - mostly with r-values greater than .25. Let’s now see how well each item loads onto the total of these scores:

library(tidyverse)
── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr     1.1.3     ✔ readr     2.1.4
✔ forcats   1.0.0     ✔ stringr   1.5.0
✔ ggplot2   3.4.4     ✔ tibble    3.2.1
✔ lubridate 1.9.3     ✔ tidyr     1.3.0
✔ purrr     1.0.2     
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag()    masks stats::lag()
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors
library(kableExtra)

Attaching package: 'kableExtra'

The following object is masked from 'package:dplyr':

    group_rows
eq_subscales_r <- data.frame(
  cor(
    eq_processed[,c(
      # Cog
      "eq14","eq15","eq29","eq34","eq35",
      
      # Social Skills
      "eq2","eq4","eq7","eq8","eq21",
      
      # Emotion
      "eq3","eq16","eq19","eq33","eq39"
      )],
    matrix(data = c(
      rowSums(eq_processed[,c("eq14","eq15","eq29","eq34","eq35")]),
      rowSums(eq_processed[,c("eq2","eq4", "eq7", "eq8", "eq21")]),
      rowSums(eq_processed[,c("eq3","eq16","eq19","eq33","eq39")])
      ), ncol = 3
    )
  )
) 

colnames(eq_subscales_r) <- c( "Cognitive", "Social", "Emotion")

eq_subscales_r %>% mutate_all(~cell_spec(
        .x, 
        color = ifelse(.x < .4, "black", "white"),
        background = ifelse(.x > .4, "red"," white"))) %>%
    kable(escape = F) %>%
    kable_styling()
Cognitive Social Emotion
eq14 0.730729714723979 0.110723735489975 0.31320589104921
eq15 0.60185382395657 0.198874708000659 0.137720182774352
eq29 0.659975490377257 0.0278299656284012 0.364451955577018
eq34 0.679619374819311 0.0226190168052008 0.395730784538071
eq35 0.587492289787915 0.122610871425514 0.242014658835711
eq2 -0.00813200484463844 0.576298173261772 0.133855862092061
eq4 0.0479274896411267 0.662778025977272 0.123591773099612
eq7 -0.0312109379222552 0.438752142983534 0.234889555117118
eq8 0.226816177181639 0.427950397740909 0.134334810841783
eq21 0.161873895295539 0.627680352407105 -0.0206828796782751
eq3 0.29722368237889 -0.00177049925158063 0.62739169422544
eq16 0.268239998089082 0.168294312535106 0.592732439474545
eq19 0.35758009314024 0.183400506448371 0.552243103025108
eq33 0.159850007854409 0.302368129555882 0.665487951247779
eq39 0.349069094803534 0.0110077011019322 0.690638221783324

This is quite dense, but in broad terms, we want to cluster items that correlate with each other into one component (AKA factor AKA subscale).

If we use a package in R we can start identifying the top 3 components and check if the questions map on to what we would expect for each of the three subscales:

eq_pca <- prcomp(eq_processed)
sort(abs(eq_pca$rotation[,1]),decreasing = T)[1:5]
     eq16      eq20      eq22      eq18      eq13 
0.2752500 0.2436235 0.2428536 0.2329153 0.2315572 
sort(eq_pca$rotation[,1],decreasing = T)[1:5]
       eq37        eq11        eq21         eq4        eq38 
 0.03183132 -0.02479062 -0.03616440 -0.04118648 -0.04219994

Component 1 involves questions 20, 33, 31, 16 and 5. These items are: - 20: I am very blunt, which some people take to be rudeness, even though this is unintentional. - 33: I usually stay emotionally detached when watching a film. - 31: Other people often say that I am insensitive, though I don’t always see why. - 16: If I say something that someone else is offended by, I think that that’s their problem, not mine. - 5: People often tell me that I went too far in driving my point home in a discussion.

I think the below are just notes or wrong

So how do we start doing this? We can make a regression to determine the loading of all the items onto the total:

eq_total <- rowSums(eq_processed)
first_component <- lm(eq_processed)
first_component$coefficients
  (Intercept)           eq2           eq3           eq4           eq5 
-0.4547242739  0.1848459062  0.0008838912  0.1848745215 -0.0496279785 
          eq6           eq7           eq8           eq9          eq10 
 0.0306243998  0.1206735873  0.0850514704 -0.1133026903  0.0828400027 
         eq11          eq12          eq13          eq14          eq15 
-0.3193175205  0.0294001228 -0.0639375439 -0.0569689598  0.1213792245 
         eq16          eq17          eq18          eq19          eq20 
-0.0949204164  0.0833283618 -0.1107909629 -0.2080609287  0.0922992241 
         eq21          eq22          eq23          eq24          eq25 
-0.0565096267  0.0101494380  0.2210134281  0.1792635433  0.0536522942 
         eq26          eq27          eq28          eq29          eq30 
 0.0636991757  0.1067968018  0.0895769174  0.1704297362  0.1226011527 
         eq31          eq32          eq33          eq34          eq35 
-0.0253840314 -0.1658941305  0.1227288110  0.2609650282  0.2561104255 
         eq36          eq37          eq38          eq39          eq40 
 0.0611831954 -0.1397851385  0.2618918575 -0.2598533042  0.1532014221

explained in smaller steps

using r package

eigen(cov(eq_processed))
eigen() decomposition
$values
 [1] 2.93449333 1.87628459 1.31590409 1.12748372 0.94180395 0.88649785
 [7] 0.79441522 0.71321627 0.65949310 0.62037952 0.57051522 0.51529423
[13] 0.47943423 0.44518243 0.42671811 0.40955342 0.38885355 0.36458886
[19] 0.34093142 0.32034798 0.28981434 0.26346324 0.25212709 0.21377123
[25] 0.19943788 0.16612460 0.15125182 0.13004091 0.12441560 0.11618794
[31] 0.11047414 0.10347115 0.08948276 0.08151707 0.06526776 0.05568862
[37] 0.05048182 0.04280785 0.03199620 0.01706141

$vectors
             [,1]        [,2]        [,3]         [,4]         [,5]
 [1,] -0.14885205  0.10112567 -0.02509082 -0.071014782  0.070127637
 [2,] -0.10091649 -0.12487961 -0.21344804  0.126134563 -0.363701783
 [3,] -0.21319619  0.06215300  0.14461470  0.111668352  0.254921340
 [4,] -0.04118648 -0.01513016 -0.42425919  0.110096470  0.018531412
 [5,] -0.17453482 -0.27933600  0.05769899 -0.012773827 -0.094372919
 [6,] -0.12317922  0.04277050  0.13093569 -0.152834097 -0.075972782
 [7,] -0.09429054 -0.12047107 -0.11068727  0.095173646  0.321122230
 [8,] -0.15532360 -0.04014554 -0.11777900 -0.162564085 -0.112894310
 [9,] -0.15175665 -0.13375762 -0.09957822  0.015337922  0.099157025
[10,] -0.11476213  0.07181521  0.20893937  0.185060033  0.179800156
[11,] -0.02479062  0.20731727 -0.11512269 -0.129484221 -0.124767585
[12,] -0.20001559 -0.08797362 -0.28406761 -0.149859215  0.179441277
[13,] -0.23155718  0.14482426  0.02484372 -0.102876589  0.133912627
[14,] -0.18108913  0.18299383 -0.01014458 -0.166832869 -0.090845908
[15,] -0.10440156  0.11709966 -0.07772977 -0.105781123  0.105817944
[16,] -0.27525004 -0.17463360  0.05807836 -0.074587632 -0.109099073
[17,] -0.12515385 -0.11189375  0.22004369 -0.070312126  0.144586699
[18,] -0.23291526 -0.05944152 -0.24972629 -0.118728554 -0.070205907
[19,] -0.13481111  0.15230707 -0.07563699  0.265291859 -0.141997509
[20,] -0.24362346 -0.36238558 -0.07453948 -0.156177866  0.025176784
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[24,] -0.05906693  0.19421309  0.15273056  0.358550925 -0.177778979
[25,] -0.09644040 -0.08117186  0.26316869 -0.140791173 -0.035738385
[26,] -0.07345369  0.18152254 -0.07792673 -0.124229730  0.065045677
[27,] -0.10514313  0.11214456 -0.02311665  0.353333232 -0.154066342
[28,] -0.21837854  0.02627596  0.18307314 -0.127036457 -0.207679119
[29,] -0.14871777  0.15334195  0.01875033  0.017018645 -0.094104526
[30,] -0.13421971 -0.19925686  0.06446130  0.102339786 -0.341431242
[31,] -0.22781047 -0.24536195  0.04637429  0.050756852  0.010756140
[32,] -0.16086212 -0.03560941 -0.12984607  0.332415006  0.208100395
[33,] -0.22804044 -0.11299201 -0.01468625  0.293444598  0.219353354
[34,] -0.17436789  0.18970876  0.03337218 -0.049325165 -0.007173955
[35,] -0.11389235  0.14957934 -0.15942072 -0.172264522  0.002624950
[36,] -0.10887641  0.25415327 -0.08877280 -0.068476179  0.041093920
[37,]  0.03183132  0.01079756 -0.09170238  0.184201107 -0.090322073
[38,] -0.04219994  0.29754522 -0.10121144 -0.115424152  0.104174103
[39,] -0.21909596  0.14852516  0.22629102  0.171059890  0.019044720
[40,] -0.16930983  0.22832068 -0.04107838  0.005916164  0.051211687
              [,6]        [,7]         [,8]          [,9]        [,10]
 [1,]  0.006119948  0.02324630  0.040691595 -0.1294096393 -0.148942216
 [2,]  0.016933565 -0.17665873  0.223096197 -0.0405096824  0.352530196
 [3,] -0.042796961  0.09290645  0.026050627  0.0395456079  0.054063729
 [4,] -0.074503190  0.16521157 -0.202173247 -0.2966167135  0.124578200
 [5,]  0.012713419 -0.06357712 -0.086066025 -0.0491824725 -0.051250400
 [6,]  0.435228223  0.06098821 -0.161284969  0.2577071219  0.117801427
 [7,] -0.271066653 -0.01194754 -0.057270194  0.0979508265 -0.044225282
 [8,]  0.134081558  0.12799009 -0.071293952 -0.2198553070 -0.451131985
 [9,]  0.207132844 -0.14029164  0.093306497  0.0656231074  0.184870075
[10,]  0.142926959 -0.18753662 -0.116615738  0.0570337168  0.013059541
[11,] -0.032643620 -0.04351784 -0.222009296  0.1315299255 -0.013179787
[12,]  0.199806055  0.13708449  0.060524975  0.1992480218 -0.132030286
[13,] -0.153104814  0.04908259  0.342429803  0.1077049680  0.149118464
[14,]  0.003958447 -0.03875740 -0.098081800 -0.0317053321  0.008479807
[15,] -0.127331352 -0.02569759 -0.006488851  0.1161201330 -0.099095263
[16,]  0.198074734  0.32323386 -0.157413533 -0.0004700701  0.258594312
[17,] -0.068202126 -0.31281778  0.085339084 -0.0076273430 -0.058200957
[18,]  0.137998888 -0.16938085  0.201376101  0.1345761916  0.016416638
[19,] -0.117115123  0.05501882 -0.138403111  0.1168033759  0.184423548
[20,] -0.309746658  0.32653654  0.092161901 -0.1094815145 -0.073249089
[21,] -0.030030064 -0.36464316 -0.109756057  0.2834541486 -0.077322145
[22,] -0.073825472 -0.14454128 -0.071059067 -0.0699442597 -0.189200852
[23,]  0.418352031  0.06055086  0.002244558  0.0325248469  0.041387139
[24,]  0.050443616  0.12239003  0.191329884  0.0599348238 -0.209461246
[25,] -0.144190583  0.06700051  0.081587883 -0.0543381124  0.206056218
[26,]  0.070323095  0.01312386  0.048277226 -0.0780515538  0.003983627
[27,]  0.084677432  0.28827864  0.182056470  0.1048929065 -0.270329635
[28,]  0.022764004 -0.15490076 -0.040686689 -0.1208785449 -0.154640753
[29,] -0.028725942  0.03954373 -0.009367995  0.1293606197 -0.006759526
[30,] -0.112625495  0.04595560 -0.210180789  0.2547197346  0.043968622
[31,] -0.060563016 -0.31190190  0.213870711 -0.0204652801 -0.004114626
[32,]  0.061696200  0.12617541  0.076618157  0.1160058088  0.013861306
[33,]  0.048341035 -0.20001719 -0.427379992 -0.2543358862 -0.115205194
[34,] -0.047060618 -0.11709317  0.050869067 -0.1546589529  0.062455555
[35,]  0.101873334  0.04138749 -0.131456373 -0.0606821789  0.037080094
[36,] -0.093811702 -0.09952521 -0.049077877 -0.1227509486 -0.019950696
[37,]  0.309862048 -0.06975852  0.245299005 -0.5309007228  0.118851107
[38,] -0.072224835  0.01754501 -0.065361115 -0.0487848696  0.308351032
[39,] -0.168248949  0.12052077 -0.146240053 -0.1386520615  0.229948767
[40,] -0.055521030  0.03278789  0.280474287 -0.0010935095 -0.095309650
             [,11]        [,12]        [,13]        [,14]       [,15]
 [1,] -0.253868134 -0.026378923  0.139382716  0.044761895  0.08948919
 [2,] -0.022810667  0.020994584 -0.077046746  0.346488286 -0.08691651
 [3,]  0.120914913 -0.072314901  0.187086205  0.301066309  0.32564864
 [4,] -0.088050052 -0.243189034 -0.267756742  0.072911971  0.03630678
 [5,] -0.072822847 -0.288424864 -0.078123589 -0.273594921  0.01349082
 [6,] -0.203816671 -0.275828938 -0.203716874  0.094047851 -0.11443579
 [7,]  0.005326963  0.006993484  0.014542079 -0.149687164 -0.15293618
 [8,]  0.045712400  0.076175741 -0.265109868  0.202428397 -0.02931851
 [9,]  0.138159936 -0.024521607 -0.160244586 -0.348243157  0.23199313
[10,]  0.264929142  0.043684988  0.050768359  0.145715076 -0.18998061
[11,]  0.110870762  0.403955313 -0.073179826  0.010223102 -0.14049182
[12,]  0.070954746  0.093634203 -0.128963760  0.049677031  0.30365893
[13,] -0.114143158 -0.269418949 -0.071534229 -0.078927689 -0.15602113
[14,] -0.173753963 -0.040394180  0.164228968  0.052666105 -0.04110498
[15,] -0.144614985  0.053256450 -0.053968648  0.066967560 -0.28890074
[16,] -0.116896928  0.230041437  0.075767587 -0.041653811 -0.25641550
[17,] -0.429776496  0.243035922 -0.261530553  0.093768297  0.17357371
[18,]  0.316419980  0.178596105  0.051927948  0.007019879 -0.21310623
[19,]  0.136542837  0.018220133  0.087625887  0.084760908  0.21910180
[20,]  0.058038535  0.002380763  0.297734072  0.115009764 -0.12710823
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[22,]  0.168649625 -0.179815303  0.017623692  0.114531283  0.11851323
[23,]  0.011044629 -0.023558273  0.317489916  0.100585874 -0.01445311
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[25,] -0.066388538  0.306500289 -0.188557761 -0.005850511  0.05072269
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[30,] -0.219397023  0.021368869  0.280512041 -0.193161328  0.25095271
[31,]  0.027619178  0.121965742 -0.060345272 -0.044247910  0.10288207
[32,] -0.110944046 -0.033972122 -0.114297377  0.183704630 -0.03051556
[33,] -0.069234258  0.128769313  0.022270597 -0.116804666 -0.16698083
[34,]  0.017632909  0.061659269  0.160647608  0.162520062  0.04542429
[35,]  0.176403011  0.121207893 -0.055449911 -0.202345939  0.33783961
[36,] -0.065575303  0.141128952  0.145660585 -0.095775914  0.10261327
[37,] -0.091821121  0.066421466  0.215964244 -0.150725673 -0.04461417
[38,] -0.227886499  0.068254196  0.004405031  0.075515277  0.06253901
[39,]  0.210024433 -0.096319225 -0.318256877 -0.058321039 -0.04186827
[40,]  0.098775253 -0.135888673 -0.007911576 -0.228873442 -0.07877680
             [,16]        [,17]       [,18]        [,19]        [,20]
 [1,] -0.094923537 -0.192238711 -0.09534877 -0.159657019 -0.263154099
 [2,] -0.121152744 -0.268133556 -0.02037691 -0.073369010 -0.361937208
 [3,]  0.136792271 -0.170694040 -0.21878266 -0.140875936 -0.102684603
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learning from https://uc-r.github.io/pca

library(tidyverse)  # data manipulation and visualization
library(gridExtra)  # plot arrangement

Attaching package: 'gridExtra'
The following object is masked from 'package:dplyr':

    combine
data("USArrests")
head(USArrests, 10)
            Murder Assault UrbanPop Rape
Alabama       13.2     236       58 21.2
Alaska        10.0     263       48 44.5
Arizona        8.1     294       80 31.0
Arkansas       8.8     190       50 19.5
California     9.0     276       91 40.6
Colorado       7.9     204       78 38.7
Connecticut    3.3     110       77 11.1
Delaware       5.9     238       72 15.8
Florida       15.4     335       80 31.9
Georgia       17.4     211       60 25.8
scaled_df <- apply(USArrests, 2, scale)

arrests.cov <- cov(scaled_df)
arrests.eigen <- eigen(arrests.cov)

arrests.cov
             Murder   Assault   UrbanPop      Rape
Murder   1.00000000 0.8018733 0.06957262 0.5635788
Assault  0.80187331 1.0000000 0.25887170 0.6652412
UrbanPop 0.06957262 0.2588717 1.00000000 0.4113412
Rape     0.56357883 0.6652412 0.41134124 1.0000000
arrests.eigen
eigen() decomposition
$values
[1] 2.4802416 0.9897652 0.3565632 0.1734301

$vectors
          [,1]       [,2]       [,3]        [,4]
[1,] 0.5358995  0.4181809 -0.3412327  0.64922780
[2,] 0.5831836  0.1879856 -0.2681484 -0.74340748
[3,] 0.2781909 -0.8728062 -0.3780158  0.13387773
[4,] 0.5434321 -0.1673186  0.8177779  0.08902432
temp_1 <- lm(scaled_df[,1] ~ scaled_df[,2])
cov(temp_1$residuals, scaled_df[,1])
[1] 0.3569992
cov(temp_1$residuals, scaled_df[,2])
[1] 5.328079e-17
str(arrests.eigen)
List of 2
 $ values : num [1:4] 2.48 0.99 0.357 0.173
 $ vectors: num [1:4, 1:4] 0.536 0.583 0.278 0.543 0.418 ...
 - attr(*, "class")= chr "eigen"
arrests.eigen
eigen() decomposition
$values
[1] 2.4802416 0.9897652 0.3565632 0.1734301

$vectors
          [,1]       [,2]       [,3]        [,4]
[1,] 0.5358995  0.4181809 -0.3412327  0.64922780
[2,] 0.5831836  0.1879856 -0.2681484 -0.74340748
[3,] 0.2781909 -0.8728062 -0.3780158  0.13387773
[4,] 0.5434321 -0.1673186  0.8177779  0.08902432
arrests.eigen$values
[1] 2.4802416 0.9897652 0.3565632 0.1734301
arrests.eigen$vectors
          [,1]       [,2]       [,3]        [,4]
[1,] 0.5358995  0.4181809 -0.3412327  0.64922780
[2,] 0.5831836  0.1879856 -0.2681484 -0.74340748
[3,] 0.2781909 -0.8728062 -0.3780158  0.13387773
[4,] 0.5434321 -0.1673186  0.8177779  0.08902432
(beep <- arrests.eigen$vectors[,1:2])
          [,1]       [,2]
[1,] 0.5358995  0.4181809
[2,] 0.5831836  0.1879856
[3,] 0.2781909 -0.8728062
[4,] 0.5434321 -0.1673186
beep <- prcomp(scaled_df)
beep$rotation <- beep$rotation * -1
beep$rotation <- beep$x * -1

biplot(beep, scale = 0)

To start

References

Lawrence, E. J., P. Shaw, D. Baker, S. Baron-Cohen, and A. S. David. 2004. “Measuring Empathy: Reliability and Validity of the Empathy Quotient.” Psychological Medicine 34 (5): 911–20. https://doi.org/10.1017/s0033291703001624.