Factor analysis

Vs. principal components

• Principal components:

  • Purely mathematical.

  • Find eigenvalues, eigenvectors of correlation matrix.

  • No testing whether observed components reproducible, or even probability model behind it.

• Factor analysis:

  • some way towards fixing this (get test of appropriateness)

  • In factor analysis, each variable modelled as: “common factor” (eg. verbal ability) and “specific factor” (left over).

  • Choose the common factors to “best” reproduce pattern seen in correlation matrix.

  • Iterative procedure, different answer from principal components.

Packages

library(ggbiplot)
library(tidyverse)
library(conflicted)
conflict_prefer("mutate", "dplyr")
conflict_prefer("select", "dplyr")
conflict_prefer("filter", "dplyr")
conflict_prefer("arrange", "dplyr")

Example

• 145 children given 5 tests, called PARA, SENT, WORD, ADD and DOTS. 3 linguistic tasks (paragraph comprehension, sentence completion and word meaning), 2 mathematical ones (addition and counting dots).

• Correlation matrix of scores on the tests:

para 1     0.722 0.714 0.203 0.095
sent 0.722 1     0.685 0.246 0.181
word 0.714 0.685 1     0.170 0.113
add  0.203 0.246 0.170 1     0.585
dots 0.095 0.181 0.113 0.585 1

• Is there small number of underlying “constructs” (unobservable) that explains this pattern of correlations?

To start: principal components

Using correlation matrix. Read that first:

my_url <- "http://ritsokiguess.site/datafiles/rex2.txt"
kids <- read_delim(my_url, " ")
kids

Principal components on correlation matrix

Turn into R matrix, using column test as column names:

kids %>% 
column_to_rownames("test") %>% 
as.matrix() -> m

Principal components:

kids.0 <- princomp(covmat = m) 

I used kids.0 here since I want kids.1 and kids.2 later.

Scree plot

ggscreeplot(kids.0)

Principal component results

• Need 2 components. Loadings:

kids.0$loadings

Loadings:
     Comp.1 Comp.2 Comp.3 Comp.4 Comp.5
para  0.534  0.245  0.114         0.795
sent  0.542  0.164         0.660 -0.489
word  0.523  0.247 -0.144 -0.738 -0.316
add   0.297 -0.627  0.707              
dots  0.241 -0.678 -0.680         0.143

               Comp.1 Comp.2 Comp.3 Comp.4 Comp.5
SS loadings       1.0    1.0    1.0    1.0    1.0
Proportion Var    0.2    0.2    0.2    0.2    0.2
Cumulative Var    0.2    0.4    0.6    0.8    1.0

Comments

• First component has a bit of everything, though especially the first three tests.

• Second component rather more clearly add and dots.

• No scores, plots since no actual data.

• See how factor analysis compares on these data.

Factor analysis

• Specify number of factors first, get solution with exactly that many factors.

• Includes hypothesis test, need to specify how many children wrote the tests.

• Works from correlation matrix via covmat or actual data, like princomp.

• Introduces extra feature, rotation, to make interpretation of loadings (factor-variable relation) easier.

Factor analysis for the kids data

• Create “covariance list” to include number of children who wrote the tests.

• Feed this into factanal, specifying how many factors (2).

• Start with the matrix we made before.

m

      para  sent  word   add  dots
para 1.000 0.722 0.714 0.203 0.095
sent 0.722 1.000 0.685 0.246 0.181
word 0.714 0.685 1.000 0.170 0.113
add  0.203 0.246 0.170 1.000 0.585
dots 0.095 0.181 0.113 0.585 1.000

ml <- list(cov = m, n.obs = 145)
kids.2 <- factanal(factors = 2, covmat = ml)

Uniquenesses

kids.2$uniquenesses

     para      sent      word       add      dots 
0.2424457 0.2997349 0.3272312 0.5743568 0.1554076 

• Uniquenesses say how “unique” a variable is (size of specific factor). Small uniqueness means that the variable is summarized by a factor (good).

• Very large uniquenesses are bad; add’s uniqueness is largest but not large enough to be worried about.

• Also see “communality” for this idea, where large is good and small is bad.

Loadings

kids.2$loadings

Loadings:
     Factor1 Factor2
para 0.867          
sent 0.820   0.166  
word 0.816          
add  0.167   0.631  
dots         0.918  

               Factor1 Factor2
SS loadings      2.119   1.282
Proportion Var   0.424   0.256
Cumulative Var   0.424   0.680

• Loadings show how each factor depends on variables. Blanks indicate “small”, less than 0.1.

Comments

• Factor 1 clearly the “linguistic” tasks, factor 2 clearly the “mathematical” ones.

• Two factors together explain 68% of variability (like regression R-squared).

• Which variables belong to which factor is much clearer than with principal components.

Are 2 factors enough?

kids.2$STATISTIC

objective 
0.5810578 

kids.2$dof

[1] 1

kids.2$PVAL

objective 
 0.445898 

P-value not small, so 2 factors OK.

1 factor

kids.1 <- factanal(factors = 1, covmat = ml)
kids.1$STATISTIC

objective 
 58.16534 

kids.1$dof

[1] 5

kids.1$PVAL

   objective 
2.907856e-11 

1 factor rejected (P-value small). Definitely need more than 1.

Places rated, again

• Read data, transform, rerun principal components, get biplot:

my_url <- "http://ritsokiguess.site/datafiles/places.txt"
places0 <- read_table(my_url)
places0 %>% 
mutate(across(-id, \(x) log(x))) -> places
places %>% select(-id) -> places_numeric
places.1 <- princomp(places_numeric, cor = TRUE)
g <- ggbiplot(places.1, labels = places$id,
       labels.size = 0.8)

• This is all exactly as for principal components (nothing new here).

The biplot

g

Comments

• Most of the criteria are part of components 1 and 2.
• If we can rotate the arrows counterclockwise:
  • economy and crime would point straight up
    • part of component 2 only
  • health and education would point to the right
    • part of component 1 only
• would be easier to see which variables belong to which component.
• Factor analysis includes a rotation to help with interpretation.

Factor analysis

• Have to pick a number of factors first.
• Do this by running principal components and looking at scree plot.
• In this case, 3 factors seemed good (revisit later):

places.3 <- factanal(places_numeric, 3, scores = "r")

• There are different ways to get factor scores. These called “regression” scores.

A bad biplot

biplot(places.3$scores, places.3$loadings,
xlabs = places$id)

Comments

• I have to find a way to make a better biplot!
• Some of the variables now point straight up and some straight across (if you look carefully for the red arrows among the black points).
• This should make the factors more interpretable than the components were.

Factor loadings

places.3$loadings

Loadings:
         Factor1 Factor2 Factor3
climate                   0.994 
housing   0.360   0.482   0.229 
health    0.884   0.164         
crime     0.115   0.400   0.205 
trans     0.414   0.460         
educate   0.511                 
arts      0.655   0.552   0.102 
recreate  0.148   0.714         
econ              0.318  -0.114 

               Factor1 Factor2 Factor3
SS loadings      1.814   1.551   1.120
Proportion Var   0.202   0.172   0.124
Cumulative Var   0.202   0.374   0.498

Comments on loadings

• These are at least somewhat clearer than for the principal components:
• Factor 1: health, education, arts: “well-being”
• Factor 2: housing, transportation, arts (again), recreation: “places to be”
• Factor 3: climate (only): “climate”
• In this analysis, economic factors don’t seem to be important.

Factor scores

• Make a dataframe with the city IDs and factor scores:

cbind(id = places$id, places.3$scores) %>% 
as_tibble() -> places_scores

• Make percentile ranks again (for checking):

places %>% 
mutate(across(-id, \(x) percent_rank(x))) -> places_pr

Highest scores on factor 1, “well-being”:

• for the top 4 places:

places_scores %>% 
slice_max(Factor1, n = 4)

Check percentile ranks for factor 1

places_pr %>% 
select(id, health, educate, arts) %>% 
filter(id %in% c(213, 65, 234, 314))

• These are definitely high on the well-being variables.
• City #213 is not so high on education, but is highest of all on the others.

Highest scores on factor 2, “places to be”:

places_scores %>% 
slice_max(Factor2, n = 4)

Check percentile ranks for factor 2

places_pr %>% 
select(id, housing, trans, arts, recreate) %>% 
filter(id %in% c(318, 12, 168, 44))

• These are definitely high on housing and recreation.
• Some are (very) high on transportation, but not so much on arts.
• Could look at more cities to see if #168 being low on arts is a fluke.

Highest scores on factor 3, “climate”:

places_scores %>% 
slice_max(Factor3, n = 4)

Check percentile ranks for factor 3

places_pr %>% 
select(id, climate) %>% 
filter(id %in% c(227, 218, 269, 270))

This is very clear.

Uniquenesses

• We said earlier that the economy was not part of any of our factors:

places.3$uniquenesses

  climate   housing    health     crime     trans   educate      arts  recreate 
0.0050000 0.5859175 0.1854084 0.7842407 0.6165449 0.7351921 0.2554663 0.4618143 
     econ 
0.8856382 

• The higher the uniqueness, the less the variable concerned is part of any of our factors (and that maybe another factor is needed to accommodate it).
• This includes economy and maybe crime.

Test of significance

We can test whether the three factors that we have is enough, or whether we need more to describe our data:

places.3$PVAL

   objective 
1.453217e-14 

• 3 factors are not enough.
• What would 5 factors look like?

Five factors

places.5 <- factanal(places_numeric, 5, scores = "r")
places.5$loadings

Loadings:
         Factor1 Factor2 Factor3 Factor4 Factor5
climate                           0.131   0.559 
housing   0.286   0.505   0.289  -0.113   0.475 
health    0.847   0.214                   0.187 
crime             0.196   0.143   0.948   0.181 
trans     0.389   0.515           0.175         
educate   0.534                                 
arts      0.611   0.564           0.172   0.145 
recreate          0.705           0.115   0.136 
econ                      0.978   0.135         

               Factor1 Factor2 Factor3 Factor4 Factor5
SS loadings      1.628   1.436   1.087   1.023   0.658
Proportion Var   0.181   0.160   0.121   0.114   0.073
Cumulative Var   0.181   0.340   0.461   0.575   0.648

Comments 1/2

• On (new) 5 factors:
• Factor 1 is health, education, arts: same as factor 1 before.
• Factor 2 is housing, transportation, arts, recreation: as factor 2 before.
• Factor 3 is economy.
• Factor 4 is crime.
• Factor 5 is climate and housing: like factor 3 before.

Comments 2/2

• The two added factors include the two “missing” variables.
• Is this now enough?

places.5$PVAL

   objective 
0.0009741394 

• No. My guess is that the authors of Places Rated chose their 9 criteria to capture different aspects of what makes a city good or bad to live in, and so it was too much to hope that a small number of factors would come out of these.

A bigger example: BEM sex role inventory

• 369 women asked to rate themselves on 60 traits, like “self-reliant” or “shy”.

• Rating 1 “never or almost never true of me” to 7 ``always or almost always true of me’’.

• 60 personality traits is a lot. Can we find a smaller number of factors that capture aspects of personality?

• The whole BEM sex role inventory on next page.

The whole inventory

Some of the data

my_url <- "http://ritsokiguess.site/datafiles/factor.txt"
bem <- read_tsv(my_url)
bem

Principal components first

to decide on number of factors:

bem.pc <- bem %>%
select(-subno) %>%
princomp(cor = T)

The scree plot

(g <- ggscreeplot(bem.pc))

• No obvious elbow.

Zoom in to search for elbow

Possible elbows at 3 (2 factors) and 6 (5):

g + scale_x_continuous(limits = c(0, 8))

but is 2 really good?

summary(bem.pc)

Importance of components:
                          Comp.1    Comp.2     Comp.3     Comp.4     Comp.5
Standard deviation     2.7444993 2.2405789 1.55049106 1.43886350 1.30318840
Proportion of Variance 0.1711881 0.1140953 0.05463688 0.04705291 0.03859773
Cumulative Proportion  0.1711881 0.2852834 0.33992029 0.38697320 0.42557093
                           Comp.6     Comp.7     Comp.8     Comp.9    Comp.10
Standard deviation     1.18837867 1.15919129 1.07838912 1.07120568 1.04901318
Proportion of Variance 0.03209645 0.03053919 0.02643007 0.02607913 0.02500974
Cumulative Proportion  0.45766738 0.48820657 0.51463664 0.54071577 0.56572551
                          Comp.11    Comp.12    Comp.13    Comp.14   Comp.15
Standard deviation     1.03848656 1.00152287 0.97753974 0.95697572 0.9287543
Proportion of Variance 0.02451033 0.02279655 0.02171782 0.02081369 0.0196042
Cumulative Proportion  0.59023584 0.61303238 0.63475020 0.65556390 0.6751681
                          Comp.16    Comp.17   Comp.18    Comp.19    Comp.20
Standard deviation     0.92262649 0.90585705 0.8788668 0.86757525 0.84269120
Proportion of Variance 0.01934636 0.01864948 0.0175547 0.01710652 0.01613928
Cumulative Proportion  0.69451445 0.71316392 0.7307186 0.74782514 0.76396443
                          Comp.21    Comp.22    Comp.23    Comp.24    Comp.25
Standard deviation     0.83124925 0.80564654 0.78975423 0.78100835 0.77852606
Proportion of Variance 0.01570398 0.01475151 0.01417527 0.01386305 0.01377506
Cumulative Proportion  0.77966841 0.79441992 0.80859519 0.82245823 0.83623330
                          Comp.26    Comp.27    Comp.28    Comp.29    Comp.30
Standard deviation     0.74969868 0.74137885 0.72343693 0.71457305 0.70358645
Proportion of Variance 0.01277382 0.01249188 0.01189457 0.01160488 0.01125077
Cumulative Proportion  0.84900712 0.86149899 0.87339356 0.88499844 0.89624921
                          Comp.31     Comp.32     Comp.33    Comp.34
Standard deviation     0.69022738 0.654861232 0.640339974 0.63179848
Proportion of Variance 0.01082759 0.009746437 0.009318984 0.00907203
Cumulative Proportion  0.90707680 0.916823235 0.926142219 0.93521425
                           Comp.35     Comp.36     Comp.37     Comp.38
Standard deviation     0.616621295 0.602404917 0.570025368 0.560881809
Proportion of Variance 0.008641405 0.008247538 0.007384748 0.007149736
Cumulative Proportion  0.943855654 0.952103192 0.959487940 0.966637677
                           Comp.39     Comp.40     Comp.41     Comp.42
Standard deviation     0.538149460 0.530277613 0.512370708 0.505662309
Proportion of Variance 0.006581928 0.006390781 0.005966449 0.005811236
Cumulative Proportion  0.973219605 0.979610386 0.985576834 0.991388070
                           Comp.43     Comp.44
Standard deviation     0.480413465 0.384873772
Proportion of Variance 0.005245389 0.003366541
Cumulative Proportion  0.996633459 1.000000000

Comments

• Want overall fraction of variance explained (``cumulative proportion’’) to be reasonably high.

• 2 factors, 28.5%. Terrible!

• Even 56% (10 factors) not that good!

• Have to live with that.

Biplot

ggbiplot(bem.pc, alpha = 0.3)

Comments

• Ignore individuals for now.

• Most variables point to 1 o’clock or 4 o’clock.

• Suggests factor analysis with rotation will get interpretable factors (rotate to 12 o’clock and 3 o’clock, for example).

• Try for 2-factor solution (rough interpretation, will be bad):

bem %>%
select(-subno) %>%
factanal(factors = 2) -> bem.2

• Show output in pieces (just print bem.2 to see all of it).

Uniquenesses, sorted

sort(bem.2$uniquenesses)

 leaderab   leadact      warm    tender  dominant    gentle 
0.4091894 0.4166153 0.4764762 0.4928919 0.4942909 0.5064551 
 forceful   strpers   compass     stand  undstand    assert 
0.5631857 0.5679398 0.5937073 0.6024001 0.6194392 0.6329347 
   soothe    affect    decide  selfsuff  sympathy     indpt 
0.6596103 0.6616625 0.6938578 0.7210246 0.7231450 0.7282742 
  helpful    defbel      risk   reliant   individ   compete 
0.7598223 0.7748448 0.7789761 0.7808058 0.7941998 0.7942910 
 conscien     happy  sensitiv     loyal  ambitiou       shy 
0.7974820 0.8008966 0.8018851 0.8035264 0.8101599 0.8239496 
 softspok  cheerful  masculin  yielding  feminine  truthful 
0.8339058 0.8394916 0.8453368 0.8688473 0.8829927 0.8889983 
  lovchil    analyt    athlet   flatter  gullible     moody 
0.8924392 0.8968744 0.9229702 0.9409500 0.9583435 0.9730607 
 childlik  foullang 
0.9800360 0.9821662 

Comments

• Mostly high or very high (bad).

• Some smaller, eg.: Leadership ability (0.409), Acts like leader (0.417), Warm (0.476), Tender (0.493).

• Smaller uniquenesses captured by one of our two factors.

• Larger uniquenesses are not: need more factors to capture them.

Factor loadings, some

bem.2$loadings

Loadings:
         Factor1 Factor2
helpful   0.314   0.376 
reliant   0.453   0.117 
defbel    0.434   0.193 
yielding -0.131   0.338 
cheerful  0.152   0.371 
indpt     0.521         
athlet    0.267         
shy      -0.414         
assert    0.605         
strpers   0.657         
forceful  0.649  -0.126 
affect    0.178   0.554 
flatter           0.223 
loyal     0.151   0.417 
analyt    0.295   0.127 
feminine  0.113   0.323 
sympathy          0.526 
moody            -0.162 
sensitiv  0.135   0.424 
undstand          0.610 
compass   0.114   0.627 
leaderab  0.765         
soothe            0.580 
risk      0.442   0.161 
decide    0.542   0.113 
selfsuff  0.511   0.134 
conscien  0.328   0.308 
dominant  0.668  -0.245 
masculin  0.276  -0.280 
stand     0.607   0.172 
happy     0.119   0.430 
softspok -0.230   0.336 
warm              0.719 
truthful  0.109   0.315 
tender            0.710 
gullible -0.153   0.135 
leadact   0.763         
childlik -0.101         
individ   0.445         
foullang          0.133 
lovchil           0.327 
compete   0.450         
ambitiou  0.414   0.137 
gentle            0.702 

               Factor1 Factor2
SS loadings      6.083   5.127
Proportion Var   0.138   0.117
Cumulative Var   0.138   0.255

Making a data frame

There are too many to read easily, so make a data frame. A bit tricky:

bem.2$loadings %>% 
unclass() %>% 
as_tibble() %>% 
mutate(trait = rownames(bem.2$loadings)) -> loadings
loadings %>% slice(1:8)

Pick out the big ones on factor 1

Arbitrarily defining \(>0.4\) or \(<-0.4\) as “big”:

loadings %>% filter(abs(Factor1) > 0.4) 

Factor 2, the big ones

loadings %>% filter(abs(Factor2) > 0.4)

Plotting the two factors

• A bi-plot, this time with the variables reduced in size. Looking for unusual individuals.

• Have to run factanal again to get factor scores for plotting.

bem %>% select(-subno) %>% 
factanal(factors = 2, scores = "r") -> bem.2a
biplot(bem.2a$scores, bem.2a$loadings, cex = c(0.5, 0.5))

• Numbers on plot are row numbers of bem data frame.

The (awful) biplot

Comments

• Variables mostly up (“feminine”) and right (“masculine”), accomplished by rotation.

• Some unusual individuals: 311, 214 (low on factor 2), 366 (high on factor 2), 359, 258 (low on factor 1), 230 (high on factor 1).

Individual 366

bem %>% slice(366) %>% glimpse()

Rows: 1
Columns: 45
$ subno    <dbl> 755
$ helpful  <dbl> 7
$ reliant  <dbl> 7
$ defbel   <dbl> 5
$ yielding <dbl> 7
$ cheerful <dbl> 7
$ indpt    <dbl> 7
$ athlet   <dbl> 7
$ shy      <dbl> 2
$ assert   <dbl> 1
$ strpers  <dbl> 3
$ forceful <dbl> 1
$ affect   <dbl> 7
$ flatter  <dbl> 9
$ loyal    <dbl> 7
$ analyt   <dbl> 7
$ feminine <dbl> 7
$ sympathy <dbl> 7
$ moody    <dbl> 1
$ sensitiv <dbl> 7
$ undstand <dbl> 7
$ compass  <dbl> 6
$ leaderab <dbl> 3
$ soothe   <dbl> 7
$ risk     <dbl> 7
$ decide   <dbl> 7
$ selfsuff <dbl> 7
$ conscien <dbl> 7
$ dominant <dbl> 1
$ masculin <dbl> 1
$ stand    <dbl> 7
$ happy    <dbl> 7
$ softspok <dbl> 7
$ warm     <dbl> 7
$ truthful <dbl> 7
$ tender   <dbl> 7
$ gullible <dbl> 1
$ leadact  <dbl> 2
$ childlik <dbl> 1
$ individ  <dbl> 5
$ foullang <dbl> 7
$ lovchil  <dbl> 7
$ compete  <dbl> 7
$ ambitiou <dbl> 7
$ gentle   <dbl> 7

Comments

• Individual 366 high on factor 2, but hard to see which traits should have high scores (unless we remember).

• Idea 1: use percentile ranks as before.

• Idea 2: Rating scale is easy to interpret. So tidy original data frame to make easier to look things up.

Tidying original data

bem %>%
ungroup() %>% 
mutate(row = row_number()) %>%
pivot_longer(c(-subno, -row), names_to="trait", 
             values_to="score") -> bem_tidy
bem_tidy

Recall data frame of loadings

loadings %>% slice(1:10)

Want to add the factor scores for each trait to our tidy data frame bem_tidy. This is a left-join (over), matching on the column trait that is in both data frames (thus, the default):

Looking up loadings

bem_tidy %>% left_join(loadings) -> bem_tidy
bem_tidy %>% sample_n(12)

Individual 366, high on Factor 2

So now pick out the rows of the tidy data frame that belong to individual 366 (row=366) and for which the Factor2 score exceeds 0.4 in absolute value (our “big” from before):

bem_tidy %>% filter(row == 366, abs(Factor2) > 0.4)

As expected, high scorer on these.

Several individuals

Rows 311 and 214 were low on Factor 2, so their scores should be low. Can we do them all at once?

bem_tidy %>% filter(
row %in% c(366, 311, 214),
abs(Factor2) > 0.4
)

Can we display each individual in own column?

Individual by column

Un-tidy, that is, pivot_wider:

bem_tidy %>%
filter(
  row %in% c(366, 311, 214),
  abs(Factor2) > 0.4
) %>%
select(-subno, -Factor1, -Factor2) %>%
pivot_wider(names_from=row, values_from=score)

366 high, 311 middling, 214 (sometimes) low.

Individuals 230, 258, 359

These were high, low, low on factor 1. Adapt code:

bem_tidy %>%
filter(row %in% c(359, 258, 230), abs(Factor1) > 0.4) %>%
select(-subno, -Factor1, -Factor2) %>%
pivot_wider(names_from=row, values_from=score)

Is 2 factors enough?

Suspect not:

bem.2$PVAL

    objective 
1.458183e-150 

2 factors resoundingly rejected. Need more. Have to go all the way to 15 factors to not reject:

bem %>%
select(-subno) %>%
factanal(factors = 15) -> bem.15
bem.15$PVAL

objective 
 0.132617 

Even then, only just over 50% of variability explained.

What’s important in 15 factors?

• Let’s take a look at the important things in those 15 factors.

• Get 15-factor loadings into a data frame, as before:

bem.15$loadings %>% 
unclass() %>% 
as_tibble() %>% 
mutate(trait = rownames(bem.15$loadings)) -> loadings

• then show the highest few loadings on each factor.

Factor 1 (of 15)

loadings %>%
arrange(desc(abs(Factor1))) %>%
select(Factor1, trait) %>%
slice(1:10)

Compassionate, understanding, sympathetic, soothing: thoughtful of others.

Factor 2

loadings %>%
arrange(desc(abs(Factor2))) %>%
select(Factor2, trait) %>%
slice(1:10)

Strong personality, forceful, assertive, dominant: getting ahead.

Factor 3

loadings %>%
arrange(desc(abs(Factor3))) %>%
select(Factor3, trait) %>%
slice(1:10)

Self-reliant, self-sufficient, independent: going it alone.

Factor 4

loadings %>%
arrange(desc(abs(Factor4))) %>%
select(Factor4, trait) %>%
slice(1:10)

Gentle, tender, warm (affectionate): caring for others.

Factor 5

loadings %>%
arrange(desc(abs(Factor5))) %>%
select(Factor5, trait) %>%
slice(1:10)

Ambitious, competitive (with a bit of risk-taking and individualism): Being the best.

Factor 6

loadings %>%
arrange(desc(abs(Factor6))) %>%
select(Factor6, trait) %>%
slice(1:10)

Acts like a leader, leadership ability (with a bit of Dominant): Taking charge.

Factor 7

loadings %>%
arrange(desc(abs(Factor7))) %>%
select(Factor7, trait) %>%
slice(1:10)

Happy and cheerful.

Factor 8

loadings %>%
arrange(desc(abs(Factor8))) %>%
select(Factor8, trait) %>%
slice(1:10)

Affectionate, flattering: Making others feel good.

Factor 9

loadings %>%
arrange(desc(abs(Factor9))) %>%
select(Factor9, trait) %>%
slice(1:10)

Taking a stand.

Factor 10

loadings %>%
arrange(desc(abs(Factor10))) %>%
select(Factor10, trait) %>%
slice(1:10)

Feminine. (A little bit of not-masculine!)

Factor 11

loadings %>%
arrange(desc(abs(Factor11))) %>%
select(Factor11, trait) %>%
slice(1:10)

Loyal.

Factor 12

loadings %>%
arrange(desc(abs(Factor12))) %>%
select(Factor12, trait) %>%
slice(1:10)

Childlike. (With a bit of moody, shy, not-self-sufficient, not-conscientious.)

Factor 13

loadings %>%
arrange(desc(abs(Factor13))) %>%
select(Factor13, trait) %>%
slice(1:10)

Truthful. (With a bit of happy and not-gullible.)

Factor 14

loadings %>%
arrange(desc(abs(Factor14))) %>%
select(Factor14, trait) %>%
slice(1:10)

Decisive. (With a bit of self-sufficient and not-soft-spoken.)

Factor 15

loadings %>%
arrange(desc(abs(Factor15))) %>%
select(Factor15, trait) %>%
slice(1:10)

Not-compassionate, athletic, sensitive: A mixed bag. (“Cares about self”?)

Anything left out? Uniquenesses

enframe(bem.15$uniquenesses, name="quality", value="uniq") %>%
  slice_max(uniq, n = 10)

Uses foul language especially, also loves children and analytical. So could use even more factors.