Chapter3 Exploratory Factor Analysis

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3.1 Eigenvalues

# correlation analysis without mvliking
cor<- cor(vars[,-1])

EV = eigen(cor)$values
EV_df = EV %>%
  # convert to df
  as.data.frame()%>%
  # name column 
  dplyr::rename(., "Eigen Value" =".") %>%
  # add a column "Factor"
  dplyr::mutate(Factor = seq(1,length(EV))) %>%
  # order columns
  dplyr::select(Factor, everything()) %>%
  # add a column "Variance"
  dplyr::mutate(`Variance %` = EV/length(EV) *100) %>%
  dplyr::mutate(`Cumulative Variance % ` = cumsum(EV/length(EV))* 100) %>%
  dplyr::mutate(., across(where(is.numeric), round, 1))

rmarkdown::paged_table(EV_df )

3.1.1 Scree Plot

psych::scree(cor, pc = TRUE, factors = FALSE,)

3.1.2 Cumulative Percentages of Variance

# Shares for the cumulative variance explained
plot(cumsum(EV/length(EV)), 
     type = "o", # type of plot: "o" for points and lines 'overplotted'
     col = "darkblue",
     pch = 16, # plot symbol: 16 = filled circle
     cex = 1, # size of plot symbols
     xlab = "Total number of factors", # a title for the x axis
     ylab = "Cumulative variance explained", # a title for the y axis 
    main= "Cumulative Variance",  
     lwd = 2) # line width
abline(v = 5, lwd = 2, col = "grey") # draw a vertical line at v = 3

3.1.3 Select Number of Factors using Kaiser Rule

nFactor <- length(which(EV > 1)) 
print(paste0(nFactor, " factors with Eigenvalue > 1" ))

## [1] "5 factors with Eigenvalue > 1"

3.2 Extract Principal Factors

EFA <- psych::fa(
  r = cor,
  # number of factors
  nfactors = nFactor,
  # principal factor
  fm = "pa",
  # maximizes the sum of the variance of the squared loadings
  rotate = "varimax"
)

3.3 Factor Loadings

columnOrder <- c("PA1", "PA2", "PA3", "PA4","PA5","Communality","Uniqueness")

EFA_loadings <- data.frame(EFA$loadings[,]
                           ) %>%
  dplyr::mutate(Uniqueness = EFA$uniquenesses,
                Communality = EFA$communality) %>%
  dplyr::mutate(., across(where(is.numeric), round, 2)) %>%
  dplyr::select(columnOrder) %>%
  arrange(., -Communality) 

EFA_loadings_DT = EFA_loadings  %>%
    tibble::rownames_to_column(., var="variable")

# PA cols
PACols = c("PA1", "PA2", "PA3", "PA4","PA5")

# color
colfunc <- colorRampPalette(c("darkblue", "lightgrey"))

# DT table 
DT::datatable(EFA_loadings_DT , 
              options = list(pageLength = 30)) %>%
  formatStyle("PA1",
              backgroundColor = styleEqual(sort(unique(EFA_loadings_DT $PA1),
                                                decreasing = TRUE),
                                           colfunc(length(
                                             unique(EFA_loadings_DT $PA1)
                                           )))) %>%
  formatStyle("PA2",
              backgroundColor = styleEqual(sort(unique(EFA_loadings_DT $PA2),
                                                decreasing = TRUE),
                                           colfunc(length(
                                             unique(EFA_loadings_DT $PA2)
                                           )))) %>%
  formatStyle("PA3",
              backgroundColor = styleEqual(sort(unique(EFA_loadings_DT $PA3),
                                                decreasing = TRUE),
                                           colfunc(length(
                                             unique(EFA_loadings_DT $PA3)
                                           )))) %>%
  formatStyle("PA4",
              backgroundColor = styleEqual(sort(unique(EFA_loadings_DT $PA4),
                                                decreasing = TRUE),
                                           colfunc(length(
                                             unique(EFA_loadings_DT $PA4)
                                           )))) %>%
  formatStyle("PA5",
              backgroundColor = styleEqual(sort(unique(EFA_loadings_DT $PA5),
                                                decreasing = TRUE),
                                           colfunc(length(
                                             unique(EFA_loadings_DT $PA5)
                                           )))) %>%
  formatStyle(PACols, color = "white")

3.4 Factor Scores

# extract rotated factor scores 
EFA.scores = factor.scores(vars[,-1], 
                           unclass(EFA$loadings))$scores 


DT_cols  = c("PA1","PA2","PA3","PA4","PA5")
subjectValue =microvan$subjnumb

EFA.scores %>%
  as.data.frame() %>%
  mutate(subjnumb = subjectValue )%>%
  select(subjnumb, everything())%>%
  head(10) %>%
  DT::datatable(caption = "Top 10 subjects",rownames = F)  %>%
  DT::formatRound(columns=DT_cols , digits=2)