# PCA and component scores based on a mix of continuous, binary and categorical variables

My question is strongly related to this one: PCA and component scores based on a mix of continuous and binary variables. I will basically use the same code, but add a new nominal feature (x6) to the data set.

I want to apply a PCA on a dataset consisting of continuous, binary and categorical variables.

# Generate synthetic dataset
set.seed(12345)
n <- 100
x1 <- rnorm(n)
x2 <- runif(n, -2, 2)
x3 <- x1 + x2 + rnorm(n)
x4 <- rbinom(n, 1, 0.5)
x5 <- rbinom(n, 1, 0.6)
x6 <- c(rep('A', 25), rep('B', 25), rep('C', 25), rep('D', 25))
data <- data.frame(x1, x2, x3, x4, x5, x6)

# Correlation matrix with appropriate coefficients
# Pearson product-moment: 2 continuous variables
# Point-biserial: 1 continuous and 1 binary variable
# Phi: 2 binary variables
# For testing purposes use hetcor function
library(polycor)
C <- as.matrix(hetcor(data=data))

# Run PCA
pca <- princomp(covmat=C)


Now in order to calculate the component scores, it was suggested to multiply the data set with the loadings L, which works fine for numerical and binary variables, but not on categorical data. The following computation causes the categorical feature to be a vector of NA´s.

scores <- data * L


How can I obtain the scores for this feature? Do I have to split it up into dummy variables to make this work?

• If you are satisfied with the continuous and binary variables, why not convert your categorical variables to binary variables? – Deathkill14 Jun 3 '14 at 15:14
• Without following your code (I'm not R user), let me put some statements though. 1) Point-biserial r = Pearson r = Phi; so, you may safely combine continuous+binary+dichotomous_nominal variables in a single classic (linear) PCA. 2) Polytomous categorical variables - you shouldn't use linear PCA with them not pre-processed. The best option would be to use categorical PCA (CatPCA, see). – ttnphns Jun 3 '14 at 15:16
• P.S. You may convert nominal variables into dummy binary (as @Deathkill14 suggested), but that will make your analysis closer to Multiple Correspondence analysis away from PCA proper. I don't recomment that way unless you get special reasons. – ttnphns Jun 3 '14 at 15:21
• Computing dummy variables works, but I was trying to avoid that since it makes the PCA result even harder to understand. Thanks ttnphns for the hint with the CatPCA, I will look into that. I hoped there is a way to calculate a PCA for all these mixed data types combined. – Paul Jun 3 '14 at 15:45

As Deathkill14 and ttnphns pointed out, it is possible to split the categorical data into binary dummy variables. A solution could look like this (using this neat code snippet):

# Generate synthetic dataset
set.seed(12345)
n <- 100
x1 <- rnorm(n)
x2 <- runif(n, -2, 2)
x3 <- x1 + x2 + rnorm(n)
x4 <- rbinom(n, 1, 0.5)
x5 <- rbinom(n, 1, 0.6)
x6 <- c(rep('A', 25), rep('B', 25), rep('C', 25), rep('D', 25))

data <- data.frame(x1, x2, x3, x4, x5)

# Create dummy variables from x6
for(level in unique(x6)){
data[paste("dummy", level, sep = "_")] <- ifelse(x6 == level, 1, 0)
}

# Correlation matrix with appropriate coefficients
# Pearson product-moment: 2 continuous variables
# Point-biserial: 1 continuous and 1 binary variable
# Phi: 2 binary variables
# For testing purposes use hetcor function
library(polycor)
C <- as.matrix(hetcor(data=data))

# Run PCA
pca <- princomp(covmat=C)

• A attempted editor argues that your final step, Calculate Scores, should be using matrix multiplication (ie, scores <- data %*% L). – gung - Reinstate Monica Apr 17 '18 at 12:06