PCA/factor analysis of mixed (quantitative + qualitative) data: inconsistent results

I have a dataset composed of 4 variables, 2 being numerical and 2 categorical (ordinal in fact). They all represent 4 types of indicators/measures of the same phenomenon . I want to analyse them in a multivariate way. I tried to apply first a PCA on the 4 variables (forcing the ordinal into numerical which is sometimes suggested), i get this graph:

then i tried to do a FAMD (factor analysis of mixed data) which was recommended with the factominer package.Unfortunately there is not a lot of documentation about it. This is the output:

And the other output (observations + levels):

my problem: the FAMD variable graph gives completely different results. It seems from the data (and the PCA) that quanti2 and quali2 should be closely related, but that's not what shows the famd's variable plot. Why so?

Moreover, on the second FAMD graph (observations), i get this "V" shape. How can i interpret it and draw conclusions about the relationship between this 4 indicators?

And of course, if you have a more clever way to analyse this dataset, please explain it!

I dput my data here:

    data <- structure(list(quanti1 = c(0.57, 0.56, 0.46, 0.63, 0.71, 0.66,
0.48, 0.39, 0.57, 0.78, 0.67, 0.63, 0.55, 0.62, 0.66, 0.5, 0.5,
0.41, 0.5, 0.46, 0.53, 0.59, 0.58, 0.66, 0.62, 0.65, 0.58, 0.62,
0.66, 0.67, 0.66, 0.59, 0.41, 0.57, 0.6, 0.42, 0.48, 0.44, 0.47
), quanti2 = c(3.01, 2.71, 2.51, 5.26, 5.36, 2.66, 3.01, 5.31,
4.71, 5.76, 7.01, 5.96, 4.01, 2.86, 5.26, 3.26, 4.51, 3.41, 2.61,
3.66, 3.01, 3.76, 4.26, 4.01, 4.76, 4.66, 2.76, 3.96, 5.01, 6.16,
7.86, 5.96, 2.51, 3.21, 5.51, 4.41, 4.01, 2.21, 2.51), quali1 = structure(c(3L,
2L, 1L, 4L, 4L, 3L, 2L, 3L, 3L, 4L, 4L, 4L, 3L, 2L, 4L, 3L, 3L,
2L, 1L, 3L, 2L, 4L, 3L, 4L, 4L, 4L, 3L, 4L, 4L, 4L, 4L, 4L, 1L,
3L, 3L, 3L, 3L, 1L, 1L), .Label = c("1", "2", "3", "4"), class = "factor"),
quali2 = structure(c(4L, 3L, 3L, 7L, 4L, 4L, 2L, 4L, 5L,
7L, 7L, 6L, 6L, 4L, 7L, 4L, 5L, 3L, 2L, 5L, 4L, 5L, 5L, 5L,
7L, 6L, 2L, 5L, 5L, 7L, 7L, 7L, 2L, 5L, 6L, 4L, 4L, 1L, 5L
), .Label = c("1", "2", "3", "4", "5", "6", "7"), class = "factor")), .Names = c("quanti1",
"quanti2", "quali1", "quali2"), row.names = c(NA, -39L), class = "data.frame")
library(FactoMineR); library(dplyr)
lapply(data, as.numeric) %>% as.data.frame %>% PCA
FAMD(data)

• I'm not R user so I'm not quite sure, but it seems that what FactorMineR is doing are Optimal scaling methods such as categorical PCA (CATPCA). At least, playing a bit with your data in SPSS I could reproduce quite similar results as yours, especially that V-shaped plot of observations. But it was so only when the quantification level was forced to be multiple nominal for all the variables or at least for the two categorical ones. – ttnphns May 28 '15 at 17:18
• (cont.) That means that the analysis totally or partly evolved into Multiple Correspondence analysis. That was your mistake. Because your two "quali" variables are ordinal you should request ordinal or spline ordinal level of quantification for them. If you do it right you will get the results which are enough similar to what you got with plain PCA (albeit generally they have not to be quite similar). – ttnphns May 28 '15 at 17:18

It is a very small sample for factor analytic procedures, but besides that I used the data to run a unidimensional factor model with lavaan. Here is the code

# load package
library(lavaan)

# lavaan requires the factors to be ordered
data$quali1 <- factor(data$quali1,ordered=T)
data$quali2 <- factor(data$quali2,ordered=T)

# fit a unidimensional model
mod1 <- 'f =~ quanti1 + quanti2 + quali1 + quali2'
# estimate model parameters
mod1.cfa <- cfa(mod=mod1, data=data, ordered=3:4, std.lv=T)
# assess model fit
fitMeasures(mod1.cfa)
# show results
summary(mod1.cfa,stan=T)


The results are clearly in favour of a unidimensional model!

But recommendations for sample size for factor analysis from statistics textbooks normally suggest at least n = 50 subjects and only if a clear factor structure is given and the model is not too complex!

• thanks for the model suggestion. I'm not familiar with CFA. it takes any kind of variable (ordered or quanti?). I read that one should do a exploratory analysis first to find out the structure of the factors: how would you do that? Can you compare different structures with CFA? – agenis May 30 '15 at 23:42