We conducted an exploratory factor analysis using the psych package with oblique rotation and found an acceptable solution with 3 factors. Now a reviewer ask me to provide the proportion of variance explained by each of these factors. Having seen other posts on this issue (What's the relationship between initial eigenvalues and sums of squared loadings in factor analysis? and Interpreting discrepancies between R and SPSS with exploratory factor analysis), I still wonder what I should provide. Did the (anonymous) reviewer mean the Eigenvalue-based proportion of variance of the principal component (method 1), although he was speaking of "each of these factors"?
library(psych) library(GPArotation) library(data.table) #load sample data from the internet for demonstration myDT <- fread('https://raw.github.com/vincentarelbundock/Rdatasets/master/csv/psych/msq.csv') myDT <- myDT[,c(7:15)] #choosing selection of variables to simpify things efa <- fa(myDT,nfactors = 3, rotate = "oblimin", fm="minres") #efa with oblique rotation propEV <- 100*efa$e.values[1:3]/length(efa$e.values) # method 1 round(propEV,2)  27.56 18.20 14.31 round(sum(propEV),2) #total  60.08
Or is it more relevant to calculate the proportion each factor explains? And which method should I choose to calculate this? The calculation based on SS-loadings in the psych package (method 2) seems to match SPSS' "Extraction Sums of Squared Loadings" (cf. this post) for unrotated factors .
propSS <- efa$Vaccounted # method 2 round(propSS,2) MR1 MR2 MR3 SS loadings 1.66 1.22 0.84 Proportion Var 0.18 0.14 0.09 Cumulative Var 0.18 0.32 0.41 Proportion Explained 0.45 0.33 0.23 Cumulative Proportion 0.45 0.77 1.00
Ziberna proposes (in his response here) to calculate mean communality for the total % of variance explained method 3, which produces similar results like method 2.
mean(efa$communalities) # method 3  0.4127141
Lorenzo-Seva (2013) states here "The reduced correlation matrix computed in most factor analysis methods is systematically non-positive definite. The typical conclusion is that the percentage of explained common variance cannot be computed in EFA." and therefore proposes to use Minimum Rank Factor Analysis instead. When I conduct this with the psych package, it seem to differ slightly from the EFA above when based on SS-loadings (method 4) and again, when based on communalities (method 5).
efa2 <- fa(myDT,nfactors = 3, rotate = "oblimin", fm="minrank") #minimum rank fa with oblique rotation propSS <- efa2$Vaccounted # method 4 round(propSS,2) MRFA1 MRFA2 MRFA3 SS loadings 1.66 1.27 0.89 Proportion Var 0.18 0.14 0.10 Cumulative Var 0.18 0.33 0.42 Proportion Explained 0.43 0.33 0.23 Cumulative Proportion 0.43 0.77 1.00 mean(efa2$communalities) # method 5  0.45381
However, the proportion of all methods 1-5 seems not to be based on common variance, all seem to have the denominator 9 (= # of items). But how does this match with the idea of factors reflecting common variance?
So the question remains what is usually reported in papers in terms of explained variance after oblique efa and how is it implemented in R?