I did a principal axis factor analysis with promax (oblique) rotation in R. Now I want to compute the explained variance for all factors of my 4-factor solution. I can extract the eigenvalues but I am not sure how they relate to the explained variance. Any suggestions?
Hi! I think the easiest way should be, to divide the eigenvalues by the number of variables.
So for example, if you used 20 variables and your first factor has eigenvalue lambda = 2, you would calculate explained variance with 2/20 = 0.10, which is 10% explained variance.
Hi Christof,
thanks for your answer. Do you have any reference for that? I am just wondering if that formula is right despite the fact that in a factor analysis all variables together do not explain 100 percent of the variance (unlike PCA). There is also the problem of correlated factors which should influence the explained variance per factor.
Hi Martin,
I didn't find a reference yet. But, if you did your principal axis factor analysis in R, you could use the fa()-function in psych-package. Here you get variance proportions as an Output (even vor PAF and Rotation). For (german) info on the usage of fa() you can read the paper Werner (2014) "Explorative Faktorenanalyse: Einführung und Analyse mit R".
And it looks like this package calculates by Eigenvalue/number of variables.
Hi Martin :)
I am not sure if I really understand your question. You can see the explained variance in the R output: for example, in my output it looks like this:
Factor1 Factor2 Factor3
SS loadings 3.098 2.816 1.495
Proportion Var 0.194 0.176 0.093
Cumulative Var 0.194 0.370 0.463
So you have sum of squared loadings as an absolute value, the proportion of variance and the cumulative variance (≠1 for PAF).
I hope it helps,
Cheers, Dorota
Hi Christof and Dorota :),
thanks for answers! I am using a slightly different function called factor.pa.ginv() from the HDMD package which can compute a factor analysis if there are more variables than observations. It just outputs the Eigenvalues, Loadings and Scores rather than any explained variance. But probably Eigenvalues/Number of variables is correct.
I hope that this helps:
The sum of all eigenvalues in a factor analysis is the total variance explained.
The eigenvalue of a factor divided by the sum of the eigenvalues is the proportion of variance explained by that factor.
For example, if the eigenvalues sum to 5, and the first factor has an eigenvalue of 1.5, then this factor explains 30% of the variance.
To Paul: what you are talking about is % of variance explained, while what the question was about is % of total variance of all the measured varaibles.
To Christoph and Dorota - the proportion of explained variance by factors compute by the print method of the psych package is wrong for oblique rotations.
To get the % of total variance explained by factor, you should compute the sum of squared structural loadings by factor and divide that by number of variables. However, you can not sum these up (in case of oblique rotations) to get the % of variance explained by all factors.
To get this, either compute the mean communality or the total % of variance explained by orthogonal factors (eg, using none or varimax rotations), which can be summed.
I really thank Aleš Žiberna as out of all the above answers his methodology gave me the correct answer, as I was working on Matlab and the available function in it does not provide eigen values for factor loadings matrix. SO his methodology was correct as it gave me correct and verified answer.
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