Factor analysis of separate groups

Question

Ex. 5.4 Everitt: Apply the factor analysis model separately to the life expectancies of men and women and compare the results

Here some code and output

library(data.table)
life <- fread('http://people.stat.sc.edu/Hitchcock/lifeex.txt')
life_M <- life[,c(1,2,3,4,5)]
life_W <- life[,c(1,6,7,8,9)]
factanal(x=life[,-1],factors =3) #for both groups

The data is about life expectancies for different countries by age and gender.

Output for men group

Call:
factanal(x = life_M[, -1], factors = 1)

Uniquenesses:
   m0   m25   m50   m75 
0.594 0.552 0.005 0.434 

Loadings:
    Factor1
m0  0.638  
m25 0.669  
m50 0.998  
m75 0.752  

               Factor1
SS loadings      2.415
Proportion Var   0.604

Test of the hypothesis that 1 factor is sufficient.
The chi square statistic is 14.45 on 2 degrees of freedom.
The p-value is 0.000728 

Output for women group

Call:
factanal(x = life_W[, -1], factors = 1)

Uniquenesses:
   w0   w25   w50   w75 
0.220 0.005 0.115 0.526 

Loadings:
    Factor1
w0  0.883  
w25 0.998  
w50 0.941  
w75 0.689  

               Factor1
SS loadings      3.134
Proportion Var   0.784

Test of the hypothesis that 1 factor is sufficient.
The chi square statistic is 52.15 on 2 degrees of freedom.
The p-value is 4.74e-12 

Some points that I noticed:

(1) I can't use more than 1 factor for 4 variables with factanal and don't know why.

(2) In both cases the chi-square test says that 1 factor is not enough, but I can't use more than 1 based in (1).

(3) For the men group one factor explains 60.4% of variance structure and in women group 78.4%. In this case what I think is that one factor is sufficient for women group but not so good for men.

(4) For both group together 3 factors explain 88.9% of variance, what makes me think: What is the advantage of split the groups?

(5) Should not the number of factors be chosen based on the ratio of explained variance and the interpretability of the factors?

Answer 1

1. As T.E.G.'s comment suggests, you can't extract more than one factor because beyond one factor, these model are not identified. With four observed indicators (D), you have D(D+1)/2 pieces of information in your variance/covariance matrix (so 10, total). Your one-factor model is identified as it has 2 degrees of freedom, because from those 10 pieces of information, you are estimating 8 things: 4 factor loadings and 4 residual variances (10-8 = 2 df). Move to a two-factor solution, and your estimation needs double, and exceed the amount of information you have.
2. I wouldn't live and die by the chi-square test because it tests a pretty unreasonable null hypothesis--that in the long run, your model-implied variance-covariance matrix will perfectly reproduce the observed variance-covariance matrix. With any meaningful amount of modelling parsimony and a half-decent sample size, this null will be regularly rejected for models that might otherwise be quite reasonable. There are other means of evaluating how many factors you need (e.g., parallel analysis, absolute or relative indexes of model fit), but with only four indicators, you are quite limited in your modelling choices.
3. (and 4). It seems like you have some evidence of a lack of measurement invariance between men and women (see Vandenberg & Lance, 2000, for an accessible review). Though you should test this possibility formally, this would mean that the underlying factor structures for these indicators is different for men and women. While your appraisal seems reasonable (the single-factor solution seems like it may be adequate for women), once again, your options are limited to test other possibilities with only four indicators.
4. See above.
5. Amount of variance explained is primarily a principal components metric, as in a common factor solution you are acknowledging that not all observed variance is meaningful true construct variance that you can account for. And while interpretability is certainly a meaningful criteria for assessing the adequacy of factor solutions, it is by no means the only one (and it's a relatively subjective one at that). Fabrigar & Wegner, 2011 have a nice accessible review of the various other empirical options (some I mentioned above). All are imperfect, so it's generally a good idea to apply different ones and select a model that multiple criteria converge on supporting.

References

Fabrigar, L. R., & Wegener, D. T. (2011). Exploratory factor analysis. Oxford University Press.

Vandenberg, R. J., & Lance, C. E. (2000). A review and synthesis of the measurement invariance literature: Suggestions, practices, and recommendations for organizational research. Organizational research methods, 3(1), 4-70.