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I have done an EFA (OLS, Oblique rotation, polychoric correlations) and CFA (DWLS, allowed factors to covary) on my data, a two factor model had the best fit in CFA, a one factor model had acceptable fit. Previous studies have identified 1 or 2 factors, theoretically it could be either. The correlation between the two factors in EFA was 0.69 The covariance between the two factors (latent variables) in CFA was 0.72 My question: Can anyone suggest a source that states what degree (number) of covariance would warrant identifying that the latent variables where highly related and should be one latent variable? Wiki said correlations above 0.70 in EFA warrant merging, another text I read said correlations above 0.85. I haven't been able to find any reference that provides a cuttof or range for covariance in CFA. Many thanks Addit: I have just realised the z score and significance seem to matter for the covariance between latent variables, these were as follows for my two factor model in CFA (conducted in r lavaan package) Estimate Std.err Z-value P(>|z|) 0.722 0.053 13.648 0.000

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  • $\begingroup$ Thank you kjetil b halvorsen, it is a great site! Is there any way to tag people that have answered questions on similar topics so that they see this question? Thank you $\endgroup$ – Leya Oct 9 '15 at 14:51
  • $\begingroup$ You can try to write a comment with @ prepending the name of the one you want to connect. $\endgroup$ – kjetil b halvorsen Oct 9 '15 at 15:08
  • $\begingroup$ @jeremy miles are you able to help with my question? Many thanks... (not sure if tag is working)... $\endgroup$ – Leya Oct 9 '15 at 15:28
  • $\begingroup$ Tag didn't work (I'm not sure I understand them) but I saw it anyway. :) $\endgroup$ – Jeremy Miles Oct 9 '15 at 16:53
  • $\begingroup$ I guess you cannot tag people who have not been active in the thread ... $\endgroup$ – kjetil b halvorsen Mar 16 at 11:11
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I think the answer is more conceptual than statistical.

You can compare the one and two factor models, and see which one fits better using CFA (note: it's the sort of thing that looks like it will work for a chi-square difference test, but it's more complex, see this paper: http://psycnet.apa.org/journals/met/13/2/150/ ).

However, the fact that you can statistically distinguish between the two factors doesn't mean that distinguishing will make any practical difference. I guess that the 0.7 cutoff is a rule of thumb that gives a guess about whether it is likely to matter. What you would really like to show is that splitting into two factors either does (or does not) alter the ability of the scale to predict (or be predicted by) some other construct. For example, you have a measure of verbal and a measure of numeric intelligence - they are highly correlated, so they could be called one factor. If is the case that one of them is more predictive of something than the other (e.g. success in a physics course, or a history course) then you have a good argument for keeping them separate. If they are not differently predictive, then you don't have evidence that it's worth separating them, and the more parsimonious solution is to keep them together - pending, of course, further evidence that they might be separate. But your theory might suggest that two factors is better (yours doesn't, as you've said).

An example is depression and anxiety - it's hard to statistically using factor analysis. But the theoretical reasons for keeping them separate are strong, and so they are usually kept separate (although when I have a scale, I've sometimes kept them together, because one fewer predictor can make your life easier.)

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  • $\begingroup$ Many thanks Jeremy, your answer was very helpful, particularly "if it is the case that one of them is more predictive of something..".. I hadn't considered this when thinking about what aspects of theory could justify decisions in CFA. $\endgroup$ – Leya Oct 10 '15 at 3:34
  • $\begingroup$ An add on Q : I know covariance and correlation are different, however does the number 0.00 - 1.00 of covariance between latent variables signify the same amount of similarity as the correlation between factors does? Secondly is the number of the z score for the covariance of import? Thirdly, if anyone can point me in the direction of a citable reference which states that covariance > 0.7 between factors is high and indicates similarity please...I can't cite wiki and am not experienced enough in stats to say it without a citation... Thanks again for your help Jeremy $\endgroup$ – Leya Oct 10 '15 at 3:42
  • $\begingroup$ If you have scaled your latent variables as having a variance of 1, then the covariance is the correlation. If you haven't, the maximum covariance could be any number at all. Second, it tells you whether the parameter is significant, and third, I've never heard that. $\endgroup$ – Jeremy Miles Oct 10 '15 at 3:43
  • $\begingroup$ Thank you. I'm not sure if latent variables were scaled. I ran fitModelTtwo <- cfa ( ModelTtwo, data= New, orthogonal = FALSE, estimator = "DWLS"). In the help section for the cfa command it says: std.lv : If TRUE, the metric of each latent variable is determined by fixing their variances to 1.0. If FALSE, the metric of each latent variable is determined by fixing the factor loading of the first indicator to 1.0. In the full command std.lv is set to FALSE so this may be the default? Not sure if estimater chosen (ie DWLS) effects this $\endgroup$ – Leya Oct 10 '15 at 4:38

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