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I'm doing a confirmatory factor analysis (CFA) in Mplus, testing a 5-factor PTSD model with responses (n=376) to a PTSD questionnaire (the Harvard Trauma Questionnaire). For some reason, the output includes an interfactor correlation of 1.6. How is this possible? The corresponding raw score correlation is 0.5. The involved factors have 2 and 3 variables, respectively. It seems plausible to me that correlations based on latent measures might overcompensate for measuring error (which might be estimated to be large, given that the small number of variables involved), resulting in a correlation estimate above 1. I haven't been able to find confirmation of this, but even if it's true, could it explain a correlation as high as 1.6?

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From input file:

DATA: FILE IS HTQdata.dat;
Variable: Names ARE pt_no agegrp3 gender country6 ethnicity htq_1_1-htq_1_16 
hscl1-hscl25 sdsf1_1 sdsf1_2 sdsf1_3;

Usevariables are htq_1_1-htq_1_16;
Missing are all (-999);

Model: Intrus by htq_1_1 htq_1_2 htq_1_3 htq_1_16;
Avoidan by htq_1_11 htq_1_15;
Numbing by htq_1_4 htq_1_5 htq_1_12 htq_1_13 htq_1_14;
DysArou by htq_1_7 htq_1_8 htq_1_10;
AnxArou by htq_1_6 htq_1_9;

Output: sampstat; stand; tech4; Mod(1);
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  • $\begingroup$ Have a look at yang's post here $\endgroup$ – user20650 Nov 5 '14 at 13:33
  • $\begingroup$ Do not create nor test factors with only 2 items. $\endgroup$ – ttnphns Nov 5 '14 at 14:27
  • $\begingroup$ @ttnphns: can you give a reference for this statement? If so, I might like to include it in my article (much appreciated). However, the most wideley tested and supported models of PTSD have at least one dimension consisting of just two variables, this includes the DSM-5 model. It would probably be perceived as arrogant or rogue to disregard most existing models based on this criteria alone. $\endgroup$ – Erik Nov 5 '14 at 16:12
  • $\begingroup$ @user20650: great reference, thanks. So correlations above 1 are well known and there's a pretty straght forward explanation. But are correlations know to go as high as 1.6, or would there be other possible explanations for this? $\endgroup$ – Erik Nov 5 '14 at 16:23
  • $\begingroup$ I cannot give an authoritative reference at this time. However, there is at least two considerations which are quite well known facts. First, theoretically FA assumes no or weak partial correlations (see my answer, for example, near the end) which logically implies that every factor driving exactly two variables should be dismissed. Second, SEM programs, as far as I know, can compute the statistics of fit and error correctly only if a factor is 3+ items rich. I'm not Mplus user but I think it should be in line with that. $\endgroup$ – ttnphns Nov 5 '14 at 17:21
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@ Erik

Actually, Mplus offers a way to restrict correaltions between latent factors. f1 WITH f2@1 should restrict the correlation between f1 and f2 to 1, if variances within both factors are fixed to 1 (by f1@1; f2@1). Otherwise, you restrict the covariance. See e.g. the question by Boliang Guo on http://www.statmodel.com/discussion/messages/9/380.html?1458238039.

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I recently spoke to a local professor in statistics, who gave an answer to the above question. I thought I'd post here for future reference. They way I initially understood Muthens comment on the Mplus forum, was that the model didn't fit in a meaningful way. However, the local professor made it clear to me, that the whole output is inadmissible. This effectively means I can't estimate whether the model fits or not. I was also told that neither Mplus nor Stata offers a way to limit these latent correlations to 1. Perhaps future editions of CFA software will feature such an option.

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    $\begingroup$ This is nice, but rather than just report that some authority said this, can you reprise their argument / reason why the "whole output is inadmissible"? $\endgroup$ – gung - Reinstate Monica Mar 25 '15 at 15:33
  • $\begingroup$ @gung: Good point - As far as I understand, the reason a latent factor correlation >1 is reported, is because it provides the best fit. Thus we can predict, that fit indices will be lowered, if we restrict the interfactor correlation. I reconsulted the link provided by user20650, and indeed Muthén did state, that the results are inadmissible. As to why the high correlation is estimated in the first place, I can only speculate. Perhaps it happens because of local dependency between an item on each of the two involved factors. $\endgroup$ – Erik Mar 26 '15 at 9:13
  • $\begingroup$ There doesn't need to be anything wrong with the model you're fitting to get a correlation > 1. That is, you can get a correlation above 1 when fitting the same model that generated the data--even when the generating model has a correlation < 1. (you can verify by simulating a model with a correlation of .99). $\endgroup$ – machow Jun 6 '16 at 22:01
  • $\begingroup$ @machow: The only way I can see this happening, is if there are high item residual correlations across the two factors. The assumption behind the model is that item residuals are uncorrelated, so they are restricted to zero. When the involved items are located in two different factors, the factor correlations will be inflated, as there are no restrictions on these. $\endgroup$ – Erik Sep 7 '16 at 7:23

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