# Factor correlations larger than 1

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?

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);

• Have a look at yang's post here – user20650 Nov 5 '14 at 13:33
• Do not create nor test factors with only 2 items. – ttnphns Nov 5 '14 at 14:27
• @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. – Erik Nov 5 '14 at 16:12
• @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? – Erik Nov 5 '14 at 16:23
• 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. – ttnphns Nov 5 '14 at 17:21