I'm currently running a Confirmatory Factor Analysis using 4 different measures of different aspects of cognition. (Using IBM AMOS).
The model doesn't fit well unless two of the residual variances (error variances) are co-varied. Once co-varied fit indices improve, however individual parameter estimates remain quite stable and significant regardless.
The issue i'm having is that the covariance is negative (-.312). Which traditionally goes against statistical convention with CFA. However, it seems logical that this would occur given the nature of the research design. All the residuals are quite high on all indicators because they measure different aspects of cognition. However, the latent factor is capturing the shared similarity amongst the indicators (this is the aim of the research, and theoretically holds). Thus, to me at least, it makes sense that two of the tasks (one is a good measure of 'planning', another 'inhibition') would share an inverse relationship in their residuals because that is reflecting the unique aspects of those tasks that the model is not accounting for.
I'd appreciate any comments on this logic, and whether it appears flawed?