Sometime ago Byrne, Shavelson and Muthen (1989) recommended examining the impact of freeing correlated errors by correlating the factor pattern coefficients and factor correlations between the baseline model with one where the measurement error has been freed. A correlation coefficient less than 1.00 (say < .90) would be an indication that the major parameters are being affected by the correlated error and this therefore would provide a strong basis (amongst other considerations) to free the correlated error in preference to constraining this to zero.
My question is this:
should it matter if I correlate the standardised or unstandardized parameters with one another when using StdYX standardisation in an MPLUS CFA. In my analysis for example, I notice no change is my measurement parameters when correlating standardized values between the baseline and respecified models but I do notice an appreciable change in the unstandardized estimates when I compare these between my baseline model and my respecified model? Should I be comparing standardized or unstandardized measurement parameters to inform my decision to free my measurement errors in the manner described by Byrne, Shavelson & Muthen (1989).