It refers to the covariance between two variables when the two variables are exogenous. It refers to the residual covariance when the two variables are endogenous.
Michael posted on Thursday, September 29, 2005 - 4:16 am
Since a WITH statement for endogenous variables refers to their residual covariance: how can I get the covariance of two endogenous variables (e.g. in a cross-lagged panel design)?
bmuthen posted on Thursday, September 29, 2005 - 5:43 am
Michael posted on Thursday, September 29, 2005 - 6:03 am
Thanks for the quick answer! Is there also a possibility to get the standard error of a covariance of two endogenous variables?
bmuthen posted on Thursday, September 29, 2005 - 6:54 am
That is not straighforward currently. This covariance is a function of several parameters and therefore the "Delta" method has to be applied to get the standard error. If you have a simple model you can use Model Constraint to define a new parameter as this function and thereby get the SE. Perhaps we should add automatic SEs for TECH4.
In testing for measurement invariance over time, it seems to me standard and reasonable to account for the association of a factor with itself over time and for the residual of an indicator with itself over time. My intuition was that it should not make a difference for model whether I model those associations as regression paths (e.g., t2 factor regressed on t1 factor) or as correlations (t1 factor with t2 factor). Yet, I am clearly wrong about that as I just tried this and see that it does impact model fit. Can you recommend a reading that might help to explain why this should make a difference? Also, I am assuming that on theoretical grounds it would be more appropriate to go with the On statements rather than With statements given the temporal ordering but not sure that is the conventional choice. Any thoughts? Thanks as always!
yes, the full model consists of 3 indicators of a factor measured at two time points. So, we have one factor at time 1 with 3 indicators and one factor at time 2 with 3 indicators. For one of the 3 indicators, its correlated residual with itself over time is notsignificant. Thus, we have in addition to the six factor loadings, one correlated residual (or regression path) from the time 1 factor to itself at time 2 and two additional correlated residuals (or regression paths) from two of the time 1 indicators to themselves at time 2. I am not sure that I follow why these other parts of the model make a difference though.
Ultimate DVs have correlated residuals by default because they are identified and are most often needed in my experience. This is due to left-out predictors. You can easily get rid of them using e.g. y1 WITH y2@0.
I and my co-authors are testing a multiple mediator SEM model as below. All factors are latent factors based on (mutually exclusive sets of) continuous indicators. X is the independent factor. Y1 and Y2 are dependent factors. M1 and M2 are theorized to mediate the relationship between X and Y1, and X and Y2 resp.
As you can see below, we allow the residual covariance between M1 and M2 to be freely estimated. We would like to preemptively avoid any reviewer questions. Specifying this (M1 WITH M2) does not change the fit statistics or the specific indirect effect statistic much (it deteriorates slightly); the results of the hypothesized relationships do not change. I would like to receive your opinion on: 1. Whether we should include M1 WITH M2. 2. If we should also include Y1 WITH Y2 or that inclusion is implicit (the Y1 WITH Y2 statistic appears in the output even if we donít include that statement).
MODEL: M1 ON X; M2 ON X; Y1 ON M1; Y1 ON M2; Y1 ON X; Y2 ON M1; Y2 ON M2; Y2 ON X; M1 WITH M2;
MODEL INDIRECT: Y1 IND M1 X; Y1 IND M2 X; Y2 IND M1 X; Y2 IND M2 X;