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Hello, I am running a SEMModell with 5 categorical items at each timepoint, 2 timepoints of measurement and 3 latent variables (2 states and 1 trait). I would like to account for measurement invariance. Therefore I set equal factor loadings between the timepoints (congeneric model, first loading fixed to 1) and the intercepts are per default fixed to zero (thus, equal). Because I am regarding only the covariance structur (no means), it is not possible in MPlus to set the thresholds equal. Is the threshold equality neccessary to have measurement invariance? Or is the metric invariance (equal factor loadings) and scalar invariance (equal intercepts) sufficient? Thanks a lot Tony 


With categorical outcomes, I would consider both thresholds and factor loadings while testing for measurement invariance. See Chapter 13 after the multiple group discussion for the models to be considered when testing for measurement invariance. Examples 5.16 and 5.17 show partial measurement invariance for categorical outcomes. These examples are for multiple groups but the same principles apply across time. 


Many thanks for your quick reply. I read Chapter 13 and tried to specify the equality of both, factor loadings and thresholds for 4pointLikert items with this input: MODEL: ETA1 BY var11@1 var21(1) var31(2) var41(3); ETA2 BY var12@1 var22(1) var32(2) var42(3); XI BY ETA1@1 ETA2@1; [var11$1 var12$1](4); [var11$2 var12$2](5); [var11$3 var12$3](6); ... (the same for the other variables). But I am not sure, if itīs the right way to do that?! Or would you recommend theta parameterization? (I tried it, but I was not able to produce the right input file...). 


I forgot about to ask, if it is really important to have equality of thresholds when I solely analyze the covariance structure of the variables (without the meanstructure)? To my knowledge the thresholds are (more) important when regarding the mean structure of the variables?! Thanks a lot, Tony 


I do not believe that measurement invariance with categorical outcomes should exclude the thresholds. The sshaped item characteristic curves that are the essence of categorical data modeling depend on both the threshold and the factor loading. Measurement invariance for categorical outcomes should consider both of these in tandem. Your input looks correct. But check TECH1 and the results to be sure that the proper equalities are there. 

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