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I have a study where the same people were exposed to advertisements both with and without an "endorser" or spokesperson. There were also several different kinds of endorsers (male vs. female and so on). The response was "change in desire for a product" on a Likert scale.

What I have done is a sort of two stage non-linear multilevel model, with the first stage being "endorser vs. not" and the second being characteristics of the endorser among those who saw the endorser. This worked fine.

But, can I combine the parameter estimates from the two stages to get a full picture of the effects? That is, is the parameter estimate for (say) a female endorser the sum of the effects for an endorser and a female?

It seems it should be, but I wanted to check.

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This is a situation where you have some conditional variables that only make sense conditional on a particular value for a previous explanatory variable. You have a binary explanatory variable indicating the presence of an endorser, and then various conditional explanatory variables for the characteristics of the endorser. These latter variables can be handled in a regression framework by including them only through interactions effects with the primary explanatory variable, excluding the main effects for these variables. This allows you to fit a single model that includes all explanatory variables without breaking this up into two stages.

For example, if you have a Participant variable and a binary Endorser variable with subsequent characteristics Sex and Age, you can model this as a non-linear multilevel model with this form:

Likert_Response ~ factor(Participant) + Endorser + Endorser:Sex + Endorser:Age

Notice that in this model form there is no main effect for Sex or Age because these are conditional variables that only make sense if there is an endorser (i.e., if the Endorser variable is one). This is similar to another question I have answered which also involved conditional variables. You should still use a model form that is appropriate for the Likert scale, so I would suggest sticking with your multilevel model, but trying to do it in one stage. If you have other conditional variables for the characteristics of the endorser, obviously you should substitute my example with this variables.

This kind of model form allows you to model the whole process in one step, avoiding a two-stage model. This is preferable, since two-stage models often have problems with conditional fitting, and it is difficult to interpret coefficients. In this model form you will get estimates for all your coefficients simultaneously, and this will include a correlation matrix for your coefficient vector. Hence, with this single-stage model you will be able to combine coefficient estimates in the standard way.

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