Interactions effects with main-effects prior values only

[Brig]

Hello,

My question is about the interpretation of 2-ways interaction effects in a choice experiment that was originally designed with prior values for the main-effects only.

Specifically, let's assume that one designs an efficient experiment and provides the algorithm with prior values for the main effects only.
Suppose also that the number of choice cards is large enough to study both main effects and two-ways effects.

My question is: what if, in the end, the analyst needs to study 2-ways interaction effects?

Is it possible to analyze 2-ways effects in an experiment with prior values for main effects only?
Would the whole design become less efficient (ie even for main effects parameters)? Or would it be that only the standard errors for the parameters of the 2-ways effects are not "optimal"?

Thanks
Baptiste

[Michiel Bliemer]

An efficient design is generated under the assumption of a specific model (e.g. mnl), utility specification, and specific parameter priors. This means that if one estimates a different model (e.g. mixed logit) with different utility functions (e.g., including interactions that were previously not specified) and with parameters that differ from the priors, then the design would lose some efficiency. How much efficiency is lost varies, but the more the specification and priors deviate, the more efficiency is lost. Losing efficiency does NOT mean that the model can no longer be estimated. Clearly revealed preference data is not optimised for any specific model and one can estimate all kinds of models with it.

So if the utility functions did not include 2-way interactions when the design was optimised, it means that you will in most cases still be able to estimate the 2-way interaction effects but the design was simply not optimised for it. If you include 2-way interaction effects when generating an efficient design, then the standard errors of the parameters of these interaction effects are also minimised, but this come to some extent at the cost of optimising the standard errors of the main effects since information needs to be distributed over more parameters. Most people do not include 2-way interaction effects at the design stage but only add them at the estimation phase for the simple reason that it is often not desirable to include all 2-way interaction effects a priori.

The interpretation of a 2-way interaction effect is the same for any type of design (orthogonal, random, efficient), no matter whether they were considered at the design stage or not.

Michiel

[Brig]

Thank you very much for the detailed reply, Prof. Bliemer.
It's also good to know what is "common practice" in that case.