ApolloChoiceModelling forum

Dear Stephane,

Unfortunately, I have another question about the scale parameter in dual response: Does it make sense to use a randomized scale parameter (e.g. lognormal distribution or lognormal distribution with Fosgerau Mabit transformation) instead of a fixed parameter? The reason is that I wish to use the same utility parameters that are estimated for the forced choices also for the dual response part. However, the dual response part asks for a willingness to participate in a demand response program (using a 7-point Likert scale) that - at least for some respondents - cannot be explained by the utility parameters of the attributes (using an ordinal regression). This seems to lead to optimization problems (or problems with the Markov chains). So making the scale parameter random would allow for a distinction betweens respondents, i. e., between those who take the attribute levels into account and those who do not. That said, I have not read any publication that uses a randomized scale parameter.

Nico

Nico

apologies for the slow response.

Theoretically, you can do what you're suggesting, but you're going down the route of random scale heterogeneity. This is a topic that has caused a lot of issues in the literature, and a lot of misunderstanding. Have a look at https://doi.org/10.1016/j.jocm.2017.03.001 and https://doi.org/10.1007/s11116-012-9394-9

Allowing for deterministic scale heterogeneity like you are doing here is quite common practice, but you should already be aware of the risks of confounding, and you should also test the model against one that allows for differences not just in scale. Once you allow for random differences, the risk of confounding just increases. Your risk is probably smaller than in a case with a single dataset, but I would still be wary

Stephane

Hi Stephane,

Thank you very much for your answer and the articles. I already knew the one with Kenneth Train, but not the other from 2012.

The problem is the more sophisticated the model, the higher its complexity. For example, when I want to "[...] test the model against one that allows for differences not just in scale.", my number of parameters doubles, as I estimate the same number of random coefficients for the forced and the free choices, respectively.

So in summary, I have several options for model specification:
1) Forced choices own random coefficients, free choices own random coefficients, no scale parameter
2) Forced choices own random coefficients, free choices own fixed coefficients, no scale parameter
3) Forced choices random coefficients, free choices same random coefficients, fixed scale parameter
4) Forced choices random coefficients, free choices same random coefficients, random scale parameter

When I estimated simple MNL models, the results indicated that the beta coefficients (or sensitivities) between the forced and free choices differed not only in absolute terms (which would suggest to simply use a scale parameter), but also (or rather) in the marginal rates of substitution. Therefore, I tend to go with 1) or 2), whereas 2) is much easier to estimate, but insights into choice behaviour is more limited.

Nico

Nico

your starting point should be specification 2 and then gradually impose equality constraints where they make sense on the basis of tests

Stephane