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When doing a Bayesian efficient design, we sometimes find priors from published literature without paying much attention on the levels of that attribute used in the literature. Here are we assuming that the preference weight (marginal impact of attribute on preference) will not change depending on the specification of levels?

For example, if I have exactly the same design but levels of one attribute vary from [2,4,6] to [1,3,5], theoretically, should I expect the preferences weight of all attributes to be exactly the same regardless of the design?

My hypothesis is that the true willingness-to-trade off between benefits and risks (or WTP) is based on respondent's profile and their preference we are trying to understand, and will not alter depending on how the DCE is designed and how the levels are specified. Then thinking about the calculation of relative importance based on preference weights and the best and worst levels (Juan Marcos 2019, the Patient), does that mean that there is an inherent benefit for those attributes which display the broadest difference in the range explored?

You are asking very good questions. I will first respond to the question about parameter values / priors, and then to the question about relative importance.

Regarding parameter values, it is important to understand that all parameters in logit models include the scale parameter. This means that even if the attribute levels are exactly the same across different studies, the parameter values may be smaller or larger depending on:
(i) the type of data, stated or revealed preference (usually SP data has less error variance and therefore larger scale and parameter values)
(ii) the number of attributes and alternatives in the model (more attributes/alternatives means more error variance and therefore smaller scale and parameter
(iii) differences in the sampled population (economic background, culture, etc)
This is discussed for example in this paper: https://www.sciencedirect.com/science/article/pii/S1755534515300877

There could also be scale effects when using [5,10,15] instead of [9,10,11] as levels or when having dominant alternatives in choice tasks, since this will often make the choice more deterministic and therefore typically increases parameter values (see for example this publication: https://www.sciencedirect.com/science/article/pii/S0191261517304228).

If there are no scale effects and the sampled population is very similar, then changing the levels from [2,4,6] to [1,3,5] will likely not influence the parameter values much, so the prior from the literature would likely not be a reasonable choice. An exception may be the price coefficient, as this coefficient depends on the currency (e.g. euro versus dollar) and the purchasing power in a country. So simply using a price coefficient from another country is risky and would at least require currency conversion.

Regarding attribute level importance, this absolutely depends on the range of the attribute levels. So you as an analyst could make an attribute much more important by simply using a wider range. This is well-known and therefore you have to interpret attribute level importance in the context of "importance within the experiment". This is mentioned in the original publication by Orme who first proposed this measure in the marketing literature.

Michiel