by **Michiel Bliemer** » Fri Jul 26, 2024 8:16 am

Scale influences choice probabilities, and hence influences design efficiency.

For example, if scale=1 and choice probabilities are 0.45 and 0.55. If scale=2 then the choice model becomes more deterministic (less error variance) and the choice probabilities would become more pronounced, e.g. 0.2 and 0.8. In other words, larger scale parameters indicate more choice certainty. Each data set has its own scale parameter and is unknown. It is only possible to estimate relative scale between data sets by estimating a joint model such as SP-RP. Otherwise, the scale parameter is absorbed by the beta parameters.

Design efficiency, as expressed in D-error etc, relies on choice probabilities. So using a different scale parameter would change the choice probabilities and therefore would change the D-error. It is therefore important to use appropriate parameters with appropriate scale. If you conduct a pilot study in the same population as your main study then you expect the scale to be similar and therefore you can used the parameters estimated from the pilot study data directly to generate an efficient design for your main study. But note that it is also known that efficient designs create more challenging choice tasks to respondents than orthogonal designs, so the scale parameter based on data from an orthogonal design may be somewhat larger than the scale parameter in a data set using an efficient design. So if you are using an orthogonal design for your pilot study and estimate model parameters, then you may want to shrink the parameters somewhat to use as priors for an efficient design, e.g. by multiplying the parameter values with 0.8 or something.

Michiel