Can the utilities or choice probabilities be weighted by population weights? Or must the weighting actually occur at the observational/individual level? I assume it must occur at the observational level, but I've seen people weight utilities. Additionally, if the weighting occurs at the observational level, will one still get accurate market share estimates when sub selecting the sample based on different attributes?
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$\begingroup$ So what would these weights of utilities represent? I know what a person's weight is: the inverse probability of selection (at least in the hypothetical settings of basic sampling books). If you add weights to choices within individuals, you will probably risk skewing the results in your modeling of how likely a person is to choose an alternative. $\endgroup$– StasKCommented Jul 17, 2015 at 13:40
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Population weights reflect the probability of selection of an individual. If you do use weights, they are attached to a person. Think about this: representing a choice model like multinomial logit in a "long" format, with one line per person-by-alternative, and an indicator for the chosen alternative, vs. "wide" format, with one line per person and the dependent variable showing just the chosen alternative, is an implementation issue, not a statistical issue. The latter one clearly has only one weight attached to it, and that is the person's weight.
I am not quite sure I would understand what the (differential) weights of the alternatives mean, at least in the context of finite population sampling.
I wrote far more extensively about analysis of complex survey data here, although that was in the context of health applications. There was a section on subsamples -- you often need to treat them carefully as you may mislead the software into misinterpreting your survey design. There are also some schools of thought that the weights are only needed for descriptive analysis, and are anywhere between useless and irrelevant for complex models... but these schools of thought may be missing a point of what the regressions with survey data are estimating... see also here. Regressions with weights and other elements of complex survey designs intend on estimating the census regression, i.e., what you would have obtained should you have run the regression for the full finite population of interest. Whether that's a relevant target quantity may be a different issue; I think a big part of the argument is that, especially for the counterfactual models, the finite population may not be that well defined. Ah well.
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$\begingroup$ Thank you for your informative response. Does weighting by individual have any implications for orthogonality of the experimental design? $\endgroup$– RTrain3KCommented Jul 17, 2015 at 15:13
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$\begingroup$ So is this a WTP study of some kind? Yes, of course weighting will probably shift things away from orthogonality. But this is a non-issue, other than for minor losses in efficiency. We needed the designs to be orthogonal in the 1950s when calculations were done by human calculators (i.e., a person in front of the mechanic calculator machine), so that the estimates will be simple sums and ratios. We are not constrained to that technology for the past 40 or so years... although of course there is still an undeniable beauty in carefully balanced and optimized designs. $\endgroup$– StasKCommented Jul 19, 2015 at 18:15
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$\begingroup$ @StasK Please, help. I am also using multinomial logit model with large weighted sample where same records represents more cases (households) than others. I am confused whether would be correct to use multinomial logit model without taking those weights into account? Thanky you very much. $\endgroup$– QuirikCommented Feb 9, 2016 at 16:35