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What is the mathematical relationship between aggregate logit and HB estimates?


This may be a very daft question and show my ignorance of bayesian statistics, however, what is the mathematical relationship between the aggregate estimate of a parameter (in this case a linear term) and the average of the individual HB estimates from a group of respondents. I ask because clearly the scale of the HB estimate is larger in magnitude than the logit estimate, yet they are almost perfectly correlated. Is there a way to workout the inflation factor?

The reason I ask is that I need to generate estimates in the magnitude of a a nested logit model for further use in a strategic transport model, but I want to take advantage of the clear benefits of using HB estimation. However, the average of individual HB parameters is to large for use in the model, so I need an equivalence with Logit parameters. Is there a clear mathematical between the terms such that I can reduce the HB estimate to the same scale as the Logit estimate?

Many thanks in advance.
asked Jul 19, 2018 by Jasha Bowe Bronze (1,745 points)

1 Answer

0 votes
Both aggregate logit and HB are based on the MNL model.  But, HB is a quite different animal, involving a balancing act between the lower-level fit (likelihood of fitting the individual-level choices) and upper-level fit (likelihood that the individual respondents seem to be drawn from the multivariate distribution of the upper-level estimates).  Furthermore, the scaling of HB parameters is affected by the prior settings (prior variances especially).

I have also found that HB leads to mean population estimates that are highly correlated with aggregate logit estimates, but with a shift in their scaling.  There is not a simple adjustment factor, such as HB population estimates are 1.5x the scale of aggregate logit estimates.
answered Jul 19, 2018 by Bryan Orme Platinum Sawtooth Software, Inc. (170,215 points)