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A friend and I are having a dis-agreement about over-dispersion in binomial/logistic regression glm modelling. We have structured our data so that each observation represents 1 Bernoulli trial (so the response is strictly either 0 or 1) and are using logistic regression to build a model.

Is it possible for there to be over-dispersion in this model (or is the scale parameter necessarily 1)? If so, why? And if we need one, what is the best scale parameter estimator to use (we'll have a very low response - sub 1%).

Thanks, Tom

TomFromWales
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1 Answers1

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If the observations are independent it is impossible to have overdispersion with Bernoulli responses. I'm not sure what made the question arise. If the regression model is improperly specified you will have a different problem. For example in a logistic model, a strong omitted covariate can cause all the $\beta$s to be biased towards zero due to non-collapsibility of the odds ratio.

Frank Harrell
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  • @Franke Harrell, I realised this is over 7 years ago, but I didn't really understand this response. Why does non-collapsibility mean that an omitted variable can lead to the coefficients in a model to be biased towards zero? – Geoff Aug 15 '23 at 16:35
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    Much has been written about this, see for example here. In a nutshell: unlike the linear model which has $\sigma$ to absorb some lack of fit from misspecifying $X$, there is no residual error in logistic and Cox models. So errors in model specification get spread to all the $\beta$s. The effect is usually to make $\hat{\beta}$ closer to zero. – Frank Harrell Aug 15 '23 at 18:11
  • Thanks for the response! Just to clarify, is the bias on the coefficients in a regression setting analgous to the table, from Gail 1986 in the link, where the pooled is not equal to the weighted average of the omitted variable: in this case, gender? – Geoff Aug 16 '23 at 09:26
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    Yes; Simpson's "paradox" is nothing more than failure to ask a specific enough question (i.e., condition on X because X matters). – Frank Harrell Aug 16 '23 at 11:11