In Chapter 10 of McElreath's Statistical Rethinking (2nd edition), he argues that the response distribution for a GLM should be chosen to maximize entropy given a set of constraints on the response variable (positive, discrete, bounded etc).

From page 312

when the outcome variable is either discrete or bounded, a Gaussian likelihood is not the most powerful choice. Consider for example a count outcome, such as the number of blue marbles pulled from a bag. Such a variable is constrained to be zero or a positive integer. Using a Gaussian model with such a variable won't result in a terrifying explosion. But it can't be trusted to do much more than estimate the average count. It certainty can't be trusted to produce sensible predictions.

The consequences of the wrong distribution for predictions makes intuitive sense. We don't want a distribution that will predict values that cannot possibly occur.

While a misspecified response distribution can cause problems with prediction, are there problems with inferences made from a model with a misspecified response distribution?

Specifically I have two questions about inferences made from a GLM with a incorrect response distribution:

  1. Will parameter estimates be biased?

  2. Will p-values for the parameters be overconfident or too conservative?

  • $\begingroup$ This looks like it's at least four questions (about parameter estimates, inferences, p-values, and posterior probabilities). Would there be some way you could focus it on a single, clear, answerable question? $\endgroup$
    – whuber
    Jun 15, 2020 at 19:22
  • 1
    $\begingroup$ I've updated to make the questions more clear/distinct, hopefully this helps. If needed I can create a separate post if the 2 questions should each be their own post $\endgroup$ Jun 15, 2020 at 20:16

1 Answer 1

  1. The parameter estimates will be different. That's a stronger condition than bias. The right GLM is the one that answers the question. If I want to know how much a very toxic drug A reduces the risk of a rare disease, I might be more justified fitting a linear regression than a logistic model. If I show the risk decreases by 0.001%, despite being a 20 fold relative risk reduction, the underwhelming effect immediately prompts the important question of overall utility. While the OR is a biased (albeit small) approximation of the relative risk reduction, I don't care. What matters is 0.001% and 20 are correct but tell different stories.
  2. It could go either way.

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