How do sufficiency statistics help in the interpretation of regression results?

One of the results why canonical link functions are widely used in GLMs is the existence of sufficiency statistics for the regression parameters, which in turn allow for:

… minimal
sufficient statistics for the regression parameter exist which allow for simple interpretation of the
results …

a statement which I do not understand.

Can someone please explain this and also give me examples of how the interpretation of regression coefficients is aided when such sufficient statistics exist, and perhaps more importantly, examples of GLMs using non-canonical links where the results are harder to interpret?

For a specific example I found this question regarding interpretation of probit regression coefficients. The response distribution is Bernoulli so the canonical link is the logit. However, when I read the answer to the question, I see that interpretation of the coefficients is still answering the question of: “how much does the probability of a 1 change when a covariate is changed, holding other covariates constant?”. This does not seem terribly different from the interpretation of the logistic regression model.

Answer

Attribution
Source : Link , Question Author : Alex , Answer Author : Community

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