For example in count data modeling, imagine modeling count data generated from dispersed negative binomial using Poisson distribution. The poisson regression fits the significant covariates and they turn out to be strongly significant. Yet the goodness of fit test fails, because the underlying distribution is wrongly assumed. Without assuming knowledge of the negative binomial underneath, how can I proceed with the Poisson regression, what are the limitations or is it even sound to proceed when the GOF test fails?
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1$\begingroup$ Usually failure of GOF tests means that if you proceed with this model, you will obtain incorrect predictions. The coefficients that it will give you will still be the best ones estimated via something like the EM algo, but using them would be wrong to estimate unknown values. $\endgroup$– FisherDisinformationAug 9, 2016 at 18:00
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$\begingroup$ If you have such data and don't know its true distribution, but you have strong suspicion of it being generated from Poisson distrib. And you fit the model to the data and GOF test on residuals fail. Then that is evidence you were wrong in your assumption. You should probably review and update your assumptions before jumping to conclusions. $\endgroup$– RodolpheJan 27, 2020 at 12:30
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It shows that one of your basic assumptions is wrong.
However, in practice it depends on how sensitive the parameter you are analyzing is with respect to the underlying assumptions. If you are looking at a strong effect (say, male wage compared to female wage), then whatever you do, the effect is there and about the correct size. One the other hand, if the effect is weakly identified, your results may end up being extremely sensitive to the distributional (and other) assumptions.
Note also that on big data you can pretty much reject every parametric assumption.
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$\begingroup$ Well, in the specific case of using Poison regression with overdispersed data, standard errores and hence hypothesis tests and CI will be wrong ... while estimates probably are OK. $\endgroup$ May 27, 2020 at 19:15