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In logistic regression, when we look at Z-values or p-values, are we making an assumption that statistics or coefficients of a predictor follows normal distribution?

Null Hypothesis is Coeficient of a predictor is zero.

When we accept/reject based on z-value being higher or lower than Z calculated at a certain confidence say 95%, we are assuming normal distribution for coefficients right? What if the distribution is not normal and we are accepting/rejecting for incorrect thresholds?

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If the assumptions hold the distribution of the estimated coefficient will be asymptotically normal. For sufficiently large samples the normal distribution should be a good approximation.

For small samples it may not be, in which case your tests may be conducted at a higher or lower significance level than you anticipate from the asymptotic approximation. In a similar manner, the actual coverage of confidence intervals for coefficients or for predicted mean values would be affected.

The question would then be "how large is large"? It will be impacted by the particular characteristics of the sample -- such as the pattern of predictor combinations and the distribution of the P(Y=1) values across those predictors.

One possibility is to use simulation of similar situations to the one you're dealing with to get some sense of the likely effect on significance level and coverage of confidence intervals.

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