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I have been running some binomial logistic regressions in R on a data set and I realised that the p-values of the estimated coefficients are not computed based upon a Normal distribution. For e.g. I have the following result from the glm() function:

    Coefficients:

                                   Estimate Std. Error z value Pr(>|z|) 
    (Intercept)                    -1.6127     0.1124 -14.347  < 2e-16 ***
    relevel(Sex, "M")F             -0.3126     0.1410  -2.216   0.0267 *  
    relevel(Feed, "Bottle")Both    -0.1725     0.2056  -0.839   0.4013    
    relevel(Feed, "Bottle")Breast  -0.6693     0.1530  -4.374 1.22e-05 ***

I previously thought that the beta hats (estimated coefficients) are asymptotically normal and the p-value should be calculated using the normal distribution. But it seems from the p-values here that a t-distribution is used to compute them (I think).

I know that the asymptotic condition does not hold when sample size is small but if so, what is the process that causes the estimator to be distributed with a Students distribution? And how do I find out the degrees of freedom for such a distribution?

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It comes from standard normal:

-0.3126/ 0.1410 =  -2.217021
2*pnorm(-2.216) = 0.0266915
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  • $\begingroup$ It actually seems to be 2*pnorm(-1*abs(estimate/std.error)). I had positive coefficients and found out that the pnorm argument needs to be negative number for this to work. $\endgroup$ – Jitendra Kulkarni May 18 '17 at 4:23
  • $\begingroup$ @JitendraKulkarni ; yes - your formula is correct (see summary.glm which uses 2 * pnorm(-abs(tvalue)) as in your comment) . But you could also use lower.tail=FALSE depending on the sign; 2*pnorm(-2.216) is the same as 2*pnorm(2.216, lower.tail=FALSE) ) $\endgroup$ – user20650 May 21 '17 at 22:13

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