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Ok, forgive my ignorance, but I keep getting confused about something at the core of GLMs. Some textbooks describe the two main parts of a GLM as the link function and the distribution of the error terms. Others describe the two main parts as the link function and the variance function, where the variance function is a description of the relationship between the mean and the variance of the response (i.e., the response distribution). But the error distribution and the response distribution seem like different things to me.

If I had an equation where $Y_i = B_0 + X_i*B_1 + e$, I can see how for any given value of $X_i$ (plus constant values of $B_0$ and $B_1$), each random variable $Y_i$ would take on whatever distribution the error term had. But does that necessarily make the overall response distribution equal to that same error distribution?

Do my questions even make sense?

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  • $\begingroup$ Your second paragraph is not entirely correct. Assume $Y$ is uniformly distributed, and the model $\beta_0+\beta_1 x_1$ fits it very well for your purposes. It may be surprising, but in this case, $\epsilon\sim\mathcal{N}(0,1)$! (This result still blows my mind. I, and many others on the site, will be perfectly happy to help you with these questions.) $\endgroup$
    – Sycorax
    Apr 1, 2014 at 2:47
  • $\begingroup$ The overall response distribution is a function of the kind of distribution (ie, normal, binomial, etc), the mean (for non-normal GLiMs where the variance is a function of the mean), & distribution of X. It is more complicated for general GLiMs, but you can get a sense from my answer here: What if residuals are normally distributed but Y is not? $\endgroup$ Apr 1, 2014 at 3:01
  • $\begingroup$ It makes sense, at least for me. I have the same question ( stats.stackexchange.com/questions/328266/… ) but I did not know and dont understand what you say here: "If I had an equation where Yi=B0+Xi∗B1+eY, I can see how for any given value of Xi (plus constant values of B0 and B1), each random variable Yi would take on whatever distribution the error term had. $\endgroup$
    – user110848
    Feb 14, 2018 at 7:33

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