It's been years since I've taken my grad school stats courses, and it's a subject I struggle with, so bear with me.

I am attempting to analyze a dataset containing two Poisson-distributed variables (they are counts). However, some of my main effect/independent variables exclude other ones (for example--one variable is 0, 1, or 2 objects, and another variable is the angular offset in degrees of those objects, so that when there are 0 objects present, there is no angular offset value), so I am subtracting mean control values (where there were 0 objects) from raw data to control for differences between control means across experiments. That way I don't have to worry about the missing (angular offset) values.

So, I had Poisson counts and subtracted control means, resulting in a Poisson-like (discrete) response variable that takes on negative values. Before I realized I couldn't treat this as a Poisson, I ran GLM with Poisson distribution. My advisor noticed that the DFs were greatly reduced, and then we realized that the negative values were being ignored.

This new response variable is obviously non-normal:

non-normal response variable

I understand now that it's not the normality of the response variable that matters, but the normality of the error terms (yet some people argue that too). Because I didn't know how else to run it (due to the nature of the variable's values, not the normality), I tried assuming a normal distribution with OLS and then GLM, and this is what I got:

I'm surprised the residuals look as decent as they do, particularly for the latter case. I can see that both have significant p-values for lack of fit (OLS) and goodness of fit (GLM), which I think means that the OLS fits poorly and the GLM fits well.

enter image description here

Here's my question: is using this GLM a valid approach, considering the weird nature of my response variable? And does the residual plot look okay, or no? To me it looks okay, but then again I have repeatedly been surprised that goodness-of-fit tests for normality have come back rejecting normality when I thought they looked normal.


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