# When does a Linear regression stop being a good fit?

I'm testing a couple of hypotheses and I have a non normally distributed continuous response variable (residuals are not normally distributed as well).

I have been given mixed suggestion on what model to use. Some say to use a Linear model regardless of the distribution, whereas others say that a Gamma model with a log link is a better fit.

I have 4 independent variables, 3 continuous and 1 nominal.

Should I go for a Gamma model? I haven't seen many papers using the gamma model so I am wondering whether it has some significant benefits over the simple linear models.

here are the scatterplots of the residuals:

• Distribution is not as important as you make it sound. The nature of relationship is the most important, the regressors, the functional form of variables etc. Sort out what matters first – Aksakal Jun 23 '20 at 14:37
• if you are considering a linear model then it's $y=X\beta+\varepsilon$, all components are important, but the main concern is whether $X\beta$ captures the main relationships, i.e. whether X has relevant variables and y depends on them linearly. if you're certain this part is taken care of, then you move to looking other stuff like distribution of $\varepsilon$, it's usually not so important anyways – Aksakal Jun 23 '20 at 15:00
• The coefficient of determination is one thing you could look at: how close to $1$ is it? But the other thing, perhaps more importantly, is what you've already done: plot the residuals. Do you see any pattern in the residuals? If so, linear regression might be missing something important. If the residuals look more random, that would be a clue that your linear regression is doing reasonably well. – Adrian Keister Jun 23 '20 at 15:37
• The error in the estimate of the (conditional) mean will approach a normal distribution centered around the true conditional mean (independent from the underlying error distribution as long as it has finite variance). Linear regression will do sufficiently well if you have many points. More fancy distributions can improve the estimates, however this is mostly when you have few points or heteroscedasticity (and the heteroscedasticity would actually not matter much is, as Aksakal mentions your underlying deterministic relationship is right). – Sextus Empiricus Jun 24 '20 at 6:01
• But possibly you wish to not describe/estimate the mean and instead some other population parameter. For instance, in your case with several dominant upwards outliers, I can imagine that the median might be more meaningful. It will depend on what relationship you wish to model. If you have a mechanistic model or some intuition about the population parameters, then you may have a preference for using the more meaningful parameters. An example how it can change: the mean of a population will be different after a logarithmic transformation. So $\mu \neq e^{\overline{\log(x)}}$ – Sextus Empiricus Jun 24 '20 at 6:08