I have two linear models, one with a log-transformed response variable, one without. I am trying to decide which is the better model before refining with stepwise etc.
Model 1 - Non-transformed model: Higher Adjusted R-Squared (0.
Model 2 - Log-transformed response variable: Lower Adjusted R-Squared, better Q-Q plot (Closer to normal).
What is more important, R-Squared value or getting closer to normality?
You have some good answers already. I'd just like to add that, aside from the particulars of your case, the question doesn't really make sense.
The question of normality is about the assumptions of the model and what happens if they are violated.
$R^2$ is a measure of how much variation in the dependent variable is accounted for by the model.
You could compare, say, the assumption of normality with the assumption of homoscedasticity. Or you could compare, say, $R^2$ with RMSE. But I don't see how you can compare across the two categories.
Issues with normality in the non-transformed model don't seem to be critical. Note that although formally a normal model is assumed, nothing in reality is really normally distributed, and the relevant question is not whether residuals are normal or not, but rather whether assumptions are violated in ways that can harm the analysis. As the residual distribution looks to have lighter tails than the normal, if anything, and isn't terribly skew, I don't see any problem with this.
The second model seems to have an issue with homoscedasticity (equal error variances) as apparent in the Scale-Location plot. I don't see such issue with the first model.
Depending on how exactly you want to use the model, it may be a good thing for interpretation and prediction to have a model that predicts the response directly rather than its log. (You can in principle run cross-validation to explore prediction quality, and in order to do this in a way that can compare different "versions" of the response, you could transform back predictions from the log-model to the original scale by taking exponentials. If you do this, chances are that the model based on the untransformed response will be quite a bit better, although in some situations predicting the log may be as good a thing as or better than predicting the original response.)
Regarding "what's more important, $R^2$ or normality", the key issue here is that it is important how normality is violated and how this plays out. In any case I'd trust cross-validation more than $R^2$. To what extent $R^2$ is informative depends on what exactly you want to do, for example prediction outside the original value range (about which $R^2$ doesn't say much). I don't see a problem with the normality assumption here; if anything the issue with homoscedasticity in the log-model looks worse.
Note also that technically assumptions for inference are invalidated by choosing analyses depending on what you see in the data. So if you first think that you want to model untransformed data, and then you see some plots and change your mind and decide to transform the response, technically model assumptions are already violated because they don't allow for such a data-dependent decision process. If there are serious issues with the first model, it may be worth to pay that price anyway, but I'd usually recommend to stick to the first model unless something really serious turns up.
The somewhat outlying observation 32 in the log model together with the overall impression of the plots makes me think that normality actually doesn't even look better in that model than in the first one.
You seem to be under the impression that normality is an assumption for linear regression. Linear regression is just a model, and it can be fit by ordinary least squares, as you did. Normality is not assumption for the coefficients to be unbiased or consistent, nor it is it an assumption required for valid inference. It is an assumption required for exact inference using the t-statistics computed from the usual ordinary least squares standard error estimates. But who says exact inference is necessary, and who says you have to use ordinary least squares standard error estimates?
By the central limit theorem, the coefficients are normally distributed in large samples under mild conditions on the error distribution, which does not have to be normal. In addition, there are robust standard errors you can compute that maintain valid inference even when other assumptions about the model, like homoscedasticity, are violated. The sandwich standard error and bootstrap standard errors are such examples.
I highly recommend this post that describes what each assumption means what it gets you.
If you violate the normality assumption, you can still fit a linear regression model, but your conclusions won't be correctly interpreted, since the model tells you what happens, conditioning on the fact that the assumptions of a simple linear model does hold true.
Now, $R^2$ did receive a lot of criticism in the literature, for example in this link. Also, your $R^2$ before and after the transformation are not really comparable, since they fit different outcomes. In the end, they are not the best measure to conclude a good fit (since they are not always even seen as a goodness-of-fit measure).
In the end, I think it is much better practice to fit a model of which the assumptions are not violated, rather than a model with violated assumptions, but that has some diagnostic measure that gives better values.
40000+170000*numberofbathroomswould not be far out. – Henry Mar 30 '24 at 00:21