# Should OLS always have a lower RMSE than Poisson Regression?

I'm working on building a predictive model for the number of singles a hitter in baseball generates over the course of a single game. Since the number of singles a hitter scores per game is count data I figured that the most ideal model would be a poisson regression. While the poisson model works fairly well, when I cross validate my model the RMSE is always lower with a gaussian assumption. This makes me feel like there is some adjustment I need to make to my data or my assumptions but I cant for the life of me figure out what that might be.

From what I can tell my data fits a poisson assumption very very well. When I plot the empirical density of singles over a poisson distribution with lambda equal to the sample mean I get an almost perfect overlap so I don't suspect zero inflation. Adjusting the link assumption from log to sqrt improves the accuracy somewhat but the gaussian model still outperforms the poisson. I'm working with around 4400 data points and only using around 5-10 predictors so I don't suspect sample size is an issue. Another potential issue I suspected was overdispersion but using a quasi-poisson and negative binomial model only increased or maintained the RMSE. Finally, when I plot the OLS residuals they clearly do not satisfy the normal assumption of a gaussian distribution. (Screenshots included below)

Why could this possibly be the case? Is the lower RMSE from a gaussian assumption suggestive of a problem with my data or model? Just for reference the difference in RMSE is usually around the .02 range. My only goal is to minimize the RMSE of my model, if OLS is better at reducing RMSE would it be good practice to just ignore the clearly wrong assumptions it makes?

• The normal linear model uses an identity link (-> additive model) while Poisson regression is usually done with log-link (-> multiplicative model). So maybe it is all about "additive" vs. "multiplicative". The distributional assumption is often not that relevant as one would think. What I don't get: are you really trying to optimize a model without doing some form of (cross-) validation? – Michael M Jul 19 '16 at 18:06
• If you want, you can try Poisson regression with identity link. It just tells that the mean of the Poisson response reacts additively when changing the covariables (depending on the data, convergence problems can occur because the counts are limited from below by 0). With validation I did not mean checking assumptions but evaluating the performance (which seems to be of key importance) on a separate validation sample or by cross-validation (as the name of our site suggests!). Otherwise it is easy to improve RMSE by adding all pairwise interactions or other forms of overfitting. – Michael M Jul 19 '16 at 18:31
• I was thinking the same, that the relationship between the predictors and response may not be proportionate to the existing amount but then adjusting the link assumption (which I did) should have fixed that right? Would you consider this model optimization and not validation? Questioning the validity of my assumptions and goodness of fit would fall under model validation yes? What would you say I should I be doing before this step? I've looked already at the quality of my predictors, multicollinearity, overfitting, bias etc. – Benjamin Whitesell Jul 19 '16 at 18:34
• Ah I see, sorry if that was unclear. I am splitting my 4400 data points into 3000 as a training set and 1400 as a test set( randomly selected each time of course) and the RMSE is the calculated on the test set. – Benjamin Whitesell Jul 19 '16 at 18:36
• I'll try a proper 5-fold cross validation or something of that sort though. Good call definitely. – Benjamin Whitesell Jul 19 '16 at 18:51

Imagine (so that the models are comparable) that we fit the same model for the mean using least squares and Poisson regression (or indeed any other model with the same model for how the mean relates to predictors)

To further simplify the notions, consider the model is for the conditional mean and linear in the predictors ($E(Y|x) = x\beta$) so that we're comparing ordinary linear regression with Poisson regression (now with the identity link because of the previous assumption).

[Further (to simplify ideas), assume the model for the mean is correct (so our estimates are unbiased). This is not necessary though]

Then your question amounts to asking "if I use an estimator that minimizes the mean square residual, will that smallest-possible mean square residual be smaller than an estimator that does anything else?"

The answer is hopefully now obvious - when you measure fit by the very criterion that one estimator minimizes it must win when compared with anything else

So the question then becomes why would you use RMSE to compare the two models?

Edit: Note that counts whose mean is near 0 will have smaller variance than counts whose mean is larger (indeed, if they're Poisson, the variance will be equal to the mean). So a large value is less "precise" than a value near zero (on average it will be further from the mean of the distribution that generated it). So why would you weight the squared deviation for all points equally?

If you want the most precise estimate (in the MSE sense, say) of the Poisson mean, you want to minimize the MSE of the estimator of that mean (i.e. minimize $E[(\hat{\mu}-\mu)^2]$ ... or equivalently, minimize its square root). That's not the same as minimizing the MSE in the data.

If your estimator is unbiased, then asymptotically (as $n\to\infty$) in in most situations the MLE will have lowest variance and so will also be asymptotically minimum MSE. In small samples (for easy cases) you can often compute the MSE and compare directly.

• Wow...it seems kinda obvious now. Thanks so much. Guess I just felt like the more 'specialized' model should always perform better. To the bigger question now though, my underlying assumption here is that on a specific day, for a specific player, 'singles' follow a distribution, and I'm trying to build a model that most accurately predicts the expected value of that distribution on that given day for that given player, which is why I felt RMSE was the best measure for model accuracy. Would you suggest something else? – Benjamin Whitesell Jul 19 '16 at 23:19
• I have briefly responded via an edit to the answer – Glen_b Jul 20 '16 at 0:23