Thursday, July 5, 2007

Refining the Prediction Model Part I - Linear vs. Logistic Regression

Recently, I came across NFL Stats, a blog similarly focused on predicting football games. Somehow, Brian, the creator, was able to achieve 63% accuracy on 2006. So I'm going to be playing catch-up for a while. Let's take a look at what separates are two systems, won't we? The biggest difference between our two systems is that he uses logistic regression for simple win/loss predictions rather than linear regression. Logistic regression tries to fit a probability distribution function between two or more classes, so the output is discrete rather than continuous.

Plugging my current dataset (as will be described in the next article) into a logistic regression model, I achieve 61.789% accuracy on average from 1997-2006, a slight uptick compared to linear regression (61.547%). The original idea behind using linear regression was that the better a team was than their opponent, the higher the margin would be in their favor. In logistic regression, we'd expect the predicted probability of winning to increase proportionally to the quality gap between two teams. So as it turns out, the predicted probabilities of winning are almost as highly correlated with the final score margin (.28357) as the predicted margins of linear regression (.28617). In this case, the predicted P(win) might be used to determine whether or not to bet on a team against the spread. Logistic regression did slightly worse in terms of classifying too many games as home team wins (76.098% to 73.672%).

R2 measures the variability in a data set (i.e. the final score margins) accounted for by a model (i.e. the input data). Unfortunately, the R2 for the linear regression model is .13813 on average. Less than 14% of the variance is accounted for by the model. NFL Stats uses this measurement in the context of how their model predicts season win totals (i.e. do the expected win totals match up with the actual win totals). From 1997-2006, the R2 for my model of VOLA stats after week 17 predicting season win totals was .80287. From 2003-2006, the R2 is .82432. For NFL Stats, the R2 is 0.85 now, I believe, though it was .75 not too long ago. So I am not completely off track with my stats. In fact, the expected win totals for 2006 were off only by 1.3085 games on average, down from 1.3523 in 2005. (San Diego, Baltimore, Indianapolis, and New England all outperformed their expected win totals in 2006.)

Originally, I said that predicting exact final scores was nearly impossible because so many non-predictable factors went into the final score. Perhaps, if even the spread is off by 10 points on average, then the final score margin is determined by enough non-predictable factors that make the problem overly difficult. From now on, I'll be testing logistic regression in addition to linear regression.

Note: For the article, I used glmval and glmfit in MATLAB to do logistic regression with a binomial model.

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Brian Burke said...

Derek,

Thanks for the hat tip. I think the explanation for the increase in r-squared from .75 to .85 was due to these 3 things:

1. I used to just use net turnover difference as a predictor. Now I use interceptions per pass attempt (interception efficiency), and similar measures for fumbles.

2. I added penalty yards to the model.

3. I also started using homoskedasticity-corrected linear regression instead of regular ordinary least squares regression. Ive been using Gretl, an open source stat software app since my SPSS licence from grad school expired.

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Homoskedasticity. Have not heard that one before.

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