# Comparison of Models for Margin and Win Prediction

I recently created two models for predicting the outcomes of matches for a particular sport, one is a linear regression model that predicts the margin of the match, and the other is a logistic regression model for predicting the probability of the outcome of the match. I noticed that occasionally the logistic model will predict team A to win (e.g. 57% chance), and the linear model will predict a margin in favour of the opposing team (say, team B wins by 1 goal) for the same match. Both models are trained on the same data and features. I'm curious to know why, and how, this difference between the two models arises. Thanks!

• Because they are different models that work differently (different assumptions - different methods), so you cannot expect them to return similar results, in general. Mar 10, 2020 at 13:00

In case I'm not missing anything from your explanation (some data and details of the model would be useful, though), I guess the reason lies in the different error functions the two models use. In the linear model, you are minimizing the squared difference between the observed and the predicted value:

$$\sum_i (y_i - \beta \textbf{x}_i - \beta_0)^2,$$

while in the logistic model you are minimizing:

$$-\sum_i \left( y_i \log p_i + (1-y_i) \log (1-p_i) \right).$$

The other way of looking at it is to say that you have an information loss in the logistic model. It doesn't care by which margin a team has won. If the prediction matches the true outcome, the error will likely be low. In the linear model, on the other hand, if you predict a win by a margin of 1, but the true win was by 100, this will cause a big error.

• I really appreciate this answer. Do you think a good metric for expressing this discrepancy between these two models would be to compare the accuracy of match outcome (win/loss) using both the linear and logistic models? Mar 10, 2020 at 13:13
• You can do that. Without the data I can only guess, but my guess is that the linear model will be more accurate. Mar 10, 2020 at 13:15
• Interesting that you would guess that. So, if I want to determine a probability of winning using the linear regression model, am I safe to assume the distribution is normal about my model and calculate the standard deviation in order to measure the probability of winning (i.e. margin greater than zero) Mar 10, 2020 at 13:35