Predicting with logit coefficients

Question

I have a very beginner question about Logit coefficients.

I am trying to predict how often the home team with win a basketball game. I did a regression analysis much like the one on this page and below are the variables and their coefficients. I haven't done a logistic regression and I'm sure how to utilize the coefficients to make predictions.

home_cumORTG     0.0766      
away_cumORTG    -0.0891      
home_cumDRTG    -0.1029      
away_cumDRTG     0.0690      
intercept        5.7061     

Assume the home and the away team both have offensive and defensive ratings of 105 what is the probability that the home team will win?

This is my best guess on how to do it but I hope it's not right because if it is my data is no good.

odds = 105 * .0766 + 105 * -.0891 + 105 * -.1029 + 105* .0690 + 5.7061 
odds / (1+odds) 

output 0.45477345837195365

Something is wrong because it is predicting that if two evenly matched teams play the home team will win less than half of the time when in reality the home team wins over 60 percent of the time.

EDIT I think this is how it should be

1/(1+(1/ math.exp(5.7061 + 105 * .0766 + 105 * -.0891 + 105 * -.1029 + 105* .0690)))

Answer 1

Assume your logistic regression model is of the following form:

$$\text{logit}(y_i)=\beta_0+\beta_1x_1+...+\beta_px_p$$

then you can make probability predictions by rewriting the equation as the following:

$$\text{logit}(y_i)=\beta_0+\beta_1x_1+...+\beta_px_p\Rightarrow y_i=\frac{e^{\beta_0+\beta_1x_1+...+\beta_px_p}}{1+e^{\beta_0+\beta_1x_1+...+\beta_px_p}}$$

where $y_i$ is now the predicted probability under your given model. Now if you replace your coefficient estimates with the $\beta$'s in my equation and the correct values of $x$ you should get what you are looking for.