Predicted Probabilities

Once the parameters of a logit model have been estimated, it is fairly simple to generate predicted probabilities that Yi = 1. To do this we need to take the logit model and solve for Pi. Raising both sides of the function for the logit model to the value e and then solving for Pi gives: Pi = 1 / (1 + exp (- (Σ βkXik))). In the standard linear, additive regression model, the simple functional form allows for an unconditional interpretation of each regression coefficient. It is the change in the expected value of Yi for a unit increase in the predictor variable, holding all other predictors constant. The functional form of the logit model does not permit this kind of unconditional interpretation. Instead, the effect of a specific predictor variable must always be evaluated at particular values of all of the other predictors. Thus, after estimating a logit model and obtaining estimated values of all of the coefficients, βk, it is only necessary to choose values of all of the predictors to obtain a predicted probability that Yi = 1.

In our model, Yi = 1 when respondents voted for Bush and 0 when they voted for the Democratic candidate (Gore or Kerry). After obtaining the estimated logit parameters we can choose values for all of our predictor variables—gender, race/ethnicity, age, religious denomination, attendance at religious services, union membership, education, and income—and generate a predicted probability that a person with those specific characteristics voted for Bush. Although we do not do so here, there are also methods that can produce a confidence interval for each predicted probability.

While logit models can be estimated with many software packages, one statistics program—Stata—makes the calculation of predicted probabilities somewhat easier. This is accomplish through additions to Stata that automate these computations. One was developed by Long and Freese, and a second by Gary King and his associates. Both are free, and there is documentation that describes the methods used. It is also possible to use these methods to obtain confidence intervals for the predicted probabilities.