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Reed College

Data @ Reed

Logit and probit

Logit and probit models are special cases of regression designed to deal with binary outcome variables. Some examples of binary outcome variables (aka dichotomous outcome variables), often coded as 0/1: a candidate won or lost an election, a plant was or was not observed in an experimental plot, a rat did or did not take a left turn in a maze.

The difference between probit and logit models lies in the underlying model for the regression. In the logit model (logistical regression), "the log odds of the outcome is modeled as a linear combination of the predictor variables." [1] In the probit model, "the inverse standard normal distribution of the probability is modeled as a linear combination of the predictors." [2]

Note: Like the rest of the Data@Reed help pages, the goal of this documentation is not to help you choose what test to run or to teach you statistics, but to help you through the mechanics of running statistical tests. For help deciding what sort of analysis to run on your data, consult with your advisor/professor. 

Logit

Using the auto dataset, let's model whether a car is foreign-made based on its mileage and weight.

sysuse auto
logit foreign mpg weight

The output gives you coefficients on the log-odds scale — a one-unit increase in mpg is associated with a change in the log odds of a car being foreign, holding weight constant. Log odds aren't very intuitive on their own, so it's common to also ask Stata to report odds ratios instead:

logit foreign mpg weight, or

Now each coefficient is interpreted as a multiplicative change in the odds of the outcome (foreign = 1) for a one-unit increase in the predictor, holding other variables constant. An odds ratio above 1 means the odds increase; below 1 means they decrease.

To get something even more interpretable — predicted probabilities — use margins:

margins, at(mpg=(15(5)35)) atmeans

This shows the predicted probability that a car is foreign at various mileage levels (15 through 35 in steps of 5), holding weight at its mean.

Additional resources:

Probit

The same model can be run as a probit instead:

probit foreign mpg weight

The coefficients here are on the z-score (probit index) scale rather than the log-odds scale, so they aren't directly comparable to the logit coefficients above — but the sign and statistical significance of each predictor will typically tell the same story. margins works the same way after probit as it does after logit:

margins, at(mpg=(15(5)35)) atmeans

In practice, logit and probit models tend to produce very similar predicted probabilities even though the coefficients look different — the choice between them is mostly a matter of convention in your field (economists tend to favor logit and probit fairly interchangeably; some other fields lean toward one or the other).

Comparing the two

A quick way to see this similarity side by side:

logit foreign mpg weight
estimates store logit_model

probit foreign mpg weight
estimates store probit_model

estimates table logit_model probit_model, star

You'll notice the coefficients differ in scale, but the predicted probabilities from margins after each model will be nearly identical.

Additional resources: