Logit and Probit Regression
For a binary outcome (yes or no; success or failure), we assign y = 0 for one outcome and y = 1 for the other, and the logit or logistic regression models E(y|X) as a nonlinear function of Xb, 1/(1+exp(-Xb)). For a fractional outcome that lies between 0 and 1, we can again assume E(y|X) = 1/(1+exp(-Xb)), and both models can be estimated using generalized linear models.
Estimates from a logit or fractional logit model are often expressed in odds ratios or log odds, a common measure of effect size for proportions. Given a proportion, fraction, or probability p, the corresponding odds are p/(1-p), and an odds ratio for two fractions p and q is p/(1-p) divided by q/(1-q). Odds ratios are multiplied together, but log odds can be added for the same effect.
Interpretation of logit estimates depends on whether coefficients are reported as effects on log odds or on odds ratios. Thus, a logit coefficient on X of 0.5 shows an increase in a fraction successful (y = 1) when X increases by one unit, and a coefficient of 0 shows no impact. On the odds ratio scale, the same coefficients would be 1.6487 and 1, so the no-impact comparison point is always 1 on the odds scale.
For a binary outcome, we assign y = 0 for one outcome and y = 1 for the other, and the probit regression models E(y|X) as cumulative normal distribution of Xb. In these regressions, coefficients have no natural interpretation and scale is arbitrary; only ratios of different coefficients are identified. Often, we seek to convert logit or probit regression results back to the probability or fraction scale, which requires computing marginal effects.