What are alternatives to the logit function for binomial or binary data?

• A. Logit and probit
• B. Probit and log-log
• C. Complementary log-log and logit
• D. Probit and complementary log-log functions

Answer: (D)

For binary outcomes, you model the probability with a link function that connects the linear predictor to p. Besides the familiar logit, two standard alternatives are probit and the complementary log-log link. Probit assumes an underlying latent normal variable, so the probability is p = Φ(Xβ), where Φ is the standard normal CDF. This gives a sigmoid relationship similar in shape to the logit but with different tail behavior. The complementary log-log link, defined by g(p) = log(-log(1-p)) = Xβ, translates to p = 1 - exp(-exp(Xβ)). This form is asymmetric and often fits situations where the event probability is very small at lower predictor values and rises quickly as the linear predictor increases, akin to hazard-like processes. Together, these two functions are common, valid alternatives to the logit for binary data, which is why they are the best pair to identify as alternatives.