What's probit regression and how does it differ from logit?

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AcadiFi · Accepted Answer

Probit regression is a GLM for binary outcomes that uses the inverse standard normal CDF as the link function, in contrast to logistic regression's logit (log-odds) link. Both predict P(Y=1|X) and produce nearly identical predicted probabilities in most datasets — the choice between them is often stylistic or theoretical rather than empirical.

Probit model: P(Y=1|X) = Φ(Xβ), where Φ is the standard normal CDF. The link function is g(p) = Φ⁻¹(p), the inverse standard normal quantile function. This maps p∈(0,1) to the real line via the normal distribution rather than the logistic distribution.

Theoretical motivation — latent variable interpretation: assume an unobserved continuous variable Y* = Xβ + ε where ε ~ N(0,1). We observe Y = 1 if Y* > 0, else Y = 0. Then P(Y=1|X) = P(Xβ + ε > 0) = Φ(Xβ). This latent variable story fits naturally in economics where agents make decisions based on unobserved utility.

Logit uses the logistic distribution for ε instead of normal, giving slightly heavier tails. For probabilities between roughly 0.1 and 0.9, logit and probit produce nearly indistinguishable results. Differences appear in extreme tails.

When to use which:

(1) Probit — theoretical grounding in normal distribution, common in economics (consumer choice, labor supply), integrates with other normal-based models (multivariate probit, Heckman selection).

(2) Logit — coefficient interpretation is easier (odds ratios), computationally simpler, standard in machine learning and biostatistics.

Estimation for both is via maximum likelihood. Coefficient magnitudes differ by roughly π/√3 ≈ 1.814 (logit coefficients larger), but marginal effects and predictions are similar. CFA candidates should recognize probit as an alternative parameterization and understand that practical differences are minimal.

What's probit regression and how does it differ from logit?

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