Answer

Logit vs probit: what is the difference, and which should you use?

Inspired by a question on Cross Validated · September 24, 2026

regressionlogistic regression

Two versions of the same idea

Both are generalized linear models for a binary outcome such as pass/fail or bought/did not buy. Each one computes a linear predictor, b0 + b1*x1 + ..., and turns it into a probability between 0 and 1 with a link function. The logit model uses the logistic curve, so the linear predictor is the log of the odds. The probit model uses the cumulative normal curve, so the linear predictor is a z-score on the standard normal scale.

A helpful way to see the difference is a hidden, continuous variable. Imagine each person has an unobserved propensity, and the event happens when that propensity crosses zero. If the random part of the propensity follows a standard normal distribution, you get probit. If it follows a logistic distribution, you get logit. The logistic distribution looks very much like the normal one, with slightly heavier tails and a standard deviation of about 1.81 instead of 1.

Why the coefficients look different

Because the two latent scales have different spreads, logit coefficients are larger. In the middle of the curve, logit estimates are usually about 1.6 to 1.8 times the probit estimates (1.6 matches the slopes of the two curves at their centre, and 1.81 matches their standard deviations). The ratio is a rule of thumb, not a law, but it is why nobody should compare a logit coefficient with a probit coefficient directly.

The fitted probabilities are what really matter, and they rarely differ by more than one or two percentage points except far out in the tails, where the logistic curve approaches 0 and 1 a little more slowly. With ordinary sample sizes, the data can almost never tell the two models apart.

Which one to choose

• Choose logit when you want interpretable effects. An exponentiated logit coefficient is an odds ratio, a quantity most readers in health, psychology and education know how to read. Probit coefficients have no such shortcut; you have to translate them into predicted probabilities or marginal effects.
• Choose logit for case-control or outcome-based sampling. When cases are over-sampled on purpose, the logit slopes (the odds ratios) are still consistent; only the intercept is off. Probit does not have this property.
• Choose probit when a normal latent variable is part of the theory. Examples include threshold models of dose and response, item response models built on the normal ogive, and econometric models that link several equations through correlated normal errors, such as bivariate probit and selection models.
• Follow your field when nothing else decides. Economists often default to probit, while most biomedical and social science journals expect logistic regression. Picking the familiar one saves your readers effort.
• Consider a third option if the curve is asymmetric. Both links are symmetric around a probability of 0.5. When the probability rises slowly and then shoots up (or the reverse), the complementary log-log link may fit better.

Do not choose between them by trying both and keeping the one with the smaller p-value. If you want an empirical check, compare the models' AIC values, but expect the difference to be small; a difference of one or two points is no reason to switch. See the page on AIC vs BIC for how to read such differences.

See it in R and Python

This simulation generates pass/fail outcomes for 500 students from a true logistic model with hours of study as the predictor, then fits both a logit and a probit model with base R's glm(). The Python tab fits the same two models by maximum likelihood with NumPy and SciPy.

set.seed(2052)
n <- 500
hours <- runif(n, 0, 10)                        # hours studied
pass  <- rbinom(n, 1, plogis(-3 + 0.6 * hours)) # true model is logistic

logit  <- glm(pass ~ hours, family = binomial(link = "logit"))
probit <- glm(pass ~ hours, family = binomial(link = "probit"))

# Coefficients differ in scale, not in story
round(rbind(logit = coef(logit), probit = coef(probit)), 3)
round(coef(logit) / coef(probit), 2)            # roughly 1.6 to 1.8

# Fit and predictions are nearly identical
round(c(AIC_logit = AIC(logit), AIC_probit = AIC(probit)), 2)
round(max(abs(fitted(logit) - fitted(probit))), 3)

# Predicted probability of passing at 2, 5 and 8 hours
new <- data.frame(hours = c(2, 5, 8))
round(rbind(logit  = predict(logit,  new, type = "response"),
            probit = predict(probit, new, type = "response")), 3)

# Logit: odds ratio per extra hour with a Wald 95% CI
b  <- coef(summary(logit))["hours", ]
round(c(b[1:4], OR = exp(b[[1]]),
        lower = exp(b[[1]] - 1.96 * b[[2]]), upper = exp(b[[1]] + 1.96 * b[[2]])), 3)
format.pval(b[[4]], digits = 3)

import numpy as np
from scipy import stats
from scipy.special import expit

rng = np.random.default_rng(2052)
n = 500
hours = rng.uniform(0, 10, n)                          # hours studied
passed = rng.binomial(1, expit(-3 + 0.6 * hours))      # true model is logistic
X = np.column_stack([np.ones(n), hours])

def fit_binary(link):
    # Maximum likelihood by Fisher scoring (what R's glm does); returns estimates, SEs, AIC
    cdf, pdf = (expit, lambda e: expit(e) * (1 - expit(e))) if link == "logit" else (stats.norm.cdf, stats.norm.pdf)
    beta = np.zeros(2)
    for _ in range(50):
        eta = X @ beta
        mu = np.clip(cdf(eta), 1e-10, 1 - 1e-10)
        w = pdf(eta) ** 2 / (mu * (1 - mu))
        info = X.T @ (X * w[:, None])
        beta = beta + np.linalg.solve(info, X.T @ ((passed - mu) * pdf(eta) / (mu * (1 - mu))))
    mu = cdf(X @ beta)
    loglik = np.sum(passed * np.log(mu) + (1 - passed) * np.log(1 - mu))
    return beta, np.sqrt(np.diag(np.linalg.inv(info))), -2 * loglik + 2 * len(beta), mu

(b_l, se_l, aic_l, mu_l), (b_p, se_p, aic_p, mu_p) = fit_binary("logit"), fit_binary("probit")

# Coefficients differ in scale, not in story
print("logit ", np.round(b_l, 3), " probit", np.round(b_p, 3))
print("ratio ", np.round(b_l / b_p, 2))                 # roughly 1.6 to 1.8

# Fit and predictions are nearly identical
print("AIC logit", round(aic_l, 2), " AIC probit", round(aic_p, 2))
print("max difference in fitted probabilities", round(np.max(np.abs(mu_l - mu_p)), 3))

# Predicted probability of passing at 2, 5 and 8 hours
new = np.column_stack([np.ones(3), [2, 5, 8]])
print("logit ", np.round(expit(new @ b_l), 3), " probit", np.round(stats.norm.cdf(new @ b_p), 3))

# Logit: odds ratio per extra hour with a Wald 95% CI
z = b_l[1] / se_l[1]
print("b", round(b_l[1], 3), "SE", round(se_l[1], 3), "z", round(z, 3), "p", 2 * stats.norm.sf(abs(z)))
print("OR", round(np.exp(b_l[1]), 3), "CI", np.round(np.exp(b_l[1] + np.array([-1.96, 1.96]) * se_l[1]), 3))

The figures below come from one seeded run of the R code. Python uses a different random number generator, so its figures differ slightly, but the pattern is the same.

• Different coefficients, same story. The logit slope is 0.708 and the probit slope is 0.414. Both intercept and slope are about 1.7 times larger on the logit scale (ratios of 1.72 and 1.71).
• Almost identical predictions. The predicted chance of passing is .104, .493 and .890 at 2, 5 and 8 hours under logit, and .106, .498 and .892 under probit. Across all 500 students, the fitted probabilities never differ by more than 0.014.
• The data cannot pick the true model. Even though the outcomes were generated from a logistic model, the probit model has a slightly lower AIC (428.32 vs 429.62). A gap that small says the two fit equally well.
• Logit gives an odds ratio for free. Each extra hour multiplies the odds of passing by 2.03, 95% CI [1.81, 2.28].

Common mistakes

• Reading probit coefficients as changes in probability. A probit slope is a change in z-score units of the latent variable. Convert it to predicted probabilities or average marginal effects before you describe its size.
• Reading odds ratios as risk ratios. An odds ratio of 2 does not mean the event is twice as likely, unless the event is rare.
• Comparing coefficients across models or links. Logit and probit coefficients are on different scales, and even within one link, adding predictors changes the scale of the others' coefficients.
• Expecting the choice to fix a poor model. Missing predictors, a wrong functional form for a continuous predictor, or ignored clustering matter far more than the link.

How to report logistic regression results in APA style (7th edition)

Say in the method section which model you fitted (for example, "a binary logistic regression" or "a probit regression"), what the outcome was coded as, and how the confidence intervals were computed. Using the logit model above as an example:

"A logistic regression showed that hours of study predicted passing, b = 0.71, SE = 0.06, z = 12.01, p < .001. Each additional hour was associated with roughly double the odds of passing, OR = 2.03, 95% CI [1.81, 2.28]."

For a probit model, report b, SE, z and p the same way, and add predicted probabilities or average marginal effects so readers can see the size of the effect. In a table, give each predictor's b, SE, p and, for logit, the odds ratio with its 95% CI, plus the sample size and a fit statistic such as AIC in a note.

Related tools and guides

• APA 7 formatter for regression
• Which statistical test should I use?
• AIC vs BIC: which should you use?
• Probit model (Wikipedia)
• Logistic regression (Wikipedia)

More answered questions

• Centering vs standardizing predictors in regression: when to do which?
• Is R-squared useful or misleading?
• AIC vs BIC: which model selection criterion should you use?

Written by AskStats with AI assistance. This is general information, not advice for your specific data or study. When the results matter, check your approach with a qualified statistician.