Predicting yes or no, with the normal distribution? (probit regression)

In brief

Can a yes/no event be predicted? Probit regression predicts it like logistic regression but uses the normal distribution instead of the logistic distribution as the link. Both almost always give the same message; "Logit (default)" stays the default.

How it works in Statable

1. "New analysis", then choose the card "Can something be predicted?"; the target has two values.
2. Choose the target column and the predictor columns.
3. "More options" → "Model" → "Probit (normal distribution)".
4. In the result b is on the scale of the normal distribution; an odds ratio exists only for logit.

Logit or probit?

Both almost always give the same message: which predictors matter, in which direction and roughly how strongly. The choice is not a matter of the result. Probit is usual in some fields (for example psychometrics and toxicology); otherwise logit is the default because odds ratios can be given there.

Reading the result

• b is on the scale of the normal distribution: it is the change in the probit value (z value) per unit of the predictor. A positive b raises the probability of the event. An odds ratio exists only for logit; Statable shows none here.
• For each b the table shows SE, z, p and the 95% interval (Wald).
• Model test: the deviance of the model is compared with that of a model without predictors (χ², df, p). AIC and the number of cases are also shown.

Trap: almost separable data

If the predictors separate the two outcomes almost or fully, the fitted probabilities of single cases are 0 or 1. The standard errors are then unreliable. Statable calculates with high precision so that such data are right too, and always shows a warning from R; a warning is never swallowed. If the calculation does not arrive at a result (no convergence), that is shown in the result too.

Difference from SPSS

The estimates b are the same as in SPSS (GENLIN). The standard errors differ by a few percent (0.5 to 6.6% in the check) because SPSS uses the Hessian and R the Fisher information. Both are usual; Statable uses R. Accordingly z, p and the intervals differ slightly.

Requirements and limits

• The target has exactly two values, the cases are independent of each other. Rows with a gap are not included; the number is shown in the result.
• Few events: as a rule of thumb about 10 cases of the rarer outcome are needed per predictor. Otherwise Statable points this out.

Reporting the result

Probit model: χ²(2) = 10.00, p = .006; b = 0.40, SE = 0.10, z = 4.00, p < .001

Typical examples

• Does someone pass the exam (yes/no) depending on study time and prior knowledge?
• Does someone drop out (yes/no) depending on motivation and age?

For specialists

R (stats): glm(y ~ ..., family = binomial("probit"), control = glm.control(epsilon = 1e-12, maxit = 100)). The high precision is needed, otherwise almost separable data differ. Checked on mtcars and birthwt against SPSS.