Probit and Complementary Log-Log Response Models

Why Use More Than One Binary-Response Link?

A binary outcome records one of two possibilities, such as whether a response occurred by a visit or whether an adverse event arose during an interval. A mixed-effects model can account for repeated outcomes from the same individual by including an individual-specific random effect in its linear predictor. The link function then maps that predictor to a conditional probability. Probit and complementary log-log links make different assumptions about how that mapping behaves.

Probit is useful when a threshold on an underlying normally distributed response is a plausible representation. The complementary log-log link is asymmetric on the probability scale and has a direct interpretation for event probabilities over intervals under a hazard model. Neither link is automatically preferable: the scientific mechanism, design, diagnostics, and predictions all matter.

Model Definition and Assumptions

The two links considered here are:

Here \(\Phi\) is the cumulative distribution function of a standard normal random variable. In a typical formulation, random effects are modeled with a mean-zero distribution, often multivariate normal, and observations from different individuals are independent conditional on their covariates and random effects. Repeated outcomes from the same individual are conditionally independent given that individual’s random effects and the covariates, unless the model specifies additional dependence.

These assumptions are substantive. The observation schedule and event definition must be clear; the link and linear predictor must be suitable; and the random-effects distribution must be a defensible representation of between-individual variation. The probit latent-variable interpretation also fixes the residual latent scale: its variance is set to one. If that variance were freely estimated, it could not be separated from the scale of the coefficients using binary outcomes alone.

Probit: A Threshold on a Latent Normal Response

The probit model can be represented by an unobserved continuous response \(Y^{*}_{ij}\). Conditional on covariates and random effects, suppose \(Y^{*}_{ij}=\eta_{ij}+\epsilon_{ij}\), where \(\epsilon_{ij}\) is standard normal. The binary response equals one when the latent response exceeds zero and equals zero otherwise. Therefore:

The last equality follows from the symmetry of the standard normal distribution. Thus a one-unit increase in a covariate with coefficient \(\beta_k\) shifts the latent response by \(\beta_k\) units, holding the other terms fixed. It does not generally change the event probability by a constant amount. For a continuous covariate, the conditional probability derivative is \(\phi(\eta_{ij})\beta_k\), where \(\phi\) is the standard normal density; the probability change depends on the current predictor and is largest near the middle of the probability range.

Complementary Log-Log: An Asymmetric Probability Scale

The complementary log-log link transforms the probability of no event, \(1-p_{ij}\), rather than treating increases and decreases in probability symmetrically. When \(\eta_{ij}\) is very negative, \(\exp(\eta_{ij})\) is small and \(p_{ij}\) is approximately \(\exp(\eta_{ij})\). As \(\eta_{ij}\) grows, the event probability approaches one. This asymmetry can be useful when a response reflects an event accumulating over time.

For example, suppose that conditional on the covariates and random effects, the event has a constant hazard \(\lambda_{ij}=\exp(\eta_{ij})\) during an interval of length one. The probability of no event in the interval is \(\exp(-\lambda_{ij})\), so the probability of at least one event is \(1-\exp(-\lambda_{ij})\), exactly the complementary log-log probability. If the interval has length \(\Delta_{ij}\), the corresponding formula is \(1-\exp\{-\Delta_{ij}\exp(\eta_{ij})\}\). Equivalently, the linear predictor includes the offset \(\log(\Delta_{ij})\) when \(\eta_{ij}\) represents the log hazard per unit time. This hazard interpretation requires the stated interval-hazard assumptions; the link can also be used as a binary-response link without claiming that mechanism.

Under this interval-hazard interpretation, a one-unit increase in a covariate with coefficient \(\beta_k\) multiplies the conditional hazard by \(\exp(\beta_k)\), holding other terms fixed. It does not multiply the event probability by that amount, because the conversion from hazard to probability is nonlinear.

Comparing the Probability Scales

A coefficient’s numerical value is tied to its link scale. The probit coefficient shifts a latent normal score; the complementary log-log coefficient shifts the log cumulative hazard in the interval-event interpretation. Comparing the coefficients directly across links is therefore not a comparison of probability differences. Predictions at relevant covariate values are often more interpretable.

Derivation and Practical Interpretation

The event-hazard formula also justifies the complementary log-log transformation directly. Let the conditional probability of no event over a unit interval be \(1-p=\exp(-\lambda)\), with \(\lambda=\exp(\eta)\). Taking the negative logarithm gives \(-\log(1-p)=\lambda\). Taking logarithms again yields \(\log\{-\log(1-p)\}=\log(\lambda)=\eta\). This two-step transformation is precisely the complementary log-log link. For interval duration \(\Delta\), replacing \(\lambda\) by \(\Delta\lambda\) gives \(\log\{-\log(1-p)\}=\log(\Delta)+\eta\), which explains the duration offset.

Both models produce probabilities conditional on \(b_i\). A population-averaged probability instead averages those conditional probabilities over the distribution of random effects. Since both inverse links are nonlinear, applying the inverse link to the predictor with \(b_i\) set to its mean does not generally equal that average. Accordingly, state whether an interpretation is conditional on an individual effect or averaged over individuals.

Diagnostics, Alternatives, and Limitations

A fitted link should be assessed against the data and the scientific structure, rather than selected only for convenience. Useful checks include comparing observed and predicted event proportions across ranges of fitted probability, covariate, and observation time; examining whether predicted probabilities are systematically too high or too low in particular regions; and assessing whether event patterns over intervals are compatible with a proposed constant-hazard interpretation. With repeated measurements, also check whether the model reproduces the observed pattern of outcomes within individuals. Such discrepancies may reflect a poor link, an inadequate predictor, a questionable random-effects distribution, or a misspecified dependence structure.

Probit and logistic links are symmetric in their central response scale and often yield similar fitted probabilities, though their coefficient scales differ. The complementary log-log link is asymmetric and may be more natural when events accumulate over intervals or when probabilities rise rapidly toward one. These are modeling considerations, not guarantees of better fit. The complementary log-log hazard interpretation is not justified merely because that link was used: interval definition, event timing, and hazard assumptions must support it.

• Boundary data: Very few events or non-events in covariate regions can make estimates unstable and predictions overly dependent on model assumptions.
• Unmodeled timing: If interval lengths differ, omitting the duration effect can distort probabilities when the model is intended to represent a constant event hazard.
• Overconfident predictions: Predictions based only on fixed effects omit between-individual variation and need not represent the population-average probability.
• Misread coefficients: A probit coefficient is not a fixed probability difference, and a complementary log-log coefficient is not a direct probability ratio.

The models also have limits. A probit threshold story may be a useful representation without being a literal biological measurement of a latent normal variable. A constant-hazard interval interpretation can be implausible when risk changes substantially within intervals. Neither link alone resolves omitted covariates, poor timing information, or an unsuitable random-effects model. Model choice should therefore combine mechanistic plausibility with probability-scale diagnostics and sensitivity to reasonable alternatives.