Probit and Complementary Log-Log Treatment Models

Why Use More Than One Binary Treatment Link?

A binary treatment model estimates the probability of treatment conditional on the covariates included in the model. The logistic link is one familiar choice, but it is not the only way to map a linear predictor to a probability. Probit and complementary log-log models use different maps, and therefore can give different predicted treatment probabilities even when they use the same covariates.

That difference matters when the fitted probabilities represent treatment decisions: a small change in a probability can have a substantial effect on the resulting inverse probability weights. The goal here is to understand how each model is fitted and how its link shapes the probabilities supplied to a weighting analysis. The link alone does not establish that a treatment model is causally adequate; the covariate history and assumptions behind the treatment mechanism remain essential.

For either link, the model assigns probability \(q_i\) to treatment and \(1-q_i\) to no treatment. The linear predictor \(\eta_i=X_i^\top\beta\) is not itself a probability. First estimate \(\beta\), then transform each fitted linear predictor through the chosen inverse link.

Likelihood and Score Equations

Assume for this derivation that the treatment indicators are conditionally independent given the observed covariates, with \(A_i\mid X_i\) Bernoulli with probability \(q_i=h(X_i^\top\beta)\). For fixed covariates, the likelihood and log-likelihood are

A maximum likelihood estimate is a value of \(\beta\) that maximizes this log-likelihood. It is generally found numerically; there is no general closed-form solution. The next derivation gives the score being solved by such an optimization.

Derivation. For one observation, differentiate its log-likelihood with respect to \(q_i\):

Since \(\partial q_i/\partial\beta=h'(\eta_i)X_i\), the chain rule gives that observation’s contribution to the score. Summing over \(i\) yields the stated expression. A finite maximum likelihood estimate, when it exists, has score zero; existence and uniqueness are not guaranteed simply because the score equation can be written. For example, complete separation can prevent a finite estimate. \(\square\)

For probit, \(h'(\eta)=\phi(\eta)\), where \(\phi\) is the standard normal density. For complementary log-log, writing \(r=\exp(\eta)\) gives \(h'(\eta)=r\exp(-r)\). Substituting the relevant derivative into the score specifies the estimating equation for each model. In practice, software maximizes the corresponding likelihood, and a fitted model’s predicted probability is obtained from its fitted linear predictor.

What the Complementary Log-Log Link Represents

The complementary log-log model is asymmetric about probability \(0.5\), unlike probit. It also has a useful exact representation in terms of a positive rate. If \(\lambda=\exp(\eta)\), then \(\lambda>0\) and the model’s probability of treatment is \(1-\exp(-\lambda)\). This representation explains a common connection to a discrete-time event model, but a binary treatment model does not automatically describe an event occurring over time.

Derivation. Starting with \(q=1-\exp\{-\exp(\eta)\}\), subtract from one to obtain \(1-q=\exp\{-\exp(\eta)\}\). Taking negative logarithms gives \(-\log(1-q)=\exp(\eta)\). Taking logarithms again gives \(\log\{-\log(1-q)\}=\eta\). Conversely, if \(\lambda=\exp(\eta)>0\), then \(q=1-\exp(-\lambda)\), as claimed. \(\square\)

For small \(\lambda\), \(1-\exp(-\lambda)\) is approximately \(\lambda\); as \(\lambda\) grows, the probability approaches one. This gives the complementary log-log link its different tail behavior from probit. A positive coefficient increases \(\eta\), and hence increases \(q\), but the probability-scale change depends on the starting value of \(\eta\).

From a Fitted Model to the Observed-Treatment Probability

For a unit with covariates \(X_i\), the model predicts \(\hat q_i\), the probability of \(A_i=1\). If the observed treatment is zero, the corresponding fitted probability is \(1-\hat q_i\); if it is one, it is \(\hat q_i\). These are the model-based probabilities for the treatment category actually observed. They are the treatment-model predictions that enter inverse probability weighting; how they are assembled into weights is a separate step.

Choosing, Checking, and Comparing the Models

With the same covariates, probit and complementary log-log models differ in their link functions, not in what the binary outcome means. Their coefficients are on different scales and should not be compared numerically as if a one-unit coefficient had the same probability effect in both models. Compare fitted probabilities over relevant covariate patterns instead. If the models are being considered as alternatives, assess whether their predictions are plausible and well calibrated for the application, rather than choosing solely by a coefficient’s size.

The model likelihood above assumes conditional independence across units. When assignments are clustered or otherwise dependent, treating every individual Bernoulli contribution as an independent likelihood contribution may be inappropriate. The treatment probability model still needs to reflect the assignment structure, and uncertainty calculations need to respect the dependence; changing the link does not solve a dependence problem.

Useful checks include examining predicted probabilities across covariate patterns, comparing predicted and observed treatment frequencies in groups, checking for extreme fitted probabilities, and looking for separation or convergence warnings. Calibration discrepancies can indicate a poor link, missing predictors, or other model misspecification. Extreme predictions can signal sparse covariate patterns or near-separation. They are also a warning that the corresponding weighting analysis may be unstable; a different link cannot create information where the data contain little treatment overlap.

The probit link may be a reasonable choice when its symmetric response curve is substantively plausible. The complementary log-log link is useful when a one-sided, asymmetric response is more plausible or when the positive-rate representation is meaningful. Neither link is universally preferable. Both rely on the specified covariates and functional form adequately representing the treatment mechanism, and both can produce poor predictions if those specifications are inadequate. The standard logistic model is another alternative; the relevant comparison is predictive behavior and substantive plausibility, not a presumption that one named link is correct.

Common Mistakes

• Treating the linear predictor as a probability. A value such as \(\hat\eta=0.4\) must be transformed: it gives \(\Phi(0.4)\) under probit and \(1-\exp\{-\exp(0.4)\}\) under complementary log-log.
• Interpreting coefficients as probability differences. A coefficient changes \(\eta\); the resulting probability change depends on the link and the starting covariates.
• Using the treatment probability when the observed treatment is zero. The probability of the observed category in that case is \(1-\hat q\), not \(\hat q\).
• Assuming that a different link fixes misspecification. A link choice cannot compensate automatically for omitted predictors, a wrongly specified functional form, dependence, or sparse data.