Link Functions For Binary Imputation Models

Why Compare Links for a Binary Imputation Model?

A binary imputation model turns a predictor-based score into a probability for the two possible outcomes. The link function determines that conversion. In a MICE update, the probability then informs a Bernoulli draw for each missing value. Logit and probit links both produce valid probabilities between zero and one, but their probability curves are not identical. Consequently, they can yield different fitted probabilities—and sometimes different imputed values—especially when the predictor score is far from zero.

The choice is not a contest between a “correct” and an inherently “incorrect” link. It is a modeling decision about how the conditional probability changes with the predictors. A useful comparison therefore asks whether the links imply materially different probabilities for the predictor patterns in the data, not merely whether their coefficient estimates have similar numerical values.

Both models assume a Bernoulli conditional distribution for \(Y_j\), with the probability determined by the chosen inverse link. They also impose a linear form for the link-transformed probability in the included predictors. These assumptions concern the conditional distribution used in the update; neither link makes a misspecified predictor set or an inappropriate linear form automatically adequate. As in any conditional imputation model, the predictors needed for the intended analysis must be represented appropriately, and the fitted model must be estimable from the observed values of \(Y_j\).

What the Two Probability Scales Mean

Under a logit model, \(\eta(X)\) is the log odds of \(Y_j=1\). Increasing a predictor by one unit, while holding the other predictors fixed, changes the log odds by its coefficient; the corresponding odds are multiplied by the exponential of that coefficient. The coefficient is not a constant change in probability: the probability change depends on the starting probability.

Under a probit model, \(\eta(X)\) is a standard-normal quantile corresponding to the conditional probability. A one-unit predictor increase changes that normal-score scale by its coefficient. Probit coefficients are not log odds ratios, and exponentiating one does not give an odds ratio. In either model, fitted probabilities are often the clearest scale for comparing the implications of a link for imputation.

At a given numerical value of \(\eta\), the inverse links can differ substantially. The following values illustrate the contrast. The table compares the links at the same linear predictor; it does not imply that fitted logit and probit models should have identical coefficients.

Near zero, the probit curve is steeper than the logit curve on the same numerical scale: their derivatives at zero are \(1/\sqrt{2\pi}\approx0.3989\) and \(1/4=0.25\), respectively. Far into either tail, the logistic distribution has heavier tails than the standard normal distribution. These shape differences explain why equal numerical linear predictors do not generally imply equal probabilities.

The coefficient scales also differ. In practice, fitted logit and probit coefficients often differ in magnitude while implying similar probabilities over the observed predictor range. A rough scale conversion is sometimes used as a rule of thumb, but it is not an exact identity and should not replace comparing predicted probabilities. In particular, coefficient signs and substantive patterns are usually more comparable than raw coefficient magnitudes.

Why the Probit Link Has a Latent-Variable Interpretation

A short derivation clarifies the probit probability scale. Suppose a latent continuous response is \(Y_j^*=\eta(X)+\varepsilon\), where \(\varepsilon\sim N(0,1)\), and the observed binary outcome is \(Y_j=1\) exactly when \(Y_j^*>0\). Then

The last equality follows from the symmetry of the standard normal distribution. The latent error variance is fixed at one to set the scale: multiplying both the latent response and its coefficients by a positive constant would leave the observed threshold outcome unchanged. Thus probit coefficients are defined on a normalized latent scale, not on the observed binary scale.

The logit model has a parallel representation using a logistic-distributed latent error, whose cumulative distribution function is \(1/\{1+\exp(-t)\}\). This gives the inverse-logit probability. The two latent error distributions have different shapes, so their tails and resulting probabilities are not exactly interchangeable.

From Fitted Probabilities to Imputed Values

For a binary MICE update, the selected link is fitted using records with observed \(Y_j\), conditional on the predictors in that update. For a missing \(Y_j\), the fitted model supplies a probability of one. In proper multiple imputation, uncertainty in model parameters is represented as well: a parameter draw gives a probability for the record, and the binary outcome is then drawn from the corresponding Bernoulli distribution. The earlier material on Bernoulli draws from logistic probabilities establishes that last probability-to-outcome step; the same Bernoulli step applies to probabilities from a probit model.

A change in link affects imputed values only through the probabilities it produces, including their changes under parameter uncertainty. If two fitted models give nearly identical probabilities for all records with missing outcomes, their imputations will usually be similar in distribution. If their probabilities differ appreciably, individual draws may differ even when both models are otherwise plausible.

When Link Choice Matters, and How to Check

Link choice is most consequential when the competing models produce meaningfully different probabilities for records whose \(Y_j\) values are missing, or when those records have predictor patterns at which the two fitted curves diverge. Differences can also matter when an imputed variable is an important predictor in later conditional updates or when the target analysis is sensitive to the number of imputed ones in particular subgroups. Similar average probabilities overall can conceal important disagreement in a subgroup or in the tails.

Useful checks focus on fitted behavior rather than on coefficient magnitudes alone. For each candidate link, inspect predicted probabilities over the observed predictor range; compare those probabilities for records with missing outcomes; and examine whether predicted probabilities align reasonably with observed proportions across relevant predictor patterns. Also check that chain summaries and the distributions of imputed values are stable enough for the intended analysis. A large number of probabilities extremely close to zero or one, large changes across links, or predictions unsupported by observed patterns should prompt investigation of the model and data.

These checks are not definitive tests that select a universally correct link. Observed outcomes are available only for some records, and apparent agreement in observed-data summaries does not guarantee that unobserved outcomes are correctly predicted. Sensitivity analysis—fitting both links and comparing resulting imputations and substantive estimates—can show whether the link choice has practical consequences. The appropriateness of the imputation model’s missing-data assumptions remains a separate issue; changing the link does not resolve a failure of those assumptions.

The main alternative to changing the link is to retain one link and improve the conditional model where its form is inadequate—for example, by reconsidering predictor coding or functional form when justified. Such changes should be substantively defensible and checked, rather than added solely to make one model’s predictions resemble the other’s. Neither link guarantees compatibility with every other conditional model in a chained-equations system.

Common Mistakes and Limitations

• Comparing coefficient magnitudes directly. A probit coefficient is on a normalized normal-score scale; a logit coefficient is a change in log odds. Compare predicted probabilities or carefully interpreted link-scale effects instead.
• Assuming the links yield the same probabilities because both are S-shaped. They agree at a score of zero but have different slopes and tails. Check probabilities at the predictor values that occur in the imputation data.
• Calling a single different draw evidence of model failure. Bernoulli draws are random. Compare predicted probabilities and repeated imputation behavior, not one pair of imputed records.
• Treating link choice as a remedy for every binary-model problem. Neither link repairs an omitted predictor, inadequate functional form, unsupported predictor patterns, or unsuitable missing-data assumptions.

A practical limitation is that data may not strongly distinguish the links over the predictor range actually observed. In that case, both may give similar imputations despite different mathematical forms. Conversely, disagreement in extreme regions may be difficult to resolve from observed outcomes if few records occur there. Report the choice and any meaningful sensitivity rather than claiming that one link is proved correct by a small numerical difference.