Choosing Response Links for Non-Gaussian Markers

Why the Response Link Matters

In a longitudinal model, the linear predictor may contain covariates, functions of time, and subject-specific random effects. The response link determines how that predictor maps to the conditional mean of the marker. Choosing a link is therefore not just a computational decision: it determines which means are possible and what a coefficient means.

The key checks are whether the mean range implied by the link agrees with the marker’s support, whether the chosen response distribution is appropriate, and whether the coefficient interpretation answers the scientific question. These checks apply to the conditional mean given the random effects. A link does not, by itself, specify the entire response distribution or guarantee a good model.

For a binary marker, the conditional mean is a probability. For an ordinal marker, a single mean often does not capture the outcome structure, so cumulative probabilities are modeled instead. For counts and continuous markers, the possible means depend on the outcome and distributional model. The earlier material on binary, probit, ordinal, count, and beta-regression models introduced these settings; here the focus is on comparing their links and coefficient scales.

What the Four Links Permit

The logit and probit links map probabilities strictly between zero and one to the whole real line. The log link maps strictly positive means to the whole real line. The identity link leaves the mean on the predictor’s scale, which has no range restriction. These mappings are useful guides, but the response distribution and the support of the observed marker must also be considered.

A finite logit or probit predictor cannot produce a probability exactly equal to zero or one. Such values can arise in degenerate situations, but they are boundary values, not finite points on either link scale. Likewise, a finite log-linked predictor cannot represent a mean of zero. A count variable that is identically zero has mean zero; ordinary nondegenerate Poisson and negative-binomial models instead have strictly positive conditional means. A log link is common for those positive means, not because every nonnegative count must have a positive mean.

The identity link is flexible in its possible mean range but can be unsafe for restricted outcomes: a fitted probability below zero or a fitted count mean below zero is not admissible. Conversely, the log link preserves positivity but does not enforce an upper bound such as one. For bounded continuous outcomes, a model must respect the outcome’s range; the link and response distribution should be chosen accordingly.

Binary and Ordinal Markers

For a binary marker coded \(Y_{ij}\in\{0,1\}\), \(\mu_{ij}=P(Y_{ij}=1\mid b_i,\mathbf{x}_{ij})\). Under a logit link, a one-unit increase in a covariate with coefficient \(\beta\), holding the other predictor terms fixed, multiplies the conditional odds by \(\exp(\beta)\). It does not generally add a fixed amount to the probability. Under a probit link, the same increase shifts the probit index by \(\beta\); the resulting probability change depends on the starting index.

For an ordinal marker with ordered categories \(1,\ldots,K\), cumulative-link models describe probabilities \(P(Y_{ij}\leq k\mid b_i,\mathbf{x}_{ij})\) for thresholds \(k=1,\ldots,K-1\). A common cumulative-logit convention is \(\operatorname{logit}\{P(Y_{ij}\leq k\mid b_i,\mathbf{x}_{ij})\}=\theta_k-\eta_{ij}\), with ordered thresholds \(\theta_1<\cdots<\theta_{K-1}\). Under this sign convention, a positive predictor coefficient shifts probability toward higher categories. The same threshold form can use a probit link, replacing the logit with \(\Phi^{-1}\).

Count and Continuous Markers

For a count marker, a log link gives \(\log(\mu_{ij})=\eta_{ij}\) and hence \(\mu_{ij}=\exp(\eta_{ij})>0\). With a one-unit increase in a covariate, the conditional mean is multiplied by \(\exp(\beta)\), provided the model’s other terms are held fixed. This interpretation is a mean ratio, not an additive increase in the expected count. The assumed count distribution must also be appropriate; choosing a log link does not resolve overdispersion or excess zeros.

For a continuous marker with an unrestricted real-valued conditional mean, an identity link allows \(\mu_{ij}=\eta_{ij}\). A coefficient is then an additive change in the conditional mean, in the marker’s measurement units. If a continuous outcome is strictly positive and a log-linked mean is appropriate, exponentiating a coefficient instead gives a multiplicative change in that mean. The outcome distribution and residual variation still need to be specified and checked; changing the link alone does not make a Gaussian error model appropriate for every positive outcome.

A Short Derivation: Coefficients and Mean Changes

The link scale explains why the same coefficient size need not imply the same mean-scale change. If \(g(\mu)=\eta\) and a covariate \(x\) enters \(\eta\) linearly with coefficient \(\beta\), then \(d\eta/dx=\beta\). Differentiating both sides with respect to \(x\) gives \(\frac{d\mu}{dx}=\frac{\beta}{g'(\mu)}\).

For the identity link, \(g'(\mu)=1\), so \(d\mu/dx=\beta\): the mean change per unit \(x\) is constant. For the log link, \(g'(\mu)=1/\mu\), so \(d\mu/dx=\beta\mu\): the absolute change depends on the current mean, while the relative change is constant because \((1/\mu)(d\mu/dx)=\beta\). For the logit link, \(g'(\mu)=1/\{\mu(1-\mu)\}\), giving \(d\mu/dx=\beta\mu(1-\mu)\). Thus its probability change varies with the current probability. This derivative argument is local; finite changes should be computed by evaluating the inverse link at both predictor values.

Choosing and Checking a Link

Use the outcome’s support as a first filter, not as the only criterion. A link should produce admissible conditional means over the predictor values that matter. Then check whether the response distribution, random-effects structure, and scientific interpretation are plausible. For binary markers, logit and probit both enforce probabilities in \((0,1)\); logit gives a direct odds-ratio interpretation, while probit gives a normal-index interpretation. For ordinal outcomes, cumulative logit or probit respects the category ordering through cumulative probabilities and thresholds. For count outcomes with strictly positive conditional means, the log link is a standard choice. For continuous outcomes with means that may be any real number, identity is often natural.

The alternatives are not interchangeable merely because they fit the same data type. A logit and probit can yield similar fitted probabilities over some ranges, but their coefficients are on different scales. An identity link can offer direct mean differences, but it can produce inadmissible means for binary or count outcomes. A log link ensures positive means, but not means below one or any other upper bound. Link selection should follow the outcome and estimand, with model fit and diagnostics used to assess the choice.

Useful checks include examining the range of fitted means, convergence warnings, and whether estimates are dominated by sparse categories or extreme predictors. For ordinal cumulative models, check that estimated thresholds are ordered and assess whether the common predictor effect across cutpoints is defensible. For count models, compare observed variability with the assumed response distribution and inspect residual patterns. For continuous models, inspect residual behavior and whether fitted means are plausible in the measurement units. These checks can reveal a poor link or a broader model misspecification; none alone proves that one link is correct.

A related mistake is to interpret every coefficient as a constant change in the observed marker. That holds for an identity-linked mean, not generally for a nonlinear link. With logit, the coefficient is a change in log odds; with probit, it is a change in the probit index; with log, it is a change in log mean. To report a probability or absolute mean difference, evaluate the inverse link at specified predictor values and state the covariate and random-effects conditions used.