Link-Scale and Response-Scale Effects

Why the Scale of an Effect Matters

A coefficient in a marginal mean model describes a change on the model’s link scale. Its practical interpretation depends on the link: the same kind of coefficient may represent a difference, a ratio, or an odds ratio. Even when a coefficient is constant on the link scale, the corresponding difference between mean responses need not be constant.

Here, \(\mu\) denotes a marginal response mean, such as the probability of an outcome in the population or the average count for a measurement occasion. We compare two covariate values while holding the other parts of the comparison fixed. This tutorial concerns translating that mean contrast between scales; it does not by itself give a causal interpretation to the contrast.

For the comparisons below, suppose a covariate changes by \(\delta\), and the model’s linear predictor changes by \(\beta\delta\). Thus the link-scale contrast is \(\beta\delta\). For a one-unit increase, \(\delta=1\), it is \(\beta\). This statement concerns the particular comparison being made; it does not assume that every covariate or every model has the same form.

Identity Link: The Coefficient Is a Mean Difference

The identity link is \(g(\mu)=\mu\), so its inverse is \(g^{-1}(\eta)=\eta\). If the linear predictor changes by \(\beta\delta\), the response mean changes by precisely that amount:

The difference does not depend on the baseline mean. For example, a coefficient of \(-2.4\) means a decrease of 2.4 response units per one-unit increase in the covariate, for comparisons to which that coefficient applies. The identity link does not guarantee that all fitted means are valid for every outcome: a model for a nonnegative response, for instance, may require attention to whether its predicted means are nonnegative.

Log Link: A Constant Ratio, Not a Constant Difference

For a positive mean, the log link is \(g(\mu)=\log(\mu)\), with inverse \(g^{-1}(\eta)=e^\eta\). A link-scale contrast of \(\beta\delta\) implies a ratio of response means:

Thus, for a fixed covariate change, the mean ratio is the same at different baseline means. The difference is instead \(\mu_1-\mu_0=\mu_0\{e^{\beta\delta}-1\}\), so it is proportional to the baseline. If \(\beta>0\), the mean increases; if \(\beta<0\), it decreases. The ratio formula is exact, not an approximation.

Logit Link: A Constant Odds Ratio, Not a Constant Probability Difference

For a binary response with mean \(\mu=P(Y=1)\), the logit link is \(g(\mu)=\log\{\mu/(1-\mu)\}\), where \(0<\mu<1\). Its inverse is the logistic function:

A link-scale contrast of \(\beta\delta\) is a log odds ratio. Exponentiating gives the odds ratio \(e^{\beta\delta}\). This does not make the probability difference constant: the inverse link is nonlinear, and a given change in log odds yields different probability changes at different baseline probabilities.

Deriving the Dependence on the Baseline

The examples illustrate a general calculation. Let \(\eta_0=g(\mu_0)\) be the baseline linear predictor, and suppose the covariate change shifts it by \(c=\beta\delta\). The response mean after the change is \(g^{-1}(\eta_0+c)\). Consequently, the response-scale difference is

This identity follows directly from the definition of the inverse link. Whether \(\Delta\mu\) depends on the baseline is determined by the shape of \(g^{-1}\). For the identity link it does not; for the log and logit links, the difference generally varies with the baseline.

Proof. By the definition of link-scale contrast, \(g(\mu_1)-g(\mu_0)=c\), so \(g(\mu_1)=g(\mu_0)+c\). Since \(g\) is one-to-one, applying its inverse gives \(\mu_1=g^{-1}\{g(\mu_0)+c\}\). Under the identity link, this is \(\mu_1=\mu_0+c\), hence the difference is \(c\). Under the log link, \(\log(\mu_1)=\log(\mu_0)+c\); exponentiation gives \(\mu_1=\mu_0e^c\), and division by the positive \(\mu_0\) gives the stated ratio. Under the logit link, write \(o_j=\mu_j/(1-\mu_j)\) for \(j=0,1\). The link equation gives \(\log(o_1)-\log(o_0)=c\). Exponentiating yields \(o_1/o_0=e^c\). Each conclusion follows from the specified link and the stated domain conditions. \(\square\)

A local change gives another view of why the baseline matters. If \(g^{-1}\) is differentiable, the derivative of the mean with respect to the linear predictor is \((g^{-1})'(\eta)\). Along a covariate change in which the linear predictor changes at rate \(\beta\), the derivative of the mean with respect to that covariate is \(\beta(g^{-1})'(\eta)\).

Derivation. Since \(\mu(x)=g^{-1}\{\eta(x)\}=g^{-1}(a+\beta x)\), the chain rule gives \(\frac{d\mu(x)}{dx}=(g^{-1})'(a+\beta x)\frac{d(a+\beta x)}{dx}=\beta(g^{-1})'\{\eta(x)\}\). For the identity inverse link, the derivative is 1. For the log inverse link, it is \(e^\eta=\mu\), so the local change is \(\beta\mu\). For the logistic inverse link, it is \(\mu(1-\mu)\), so the local change is \(\beta\mu(1-\mu)\). These last two derivatives depend on the current mean. \(\square\)

Interpreting Effects Carefully

For marginal mean models, the means being compared are marginal means, not automatically subject-specific conditional means. The scale translation does not change which mean the model represents. Nor does a coefficient, by itself, establish that changing a covariate would cause the response to change: causal claims require additional design and assumptions.

When reporting an effect, name both the scale and the comparison. For example, “the mean is multiplied by \(e^\beta\) per unit” is a log-link interpretation; “the odds are multiplied by \(e^\beta\) per unit” is a logit-link interpretation. If a response-scale difference is important, calculate it at a stated baseline mean or probability rather than treating it as constant.

Common Mistake

Wrong reasoning: “The coefficient is \(\beta\), so the response mean increases by \(\beta\) everywhere.” This is valid for a coefficient contrast under the identity link, but not in general. Under a log link, \(\beta\) is a log mean ratio for a one-unit increase; under a logit link, it is a log odds ratio. Neither generally equals the change in the mean or probability.

Correction: Identify the link, transform the coefficient to its link-appropriate comparison, and use the baseline mean when calculating a response-scale difference. Report the covariate change and units as well: a coefficient per one unit cannot be interpreted as a per-ten-unit effect without multiplying the link-scale contrast by ten.