Common Errors In Link-Scale Interpretation

Keep the Scales Separate

A generalized linear model (GLM) connects covariates to a response mean through a sequence of quantities. Many interpretation errors occur when these quantities are treated as interchangeable. The linear predictor is not automatically the mean, a coefficient is not automatically an additive change in the mean, and the link function is not the inverse link. Keeping these distinctions explicit is essential whenever a model uses a nonlinear link.

For observation $i$, write $\mu_i=E[Y_i\mid x_i]$ for the conditional mean, $g$ for the link, and $\eta_i=x_i^\mathsf{T}\beta$ for the linear predictor. The GLM mean structure is $g(\mu_i)=\eta_i$. When $g$ is one-to-one on the relevant mean range, its inverse gives $\mu_i=g^{-1}(\eta_i)$. Here $x_i$ is the covariate vector and $\beta$ is the coefficient vector.

Definition: The linear predictor is $\eta_i=x_i^\mathsf{T}\beta$. The conditional mean is $\mu_i=E[Y_i\mid x_i]$. A link $g$ satisfies $g(\mu_i)=\eta_i$; the inverse link $g^{-1}$ maps the predictor back to the mean, so $\mu_i=g^{-1}(\eta_i)$, when the inverse is defined.

These quantities can coincide under the identity link, but that is a special case, not a general GLM rule. The link and its inverse also have different jobs: $g$ transforms a mean to the predictor scale, while $g^{-1}$ transforms a predictor back to the mean scale.

What a Coefficient Says—and What It Does Not

Holding the other covariates fixed, changing $x_j$ by $\Delta$ changes the linear predictor by $\beta_j\Delta$. In particular, a one-unit increase changes $\eta$ by $\beta_j$. This is the direct coefficient interpretation that applies across links. Its response-scale interpretation depends on the inverse link.

Theorem: Suppose $g$ is one-to-one and differentiable on the relevant mean range, and $g'(\mu)\ne 0$. For a differentiable covariate $x_j$, with the other covariates held fixed, $$ \frac{\partial \mu}{\partial x_j}=\frac{\beta_j}{g'(\mu)}. $$ Thus a coefficient is a constant slope on the predictor scale; the local slope on the mean scale generally depends on the current mean.

Proof. The mean is defined implicitly by $g(\mu)=\eta=x^\mathsf{T}\beta$. Differentiate both sides with respect to $x_j$, holding other covariates fixed. By the chain rule, the left side is $g'(\mu)\,\partial\mu/\partial x_j$, and the right side is $\beta_j$. Since $g'(\mu)\ne0$, division gives the stated expression. $\square$

For a finite covariate change, the exact mean contrast is

\mu(x_j+\Delta)-\mu(x_j) =g^{-1}(\eta+\beta_j\Delta)-g^{-1}(\eta).

This expression makes the dependence on the starting predictor $\eta$ visible. Under the identity link, it simplifies to $\beta_j\Delta$, an additive mean change. Under a nonlinear inverse link, it will generally not simplify to that.

Worked Examples

Worked Example: A Log-Link Coefficient Is Not an Additive Mean Change

Suppose a Poisson GLM has a log link and predictor $\eta= \log(2)+x\log(1.5)$. Compare the fitted means at $x=0$ and $x=1$.

At $x=0$, $\eta=\log(2)$, so the inverse link gives $\mu=\exp(\log(2))=2$. At $x=1$, $\eta=\log(2)+\log(1.5)=\log(3)$, so $\mu=\exp(\log(3))=3$. The mean difference is $3-2=1$, whereas the coefficient is $\log(1.5)$, approximately $0.4055$. The coefficient is not the additive mean change.

The link-scale change is $\log(1.5)$. Exponentiating it gives $\exp(\log(1.5))=1.5$, so a one-unit increase in $x$ multiplies the mean by $1.5$. Indeed, $3/2=1.5$. This is a multiplicative interpretation, not an additive one.

Worked Example: A Logit Coefficient Is Not a Probability Difference

Suppose a Bernoulli GLM has $\eta=-1+0.8x$ and uses the logit link $g(\mu)=\log\{\mu/(1-\mu)\}$. Find the means at $x=0$ and $x=1$, and interpret the coefficient.

The inverse link is $g^{-1}(\eta)=1/(1+\exp(-\eta))$. At $x=0$, $\eta=-1$, giving $\mu=1/(1+\exp(1))\approx 1/(1+2.7183)\approx0.2689$. At $x=1$, $\eta=-0.2$, giving $\mu=1/(1+\exp(0.2))\approx1/(1+1.2214)\approx0.4502$. The probability difference is approximately $0.4502-0.2689=0.1813$, not $0.8$.

The coefficient $0.8$ is the change in log odds for a one-unit increase in $x$. The odds at $x=0$ are $\exp(-1)$; at $x=1$ they are $\exp(-0.2)$. Their ratio is $\exp(-0.2)/\exp(-1)=\exp(0.8)\approx2.2255$. Thus the odds are multiplied by about $2.23$. The coefficient is neither a probability difference nor a direct odds ratio; exponentiating it gives the odds ratio.

Worked Example: The Link Is Not Its Inverse

Consider a Bernoulli model with the probit link $g(\mu)=\Phi^{-1}(\mu)$, where $\Phi$ is the standard normal cumulative distribution function. If a covariate pattern gives $\eta=1$, find the fitted mean and explain why $\eta$ cannot be read as a probability.

The inverse link is $\Phi$, so $\mu=\Phi(1)\approx0.8413$. The predictor is $1$, but the fitted probability is about $0.8413$, not $1$. In general, the link takes a probability to the predictor scale, while the inverse link takes the predictor back to a probability.

The probit link is noncanonical for a Bernoulli response: the canonical link is the logit. Nevertheless, the probit inverse maps every finite real predictor to a value strictly between 0 and 1, so it provides valid Bernoulli means. This example also shows why “noncanonical” does not mean “invalid.”

Two Further Checks on Interpretation

The derivative result has an exact finite-change counterpart. If the predictor changes by a fixed amount, the mean change is determined by applying the inverse link at both endpoints, not by copying the predictor change onto the response scale.

Result: If the other covariates are fixed and $x_j$ changes by $\Delta$, then the predictor changes from $\eta$ to $\eta+\beta_j\Delta$, and the mean changes from $g^{-1}(\eta)$ to $g^{-1}(\eta+\beta_j\Delta)$. The mean contrast equals $\beta_j\Delta$ for all such changes under the identity link. For a general nonlinear inverse link, it need not equal $\beta_j\Delta$.

Derivation. The linear predictor is a sum of the covariate contributions. Replacing $x_j$ with $x_j+\Delta$ changes only its contribution, from $x_j\beta_j$ to $(x_j+\Delta)\beta_j$, so the new predictor is $\eta+\beta_j\Delta$. Applying the inverse link at the old and new predictors gives the two means and their difference. If $g$ is the identity, $g^{-1}(u)=u$, and the difference is $(\eta+\beta_j\Delta)-\eta=\beta_j\Delta$. With a nonlinear inverse link, this cancellation does not generally occur. $\square$

A second check concerns canonicality. In an exponential-family model, the canonical link sets the linear predictor equal to the natural parameter. This is a property of the link’s relationship to the response family; it is not a rule that every valid GLM must use that link. As the probit example illustrates, a noncanonical link can still map predictors to allowable means. Canonicality may have consequences for estimation, but it does not by itself determine how to interpret a coefficient on the mean scale.

Common Mistakes and Why They Matter

Calling $\eta$ the mean. The mistaken reasoning is that the model predicts $\eta$, so $\eta=E[Y\mid x]$. The correction is $g(\mu)=\eta$, hence $\mu=g^{-1}(\eta)$. Equality holds for the identity link, not automatically for other links.
Calling every coefficient an additive mean effect. The mistaken reasoning is that increasing a covariate by one adds $\beta_j$ to the response mean. It adds $\beta_j$ to the predictor. Only under the identity link does that also add $\beta_j$ to the mean.
Confusing a link with its inverse. The mistaken reasoning is to apply $g$ to $\eta$ to recover the mean. The correction is to apply $g^{-1}$ to $\eta$; $g$ acts in the other direction, from mean to predictor.
Assuming canonical links are compulsory. The mistaken reasoning is that a noncanonical link cannot define a GLM. Canonicality means the predictor equals the family’s natural parameter. A noncanonical link can still be used when its mean mapping is appropriate and valid.

These are not merely wording issues. Misreading a log-link coefficient as an additive count change, for example, can produce misleading conclusions about practical impact. Likewise, reporting a logit coefficient as a probability difference confuses a change in log odds with a change in probability. Before interpreting any coefficient, identify the link, write the inverse-link equation, and state which scale the proposed effect belongs to.

Check Your Understanding

For each question, distinguish the predictor-scale statement from the mean-scale statement.

If $g(\mu)=\eta$ and $g$ is one-to-one, which function recovers $\mu$ from $\eta$?
Under a log link, what does a one-unit coefficient $\beta_j$ imply about the ratio of means, holding other covariates fixed?
Under a logit link, is $\beta_j$ a probability difference, a log-odds difference, or an odds ratio? What transformation gives the odds ratio?
Why does the identity link permit an additive mean interpretation of a coefficient that does not generally hold for nonlinear links?
Does a noncanonical link necessarily produce invalid means? Give the reasoning illustrated by the probit example.