Common Interpretation Errors With Alternative Links

Why Link Interpretation Requires Care

In a binary generalized linear model, the conditional success probability \(p(x)=P(Y=1\mid X=x)\) is related to a linear predictor \(\eta(x)=x^\mathsf{T}\beta\) by \(g\{p(x)\}=\eta(x)\), where \(g\) is the link function. A coefficient therefore describes a change on the link scale. Its interpretation on the probability scale depends on the chosen link.

A frequent source of error is to see a coefficient and automatically exponentiate it, then call the result an odds ratio. That interpretation is exact for the logit link, but not for probit, identity, or most other links. Even when exponentiation has a useful interpretation, it may describe a risk ratio or a transformed probability measure rather than an odds ratio. Probability changes also depend on the starting probability for nonlinear links.

Suppose two profiles differ by one unit in a continuous covariate \(x_j\), with all other covariates held fixed. Their linear predictors differ by \(\beta_j\). For a binary covariate, the same coefficient describes the change from its reference category to the category coded as one, again holding the other model terms fixed. What that predictor change implies for risks, odds, or their ratios is determined by the link.

What a Coefficient Means Under Different Links

Here \(\Phi\) is the standard normal cumulative distribution function. In the complementary log-log row, the multiplied quantity is a transformation of the event probability; it is not the odds or the risk itself. For a log-link model, probabilities must remain between zero and one over the covariate values where the model is used, so the claimed risk-ratio interpretation applies only to feasible predictions.

Logit Coefficients: Odds Ratios, Not Risk Differences

For the logit link, exponentiating a coefficient has an exact and standard interpretation as an odds ratio. The coefficient does not, in general, equal a risk difference, and its exponential is not a risk ratio.

To verify this, write the logit model at the two profiles as \(\log\{p_0/(1-p_0)\}=\eta_0\) and \(\log\{p_1/(1-p_1)\}=\eta_0+\beta_j\). Subtracting gives

Exponentiating both sides proves that the odds ratio is \(\exp(\beta_j)\). The proof concerns odds; it does not imply that \(p_1-p_0=\beta_j\) or that \(p_1/p_0=\exp(\beta_j)\). \(\square\)

Probit Coefficients Are Normal-Score Shifts

Under the probit link, \(p(x)=\Phi\{\eta(x)\}\). A coefficient shifts the argument of the normal distribution function. It is measured in standard-normal score units, not in odds units. There is no general rule that exponentiating a probit coefficient produces an odds ratio.

Exponentiation Has Different Meanings for Other Links

A useful way to avoid overgeneralizing is to exponentiate the model equation—not to assume in advance that the result is an odds ratio. With a log link, \(\log(p)=\eta\), so for profiles differing by one unit in \(x_j\), \(p_1/p_0=\exp(\beta_j)\), provided the risks are positive and the model predictions are feasible. This is a risk ratio. With a complementary log-log link, exponentiating instead gives a ratio of \(-\log(1-p)\), not a ratio of odds or risks.

The identity link is different again: \(p=\eta\), so a one-unit increase in \(x_j\) changes the modelled risk by \(\beta_j\), provided both predictions are valid probabilities. For example, a coefficient of \(0.04\) corresponds to a four-percentage-point risk increase for that contrast, not a four-percent relative increase. But identity-link predictions can fall outside \([0,1]\), so the interpretation cannot be applied indiscriminately to every possible covariate profile.

Local Probability Effects Are Not Constant Risk Differences

For nonlinear links, even a fixed coefficient generally does not produce the same probability change at every starting risk. Let \(h=g^{-1}\) be the inverse link, assumed differentiable at the relevant predictor values. For a continuous covariate \(x_j\), with other covariates held fixed, \(p(x)=h\{\eta(x)\}\) and \(\partial\eta/\partial x_j=\beta_j\). The chain rule gives

This is the local probability effect: the instantaneous rate of change in risk for a small change in \(x_j\). Its derivation requires differentiability of \(h\) at \(\eta(x)\) and a linear predictor whose derivative with respect to \(x_j\) is \(\beta_j\). For a finite change from \(x_j=a\) to \(x_j=b\), the fundamental theorem of calculus gives, when \(h'\) is continuous on the predictor interval,

Unless \(h'\) is constant over that interval, the finite risk difference is not generally \((b-a)\beta_j\). Under a logit link, \(h'(\eta)=p(1-p)\), so the local effect is \(\beta_jp(1-p)\). It depends on the current risk and is largest in magnitude at \(p=0.5\), not constant across profiles. Under probit, \(h'(\eta)=\phi(\eta)\), where \(\phi\) is the standard normal density, which also varies with the predictor. By contrast, for the identity link \(h'(\eta)=1\), so the local effect is \(\beta_j\) wherever the model is valid.

Common Interpretation Errors

• Wrong: “The probit coefficient is 0.5, so the odds ratio is \(\exp(0.5)\).” Correction: A probit coefficient shifts the normal-score index. Calculate the probabilities from \(\Phi(\eta)\), then form an odds ratio if one is needed.
• Wrong: “Exponentiating any coefficient gives an odds ratio.” Correction: Under logit it gives an odds ratio; under a log link it gives a risk ratio; under complementary log-log it gives a ratio of \(-\log(1-p)\). For links such as probit and identity, exponentiation does not supply those standard interpretations.
• Wrong: “A coefficient of 0.04 always means a four-percentage-point increase in risk.” Correction: That direct interpretation applies to an identity-link coefficient for the specified contrast, where predictions are valid. For nonlinear links, use the inverse link to calculate a finite probability contrast or its derivative for a local effect.
• Wrong: “A local probability effect can be reported as the risk difference for any size of change.” Correction: A derivative describes an infinitesimal change at a particular covariate profile. For a finite change, compute the two fitted probabilities and subtract them.