How Link Choice Changes Mean Curves

One Straight Predictor, Three Mean Curves

A generalized linear model uses a linear predictor, but the mean response need not be linear in a covariate. The inverse link transforms that predictor into the mean. Consequently, holding the predictor fixed while changing only the inverse link can produce curves with different shapes, ranges, and covariate effects. This comparison is useful when deciding what a model says about changes in the mean, not just how its coefficients look on the predictor scale.

Let \(x\) be a continuous covariate and use the same linear predictor for all three links: \(\eta(x)=\beta_0+\beta_1x\). Here \(\beta_0\) and \(\beta_1\) are fixed coefficients. We compare identity, log, and logit inverse links. For the logit, write \(p(x)\) for the resulting mean.

The identity curve is a straight line, the log-link curve is exponential in \(x\), and the logit-link curve is an S-shaped curve. These shape descriptions concern the mean as a function of \(x\); the predictor itself remains a straight line. The links have different allowable mean ranges: the identity inverse link can take any real value, the log inverse link is positive, and the logit inverse link lies strictly between zero and one.

Worked Comparison at Specific Covariate Values

Deriving Slope and Curvature

The first derivative of a mean curve describes its local rate of change per unit increase in \(x\). The second derivative describes how that rate itself changes. These derivatives make precise the differences suggested by the curve shapes. The following calculations assume \(x\) is continuous and \(\beta_0,\beta_1\) are fixed.

For the identity link, differentiating \(\mu_I(x)=\beta_0+\beta_1x\) gives \(\mu_I'(x)=\beta_1\), and differentiating again gives \(\mu_I''(x)=0\).

For the log link, \(\mu_L(x)=\exp(\beta_0+\beta_1x)\). The chain rule gives \(\mu_L'(x)=\beta_1\exp(\beta_0+\beta_1x)=\beta_1\mu_L(x)\). Differentiating this expression once more gives \(\mu_L''(x)=\beta_1^2\mu_L(x)\). Since \(\mu_L(x)>0\), the curve is strictly convex when \(\beta_1\ne0\), and is constant when \(\beta_1=0\).

For the logit link, let \(p(x)=e^{\eta(x)}/[1+e^{\eta(x)}]\). Differentiating with respect to \(\eta\) yields \(dp/d\eta=p(1-p)\), while \(d\eta/dx=\beta_1\). Thus \(p'(x)=\beta_1p(1-p)\). Differentiating again and applying the product rule gives \(p''(x)=\beta_1^2p(1-p)(1-2p)\).

Because \(0<p<1\), the sign of the logistic curve’s curvature is the sign of \(1-2p\), provided \(\beta_1\ne0\). Its curvature is positive where \(p<1/2\), zero where \(p=1/2\), and negative where \(p>1/2\). Since \(p=1/2\) exactly when \(\eta=0\), the curve changes curvature at the predictor value zero. If \(\beta_1\ne0\), this occurs at \(x=-\beta_0/\beta_1\). If \(\beta_1=0\), all three curves are constant and there is no covariate-driven curvature.

The slope formulas also show how quickly the means change. For a positive \(\beta_1\), the identity slope is constant, while the log-link slope grows with its mean. The logistic slope is positive but depends on \(p\); its magnitude is largest at \(p=1/2\), because \(p(1-p)=1/4-(p-1/2)^2\leq1/4\). Thus its maximum absolute slope is \(|\beta_1|/4\), attained at \(\eta=0\).

Covariate Changes: Additive, Multiplicative, and Odds-Scale

A slope is a local effect. For a finite covariate change \(\Delta\), each link also gives a useful exact comparison. The identity link makes the mean difference constant. The log link makes the ratio of means constant. The logit link makes the odds ratio constant. These statements follow directly from the shared linear predictor.

For the identity link, substitution gives \(\mu_I(x+\Delta)=\beta_0+\beta_1(x+\Delta)\). Subtracting \(\mu_I(x)=\beta_0+\beta_1x\) leaves \(\beta_1\Delta\). For the log link, the ratio is \(\exp\{\beta_0+\beta_1(x+\Delta)\}/\exp\{\beta_0+\beta_1x\}=\exp(\beta_1\Delta)\).

For the logit link, the odds are \(p(x)/[1-p(x)]=\exp\{\eta(x)\}\), by substituting the logistic expression for \(p(x)\). Therefore the odds ratio is \(\exp\{\eta(x+\Delta)-\eta(x)\}=\exp(\beta_1\Delta)\). This is a constant odds ratio, not a constant difference in probabilities: the probability difference generally depends on the starting value of \(p(x)\).

Reading the Comparison Carefully

The table concerns the mathematical consequences of each inverse link for a given predictor. It does not by itself establish that a particular link is appropriate for a response: the response’s support and the model’s purpose matter too. For example, a probability mean must be in \([0,1]\), whereas the identity inverse link is unrestricted. The logit inverse link produces values strictly inside that interval for finite predictors; it does not produce exact boundary values at finite \(x\).

Common mistake: “The coefficient \(\beta_1\) is 0.3, so a two-unit increase in \(x\) raises the mean by 0.6 for any link.” This is correct for the identity link, because it gives a mean difference. It is not the log-link interpretation: there, the mean is multiplied by \(e^{0.6}\). Nor is it the logit-link interpretation: there, the odds are multiplied by \(e^{0.6}\), while the probability difference depends on the starting probability. Always identify the scale on which the change is constant.