Interpreting Coefficients On The Link Scale

What a Coefficient Changes

In a generalized linear model, a regression coefficient first describes a change in the linear predictor. It does not automatically describe a change of the same size in the conditional response mean. This distinction matters because the model connects the linear predictor to the mean through a link function.

Write the conditional mean at covariate value \(x\) as \(\mu(x)=E[Y\mid x]\). A GLM specifies a link \(g\) and a linear predictor \(\eta(x)\), related by \(\eta(x)=g(\mu(x))\). Equivalently, when the inverse link is defined, \(\mu(x)=g^{-1}(\eta(x))\). The coefficient interpretation developed here concerns changes in \(\eta\), with the other predictor terms held fixed.

Definition: In a model with linear predictor \(\eta=\beta_0+\beta_1 z_1+\cdots+\beta_p z_p\), the coefficient \(\beta_j\) is the change in the linear predictor for a one-unit increase in \(z_j\), holding the other \(z\)-terms fixed. This direct interpretation applies when \(z_j\) enters as a separate linear term; if the model contains transformed terms or interactions, interpret the specified predictor expression by comparing its values.

The phrase “one-unit increase” refers to the units of the predictor term as it appears in the model. If a variable is measured in tens, its coefficient is per ten units of the original measurement. If the model includes an interaction, the change can depend on the value of the interacting variable. In each case, the calculation begins by writing the linear predictor at the two covariate settings and subtracting.

The One-Unit Change Result

Suppose a covariate \(x_j\) enters the predictor linearly, with no interaction involving \(x_j\), and the other predictor values remain fixed. Both settings must be admissible for the model. The coefficient’s interpretation follows by direct subtraction.

Theorem: If \(\eta(x)=\beta_0+\beta_jx_j+\sum_{k\ne j}\beta_kx_k\), then changing \(x_j\) from \(a\) to \(a+1\), while keeping every \(x_k\), \(k\ne j\), fixed, changes the linear predictor by exactly \(\beta_j\). More generally, a change of \(\Delta\) units in \(x_j\) changes it by \(\beta_j\Delta\).

Write the starting predictor.
At \(x_j=a\), retain the fixed values of all other terms.

Write the changed predictor.
At \(x_j=a+\Delta\), replace only the \(x_j\) term.

Subtract.
The fixed terms cancel, leaving \(\beta_j\Delta\).

Proof. Let \(C=\beta_0+\sum_{k\ne j}\beta_kx_k\), which is unchanged because all the other predictor values are fixed. At the starting value, \(\eta(a)=C+\beta_ja\). At the changed value, \(\eta(a+\Delta)=C+\beta_j(a+\Delta)\). Therefore,

\eta(a+\Delta)-\eta(a) =\{C+\beta_j(a+\Delta)\}-\{C+\beta_ja\} =\beta_j\Delta.

Setting \(\Delta=1\) gives the one-unit result. \(\square\)

Worked Example: A Continuous Covariate in the Predictor

Suppose a model has linear predictor \(\eta=-0.8+0.35x-0.2z\). Compare two cases that have the same \(z=3\), but \(x=4\) and \(x=5\).

At \(x=4\), the predictor is \(\eta=-0.8+0.35(4)-0.2(3)=-0.8+1.4-0.6=0\).

At \(x=5\), it is \(\eta=-0.8+0.35(5)-0.2(3)=-0.8+1.75-0.6=0.35\). The change is \(0.35-0=0.35\), exactly the coefficient of \(x\). The interpretation is conditional on holding \(z\) fixed. It says that this one-unit increase in \(x\) raises the linear predictor by \(0.35\); it does not, by itself, say that the mean rises by \(0.35\).

What Changes When the Predictor Term Changes

A coefficient’s interpretation depends on how the corresponding term is written in the model. A categorical predictor commonly appears through an indicator, so its coefficient compares the indicated category with the reference category. A rescaled or transformed covariate has a coefficient per unit of that specified term. An interaction means that the change associated with one covariate can depend on another covariate.

Worked Example: A Categorical Predictor

Let \(T=0\) denote the reference group and \(T=1\) denote a treatment group. Suppose \(\eta=-1.1+0.4T+0.03A\), where \(A\) is age. Compare the groups at the same age, \(A=50\).

For the reference group, \(\eta=-1.1+0.4(0)+0.03(50)=-1.1+1.5=0.4\). For the treatment group, \(\eta=-1.1+0.4(1)+0.03(50)=0.8\). The difference is \(0.8-0.4=0.4\), the coefficient of \(T\). Thus, at a fixed age, the treatment indicator changes the linear predictor by \(0.4\) relative to the reference group. It is a link-scale group contrast, not automatically a mean difference of \(0.4\).

Result: For a predictor with an interaction, the coefficient of one covariate alone need not equal its change in the linear predictor at every value of the other covariate. The relevant change is found by substituting the two covariate settings into the full predictor and subtracting.

Worked Example: An Interaction Makes the Change Conditional

Consider \(\eta=0.5+0.2x-0.3z+0.1xz\). Compare \(x\) with \(x+1\), holding \(z\) fixed. The terms that involve \(x\) are \(0.2x+0.1xz\). At \(x+1\), these become \(0.2(x+1)+0.1(x+1)z\). Subtracting the original predictor gives

\eta(x+1,z)-\eta(x,z) =0.2+0.1z.

When \(z=0\), the one-unit increase in \(x\) changes \(\eta\) by \(0.2+0.1(0)=0.2\). When \(z=4\), it changes \(\eta\) by \(0.2+0.1(4)=0.6\). The coefficient \(0.2\) is the change in \(\eta\) for a one-unit increase in \(x\) specifically at \(z=0\); it is not the change at every value of \(z\).

Why a Link-Scale Change Is Not a Mean-Scale Change

The model relates the conditional mean to the predictor by \(\mu=g^{-1}(\eta)\). Thus, if a covariate change shifts \(\eta\) by \(\beta_j\), the corresponding mean contrast is \(g^{-1}(\eta+\beta_j)-g^{-1}(\eta)\). That contrast depends on the inverse link and, in general, on the starting value of \(\eta\). It should not be reported as \(\beta_j\) without justification.

Key distinction: A coefficient for a linear predictor term gives an additive change on the link scale. The mean-scale contrast is the difference between the inverse-link values at the two predictor settings. These are different quantities unless the link and settings make them equal.

Worked Example: Same Coefficient, Different Mean Contrasts

Take a normal response with conditional mean \(\mu\) and the valid one-to-one link \(g(\mu)=\mu^3\), whose inverse is \(g^{-1}(\eta)=\sqrt[3]{\eta}\). Let \(\eta=1+2x\). The coefficient of \(x\) is \(2\), so each one-unit increase in \(x\) changes the predictor by \(2\).

From \(x=0\) to \(x=1\), the predictor goes from \(1\) to \(3\), and the mean goes from \(\sqrt[3]{1}=1\) to \(\sqrt[3]{3}\approx1.442\). The mean contrast is approximately \(1.442-1=0.442\), not \(2\).

From \(x=1\) to \(x=2\), the predictor goes from \(3\) to \(5\), and the mean goes from approximately \(1.442\) to \(\sqrt[3]{5}\approx1.710\). This mean contrast is approximately \(1.710-1.442=0.268\), while the predictor contrast is again \(2\). The example verifies that the same coefficient can correspond to different mean contrasts at different starting settings.

More formally, if \(h=g^{-1}\) and the predictor changes by \(\delta\), the mean contrast is \(h(\eta+\delta)-h(\eta)\). This equals \(\delta\) only when the values in question satisfy that equality; it does not follow from the coefficient interpretation. The example above gives \(\delta=2\) but mean contrasts approximately \(0.442\) and \(0.268\).

Common Mistake

Incorrect reasoning: “The estimated coefficient is \(0.35\), so a one-unit increase in the covariate raises the expected response by \(0.35\).” This conclusion treats the coefficient as if it were necessarily on the response-mean scale.
Correction: First identify the modeled predictor term and state its change in \(\eta\), holding the other terms fixed. Then, if a mean-scale interpretation is needed, compare the inverse-link values at the two predictor settings. A coefficient alone does not supply a constant mean difference for a nonlinear inverse link.
Incorrect reasoning: “The coefficient of \(x\) is the effect of \(x\), even though the model includes \(xz\).” The interaction makes the predictor change depend on \(z\).
Correction: Substitute both settings into the complete predictor. In the interaction example, the one-unit change is \(0.2+0.1z\), not \(0.2\) for all \(z\).

Key takeaway: Read a coefficient in the predictor as a change in the linear predictor for a specified covariate change, conditional on fixed values of the other predictor terms. Check transformations and interactions in the actual model expression, and do not equate a link-scale change with a mean-scale change.

Check Your Understanding

For each question, distinguish the predictor change from any claim about the response mean.

If \(\eta=2-0.6x+0.1z\), what is the change in \(\eta\) when \(x\) increases by two units and \(z\) is fixed?
A binary indicator has coefficient \(-0.25\). State its link-scale interpretation relative to the reference category, holding other terms fixed.
In \(\eta=1+0.5x+0.2xz\), calculate the change in \(\eta\) for a one-unit increase in \(x\) when \(z=3\).
Why does knowing that a coefficient is \(0.4\) not generally establish that the response mean changes by \(0.4\)?
A predictor uses \(x/10\) rather than \(x\). What covariate change corresponds to a one-unit increase in the modeled term \(x/10\)?