Interpreting Log-Link Coefficients As Mean Ratios

Why a Log-Link Coefficient Is a Ratio

In a gamma regression model, the response is positive and its conditional variability typically increases with its conditional mean. A log link gives a direct way to model that mean while guaranteeing that fitted means are positive. Its coefficients are on the log-mean scale; to interpret them in the response scale, exponentiate the relevant coefficient or contrast. The result is a ratio of conditional means.

Let \(Y_i\) be a positive response and let \(x_i=(x_{i1},\ldots,x_{ip})^\top\) be its vector of predictors, including any intercept column. Write \(\mu_i=E[Y_i\mid x_i]\). In a gamma regression with a log link, the mean model is

where \(\beta=(\beta_1,\ldots,\beta_p)^\top\) is the coefficient vector. The gamma distribution specifies a conditional distribution for \(Y_i\) with this positive mean; the ratio interpretation below follows from the log-link mean model. It does not depend on a particular value of the gamma dispersion parameter.

The percentage is calculated relative to the original conditional mean. It is not generally \(100\beta_j\%\): that is only an approximation when \(\beta_j\) is small. A positive coefficient means a larger conditional mean for the specified increase in the predictor; a negative coefficient means a smaller one.

Deriving Mean Ratios and Percentage Changes

The basic interpretation follows by comparing two predictor settings that differ only in \(x_j\). Suppose the first setting has \(x_j=a\), the second has \(x_j=a+1\), and every other predictor has the same value in both settings. Denote their conditional means by \(\mu(a)\) and \(\mu(a+1)\).

To prove the result, write the predictor contributions other than \(x_j\) as \(c\), which is unchanged between the two settings. Then \(\log\{\mu(a)\}=c+a\beta_j\) and \(\log\{\mu(a+1)\}=c+(a+1)\beta_j\). Subtracting the first equation from the second gives \(\log\{\mu(a+1)\}-\log\{\mu(a)\}=\beta_j\). Since the difference of logarithms is the logarithm of a ratio, exponentiating gives \(\mu(a+1)/\mu(a)=\exp(\beta_j)\). Subtracting one from this ratio gives the relative change, \(\exp(\beta_j)-1\). Multiplying that relative change by 100 expresses it as a percentage. \(\square\)

More generally, compare a covariate value \(a\) with \(a+\delta\), where \(\delta\) can be any real-valued change. Under the same condition that other predictors remain fixed, the ratio is \(\exp(\delta\beta_j)\). This finite-change formula is especially useful for continuous predictors: it avoids treating a one-unit increase as the only meaningful comparison.

Continuous Predictors: Finite Changes and Local Effects

For a continuous predictor such as age, dose, or time, \(\beta_j\) is the change in log conditional mean per one-unit increase. The model implies a constant multiplicative factor for each one-unit increase, provided the other predictors are fixed and the model has no interaction involving \(x_j\). For a change of \(\delta\) units, the factor is \(\exp(\delta\beta_j)\), not generally \(\exp(\beta_j)\).

The finite-change formula follows by dividing the means: \(\exp\{c+\beta_j(x_j+\delta)\}/\exp(c+\beta_jx_j)=\exp(\delta\beta_j)\). For the derivative, apply the chain rule to \(\mu(x_j)=\exp(c+\beta_jx_j)\): \(d\mu/dx_j=\exp(c+\beta_jx_j)\beta_j=\beta_j\mu(x_j)\). Dividing by the positive \(\mu(x_j)\) gives the proportional rate of change \(\beta_j\). This establishes both statements. \(\square\)

The proportional rate is also the elasticity when \(x_j\) is measured on a ratio scale: \((x_j/\mu)(d\mu/dx_j)=x_j\beta_j\). It is not generally constant across values of \(x_j\). In contrast, the log-mean slope \(d\log(\mu)/dx_j=\beta_j\) is constant in this model.

Contrasts Between Predictor Settings

The coefficient-by-coefficient rule is a special case of a general contrast rule. For any two predictor vectors \(x_A\) and \(x_B\), let \(\mu_A=\exp(x_A^\top\beta)\) and \(\mu_B=\exp(x_B^\top\beta)\). Their conditional mean ratio is

This follows because \(\mu_A/\mu_B=\exp(x_A^\top\beta)/\exp(x_B^\top\beta)\), and division of exponentials subtracts their exponents. Thus, a comparison that changes several predictors uses the sum of their coefficient-weighted changes, \((x_A-x_B)^\top\beta\), before exponentiation. Any common intercept or unchanged predictor contributions cancel.

For a categorical predictor represented by indicator variables, this rule gives the adjusted ratio between two categories. The coefficient for a category compares that category with the reference category only when the model’s coding makes the difference in design vectors equal to one in that coefficient’s column and zero in the other relevant columns. For a comparison between two nonreference categories, subtract their relevant coefficients before exponentiating.

Interactions, Uncertainty, and Interpretation Limits

When the model includes an interaction, the effect associated with one predictor can depend on another predictor’s value. The general design-vector contrast still applies, but a single coefficient may no longer be the mean-ratio effect for every covariate setting. For example, if the linear predictor contains \(\beta_1x_1+\beta_2x_2+\beta_{12}x_1x_2\), increasing \(x_1\) by one unit at fixed \(x_2\) changes the log mean by \(\beta_1+\beta_{12}x_2\). The corresponding mean ratio is \(\exp(\beta_1+\beta_{12}x_2)\). State the value of the interacting predictor when interpreting that comparison.

For estimation, replace the unknown coefficients by their estimates. If a contrast has estimated log ratio \(\hat\theta=(x_A-x_B)^\top\hat\beta\), then its estimated mean ratio is \(\exp(\hat\theta)\). A confidence interval for the log ratio, \([L,U]\), transforms to \([\exp(L),\exp(U)]\) for the ratio because the exponential function is increasing. To express that interval as percentage changes, transform both endpoints as \(100\{\exp(L)-1\}\%\) and \(100\{\exp(U)-1\}\%\).

These are interpretations of conditional means under the specified model. They are not automatically causal effects: a causal interpretation requires appropriate study design and assumptions beyond the log-link equation. Nor do mean ratios describe how each observed response changes. Gamma responses vary around their conditional means, so individual outcomes need not follow the mean ratio.

Common Mistakes

• Reporting the coefficient as the exact percentage change. A coefficient of 0.20 gives a ratio of \(\exp(0.20)\), so the exact increase is \(100\{\exp(0.20)-1\}\%\approx22.1\%\), not 20%.
• Forgetting the size of a continuous change. A five-unit increase with coefficient \(\beta_j\) has ratio \(\exp(5\beta_j)\), not \(\exp(\beta_j)\).
• Using the wrong reference category. A dummy-variable coefficient compares its category to the designated reference; comparing two nonreference categories requires a coefficient difference.
• Interpreting a conditional mean ratio as an individual-level or causal effect. The model compares expected responses at specified predictor values. Individual outcomes vary, and causality requires additional assumptions.
• Ignoring interactions. If a predictor interacts with another variable, its mean-ratio effect depends on that variable’s value; report the full contrast for the setting being compared.