Choosing And Comparing Gamma Links In Practice

Why Link Choice Is a Practical Decision

For a Gamma regression, the response is positive, but that fact alone does not determine the link. A log link, an identity link, and a reciprocal-mean link can each describe a positive conditional mean while imposing different shapes and interpretations. The practical question is not simply which link is familiar; it is which mean pattern is scientifically plausible, valid over the intended prediction range, and adequately supported by the data.

Let \(Y_i\) be a positive response for observation \(i\), with conditional mean \(\mu_i=E[Y_i\mid x_i]\) and linear predictor \(\eta_i=x_i^\mathsf{T}\beta\). In the usual Gamma mean-variance specification, \(\operatorname{Var}(Y_i\mid x_i)=\phi\mu_i^2\), where \(\phi>0\) is the dispersion parameter. The link specifies how \(\eta_i\) determines \(\mu_i\); it does not change the positive support of the response.

These links are not interchangeable descriptions of the same coefficient. They specify different ways for the conditional mean to change with predictors. A meaningful comparison therefore considers both the implied curve and the range over which that curve is valid.

What the Coefficients Say

For a single quantitative predictor \(x\), write \(\eta(x)=\beta_0+\beta_1x\). The coefficient \(\beta_1\) has a distinct interpretation under each link. A practical comparison should keep that distinction explicit.

For the log link, \(\mu(x)=\exp(\beta_0+\beta_1x)\). Thus $$ \frac{\mu(x+d)}{\mu(x)} =\frac{\exp(\beta_0+\beta_1(x+d))}{\exp(\beta_0+\beta_1x)} =\exp(\beta_1d). $$ For the identity link, \(\mu(x)=\beta_0+\beta_1x\), so \(\mu(x+d)-\mu(x)=\beta_1d\). For the reciprocal-mean link, set \(a=\beta_0+\beta_1x\) and \(c=\beta_1d\). Provided \(a>0\) and \(a+c>0\), $$ \mu(x+d)-\mu(x) =\frac{1}{a+c}-\frac{1}{a} =-\frac{c}{a(a+c)}. $$ So a positive \(\beta_1\) decreases the mean as \(x\) increases, as long as the reciprocal predictor remains positive. The absolute change is not constant: it depends on both the starting reciprocal mean and the change in it.

Check Validity Before Comparing Fit

A Gamma conditional mean must be positive. This requirement is automatic with a log link, but not with the other two links. For an identity-link model, every fitted predictor value must satisfy \(\eta_i>0\). For a reciprocal-mean model, every fitted value must also satisfy \(\eta_i>0\), because \(1/\eta_i\) must be positive and finite. A zero reciprocal predictor does not give a finite mean, and a negative one gives an inadmissible negative mean.

To prove this, first suppose \(\eta(L)>0\) and \(\eta(U)>0\). Every \(x\in[L,U]\) can be written as \(x=(1-t)L+tU\) for some \(t\in[0,1]\). Affinity gives $$ \eta(x)=(1-t)\eta(L)+t\eta(U). $$ This is a weighted average of two positive numbers, with nonnegative weights summing to one, so \(\eta(x)>0\). Conversely, if \(\eta(x)>0\) throughout the interval, it is positive at \(L\) and \(U\) because they belong to the interval. This proves the claim. \(\square\)

The endpoint check is useful for a continuous predictor over a stated interval, but it does not certify predictions outside that interval. With several predictors, check every observed design point and each proposed prediction profile; a convenient one-dimensional endpoint rule need not apply to an arbitrary prediction region.

Compare Mean Curves and Residual Patterns

For a given data set, fit candidates on the same response observations and inspect their fitted means over the observed predictor range. Look for systematic structure in residuals against the predictor and fitted values: curvature, a trend, or changing spread can indicate that the mean curve or the assumed mean-variance relationship is inadequate. A residual pattern is evidence to investigate, not an automatic verdict that one link is correct.

For Gamma data, the variance grows with the square of the mean under the usual variance specification. Raw residuals \(y_i-\widehat{\mu}_i\) can therefore have larger spread at larger fitted means even when the model is appropriate. Compare residual plots with suitable scaling, such as Pearson residuals \((y_i-\widehat{\mu}_i)/(\sqrt{\widehat{\phi}}\widehat{\mu}_i)\), and assess whether any pattern remains. Mean-variance fit and link choice are related diagnostic questions, but they are not the same question.

A Practical Comparison Strategy

A link should be selected by combining the scientific meaning of the effect, fitted-mean validity, diagnostic evidence, and the prediction task. The following sequence keeps those considerations separate enough to make the comparison interpretable.

When candidate models use the same response data and Gamma likelihood, likelihood-based criteria can help compare relative fit, provided the models are fitted and compared under compatible likelihood conventions. Such a criterion does not replace the validity checks or explain whether a coefficient has the interpretation the study needs. Likewise, a small improvement in fit should not justify implausible predictions or a mean curve that conflicts with the design and subject matter.

Common Mistake

A common mistake is to choose the identity link because its fitted means are positive for all observed cases, then assume that it is safe for any future prediction. Positivity at observed design points establishes validity only at those points. A linear predictor can become zero or negative outside the range, as the worked example showed. The correction is to check the full range of intended predictions and state clearly where the fitted mean remains admissible. The reverse shortcut is also unsafe: the log link guarantees positive means, but positivity alone does not establish a good fit or sensible extrapolation.