Interpreting Inverse-Gaussian Regression Coefficients

From Link-Scale Coefficients to Mean Changes

An inverse-Gaussian regression coefficient describes a change in the linear predictor, not generally a fixed change in the response mean. To interpret it on the response scale, first apply the inverse link. The resulting change may be additive, multiplicative, or nonlinear, and it can depend on the starting covariate values.

Write \(x\) for a covariate vector and \(\beta\) for the coefficient vector. The linear predictor is \(\eta(x)=x^\top\beta\), and the conditional mean is \(\mu(x)=E[Y\mid x]=g^{-1}(\eta(x))\). Here \(Y\mid x\) has an inverse-Gaussian distribution with positive mean \(\mu(x)\). All interpretations below concern this conditional mean; they do not say that an individual observed response changes by the same amount.

For a one-unit increase in covariate \(x_j\), holding all other covariates fixed, the linear predictor changes by \(\beta_j\). For a change of \(\delta\) units, it changes by \(\beta_j\delta\). The inverse link determines how that predictor change translates into a mean difference or ratio.

Exact Contrasts Under Four Links

For the log link, \(g(\mu)=\log\mu\), so \(\mu=\exp(\eta)\). For the identity link, \(g(\mu)=\mu\), so \(\mu=\eta\). For the reciprocal link, \(g(\mu)=1/\mu\), so \(\mu=1/\eta\). Under the canonical inverse-Gaussian link, the natural parameter is \(\eta=-1/(2\mu^2)\), giving \(\mu=(-1/(2\eta))^{1/2}\). The canonical convention here uses this natural parameter itself, including its negative sign and factor of \(1/2\).

Proof. Under the log link, \(\mu_A=\exp(\eta_A)\) and \(\mu_B=\exp(\eta_A+d)\). Dividing gives \(\mu_B/\mu_A=\exp(d)\); subtracting gives \(\mu_A\{\exp(d)-1\}\). Under the identity link, \(\mu_A=\eta_A\) and \(\mu_B=\eta_A+d\), so subtraction gives \(d\), and division gives \((\eta_A+d)/\eta_A\).

Under the reciprocal link, \(\mu_A=1/\eta_A\) and \(\mu_B=1/\eta_B\), where \(\eta_B=\eta_A+d\). Thus \(\mu_B/\mu_A=\eta_A/\eta_B\), and \[ \mu_B-\mu_A=\frac{1}{\eta_B}-\frac{1}{\eta_A} =\frac{\eta_A-\eta_B}{\eta_A\eta_B} =-\frac{d}{\eta_A\eta_B}. \] Under the canonical link, \(\mu_A=(-1/(2\eta_A))^{1/2}\) and \(\mu_B=(-1/(2\eta_B))^{1/2}\). Their ratio is \((\eta_A/\eta_B)^{1/2}\); this is real and positive because both predictors are negative. Subtracting the means gives the stated difference. The domain conditions ensure the displayed means are positive and defined. \(\square\)

These formulas apply to any finite contrast, including a change in several covariates at once. In particular, for a one-unit change in \(x_j\), set \(d=\beta_j\); for a \(\delta\)-unit change, set \(d=\beta_j\delta\). For a categorical indicator, the coefficient compares the group coded 1 with the reference group coded 0, conditional on the other covariates.

Worked Examples

Local Slopes and Domain Checks

A coefficient can also be translated into a local rate of change in the conditional mean. This derivative describes the effect of an infinitesimally small covariate change at a specified point. It is not a replacement for the exact finite-contrast formulas when the covariate change is substantial.

Derivation. Since \(\eta=x^\top\beta\), its derivative with respect to \(x_j\) is \(\beta_j\). The chain rule therefore gives \(\partial\mu/\partial x_j=(dg^{-1}/d\eta)\beta_j\). For the log link, \(d\exp(\eta)/d\eta=\mu\); for identity, the derivative is 1. For the reciprocal link, \(d(1/\eta)/d\eta=-1/\eta^2=-\mu^2\). For the canonical link, \(\mu=(-2\eta)^{-1/2}\), so \[ \frac{d\mu}{d\eta}=(-2\eta)^{-3/2}=\mu^3, \] where the equality follows from \(\mu=(-2\eta)^{-1/2}>0\). Multiplying each derivative by \(\beta_j\) proves the result. \(\square\)

The derivative signs help explain direction, but the predictor domain must still be checked over the contrast. The reciprocal link requires \(\eta>0\), and the canonical link requires \(\eta<0\), at every covariate value being interpreted. The identity link requires a positive fitted mean. A coefficient can point in a direction that eventually reaches an invalid predictor; a local slope alone does not guarantee that a large finite extrapolation is valid.

Common Interpretation Pitfalls

• Reading every coefficient as a mean difference. This is correct for the identity link, but not for log, reciprocal, or canonical links. Apply the inverse link before interpreting a response-scale effect.
• Calling a log-link coefficient a percentage directly. The exact percentage change for a one-unit increase is \(100\{\exp(\beta_j)-1\}\%\), not \(100\beta_j\%\). The latter is only an approximation when \(\beta_j\) is small.
• Using a single ratio for every link. The log-link ratio is independent of the starting covariates. For identity, reciprocal, and canonical links, the mean ratio generally depends on the starting predictor or mean.
• Ignoring the link domain. A reciprocal predictor must be positive and a canonical predictor must be negative. An algebraic contrast formula is not a valid model interpretation if either endpoint violates its domain.
• Making a causal claim from a regression coefficient. These formulas describe model-based conditional means. A causal interpretation requires additional design or identification assumptions.