Inverse-Gaussian Deviance Contributions

From Likelihood to Deviance

A residual records how an observed response differs from its fitted mean, but a raw difference alone does not reflect the response model’s likelihood. For an inverse-Gaussian response, the likelihood supplies a model-based measure of discrepancy: the unit deviance. Its signed square root gives a residual whose sign indicates the direction of the discrepancy and whose square is the unit deviance.

The derivation compares two likelihoods for the same observed response: the likelihood under the fitted mean, and the likelihood under a saturated model that fits that response exactly. Keeping the dispersion factor visible in this comparison is important: the conventional deviance and the likelihood-ratio statistic differ by that factor.

Deriving the Unit Deviance

Take the logarithm of the density. When comparing fitted and saturated likelihoods for the same \(y\) and \(\phi\), the density’s normalizing term is identical in both cases. At the saturated mean, the exponential term is one because \(y-\mu_{\mathrm{sat}}=0\). Thus the log-likelihood difference is determined entirely by the fitted model’s exponential term.

For one observation, write \(\ell(\mu;y,\phi)=\log f(y;\mu,\phi)\), and let \(\ell_{\mathrm{sat}}(y,\phi)=\ell(y;y,\phi)\). The two log-likelihoods are

Subtracting gives \(2\{\ell_{\mathrm{sat}}-\ell(\mu;y,\phi)\}=(y-\mu)^2/(\phi\mu^2y)\). Therefore the conventional inverse-Gaussian unit deviance is

This derivation proves the result for every \(y>0\), \(\mu>0\), and \(\phi>0\). In particular, \(d(y,\mu)\geq 0\), since its numerator is a square and its denominator is positive; it is zero exactly when \(y=\mu\). The unscaled likelihood-ratio statistic is \(2\{\ell_{\mathrm{sat}}-\ell\}=d(y,\mu)/\phi\). For independent observations, the conventional model deviance sums their unit deviances, while the corresponding twice-log-likelihood contrast is that sum divided by \(\phi\) when the observations share a common dispersion.

Signed Deviance Residuals

The unit deviance measures discrepancy without recording whether the fitted mean is too high or too low. The signed deviance residual attaches the direction of the raw residual \(y-\mu\) to the square root of the unit deviance. Define \(\operatorname{sign}(0)=0\).

To verify the simplified expression, positivity of \(\mu\) and \(y\) gives \(\sqrt{d(y,\mu)}=|y-\mu|/(\mu\sqrt{y})\). Multiplication by \(\operatorname{sign}(y-\mu)\) replaces the absolute value with \(y-\mu\), including the zero case. Squaring the resulting residual returns \((y-\mu)^2/(y\mu^2)=d(y,\mu)\). Thus the residual retains the unit deviance’s magnitude information while restoring the direction.

For a fitted model, the observation is \(y_i\) and its fitted response is \(\hat\mu_i\). The residual is therefore \(r_{D,i}=(y_i-\hat\mu_i)/(\hat\mu_i\sqrt{y_i})\). The conventional deviance residual uses the unit deviance, not the likelihood contrast divided by \(\phi\). If one instead wants a signed square root of the dispersion-scaled likelihood contrast, it is \(r_{D,i}/\sqrt{\phi}\). These quantities should not be interchanged without stating which scale is being used.

Worked Calculations

Interpretation and Common Pitfall

A deviance residual is not a raw residual: its denominator depends on both the fitted mean and the observed response. Nor is it generally the Pearson residual, which for an inverse-Gaussian response uses the model variance \(\operatorname{Var}(Y)=\phi\mu^3\). The deviance residual instead comes from the saturated-versus-fitted likelihood comparison. These residuals answer related but distinct questions and need not have the same numerical values.

• Pitfall: dropping \(\mu^2\) from the denominator. The inverse-Gaussian unit deviance is \((y-\mu)^2/(y\mu^2)\), not a squared difference divided by \(y\mu\). The \(\mu^2\) follows directly from the exponent in the inverse-Gaussian density.
• Pitfall: treating the likelihood contrast as the unit deviance. The twice-log-likelihood contrast is \(d(y,\mu)/\phi\); the conventional unit deviance is \(d(y,\mu)\). Check whether a calculation asks for the conventional deviance or the dispersion-scaled likelihood contrast.
• Pitfall: taking only a positive square root. The unsigned square root loses whether the response is above or below its fitted mean. Use \(\operatorname{sign}(y-\mu)\sqrt{d(y,\mu)}\).

The deviance is useful because it turns the fitted-versus-saturated likelihood difference into an additive discrepancy measure across observations. Its signed residuals also preserve the direction of each observation’s departure from its fitted mean. Both rely on positive observed responses and positive fitted means; the inverse-Gaussian support and mean domain make these conditions essential, not optional.