Choosing Between Gamma and Inverse-Gaussian Models

Why Compare These Two Families?

A strictly positive response can be strongly right-skewed, making a Gaussian model a poor description of its conditional distribution. The gamma and inverse-Gaussian families are two common alternatives: both have positive support and allow variance to increase with the mean, but they imply different rates of variance growth. That difference gives a practical starting point for choosing between them.

Neither family should be selected just because a response is positive or skewed. The choice should be consistent with the observed relationship between mean and variance, and the fitted model should not leave systematic patterns in residuals. These are pieces of evidence, not mechanical rules: sampling noise, small samples, and an incorrect model for the mean can all make the evidence misleading.

Mean, Variance, and Shape

Let \(Y\) be a strictly positive response with conditional mean \(\mu=E[Y]\) and dispersion parameter \(\phi>0\). In the parameterizations used here, the gamma and inverse-Gaussian variance functions are as follows.

The same symbol \(\phi\) has the same role in each variance function, but its numerical value need not be the same when fitting the two families. A dispersion value belongs to a particular family and its parameterization. To compare the families fairly, it is often useful to compare their variances at a reference mean, rather than assume their fitted dispersions match.

The gamma variance grows quadratically with the mean; the inverse-Gaussian variance grows cubically. Thus, if the mean doubles while dispersion stays fixed, the gamma variance is multiplied by four, whereas the inverse-Gaussian variance is multiplied by eight. The following result makes this contrast precise.

Justification. For the gamma family,

For the inverse-Gaussian family, the same calculation gives

These ratios concern the variance at two means under a fixed-dispersion model. They do not say that every individual observation will follow a smooth variance curve, nor that variance estimates from small groups will reveal the curve exactly. \(\square\)

The families also differ in standardized skewness. For the gamma family, skewness is \(2\sqrt{\phi}\), independent of the mean. For the inverse-Gaussian family, it is \(3\sqrt{\phi\mu}\), so skewness increases with the mean when dispersion is fixed. These formulas describe the distributions implied by the models; they are not guarantees that a particular sample will visibly display the difference.

Derivation. For a gamma variable with shape \(1/\phi\) and scale \(\phi\mu\), its cumulant-generating function is \(K(t)=-(1/\phi)\log(1-\phi\mu t)\), for \(t<1/(\phi\mu)\). Differentiating at zero gives the second and third cumulants, \(\kappa_2=\phi\mu^2\) and \(\kappa_3=2\phi^2\mu^3\). Since skewness is the third central moment divided by the variance to the power \(3/2\), it is

For an inverse-Gaussian variable in this parameterization, the cumulant-generating function is \(K(t)=(1/(\phi\mu))(1-\sqrt{1-2\phi\mu^2t})\), for \(t<1/(2\phi\mu^2)\). Differentiating twice and three times at zero gives \(\kappa_2=\phi\mu^3\) and \(\kappa_3=3\phi^2\mu^5\). Therefore,

The cumulants used here exist because the moment-generating functions are finite in an open interval containing zero. \(\square\)

Using Mean–Variance Patterns

When several groups or covariate regions have different fitted means, their response variability can provide evidence about the family. The basic comparison is whether a variance proportional to \(\mu^2\) or to \(\mu^3\) better describes the data, allowing for an unknown dispersion. In an exploratory summary, one can compare ratios of estimated variance to the corresponding power of the estimated mean. A roughly stable ratio across groups is compatible with that variance function.

This comparison needs care. Group variances are themselves noisy estimates, especially when groups contain few observations. Also, these summaries are most interpretable when groups are reasonably comparable and their means are estimated sensibly. A trend in group variances can reflect an omitted predictor, dependence, or a misspecified mean structure, rather than the wrong choice between gamma and inverse-Gaussian responses.

Residual Behavior as a Second Check

A variance function affects how large a response’s deviation from its fitted mean is expected to be. For a fitted mean \(\hat{\mu}_i>0\), a Pearson residual under a candidate family is \(r_{Pi}=(y_i-\hat{\mu}_i)/\sqrt{V(\hat{\mu}_i)}\), where \(V(\hat{\mu}_i)\) is that family’s variance function evaluated at the fitted mean, including its dispersion. Under the gamma model this denominator is \(\sqrt{\phi}\hat{\mu}_i\); under the inverse-Gaussian model it is \(\sqrt{\phi}\hat{\mu}_i^{3/2}\).

Proof. Conditional on \(x_i\), the denominator is fixed. The conditional expectation of the residual is

For the squared residual, the definition of conditional variance gives

This identity relies on the true conditional mean and variance, not estimates fitted from the same data. Fitted Pearson residuals will vary, and their squared values need not average exactly one. \(\square\)

As a practical check, look for residuals that remain centered around zero and do not show a systematic increase or decrease in spread as fitted means change. A variance function that is too small at large means can leave large absolute standardized residuals there; one that is too large can shrink them. A visible pattern is a reason to reconsider the model, but residual behavior alone may not identify whether the problem is the family, the mean structure, or another feature of the data.

Common Mistakes and a Practical Choice

• Choosing a family just because the response is positive. Both candidates have positive support. Positivity alone does not distinguish their variance functions or establish that either conditional distribution is adequate.
• Assuming equal dispersion means equal variability. The variance includes both \(\phi\) and a different power of \(\mu\) for each family. Compare implied variances at relevant means.
• Treating a noisy variance pattern as a verdict. Group variance estimates can fluctuate substantially. Use the pattern as evidence alongside model-based residual behavior.
• Blaming the family for every residual trend. A trend can also signal an inadequate mean model or unaccounted-for structure. A family change cannot automatically fix those problems.

A useful sequence is to check that the response is strictly positive, compare the observed variance growth with the quadratic and cubic patterns, fit plausible candidates, and inspect whether their residuals show systematic changes in center or spread across fitted means. Prefer a family only when its implied variance behavior and residual patterns are reasonably consistent with the data and the scientific setting. The choice remains provisional when evidence is weak or competing explanations remain plausible.