Choosing a Family for the Scientific Mean

Start With the Mean the Study Needs

A response family is not merely a convenient shape for a histogram. In a generalized linear model, it makes assumptions about the response’s possible values and its variability around a conditional mean. Those assumptions should be plausible for the process that generated the measurements. The first question is therefore scientific: what average response does the study aim to describe?

For a response \(Y_i\) and predictors \(x_i\), write its conditional mean as \(\mu_i=E[Y_i\mid x_i]\). This is an average on the response’s stated measurement scale. For instance, if \(Y_i\) is concentration in milligrams per litre, then \(\mu_i\) is expected concentration in those units. A family choice should respect what this mean represents, what values \(Y_i\) can take, and how variation is expected to change as \(\mu_i\) changes.

Definition: For a candidate response family indexed by its mean \(\mu\) and a dispersion parameter \(\phi>0\), a mean–variance specification states the conditional expectation \(E[Y\mid x]=\mu\), the conditional variance \(\operatorname{Var}(Y\mid x)=\phi V(\mu)\), and the support of \(Y\mid x\). The function \(V(\mu)\) describes how the expected variance changes with the mean; the support describes which response values the family permits.

For common examples, a Gaussian family has \(V(\mu)=1\), a Gamma family has \(V(\mu)=\mu^2\), and an inverse-Gaussian family has \(V(\mu)=\mu^3\). These are different scientific variability assumptions, not interchangeable labels for “continuous data.” In each case, the mean must also be in a range where the family is defined: Gaussian means can be any real number, while Gamma and inverse-Gaussian means must be positive.

This comparison concerns the response distribution conditional on predictors. It is separate from the choice of link that relates \(\mu_i\) to predictors. A plausible mean–variance pattern cannot rescue a scientifically unsuitable relationship for the mean, and a convenient link cannot make an impossible response value plausible.

Three Questions for Screening a Family

A useful initial screen connects the measurement process to the family’s assumptions. It does not prove that a family fits the data; it identifies assumptions that are credible enough to assess further.

What does the mean represent?
Specify the expected response conditional on the predictors, in the original scientific units. Check that this is the average relevant to the research question.

What values can the response take?
Check whether negative, zero, or only strictly positive values are scientifically possible. A family whose support excludes plausible observations is not a suitable description without further justification.

How should variability change as the mean changes?
Use knowledge of the measurement or response process to consider whether the standard deviation is roughly constant, grows in proportion to the mean, or grows more rapidly.

The variance patterns have direct interpretations. Under Gaussian constant variance, the standard deviation does not change with the mean. Under Gamma variance, the coefficient of variation—the standard deviation divided by the mean—is constant. Under inverse-Gaussian variance, the standard deviation grows as the mean to the power \(3/2\), when \(\phi\) is fixed. These are expectations about repeated responses at given predictor values, not promises about every observation.

Worked Comparisons

Worked Example: Additive Measurement Error

A laboratory records the difference between a sample’s measured signal and a fixed reference signal. Differences can be positive or negative. At two settings, the expected differences are 12 and 20 units. Instrument calibration suggests that repeated readings at either setting have a standard deviation near 2 units.

A Gaussian candidate has constant variance \(\phi\). Setting \(\phi=2^2=4\) gives variance 4 at both means:

\operatorname{Var}(Y\mid x_1)=4,\qquad \operatorname{Var}(Y\mid x_2)=4.

The support of the Gaussian distribution allows negative differences, as this measurement does. The constant-variance assumption also matches the stated additive-error mechanism: the instrument’s error scale is approximately the same at both settings. This makes the Gaussian family a scientifically reasonable candidate. It does not establish that the full conditional distribution is Gaussian; that remains an empirical question.

Worked Example: Proportional Variation in a Positive Measurement

A biological assay measures a strictly positive concentration. Suppose scientific knowledge suggests that variability is multiplicative: samples with twice the expected concentration tend to have twice the standard deviation. Consider expected concentrations of 10 and 20 milligrams per litre, with a coefficient of variation of 0.25.

At mean 10, the implied standard deviation is \(0.25(10)=2.5\), so the variance is \(2.5^2=6.25\). At mean 20, the standard deviation is \(0.25(20)=5\), and the variance is \(5^2=25\). The variance-to-mean-squared ratios agree:

\frac{6.25}{10^2}=0.0625,\qquad \frac{25}{20^2}=0.0625.

Thus a Gamma mean–variance specification with \(\phi=0.0625\), giving \(\operatorname{Var}(Y\mid x)=0.0625\mu^2\), matches these two expected variance benchmarks. Its positive support is also consistent with strictly positive concentrations. The calculation checks the scientific variance assumption; it does not show that every aspect of the Gamma distribution is appropriate.

Worked Example: Variability That Rises More Rapidly With the Mean

Suppose a strictly positive response is a duration, and process knowledge suggests that the variance grows roughly as the cube of its conditional mean. Compare two settings with means 2 and 4 time units. For an inverse-Gaussian candidate, let \(\phi=0.1\), so that \(\operatorname{Var}(Y\mid x)=\phi\mu^3\).

At mean 2, the predicted variance is \(0.1(2^3)=0.1(8)=0.8\), with standard deviation \(\sqrt{0.8}\), approximately 0.894. At mean 4, the predicted variance is \(0.1(4^3)=0.1(64)=6.4\), with standard deviation \(\sqrt{6.4}\), approximately 2.530. Doubling the mean multiplies this variance by \(6.4/0.8=8\), as the cubic relationship requires.

This candidate is worth considering if a cubic variance increase is scientifically credible. If the actual process instead suggests a constant coefficient of variation, the Gamma variance pattern would be a more natural starting point. Positivity alone does not distinguish these choices.

Why Measurement Units Matter

A scientific response has a measurement scale and units. A family’s variance assumption should remain interpretable when those units change. The transformation of the mean and variance gives a precise check.

Result: If \(Z=a+bY\) for constants \(a\) and \(b>0\), then \(E[Z\mid x]=a+b\mu\) and \(\operatorname{Var}(Z\mid x)=b^2\operatorname{Var}(Y\mid x)\). In particular, if \(a=0\) and \(\operatorname{Var}(Y\mid x)=\phi\mu^p\), then \(Z\) has the same power \(p\) variance pattern in its new mean, with dispersion \(b^{2-p}\phi\).

Proof. Conditional expectation is linear, so \(E[a+bY\mid x]=a+bE[Y\mid x]=a+b\mu\). Adding a constant does not change variance, and multiplying by \(b\) multiplies variance by \(b^2\); hence \(\operatorname{Var}(a+bY\mid x)=b^2\operatorname{Var}(Y\mid x)\). For \(a=0\), the new mean is \(\nu=b\mu\), so \(\mu=\nu/b\). Substituting into the variance gives

b^2\phi\mu^p =b^2\phi\left(\frac{\nu}{b}\right)^p =b^{2-p}\phi\nu^p.

The power pattern is preserved under a change of units with no shift, while the numerical dispersion changes. A shift \(a\ne0\) changes the mean’s origin: the transformed variance becomes \(b^2\phi((\nu-a)/b)^p\), which is generally not a pure power of \(\nu\). Thus, the response’s zero point matters as well as its units. The result follows. \(\square\)

This also explains why dispersion numbers should not be interpreted without their measurement scale. A variance coefficient attached to measurements in one unit system will generally have a different numerical value after conversion, even though the underlying variability has not changed.

Interpreting the Mean–Variance Assumptions

Result: Under the Gamma mean–variance specification \(\operatorname{Var}(Y\mid x)=\phi\mu^2\), with \(\mu>0\) and \(\phi>0\), the conditional coefficient of variation is \(\sqrt{\phi}\), independent of \(\mu\).

Derivation. The conditional standard deviation is the positive square root of the variance:

\operatorname{SD}(Y\mid x) =\sqrt{\operatorname{Var}(Y\mid x)} =\sqrt{\phi\mu^2} =\sqrt{\phi}\,\mu,

where the last equality uses \(\mu>0\). Dividing by the conditional mean gives \(\operatorname{SD}(Y\mid x)/\mu=\sqrt{\phi}\). Therefore, equal relative variability across predictor settings is exactly the Gamma variance assumption, not equal absolute variability. \(\square\)

The Gaussian and inverse-Gaussian patterns can be read just as directly. If \(\operatorname{Var}(Y\mid x)=\phi\), then the conditional standard deviation is \(\sqrt{\phi}\), independent of the mean. If \(\operatorname{Var}(Y\mid x)=\phi\mu^3\) for \(\mu>0\), then the standard deviation is \(\sqrt{\phi}\mu^{3/2}\). These consequences help translate an abstract variance function into a claim about the scientific process.

Common Mistake: Choosing by Positivity Alone

A common but incomplete argument is: “The response is positive, so a positive-support family must be the right one.” Positivity can rule out some candidates when negative responses are genuinely impossible, but it cannot distinguish positive-support families from each other. Gamma and inverse-Gaussian models both require positive means and describe positive responses, yet their variance functions make different predictions as the mean changes.

The correction is to check support and variability together, and to define the target mean in context. Ask whether the scientific process suggests roughly constant absolute spread, constant relative spread, or another pattern. Then treat a compatible family as a candidate for further assessment—not as a conclusion established by the response scale alone.

Key takeaway: Choose a candidate family by aligning its support and mean–variance assumptions with the scientific response process and measurement scale. The family should make sense for the conditional mean the study needs to interpret; empirical checks are still needed to assess whether the candidate is adequate.

Check Your Understanding

Use the support, mean, and expected variability together in each answer.

A response can be negative, and its repeated-measurement standard deviation is approximately constant as the mean changes. Which of the Gaussian, Gamma, and inverse-Gaussian variance patterns best matches that description? Explain.
Under a Gamma mean–variance specification with \(\phi=0.04\), what are the coefficient of variation and variance when \(\mu=15\)? Show the calculation.
A response’s mean doubles, and process knowledge suggests that its variance should increase by a factor of eight. Which of the three power patterns discussed is consistent with that prediction?
Why does converting a positive response from one unit to another change the numerical dispersion parameter even when it preserves a power variance pattern?
Why is “the response is positive” not enough, on its own, to choose between Gamma and inverse-Gaussian candidates?