Identity-Link Effects And Validity Of Fitted Means

What Identity-Link Coefficients Describe

For a gamma response, the conditional mean must be positive. With an identity link, the linear predictor itself is the conditional mean, so a change in a predictor corresponds to an additive change in that mean—not a ratio or a change in its reciprocal. This makes the coefficients direct to interpret, but it also imposes an important restriction: the fitted mean must stay strictly positive wherever the model is used.

Write the identity-link mean model for covariate vector \(x\) as \(\mu(x)=E[Y\mid x]=\eta(x)=\beta_0+\sum_{j=1}^p\beta_jx_j\). Here \(\beta_0\) is the intercept and \(\beta_j\) is the coefficient of covariate \(x_j\). After fitting the model, the corresponding estimated mean at covariate value \(x\) is \(\hat\mu(x)=\hat\beta_0+\sum_{j=1}^p\hat\beta_jx_j\). The mean and coefficients are in the response’s units when the predictors are expressed in their stated units.

The positivity condition is strict. A gamma distribution has positive support and a positive mean; zero is not an admissible gamma mean. Thus, checking whether the mean is merely nonnegative is not enough.

Additive Effects and Contrasts

For a model with no interactions, \(\beta_j\) is the change in the conditional mean associated with a one-unit increase in \(x_j\), holding the other covariates fixed. More generally, a change of \(d\) units in \(x_j\) changes the modeled mean by \(d\beta_j\), provided the other predictors do not change. The units are those of the response per unit of \(x_j\).

Proof. Let \(x\) denote the original covariate vector and \(x'\) the vector after replacing \(x_j=a\) by \(a+d\), with all other coordinates unchanged. Subtract the two model means:

The identical subtraction applies to the fitted mean function. This is an algebraic property of the identity-link linear predictor; it does not assert that the model is valid at either covariate value. Both means must still be positive for those values to be admissible. \(\square\)

A binary predictor is interpreted by comparing its two coded values, holding the other covariates fixed. For a 0/1 indicator with no interaction, its coefficient is the conditional mean difference between the group coded 1 and the reference group coded 0. If there is an interaction, the difference generally depends on the interacting covariate, so the indicator’s coefficient alone describes the contrast only at the value where that covariate is zero.

Interactions Change the Meaning of a Coefficient

An identity-link model can include products of predictors. For example, \(\mu(x,z)=\beta_0+\beta_1x+\beta_2z+\beta_3xz\). The effect of changing \(z\) from 0 to 1, holding \(x\) fixed, is then \(\beta_2+\beta_3x\), not just \(\beta_2\). The interaction coefficient describes how the additive contrast changes per unit increase in \(x\).

Checking Positivity on a Covariate Region

A fitted value at each observed covariate pattern is an important check, but it is not always sufficient. If the model will be used to predict at other combinations, those intended combinations must also be checked. In particular, data may contain observations at certain combinations of covariates without covering every combination in a plausible prediction region.

For an affine mean function over a rectangular region, positivity can be checked at the corners. This gives a convenient finite check when each continuous covariate has a specified interval and there are no other constraints on the region.

Proof. Every corner belongs to \(R\), so positivity everywhere in \(R\) implies positivity at every corner. For the converse, take any \(x\in R\). For each coordinate with \(a_j<b_j\), define \(t_j=(x_j-a_j)/(b_j-a_j)\), which lies in \([0,1]\). The point \(x\) is a weighted average of the corners \(v\) of \(R\): the weight for corner \(v\) is the product, across coordinates, of \(t_j\) when that corner uses \(b_j\), and \(1-t_j\) when it uses \(a_j\). These weights are nonnegative, sum to 1, and reproduce each coordinate of \(x\). (A coordinate with \(a_j=b_j\) is fixed and contributes no choice of corner.) Since \(m\) is affine, it preserves such weighted averages. Therefore \(m(x)\) is the same weighted average of the values \(m(v)\) at the corners. If all corner values are strictly positive, their weighted average is strictly positive. This proves the converse. \(\square\)

For categorical predictors, check every relevant category combination together with the continuous covariate values of interest. If the covariate region is constrained rather than rectangular—for example, some combinations cannot occur—the corner result for a full rectangle does not automatically apply to that smaller region. Check the relevant set itself or use a valid bound for that set.

Validity Is Part of Interpretation

An identity-link coefficient is an additive effect in the modeled mean, but that interpretation is useful only at covariate values where the mean model is admissible. A coefficient’s sign does not by itself determine validity: a negative slope can be compatible with positive means over a limited range, while a positive slope can still accompany a negative fitted mean if the intercept or other terms make it so. The complete linear predictor must be evaluated.

Also distinguish the range used to fit the model from the range where predictions are intended. Checking positivity at all observed points confirms that the fitted values there can serve as gamma means. It does not guarantee positivity at new covariate combinations, nor does it validate extrapolation beyond the range of the data. When a fitted mean is nonpositive, it is not a small or unusual gamma mean; it lies outside the allowed mean domain.

• Interpret the contrast at the right values. With interactions, state the covariate values at which a group difference or slope is being described.
• Check the whole relevant set. Include planned prediction points and combinations, not only the observed rows.
• Require strict positivity. A fitted value of zero is invalid just as a negative fitted value is.

Common mistake: “The fitted means at every observed row are positive, so the model is valid for any prediction.” This conclusion extends the check beyond what it establishes. The correction is to specify the covariate region where predictions will be made and verify positivity throughout that region; for a rectangular region and an affine mean, checking every corner is sufficient. Another mistake is to call an indicator coefficient the treatment effect even when it interacts with a covariate. In that case, compute the contrast as a function of the interacting covariate and report its value at the covariate levels of interest.