Marginal Mean Specifications for Continuous Outcomes

Marginal Means for Continuous Clustered Responses

A continuous outcome measured repeatedly on the same person, or on several members of the same group, can have a mean that changes with treatment, time, and other covariates. A marginal mean model describes that change directly: it specifies the average response for each covariate pattern without making the mean conditional on the other responses in the cluster.

Two useful choices are the identity link, which models the mean itself as a linear predictor, and a nonlinear link, such as the log link, which models a transformation of the mean. The choice changes both the form of the fitted means and the interpretation of covariate effects.

The predictor \(x_{ij}^{\mathsf T}\beta\) may include an intercept, indicators, time terms, interactions, or other chosen functions of covariates. The model concerns the mean, not the entire distribution of \(Y_{ij}\). In particular, specifying a mean does not by itself specify a variance or a within-cluster dependence pattern.

Identity-Link Models

With the identity link \(g(\mu)=\mu\), the marginal mean is linear in the predictor:

This link is natural when additive differences in the original outcome units are scientifically meaningful. A coefficient for a binary treatment, when there are no interactions involving treatment, is the adjusted difference in marginal means between treatment groups. A coefficient for a continuous covariate is the change in the mean per unit increase in that covariate, holding the other modeled covariates fixed.

Because a linear predictor can take any real value, an identity-link specification does not automatically keep fitted means positive. Whether that is acceptable depends on the outcome and on the covariate patterns over which predictions are needed. A model whose fitted mean is negative for a pattern where negative values are impossible is not a satisfactory mean specification for that use.

Log-Link Models for Positive Means

For a response whose mean must be positive, a common nonlinear choice is the log link \(g(\mu)=\log(\mu)\). Its inverse link is the exponential function:

Every finite linear predictor then gives a positive fitted mean. The model is linear in the log of the mean, not in the outcome itself. Consequently, coefficients generally do not represent additive changes in the original outcome units. For a binary covariate with coefficient \(\beta_A\) and no interactions involving it, changing that covariate from 0 to 1 multiplies the mean by \(\exp(\beta_A)\). This is a ratio of means; the corresponding difference depends on the starting mean.

How Link-Scale Effects Become Response-Scale Effects

A coefficient is first interpreted on the scale specified by the link. To answer a scientific question about the original outcome, use the inverse link to obtain fitted means and then take the requested difference, ratio, or other contrast. Under a log link, this translation can be written exactly.

Proof. By the inverse log link, \(\mu(x_1)=\exp(x_1^{\mathsf T}\beta)\) and \(\mu(x_0)=\exp(x_0^{\mathsf T}\beta)\). Dividing the first equality by the second and using \(\exp(a)/\exp(b)=\exp(a-b)\) gives

Subtracting the two fitted means gives the stated difference directly. The difference is not generally a constant determined by the coefficient contrast alone, since it also depends on the covariate pattern through the baseline mean. \(\square\)

For a continuously varying covariate \(z\), if the predictor is differentiable in \(z\), the response-scale slope follows from the chain rule. In a log-link model, \(\mu(z)=\exp(\eta(z))\), so

Thus, if \(z\) enters the predictor linearly with coefficient \(\beta_z\), the instantaneous response-scale change is \(\beta_z\mu(z)\), not simply \(\beta_z\) outcome units. For a finite change \(\Delta\), the mean ratio is \(\exp(\beta_z\Delta)\), provided no other model terms change.

Interactions and Mean Contrasts Under the Identity Link

Under an identity link, a treatment-by-time interaction has a direct additive interpretation. In the model \(\mu=\beta_0+\beta_A A+\beta_t t+\beta_{At}At\), the treatment-minus-control mean difference at time \(t\) is \(\beta_A+\beta_{At}t\). The interaction coefficient is the change in that difference per unit of time.

Derivation. Setting \(A=0\) gives \(\mu(0,t)=\beta_0+\beta_tt\), whose slope with respect to \(t\) is \(\beta_t\). Setting \(A=1\) gives \(\mu(1,t)=\beta_0+\beta_A+(\beta_t+\beta_{At})t\), whose slope is \(\beta_t+\beta_{At}\). Subtracting the control slope from the treatment slope gives \(\beta_{At}\). Subtracting the two means at the same \(t\) gives

These are additive effects in the outcome's original units. With a log link and the same predictor terms, the corresponding treatment-to-control ratio is \(\exp(\beta_A+\beta_{At}t)\): the interaction changes the log ratio per unit time. \(\square\)

Choosing and Reporting a Mean Specification

The link should reflect the scale on which a linear predictor is useful and the range of means that is plausible. An identity link makes additive changes straightforward but may predict means outside a scientifically possible range. A log link guarantees positive means and makes multiplicative comparisons straightforward, but its coefficients are not additive outcome-unit effects. Neither link is automatically right for every continuous outcome.

A report should state the modeled mean, identify the link, and give the covariate pattern for each interpretation. For example, “the treatment-to-control ratio at year two is \(\exp(\beta_A+2\beta_{At})\)” is more informative than saying only that “the treatment coefficient is an effect.” If the scientific question concerns an absolute difference, calculate that difference from the fitted means at the stated covariate values. If it concerns a relative change, report the corresponding ratio or percentage.

A related error is to treat a log-link mean model as a model for \(\log(Y_{ij})\). The specification here is \(\log\{E[Y_{ij}\mid X_i]\}=x_{ij}^{\mathsf T}\beta\); it does not assert that the observed outcome has been logged or that its conditional distribution is lognormal. Keeping the mean specification distinct from assumptions about the full response distribution makes the interpretation precise.