Exponential-Family Longitudinal Submodels

Why Specify the Longitudinal Submodel Carefully?

In a joint model, the longitudinal component describes repeated measurements over time while accounting for subject-specific features that may also be relevant to the event process. A Gaussian linear mixed model is not appropriate for every repeated response: measurements may be strongly skewed, restricted to a particular range, or better represented by a non-Gaussian sampling distribution. Exponential-family longitudinal submodels provide a common framework for specifying such responses without requiring the trajectory to be linear on the response scale.

The central distinction is between the conditional mean of a response and its observed value. We model the mean given covariates and subject-level random effects, then specify a distribution whose variance is compatible with that mean. A link function connects the mean to a predictor, which can include the spline and random-effect terms used to represent individual trajectories.

Likelihood, Mean, Variance, and Link

A common exponential-family form for a single response is

Here \(\theta\) is the natural parameter, \(\phi\) is a dispersion parameter, and \(a\), \(b\), and \(c\) define the family and its parameterization. Differentiating the log normalizing structure gives

For a generalized linear model, write the mean-dependent part of the variance as a variance function \(V\). Under the usual dispersion parameterization, \(\operatorname{Var}(Y\mid\cdot)=\phi V(\mu)\). This separates three modeling choices: the response family, the relationship between variance and mean, and the link that transforms the mean to the predictor scale. A canonical link makes the linear predictor equal to the natural parameter, but canonical links are not required.

For repeated responses, a typical predictor is

The vectors \(x_{ij}\) and \(z_{ij}\) contain fixed-effect and random-effect covariates, respectively. The function \(h(t_{ij})\) is a chosen basis for time, such as a spline basis, and \(\gamma\) contains its coefficients. Depending on the model, some time terms may instead be represented among the fixed or random effects. The key point is that these terms define the predictor; the inverse link \(g^{-1}\) maps it to the conditional mean, \(\mu_{ij}=g^{-1}(\eta_{ij})\).

A standard likelihood assumption is that repeated responses are conditionally independent given the random effects and covariates. Under that assumption,

This is a conditional likelihood, not a claim that responses from the same subject are marginally independent. Integrating over shared random effects generally induces marginal dependence. If residual association remains after conditioning on the modeled effects, the product assumption may be inadequate and the longitudinal dependence structure needs reconsideration.

Choosing a Link and Matching the Variance

A link must map the allowed range of the conditional mean to the real line, so that \(\eta_{ij}\) can vary freely while \(g^{-1}(\eta_{ij})\) remains an admissible mean. The response family and its variance function should also reflect how spread changes with the mean. These choices are related but distinct: choosing a link alone does not specify a complete response distribution.

For example, for positive measurements a log link is one possible way to keep the conditional mean positive. If the conditional distribution is Gamma with mean \(\mu\) and dispersion \(\phi\), its variance is \(\phi\mu^2\). This family is an illustration of how a mean-variance relationship can differ from the constant-variance Gaussian case; it is not a general rule that every positive response must use this distribution or link.

The model is conditional on \(b_i\), so its mean trajectory is not generally the same as the population-average trajectory after random effects are integrated out. In a nonlinear link model, \(E\{g^{-1}(\eta_{ij})\}\) is generally not \(g^{-1}\{E(\eta_{ij})\}\). Consequently, a coefficient or predicted mean on the conditional scale should not automatically be reported as a population-average effect.

Justification: Why the Exponential-Family Variance Matters

The connection between the likelihood and its variance follows directly from the exponential-family form. For fixed \(\phi\), the log density, apart from terms not involving \(\theta\), is \(\{y\theta-b(\theta)\}/a(\phi)\). Its first derivative with respect to \(\theta\) is \(\{y-b'(\theta)\}/a(\phi)\). The expected score is zero at the model mean, which requires \(E(Y)=b'(\theta)\). Differentiating again gives \(-b''(\theta)/a(\phi)\). The information identity equates the variance of the score to the negative expected second derivative; since the score is \(\{Y-b'(\theta)\}/a(\phi)\), this yields \[ \frac{\operatorname{Var}(Y)}{a(\phi)^2} =\frac{b''(\theta)}{a(\phi)}, \qquad \operatorname{Var}(Y)=a(\phi)b''(\theta). \] Thus, once the family is chosen, its mean and variance are linked by the same cumulant function. A variance pattern that is incompatible with the data can signal a misspecified family or dispersion structure, even if the mean trajectory looks reasonable.

Diagnostics, Alternatives, and Limitations

Diagnostics should examine both the mean structure and the distributional assumptions. Conditional residuals, such as Pearson residuals \(r_{ij}=(y_{ij}-\widehat{\mu}_{ij})/\sqrt{\widehat{\phi}V(\widehat{\mu}_{ij})}\), can be plotted against time, fitted means, and covariates. Systematic trends may point to a missing time pattern or covariate effect; changing residual spread may indicate a poor variance function; extreme residuals may reveal outliers or a family that does not describe the response tails. In a joint model, these checks should be interpreted with care because random effects are estimated, not known, and ordinary residual patterns need not have their textbook behavior.

The main alternative is not a different link alone, but a different complete model specification. A Gaussian linear mixed model is a natural choice when conditional normality and approximately constant variance are defensible. A non-Gaussian exponential-family model can represent a different response distribution and mean-variance relationship. A quasi-likelihood approach specifies a mean and variance relationship without a full response likelihood; it may be useful for estimating mean effects, but does not supply the same fully specified likelihood contribution for joint likelihood-based inference. The choice should follow the measurement scale, scientific meaning, and diagnostics rather than convenience alone.

Limitations remain even with a suitable family. A chosen link may make effects difficult to interpret on the response scale; a correct conditional mean does not guarantee a correct variance or tail behavior; and random effects do not automatically capture every form of within-subject dependence. In a joint analysis, misspecification of the longitudinal submodel can affect estimation of shared subject-level quantities and their association with the event process. The submodel should therefore be treated as a set of explicit, assessable assumptions.

How the Longitudinal Likelihood Enters a Joint Model

Conditional on the subject-level random effects, the longitudinal contribution is the product of the visit-specific response densities. A joint model combines that contribution with the subject's event-time contribution conditional on the same shared effects, then integrates over their distribution. Schematically, for subject \(i\), the integrated contribution has the form

Here \(T_i\) and \(\delta_i\) denote the observed event or censoring time and event indicator, and \(f(T_i,\delta_i\mid b_i)\) represents the conditional event-time contribution. This expression emphasizes why the longitudinal likelihood must be specified conditionally and consistently: the same random effects connect its repeated responses to the event process. It does not imply that the event process causes a particular observed biomarker value.