Sensitivity to Latent-Effect Distributions and Link Functions

Why Separate These Sensitivity Questions?

A shared-parameter model connects a repeated outcome process and dropout through latent effects. The conclusions can therefore depend on two different modeling choices: how those latent effects are distributed across individuals, and how a latent effect changes the dropout hazard. These choices are related in the likelihood, but they are not the same assumption. Varying both at once makes it difficult to tell what caused a change in the results.

The aim here is a controlled sensitivity analysis. First compare latent-effect distributions with the dropout link fixed. Then compare dropout links with the latent-effect distribution fixed. Finally, fit combinations of the choices to see whether their effects interact. This is a way to assess dependence on model specification; it does not establish which untestable specification is true.

Write \(Y_{it}\) for the outcome of individual \(i\) at time \(t\), \(D_{it}=1\) if dropout occurs in interval \(t\), and \(b_i\) for an individual-level latent effect. Let \(D_{it}=0\) denote remaining under observation through interval \(t\). A representative discrete-time shared-parameter model has an outcome distribution conditional on \(b_i\), and a dropout hazard \(h_{it}=P(D_{it}=1\mid D_{i,t-1}=0,Y_{i,\mathrm{obs}},b_i)\). The observed-data likelihood integrates over the distribution of \(b_i\), as in the earlier material on integrating random effects in the observed-data likelihood.

For example, a dropout model may have \(h_{it}=g^{-1}(\eta_{it})\), where \(\eta_{it}=\gamma_0+\boldsymbol{\gamma}^{\mathsf T}\mathbf{x}_{it}+\alpha b_i\). Here \(g^{-1}\) is the inverse link, \(\mathbf{x}_{it}\) contains observed predictors, and \(\alpha\) describes the association of the latent effect with dropout on the linear-predictor scale. A distributional comparison changes the distribution of \(b_i\); a link comparison changes \(g^{-1}\). The outcome model, covariates, and time structure should otherwise be held consistent unless there is a stated reason to vary them too.

Why Equal Moments Do Not Ensure Equal Hazards

The population-average dropout hazard is an average of conditional hazards, not generally the hazard evaluated at the average latent effect. For a fixed linear predictor apart from \(b_i\), it is \(E\{g^{-1}(\eta+\alpha b_i)\}\). Because inverse links are nonlinear, this expectation can depend on features of the distribution beyond its mean and variance. Skewness, tail weight, or multiple modes can matter, particularly when the link is steep over values with appreciable probability.

A short Taylor expansion makes the role of distributional shape explicit. Put \(f(x)=g^{-1}(x)\) and suppose the derivatives and moments used below exist. Expanding \(f(\eta+\alpha b_i)\) around \(\eta+\alpha\mu_b\), where \(\mu_b=E(b_i)\), and taking expectations gives, to second order,

The linear term vanishes because \(E(b_i-\mu_b)=0\). Higher-order terms involve higher moments, such as skewness and kurtosis. Thus matching the first two moments removes neither the curvature effect nor the influence of distributional shape on the exact marginal hazard.

Varying the Dropout Link

A link determines how changes in the linear predictor translate into conditional hazards. Two common discrete-time choices are the logit link, with inverse link \(\operatorname{expit}(\eta)=1/\{1+\exp(-\eta)\}\), and the complementary log-log link, with inverse link \(1-\exp\{-\exp(\eta)\}\). The latter is often useful when discrete intervals represent grouped event times under a proportional-hazards interpretation. Their functional forms differ, so the same numerical predictor need not imply the same hazard.

A fair link comparison keeps the distribution of \(b_i\), the outcome model, covariates, time indexing, and other structural choices fixed. It should also make clear whether parameters are merely re-estimated under each link or whether some predictors are held at specified values for a conditional comparison. Coefficients on different link scales are not directly comparable as if they had the same units.

If the scientific question concerns differences in marginal dropout risk, compute or estimate those risks under each fitted model by integrating over its latent-effect distribution. A conditional hazard at \(b_i=0\) is not generally the population-average hazard. When comparing links, consider reporting risks or other quantities on a common, interpretable scale alongside the link-scale coefficients.

Crossed Comparisons and a Worked Sensitivity Analysis

A crossed design fits every combination of the selected distributions and links. For instance, use a normal and a two-point latent-effect distribution, each with a logit and a complementary log-log dropout link. The two distribution contrasts at fixed link show distribution sensitivity; the two link contrasts at fixed distribution show link sensitivity. If those contrasts differ, the effect of one choice depends on the other, indicating an interaction in the sensitivity analysis.

In a real analysis, alternatives should be scientifically plausible and estimable. For a distribution comparison, specify how location and scale are handled. A common strategy is to impose a mean-zero latent effect and estimate its variance under each candidate distribution; if scales are instead fixed, explain why. For a link comparison, refit parameters under each link rather than interpreting a coefficient from one link as though it were on the other link’s scale. Record convergence and integration accuracy for every fit, since an apparent sensitivity can be caused by numerical failure.

Assumptions, Diagnostics, and Failure Modes

These comparisons inherit the assumptions of the shared-parameter model. In particular, the chosen outcome and dropout models must be appropriate conditional on their covariates and latent effects; the latent effects must be integrated using the specified distribution; and any conditional-independence assumptions used to construct the likelihood must be defensible. The sensitivity analysis varies selected assumptions; it does not repair a misspecified outcome model, an omitted predictor, or a poorly identified association structure.

Some checks are especially important when varying distributions and links:

• Check comparable parameterization. Confirm that the latent effect has a consistent meaning and scale across candidate distributions. Centering and variance constraints should be stated, not silently changed.
• Check integration and optimization. Compare numerical integration settings, quadrature stability, convergence information, and parameter estimates from reasonable starting values. A fit that fails to converge is not evidence of sensitivity to that scientific model.
• Check extreme latent-effect regions. Heavy tails or discrete support points can place probability where the hazard is near zero or one. Inspect whether estimates, standard errors, or marginal risks are driven by a small amount of tail mass.
• Check identification. The dropout intercept, latent-effect scale, and association parameter can compensate for one another. A boundary estimate, a nearly flat likelihood, or large changes in these parameters across fits may signal weak identification rather than a stable distinction between models.
• Check the target scale. Compare estimates of the same scientific quantity, such as a marginal risk difference or adjusted outcome contrast. Raw coefficients from different links are on different scales.

Sensitivity results should be summarized by both the chosen model and the substantive conclusion: direction, magnitude, uncertainty, and whether the conclusion changes across plausible specifications. If a key conclusion changes, report that dependence directly. If it remains similar, state the range of estimates and uncertainty rather than implying that sensitivity analysis proves robustness to every possible model.

Comparison and Limitations

A single preferred parametric model is efficient and can be interpretable, but it conceals dependence on its distributional and link assumptions. A crossed sensitivity analysis makes that dependence visible and helps locate its source. Flexible latent-effect distributions can represent skewness or heavy tails, but add parameters and may be weakly identified. Alternative links allow different hazard shapes, but can change coefficient interpretation and may be difficult to distinguish with limited dropout information. No comparison can identify a unique model if the observed data provide little information about the unobserved outcomes or latent heterogeneity.

A common mistake is to change the random-effect distribution and the dropout link simultaneously, observe a different estimate, and attribute the entire change to “non-normal random effects.” The comparison does not justify that explanation: the link changed too. The correction is to fit controlled comparisons, first varying one specification at a time and then crossing the choices. Another mistake is to infer equal marginal dropout from equal latent-effect means and variances. The worked example shows why this fails: nonlinear averaging depends on distributional shape as well.

These sensitivity fits are model-based analyses of the same observed data, not independent replications. Their spread is not a confidence interval for uncertainty over all possible models. It is evidence about how the chosen conclusions respond to the specific, defensible alternatives that were examined.