Why Check the Random-Effects Distribution?
In a joint model, random effects represent subject-to-subject heterogeneity that is not captured by measured covariates. Their assumed distribution affects more than the longitudinal fit: it also affects how subject-specific trajectories are estimated and, when the event model depends on those trajectories or random effects, how event risk is modeled. A plausible mean trajectory and acceptable overall residual summaries therefore do not guarantee that the assumed distribution of heterogeneity is adequate.
The diagnostic question is not simply whether a histogram of estimated random effects looks normal. Subject-specific estimates are uncertain and typically pulled toward the population mean. The event and longitudinal data both contribute information about them, often to different degrees. A useful check asks whether the observed subject-level evidence is compatible with the distribution implied by the fitted joint model, allowing for this uncertainty and shrinkage.
For example, write a longitudinal submodel as \(Y_{ij}=m_i(t_{ij};b_i)+\epsilon_{ij}\), where \(Y_{ij}\) is the marker for subject \(i\) at time \(t_{ij}\). A survival submodel may have hazard \(h_i(t\mid b_i)\), with the random effects entering directly or through the marker trajectory. The joint model specifies the distribution of the observed data by integrating over \(b_i\):
A distribution check is most interpretable when the other model components are reasonably specified: the longitudinal mean and residual variance, the event-time model, the association structure, and the censoring assumptions. If one of these is wrong, unusual subject-level patterns can be incorrectly attributed to \(p(b_i)\). The check is therefore evidence about the fitted joint model, not an isolated test of a distribution in a vacuum.
Why Empirical Bayes Estimates Need Care
Even when the model assumes normally distributed random effects, the empirical Bayes estimates will not generally have that same normal distribution. Subjects with few measurements or little event information have uncertain conditional distributions, so their estimates tend to lie closer to the population mean than their underlying random effects. Subjects with more informative records may have less shrinkage. The distribution of the estimates is consequently shaped by the random-effects distribution, measurement schedules, residual variation, event information, and estimation method.
Worked Example: Seeing Shrinkage in a Random-Intercept Model
Suppose a fitted model has a random intercept \(b_i\sim N(0,1)\), with marker measurements \(Y_{ij}=10+b_i+\epsilon_{ij}\), where independent errors have variance \(1\). Consider a hypothetical subject with four measurements \(11.2,\ 10.8,\ 11.0,\ 11.0\). The deviations from the fixed mean 10 are \(1.2,\ 0.8,\ 1.0,\ 1.0\), so their average is \(1.0\).
For this normal-normal model, the conditional distribution of \(b_i\) given the four measurements has variance and mean
Thus the empirical Bayes posterior mean is \(0.8\), not the raw average deviation \(1.0\). Its posterior standard deviation is \(\sqrt{0.2}=0.4472\). The estimate is pulled toward the population mean zero because the model treats the four measurements as noisy evidence about a random effect. A histogram of such posterior means will generally be narrower than the assumed \(N(0,1)\) population distribution. That difference alone is not evidence of a failure of normality.
In a joint model, the conditional distribution uses the event record as well as the marker record. Formally, if \(\tilde T_i\) is the observed event or censoring time and \(\delta_i\) indicates whether the event was observed, then the posterior is proportional to the product of the longitudinal likelihood, the event contribution, and the random-effects density:
This expression assumes the model’s stated conditional structure, including conditional independence of the longitudinal and event processes given the shared random effects and covariates when that is the chosen joint-model specification. Censoring must also be handled as specified by the fitted model. The event record can shift or sharpen the posterior for \(b_i\); treating an estimate based only on the marker data as if it were the joint-model empirical Bayes estimate can therefore be misleading.
A Simulation-Based Distribution Check
A practical approach is to compare a chosen statistic of the observed empirical Bayes summaries with the same statistic calculated from data simulated under the fitted model. Each simulated data set should preserve the relevant design features, such as the number of subjects, their covariates, measurement times, and censoring scheme. It should generate random effects from the fitted distribution, then generate marker and event data under the fitted joint model. Refitting the model to each simulated data set is important when the observed analysis estimates the parameters: it incorporates the estimation process into the reference comparison.
The statistic might be the spread or skewness of posterior means, a tail count, a correlation between random-intercept and random-slope estimates, or a feature of residuals grouped by an empirical Bayes summary. It should be selected to target a plausible departure, rather than chosen after inspecting many results and reporting only the most unusual one. The observed and simulated statistics must be defined in the same way.
Choose a summary that reflects the distributional concern, such as excess tail estimates or an unusual association between random-effect components.
Simulate complete longitudinal and event records from the fitted joint model, retaining the observed design and censoring setup as appropriate.
Refit the model and calculate the chosen empirical Bayes or residual summary in every simulated data set, just as for the observed data.
Locate the observed statistic in the simulated reference distribution. An unusual value flags a lack of compatibility with the fitted model; it does not identify the cause by itself.
Worked Example: Comparing Tail Counts With Simulations
Suppose a hypothetical analysis has 30 subjects. The diagnostic statistic is the number of their empirical Bayes random-intercept estimates whose absolute values exceed \(1.5\) times the fitted population standard deviation. The observed count is 8. This statistic measures the frequency of relatively extreme estimated intercepts, but its reference distribution must account for shrinkage and the subjects’ information levels.
The fitted model is used to generate 999 simulated data sets with the same covariates, measurement schedules, and censoring setup. The model is refitted to each data set, and the count is recalculated. In 7 of the 999 simulations, the count is at least 8. With the common plus-one estimate of a one-sided simulation tail probability,
The observed tail count is unusual relative to this fitted-model simulation reference: only about 0.8% of the simulated statistics were at least as large. This is evidence that the fitted model does not readily reproduce this feature. It is not a 0.8% probability that the model is correct or incorrect, and it does not establish that the random-effects distribution is the source of the discrepancy. A misspecified trajectory, event model, residual variance, or dependence structure could also produce an unusual count.
A graphical comparison can complement a scalar statistic. Plot the observed empirical Bayes summaries against summaries from simulated-and-refitted data sets, or compare observed and simulated quantiles. The key is that the simulations reproduce the estimation process. Drawing a normal curve with the fitted random-effects variance and comparing it directly to posterior means does not do so.
Residual Patterns and What They Can Tell You
Distributional misspecification can leave structure in longitudinal residuals. For example, if the fitted random-effects distribution assigns too little probability to subjects with unusually high or low trajectories, the model may systematically underpredict some subjects and overpredict others. Residuals grouped by empirical Bayes intercept or slope can help reveal such patterns. However, residuals are not independent evidence of the random-effects distribution: they are calculated using fitted trajectories, and the same records helped estimate the empirical Bayes summaries.
Worked Example: Interpreting Residuals Across Empirical Bayes Groups
In a hypothetical fitted model, subjects are divided into three groups according to their empirical Bayes random-intercept estimates. The average longitudinal residuals, in marker units, are \(-0.40\) for the low-estimate group, \(0.02\) for the middle group, and \(0.40\) for the high-estimate group. The end-to-end difference is
This pattern is a warning: the fitted conditional trajectories may leave residual structure that varies with estimated subject level. One possible explanation is that the assumed random-intercept distribution or its tails do not represent heterogeneity adequately. But the pattern could also arise from an incorrect fixed time trend, an omitted covariate, or a wrong association between the marker and event process. The diagnostic should prompt targeted checks of these alternatives and a comparison with model-based simulated residual patterns; it is not a stand-alone test that the random effects are nonnormal.
Useful diagnostic questions include whether empirical Bayes estimates show unexpected skewness or clusters, whether random-intercept and random-slope summaries show an implausible relationship, and whether residual means or spreads change systematically across estimate groups. A concern is more persuasive when a feature is both scientifically interpretable and unusual under simulations that include the actual design. Conversely, a smooth-looking plot does not establish that the assumed distribution is correct, especially with few subjects or weak information per subject.
Alternatives, Failure Modes, and Limits
| Approach | What it can show | Main limitation |
|---|---|---|
| Histogram or normal Q-Q plot of empirical Bayes estimates | Large-scale skewness, clustering, or extreme estimates may be visible. | Ignores unequal posterior uncertainty and shrinkage; the estimates are not direct draws from the population distribution. |
| Model-based simulation comparison | Checks whether a chosen feature of the estimates or residuals is plausible under the fitted analysis. | Depends on the rest of the joint model and can have limited power with few subjects or weak information. |
| Alternative random-effects distributions | Shows whether fit or predictions are sensitive to the distributional assumption. | More flexible distributions can be difficult to estimate and do not automatically resolve misspecification elsewhere. |
The normality assumption is not equally consequential for every purpose. It can matter substantially when predicting individual trajectories or integrating random effects into event-risk predictions; for some population-average summaries, modest deviations may have less effect. This does not make distributional checks optional, but it means that a flagged departure should be judged in light of the intended estimand and predictions.
Common failure modes include comparing empirical Bayes estimates directly with the assumed population density, ignoring the event contribution to their posterior distributions, and interpreting a small simulation tail probability as proof of a particular alternative. Another is to simulate data but not refit the model, thereby comparing quantities with different amounts of parameter-estimation uncertainty. Finally, when there are few subjects, the data may not contain enough information to distinguish a normal distribution from a moderately heavy-tailed or skewed alternative. A lack of visible departure is not evidence of exact normality.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Why can the empirical Bayes estimates have less spread than the assumed population random-effects distribution?
- A simulation check uses \(B=499\) data sets, and 4 simulated statistics are at least as extreme as the observed statistic. Calculate the plus-one tail probability.
- In a random-intercept model with prior variance 1 and residual variance 1, a subject has two measurements with deviations from the fixed mean of 1 and 1. What is the posterior mean of the random intercept?
- Why does an unusual residual trend across empirical Bayes groups not prove that the random-effects distribution is misspecified?
Solutions. (1) Each estimate is a conditional summary with finite uncertainty and is generally pulled toward the population mean; the degree of shrinkage varies with the subject’s information. Thus the estimates are not direct draws from the population distribution. (2) Here \(r=4\) and \(B=499\), so \(\hat p_{\mathrm{sim}}=(r+1)/(B+1)=5/500=0.010\). The observed statistic is unusual under the fitted-model simulations, but that result is not the probability that the model is false. (3) The posterior variance is \((1+2)^{-1}=1/3\), and the posterior mean is \((1/3)(1+1)=2/3\), approximately \(0.667\) random-intercept units. (4) The residual trend could also reflect an incorrect longitudinal mean, omitted covariates, or misspecification in the event or association component. The pattern identifies a model-fit concern, not its unique cause.