Conditional and Unconditional Parametric Bootstraps

Why the Bootstrap Must Specify What Is Held Fixed

In a nonlinear mixed-effects model, a subject’s observations vary for at least two reasons: the subject has a latent random effect, and observations have residual error. A parametric bootstrap can simulate both sources of variation, or it can hold subject-specific random effects fixed and simulate only residual variation. These choices answer different questions. The word conditional refers to conditioning on subject-specific effects; unconditional refers to drawing new effects from the population distribution as well as generating residual error.

This distinction matters when the bootstrap is used to study uncertainty in population parameters. If a new study would recruit a new set of subjects from the same population, the subjects’ random effects would vary across repeated studies. Holding estimated effects fixed omits that component of repeated-sampling variation. Conversely, if the question concerns repeated measurements on these particular subjects, conditioning on their effects may be relevant. The bootstrap scheme should follow the target question, not be selected merely because one option is computationally convenient.

Here \(t_{ij}\) and \(x_{ij}\) denote the observation time and covariates, \(\theta\) the population parameters, \(\Omega\) the random-effects covariance matrix, and the residual model specifies the distribution of \(\epsilon_{ij}\). The notation \(N(0,\Omega)\) means a multivariate normal distribution with mean vector zero and covariance matrix \(\Omega\). The fitted model, including its distributional assumptions, is the data-generating model for either bootstrap. These procedures therefore depend on that model being a reasonable description of the population and residual variation.

In practice, “holding effects fixed” usually means fixing their fitted empirical Bayes estimates, not conditioning on known true values. Those estimates are calculated from the observed data and are shrunken toward the population mean to a degree that depends on the information available for each subject. Thus a conditional bootstrap is a plug-in approximation to a conditional sampling distribution, not an experiment with perfectly known latent effects.

What Variation Enters Each Bootstrap?

For each replicate, keep the fitted population parameters and the study’s subject records as the simulation basis. In a conditional replicate, use \(\hat{\eta}_i\) for subject \(i\); in an unconditional replicate, draw \(\eta_i^*\) independently across subjects from \(N(0,\hat{\Omega})\), unless the model specifies a different dependence structure. Generate residual errors according to the fitted residual model, construct the observations, and refit the same model. The replicate’s refitted parameters are then one draw from the bootstrap distribution under the chosen simulation scheme.

The conditional scheme contains residual variability and the consequences of refitting the model, but it does not include fresh between-subject random-effect variation. The unconditional scheme contains both sources. The distinction is about the simulated data-generating mechanism; it does not mean that a conditional bootstrap somehow avoids refitting the model or that an unconditional bootstrap estimates random effects instead of population parameters.

A Variance Decomposition

A simple random-intercept model makes the contrast explicit. Suppose there are \(m\) subjects and \(n\) observations per subject, with \(y_{ij}=\mu+b_i+\epsilon_{ij}\), where \(b_i\sim N(0,\tau^2)\), \(\epsilon_{ij}\sim N(0,\sigma^2)\), and all random effects and residual errors are mutually independent. Assume these variances are known for this derivation, and estimate \(\mu\) by the grand mean \(\bar{y}\). This is a simplified model used to show the variance contribution; it is not a general formula for every NLME estimator.

If the \(b_i\) are held fixed, the only random part of \(\bar{y}\) in a bootstrap replicate is the average residual error. Independence and the residual variance assumption give the conditional variance. If instead the random effects are redrawn, their contribution must be included as well:

The second equality follows by adding the variance of \(m^{-1}\sum_i b_i\), which is \(\tau^2/m\), to the residual contribution. Equivalently, the law of total variance gives \(\operatorname{Var}(\bar{y})=E\{\operatorname{Var}(\bar{y}\mid b)\}+\operatorname{Var}\{E(\bar{y}\mid b)\}\). The conditional distribution is centered at \(\mu+m^{-1}\sum_i b_i\), not necessarily at the population mean \(\mu\). This explains why conditioning can change both the spread and, in nonlinear fits, the center of the resulting bootstrap estimates.

Reading Bootstrap Replicates in Context

A practical bootstrap repeats the simulation and refitting many times. The resulting distribution of an estimate reflects the specified data-generating mechanism and the fitting procedure. It is not a distribution of the unknown parameter itself. A wider unconditional distribution may be appropriate for inference about population parameters across newly sampled subjects, but that width does not imply that the fitted model is worse. A narrower conditional distribution may describe repeated residual measurements on the represented subjects more closely, but it can omit important uncertainty for a new-cohort question.

Assumptions, Diagnostics, and Limitations

Both procedures are parametric: their validity depends on the fitted model being an adequate generator for the data. The structural model, random-effects distribution, residual error model, and independence assumptions must be plausible for the intended analysis. The simulations should also reflect the data structure being analyzed. If the observed design is held fixed, use the same records, covariates, and observation schedule as the original data; the separate question of how to preserve designs in more complex settings requires deliberate attention.

Diagnostics should compare the conditional and unconditional results with the target question and inspect whether the simulations behave plausibly. Useful checks include whether estimates are systematically shifted from the original fit, whether the spread changes substantially between schemes, whether variance or correlation estimates approach boundaries, and whether the fitted model repeatedly struggles to describe simulated data. For nonlinear models, a large difference in centers can arise because the model response is nonlinear in \(\eta_i\), because conditional effects are estimated with shrinkage, or because refitting interacts with the simulated data. It should prompt investigation rather than automatic selection of whichever result is preferred.

The conditional approach can be useful when inference is explicitly conditional on the observed cohort’s latent effects, but empirical Bayes estimates are not known effects. Their estimation uncertainty and shrinkage mean that fixing them can misrepresent even that conditional target. The unconditional approach incorporates population heterogeneity by drawing effects, but it remains conditional on fitted parameter values as the bootstrap generator; it does not eliminate model misspecification or finite-simulation error. Neither method by itself resolves uncertainty caused by a poorly identified model.

A subject-level nonparametric bootstrap instead resamples observed subjects and their records, while Wald intervals use an estimated local covariance approximation. These alternatives target uncertainty differently and have their own assumptions. The parametric bootstrap’s advantage is that it uses the fitted NLME model to generate complete replicate data; its corresponding limitation is dependence on that model. The earlier material on parameter uncertainty and parametric bootstrap intervals provides the broader setting for interpreting refitted parameter distributions.

A Common Mistake