Nested Simulation With Parameter and Subject Draws

Why Nest the Simulation?

A simulated outcome from a fitted nonlinear mixed-effects model can vary for more than one reason. The fitted parameters are uncertain, and subjects in the modeled population differ through their random effects. If simulated observations include residual error, that adds another source of variation. A useful simulation design keeps these sources distinct rather than blending them into a single unexplained cloud of outcomes.

Nested simulation does this with two loops. In the outer loop, draw a plausible fitted parameter set. In the inner loop, use that same parameter set to simulate multiple subjects from the population model. Thus, variation between outer draws reflects parameter uncertainty, while variation within an outer draw reflects subject-level variability, and any residual error included in the outcome model.

The source of the outer draws matters. They might be obtained from a parameter covariance approximation, a bootstrap, or Bayesian posterior draws. Each source represents uncertainty in a particular way; nesting does not make those interpretations interchangeable. The key is that a parameter set is selected at the outer level, then used consistently for the inner population simulation.

The Nested Structure and Its Variance

Write the simulated outcome for outer draw \(m\) and inner subject \(j\) as \(Y_{mj}\). A general representation is \(Y_{mj}=g(\theta^{(m)},\eta_{mj},\epsilon_{mj})\), where \(g\) is the model prediction, \(\eta_{mj}\) is a subject-level random effect, and \(\epsilon_{mj}\) is residual error when simulating observations rather than the latent model prediction. The conditional distributions of the random effects and residuals are those specified by the fitted model at \(\theta^{(m)}\).

Define \(a(\theta)=E(Y\mid\theta)\), the conditional mean outcome, and \(v(\theta)=\operatorname{Var}(Y\mid\theta)\), the conditional variance across simulated subjects and any included residual errors. The law of total variance gives:

The first term measures how the conditional mean changes across plausible parameter sets. The second averages the conditional population variance over those parameter sets. This identity follows by writing \(Y-E(Y)=[Y-E(Y\mid\theta)]+[E(Y\mid\theta)-E(Y)]\), squaring, and taking expectations. The cross-product has expectation zero because \(E[Y-E(Y\mid\theta)\mid\theta]=0\). The two remaining squared terms have expectations \(E\{\operatorname{Var}(Y\mid\theta)\}\) and \(\operatorname{Var}\{E(Y\mid\theta)\}\), respectively.

This decomposition is a useful guide to what the simulation estimates, not a claim that the two contributions can always be cleanly identified from a small finite simulation. In practice, the inner draws estimate conditional means and variances, while the outer draws show how those conditional summaries change with the parameter set.

Nonlinear Models Can Change Both Components

In an NLME model, \(a(\theta)\) and \(v(\theta)\) need not have simple forms. A parameter can affect the mean response, the spread of subject effects, or both; nonlinear transformations can also make the conditional mean differ from the response evaluated at an average random effect. The nested design accommodates this by evaluating the model separately at each outer parameter draw and simulating subjects conditionally under that draw.

Why the Outer Draw Is Shared Within an Inner Cohort

The outer parameter draw represents uncertainty about the fitted model, not an independently redrawn model for each subject in the same simulated cohort. Holding \(\theta^{(m)}\) fixed across the inner subjects allows the simulation to represent a population generated under one plausible parameter set. If a new independent parameter draw is made for every subject, parameter uncertainty is inadvertently mixed into individual variation. That may leave the marginal spread of a single outcome similar, but it changes the joint behavior of outcomes in a cohort.

Practical Checks, Alternatives, and Limitations

Before interpreting a nested simulation, check that the loop structure matches the intended question. A useful conceptual flow is:

Several patterns can reveal a problem. If outcomes vary across inner subjects but are identical across all outer draws, the parameter draws may not be reaching the prediction calculation. If inner subjects are identical within an outer draw despite a model with nonzero random-effect variance, subject-level effects may not have been sampled or applied. If different subjects appear to have independently changing fixed-effect parameter sets, the nesting has likely been lost. Also check that simulated parameters remain valid for the model and that the simulated outcome units and scale match the intended quantity.

Nested simulation differs from plug-in simulation, which fixes parameters at their estimates and therefore omits variation across plausible fitted parameter sets. It also differs from a single-layer simulation that produces outcomes from just one parameter set: such a simulation can describe subject variability conditional on that set, but cannot show the impact of parameter uncertainty. A nested design represents both components and their shared-parameter structure. Its limitations remain important: the result depends on the quality of the fitted model and the chosen uncertainty distribution; too few outer or inner draws give unstable summaries; and uncertainty sources not included in the outer draws or the subject model are not automatically accounted for.

A common mistake is to say, “Every subject should get a fresh parameter draw because every subject is different.” That reasoning confuses uncertainty about the fitted population model with variability among subjects under a given model. The correction is to draw the uncertain model parameters at the outer level, then draw subject effects at the inner level. Subject differences are generated by the random-effects distribution, not by assigning each subject a new fitted model.