What Varies When Fitted Parameters Are Held Fixed?
A fitted NLME model can be used to generate different plausible data sets even when its estimated population parameters are held unchanged. The reason is that a simulation may draw new subject-level random effects and new residual errors. These draws represent process variability under the model: differences among subjects and unexplained differences among observations. They are not new estimates of the population parameters.
This distinction matters when interpreting simulated distributions. A wide range of simulated concentrations may reflect substantial subject-to-subject variability, residual error, or both. It does not, by itself, show that the fitted population parameters are uncertain. In a plug-in simulation, those parameters are treated as known constants at their fitted values.
Model and Assumptions
Write the observation for subject \(i\) at record \(j\) as
Here \(t_{ij}\) denotes the record time, \(d_i\) collects the subject's known design information, \(\theta\) denotes population parameters, \(\eta_i\) is the subject's random-effect vector, \(\phi_i\) is the subject-specific parameter vector, and \(\epsilon_{ij}\) is residual error. The function \(f\) is the structural model, and \(g\) maps population parameters and random effects to individual parameters. For a fitted model, the plug-in simulation fixes the estimated values \(\hat\theta\) and the estimated variance parameters, such as \(\hat\Omega\) for random effects and parameters specifying residual error.
The precise distributions and dependence structure must match the fitted model. A common case is \(\eta_i\sim N(0,\Omega)\), with independent random-effect draws across newly generated subjects. Residual errors may be normal with a variance that depends on the predicted value, or may follow another specified residual model. A typical assumption is that residual errors are independent across records and independent of the random effects, conditional on the model and design. If the fitted model specifies a different structure, the simulation must preserve it.
The key operation is to fix the estimated distribution parameters but still draw from the distributions they specify. For example, fixing \(\hat\Omega\) does not mean setting every \(\eta_i\) to zero: it means using the same estimated covariance matrix to generate new random effects. Likewise, fixing estimated residual-error parameters does not mean setting every \(\epsilon_{ij}\) to zero.
Use the fitted population parameters, random-effect distribution parameters, and residual-error parameters as constants.
For a new subject, draw \(\eta_i\) from the fitted random-effect distribution. For multiple records on that subject, use that subject's same draw throughout.
Evaluate the structural model using the subject's parameters and design, then generate residual errors according to the fitted residual model.
Identify which random quantities were drawn. Do not attribute the spread to parameter uncertainty if the fitted parameters were never varied.
Why the Sources of Variation Add Up Differently
A simple additive model makes the distinction clear. Suppose a single observation has the form \(Y=\theta+\eta+\epsilon\), where \(\theta\) is fixed, \(\eta\) has variance \(\omega^2\), and \(\epsilon\) has variance \(\sigma^2\). If the two random terms are independent, then
The variance includes the random effects and residual error, but not variation in \(\theta\), which was fixed. This follows by expanding the variance:
because independence implies \(\operatorname{Cov}(\eta,\epsilon)=0\). If the simulation instead holds a particular subject's \(\eta\) fixed, only the residual term varies across records, so the conditional variance is \(\sigma^2\). This is a different simulation target, not a contradiction.
Worked Example: One Observation From a New Subject
Suppose a hypothetical model for a single concentration is \(Y=10+\eta+\epsilon\) mg/L. The fitted mean is fixed at 10 mg/L, the random effect is \(\eta\sim N(0,4)\) in squared concentration units, and the residual error is \(\epsilon\sim N(0,1)\) in squared concentration units. Assume the two terms are independent. Find the predictive mean, standard deviation, and central 95% interval for one simulated observation from a new subject.
The fixed fitted mean is 10 mg/L. Independence gives the variance of the sum as
Thus the standard deviation is \(\sqrt{5}=2.236\) mg/L. Under these normality assumptions, \(Y\) is normally distributed with mean 10 mg/L and variance \(5\;(\mathrm{mg/L})^2\). Its central 95% interval is approximately
This interval describes the model-based distribution of observations for new subjects under the fixed fitted parameters. It includes both random-effect and residual variation. It does not express uncertainty about whether the fitted mean is exactly 10 mg/L.
Repeated Records and the Same Subject Effect
When several records come from one subject, the random effect is ordinarily drawn once for that subject and shared across the subject's records. Drawing a fresh random effect at every record would change the model: it would discard the within-subject dependence induced by the shared subject effect. Under the additive model, with independent residual errors, repeated observations have covariance due to that common random effect.
Worked Example: Two Records From One Subject
Continue with \(Y_{ij}=10+\eta_i+\epsilon_{ij}\) mg/L, where \(\operatorname{Var}(\eta_i)=4\;(\mathrm{mg/L})^2\) and independent residual errors each have variance \(1\;(\mathrm{mg/L})^2\). For one simulated subject, calculate the variance of the average of two records and the covariance between the records.
The same \(\eta_i\) appears in both observations:
The random-effect contribution to the average has variance 4. Because the two residual errors are independent, the variance of their average is \((1+1)/4=0.5\). Therefore
For the two records, the shared random effect contributes covariance 4, while independent residual errors contribute zero covariance. Hence \(\operatorname{Cov}(Y_{i1},Y_{i2})=4\;(\mathrm{mg/L})^2\). If a simulation draws a new \(\eta\) independently for each record, this covariance disappears and the simulated records no longer represent the stated subject-level model.
Choosing What Is Conditioned On
Not every simulation is intended to represent new subjects. A simulation may instead hold an individual's random effect fixed, for example when generating repeated observations conditional on a specified individual parameter value. Such a simulation includes residual variability but not new between-subject variation. The target must therefore be stated before interpreting the output.
Worked Example: New Subjects Versus a Fixed Individual
Suppose a hypothetical one-record model is \(Y=8+\eta+\epsilon\) mg/L, with fixed fitted parameters, \(\eta\sim N(0,2.25)\), and \(\epsilon\sim N(0,0.64)\), independently. Compare the standard deviation for a new subject with that for records simulated at a fixed individual effect \(\eta=0.6\).
For a new subject, both random terms are drawn. The variance is \(2.25+0.64=2.89\;(\mathrm{mg/L})^2\), so the standard deviation is \(\sqrt{2.89}=1.700\) mg/L. The mean over new subjects and residual errors is 8 mg/L.
For a fixed individual effect \(\eta=0.6\), the simulated observation is \(Y=8+0.6+\epsilon\). Its conditional mean is 8.6 mg/L, and only residual error varies, giving standard deviation \(\sqrt{0.64}=0.800\) mg/L. The smaller spread is expected: this simulation conditions on the individual effect rather than drawing it anew. It does not include uncertainty in the fixed fitted parameters, and treating an estimated individual effect as fixed also does not represent uncertainty about that individual effect.
Diagnostics, Alternatives, and Limitations
A simulation should be checked against its intended target and the fitted model's assumptions. Useful checks include verifying that random effects are drawn once per new subject, that their empirical means and covariance are broadly compatible with the specified distribution, and that residual errors follow the intended residual model. Compare simulated summaries and trajectories with model-implied expectations, and inspect whether the generated values are plausible on the response scale. Large simulation runs may still show small discrepancies from theoretical summaries because of Monte Carlo variation.
Common warning signs include identical subject trajectories when between-subject random effects were intended, implausibly large or small within-subject scatter, incorrect correlations between records, or impossible values from an unsuitable residual distribution. A mismatch can come from the model itself, an implementation error, or a simulation target that was not clearly defined. Increasing the number of replicates may reduce Monte Carlo noise, but it cannot fix a wrong random-effect draw scheme or an inappropriate error model.
| Simulation choice | What varies | What the spread represents |
|---|---|---|
| New subjects, new records | Random effects and residual errors | Between-subject and within-subject process variability, conditional on fitted parameters |
| Fixed individual effect, new records | Residual errors | Within-subject process variability conditional on that effect |
| Typical subject with errors set to zero | Nothing stochastic in these components | A model prediction, not a predictive distribution |
| Fitted parameters also varied | Parameter values as well as process draws | Can represent parameter uncertainty, depending on how parameter variation is generated |
The main alternative to a plug-in simulation is one that also varies the fitted population parameters. That adds a distinct source of uncertainty; it is not achieved merely by generating more random effects or residual errors. The method for generating parameter values is a separate modeling and inferential choice. A plug-in simulation is useful for describing variability under the fitted model, but it can understate total predictive uncertainty when parameter uncertainty is material.
The interpretation is also conditional on the fitted model being an adequate representation of the data-generating process. Simulated variability does not automatically account for model misspecification, unmodeled covariate variation, missing data, or uncertainty in a chosen structural model. Nor does a finite set of simulated replicates reproduce the exact model distribution: its empirical summaries have Monte Carlo error.
Common Mistake
A frequent mistake is to say, “I generated many subjects, so the simulation includes uncertainty in the population estimates.” This is incorrect. If every replicate uses the same fitted parameter values, generating more subjects only samples the fitted random-effect and residual distributions more thoroughly; it does not vary those estimates. The correction is to list explicitly which quantities are redrawn and which are fixed, then describe the resulting spread only in terms of the quantities that vary.
Check Your Understanding
For each question, identify the uncertainty source and show the calculation where needed. Solutions follow each question.
- A new-subject simulation draws random effects from a fitted distribution but sets residual errors to zero. Which source of process variability is represented? Solution: It represents between-subject variation from the random effects, conditional on the fitted parameters. It omits residual variability.
- In \(Y=12+\eta+\epsilon\), suppose independent \(\eta\) and \(\epsilon\) have variances 9 and 4, respectively, and all fitted parameters are fixed. Find the standard deviation for one new-subject observation. Solution: The variance is \(9+4=13\), so the standard deviation is \(\sqrt{13}=3.606\) in the response units. This includes both process components, not parameter uncertainty.
- Two records for one subject share a random effect with variance 2; their independent residual errors each have variance 3. Find the covariance between the records. Solution: The shared random effect contributes 2 to the covariance. The independent residual errors contribute zero, so the covariance is 2 in squared response units.
- A simulation uses the same fitted population parameters in every replicate but draws a new random effect and residual error each time. Does the variation across replicates measure parameter uncertainty? Solution: No. The fitted parameters are fixed. Variation across replicates represents the random effects and residual errors drawn under those fixed values.
- For repeated records on a fixed individual, the individual effect is held at 0.5 and residual errors are redrawn. Which variability is included? Solution: Only residual variability is included. The individual effect determines the conditional prediction but does not vary across the simulated records.