Why Resample Subjects?
An NLME data set often contains many observations per subject. Those observations are linked by the subject’s shared latent effects and by their common history, so resampling individual concentration records would generally break the structure the model is intended to describe. A subject-level nonparametric bootstrap instead resamples whole subject records, refits the model, and examines how the estimates vary across the resulting data sets.
The result is a collection of estimates intended to approximate the sampling distribution of the fitted parameters under repeated sampling of subjects from the target population. This tutorial focuses on constructing and checking those replicates. The introductory material on parameter uncertainty distinguishes uncertainty in estimated parameters from other sources of variability; here the emphasis is on the resampling algorithm itself.
Model and Resampling Assumptions
Write the original data as \(Z_1,\ldots,Z_n\), where each \(Z_i\) is one subject’s record. Let \(\theta\) be the model’s population-level parameter vector, including whichever fixed effects, random-effects distribution parameters, and residual-error parameters are part of the specified model. A fitted estimate is denoted by \(\hat{\theta}\). The precise parameterization and estimation method must be held fixed across bootstrap fits.
For example, if subject \(i\) has outcome vector \(Y_i\), design information \(D_i\), and latent random effect \(b_i\), a marginal likelihood contribution has the general form
The ordinary subject-level bootstrap is based on treating the subject records as independent draws from a common population, or on an appropriate exchangeability assumption for the records in the sampling scheme. Exchangeability means that the joint distribution of the subject records is invariant to their ordering; it does not mean that observations within one subject are independent. Separately, the bootstrap algorithm assigns probability \(1/n\) to each observed subject record when drawing an index.
The empirical distribution of the observed records acts as a plug-in approximation to the unknown subject population. This procedure is most directly justified when subjects are independent and identically distributed from the population relevant to the intended inference, and when the fitted estimator is sufficiently stable under small changes in the sample. If subjects were recruited under groups or other design constraints, simple resampling of all records without regard to those features may not represent the sampling process. The resampling scheme must match the target population and study design.
The Algorithm
For each replicate, draw \(n\) indices independently with replacement from \(\{1,\ldots,n\}\). Copy each selected subject’s entire record into the replicate data set, assigning a distinct replicate subject identifier to each copy. Then refit the same model, using the same structural model, covariate relationships, parameter constraints, estimation method, and relevant fitting settings as for the original fit.
Record the model specification, estimation settings, original estimate \(\hat{\theta}\), and subject-level records that form the resampling units.
For replicate \(b\), draw \(I^{*(b)}_1,\ldots,I^{*(b)}_n\) independently, with \(P(I^{*(b)}_j=i)=1/n\) for every observed subject \(i\).
For each selected index, include all observations and associated subject-level information in that subject’s record. A selected record may appear more than once; each occurrence is a separate bootstrap copy.
Fit the same NLME model to the \(n\) bootstrap subject copies and save the estimate \(\hat{\theta}^{*(b)}\), along with fit diagnostics.
Repeat for \(b=1,\ldots,B\). The resulting collection \(\hat{\theta}^{*(1)},\ldots,\hat{\theta}^{*(B)}\) approximates the estimator’s sampling distribution through the empirical subject population.
A duplicated record is treated as a repeated draw of a subject record, not as extra observations belonging to one original subject with a shared latent effect. Under the marginal likelihood, a bootstrap sample with selected indices \(i_1,\ldots,i_n\) has likelihood
Thus, if one original index is selected twice, its marginal likelihood contribution appears twice. The two copies occupy two subject slots in the resampled data set. This is distinct from combining their rows under one subject identifier, which would define a different data structure and likelihood.
Why the Resampling Scheme Matches the Target
The justification is a plug-in argument. If the unknown population distribution of subject records is \(F\), the observed empirical distribution is
where the indicator is interpreted componentwise or through the relevant measurable sets for the record. The bootstrap draws records from \(F_n\) as an approximation to drawing new subjects from \(F\). Applying the same estimator \(T\) to each bootstrap sample gives \(\hat{\theta}^{*}=T(Z_1^*,\ldots,Z_n^*)\), intended to approximate the variation of \(T(Z_1,\ldots,Z_n)\) around its population target.
The sampling step is with replacement because each bootstrap draw is made from the same empirical distribution: after one subject is drawn, that record remains eligible for the next draw. Without replacement, every replicate would contain the original records exactly once, so resampling would not represent the variation in which subjects appear in a new sample.
Worked Example: Drawing Complete Subject Records
Suppose a hypothetical study has four subjects. Each record contains the subject’s dose history and three measured concentrations. The original subject records are labeled A, B, C, and D. One bootstrap draw of four indices is \((2,4,2,1)\).
The replicate contains subject B’s complete record, subject D’s complete record, a second copy of subject B’s complete record, and subject A’s complete record. Subject C is absent. The copied B records receive separate replicate identifiers, such as B-copy-1 and B-copy-2, so each is a separate subject-level contribution in the refitted model.
The replicate still has four subject copies, but only three distinct original records. It does not contain four randomly selected concentration observations. If B had three concentration observations, all three are included in each of the two B copies; none is sampled separately from the others.
The draw follows the algorithm because each of its four indices is selected from the original four with replacement. Its probability under this particular scheme is \((1/4)^4=1/256\). This calculation describes the bootstrap’s random index draw, not a probability statement about whether the original subjects are exchangeable.
Worked Example: Refit With a Repeated Subject Index
Consider three hypothetical subjects with marginal likelihood contributions \(L_1(\theta)\), \(L_2(\theta)\), and \(L_3(\theta)\). A bootstrap draw has indices \((1,1,3)\), so the bootstrap likelihood is \(L^*(\theta)=L_1(\theta)^2L_3(\theta)\). Suppose that at two candidate parameter vectors, the original subjects’ log-likelihood contributions are:
| Candidate | \(\log L_1\) | \(\log L_2\) | \(\log L_3\) |
|---|---|---|---|
| \(\theta_A\) | \(-2.0\) | \(-1.2\) | \(-2.4\) |
| \(\theta_B\) | \(-1.7\) | \(-1.8\) | \(-2.0\) |
At \(\theta_A\), the replicate log likelihood is \(2(-2.0)+(-2.4)=-6.4\). At \(\theta_B\), it is \(2(-1.7)+(-2.0)=-5.4\). Thus \(\theta_B\) has the larger log likelihood of these two candidates for this replicate. That comparison alone does not identify the fitted estimate: optimization must evaluate the objective over the parameter space, with all model parameters estimated as specified.
The contribution \(L_1\) appears twice because subject 1 was drawn twice. Treating the two copies as one subject with duplicate rows would instead alter the subject’s record and would not implement the stated bootstrap likelihood.
Collecting and Summarizing Replicate Estimates
Each successful replicate gives a vector estimate. For a scalar parameter, the collection of replicate estimates can be inspected for spread, skewness, multiple modes, boundary accumulation, and unusual values. For a vector of parameters, dependence also matters: changes in two parameters may occur together, so separate marginal summaries do not describe the full joint distribution.
For example, the bootstrap standard deviation of a scalar parameter can be summarized descriptively by the sample standard deviation of the \(B\) replicate estimates:
This number measures the spread among the bootstrap estimates in the units of the parameter. It is not itself a confidence interval. Confidence-interval construction and the Monte Carlo precision of bootstrap limits require additional choices.
Worked Example: Summarizing Bootstrap Estimates of Clearance
Suppose eight successful hypothetical subject-level bootstrap fits give clearance estimates, in L/h, of \(5.1, 5.4, 4.9, 5.8, 5.2, 6.0, 5.0,\) and \(5.6\). Their sum is \(43.0\) L/h, so the replicate mean is \(\bar{\theta}^{*}=43.0/8=5.375\) L/h.
The squared deviations from 5.375 are \(0.075625, 0.000625, 0.225625, 0.180625, 0.030625, 0.390625, 0.140625,\) and \(0.050625\), with sum \(1.095\;(\mathrm{L/h})^2\). Therefore,
The eight fits show a bootstrap spread of about \(0.396\) L/h. Because there are only eight replicates in this illustration, this spread is a rough numerical summary; it should not be mistaken for a precise estimate of the full sampling distribution.
Diagnostics, Alternatives, and Limitations
A bootstrap replicate must be a valid refit of the intended model, not merely a number returned by an optimizer. Review convergence status, parameter boundaries, gradient or optimization diagnostics, and whether the estimates are scientifically and numerically plausible. Also check that each copied subject has a unique replicate identifier and retains all observations belonging to that record. A suspiciously narrow distribution can result from resampling the wrong unit or accidentally omitting between-subject variation.
The subject-level nonparametric bootstrap and common alternatives differ in how they represent uncertainty:
| Approach | Source of approximation | Useful distinction |
|---|---|---|
| Subject-level nonparametric bootstrap | Resampling observed subject records and refitting | Uses the empirical distribution of observed subjects rather than generating new records from a fully specified outcome model |
| Wald approximation | Local curvature of the likelihood near the estimate | Does not repeatedly resample subjects; can be sensitive to local normality and boundary behavior |
| Profile likelihood | Likelihood values as a parameter is constrained and nuisance parameters are reoptimized | Uses the fitted likelihood rather than an empirical resampling distribution |
| Parametric bootstrap | New records generated from a fitted model | Relies on the data-generating model used for simulation, rather than resampling only observed subject records |
The nonparametric procedure cannot create subject patterns absent from the observed sample. Its behavior can be unstable in small samples, particularly when a few subjects strongly influence the fit or when parameter estimates are near boundaries. It also does not correct model misspecification: repeatedly fitting the same wrong structural or error model does not make that model correct.
A common mistake is to resample individual concentration observations because there are more rows than subjects. That treats measurements from one subject as if they were independent sampling units and can destroy the dependence represented by the random effects. The correction is to draw subject indices and carry each selected subject’s entire record into the replicate. A separate mistake is to call equal draw probabilities “exchangeability.” Equal probabilities \(1/n\) specify the resampling rule; exchangeability is invariance of the joint distribution of subject records to their ordering.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In a study with 60 subjects and five concentrations per subject, what is sampled in one ordinary subject-level bootstrap replicate?
- Why can the same original subject appear twice in a replicate, and how should the two copies be represented when fitting?
- Suppose the original subject records are not exchangeable under the study’s sampling scheme. What concern does this raise for the ordinary subject-level bootstrap?
- Eight bootstrap clearance estimates have a sum of 43.0 L/h. What is their mean, and what does that mean represent?
- Does a subject-level nonparametric bootstrap generate new concentration profiles from the fitted NLME model? Explain.
Solutions. (1) Draw 60 subject indices with replacement and copy each selected subject’s complete five-observation record and associated information. (2) Drawing is with replacement, so duplicates are possible; give each occurrence a separate replicate subject identifier so it contributes as a separate subject copy. (3) The resampling distribution may not represent the actual sampling process or target population; a resampling scheme appropriate to that design is needed. (4) The mean is \(43.0/8=5.375\) L/h, the average of those eight replicate estimates, not the original estimate or a confidence limit. (5) No. It resamples observed subject records; generating new records from a fitted model is a parametric bootstrap.