Why Boundaries Change Bootstrap Behavior
Bootstrap uncertainty is often summarized by the spread and quantiles of estimates from repeated resampled data sets. When an estimate can vary freely over the real line, a roughly symmetric distribution may be a reasonable approximation. Many population PK parameters, however, are restricted: a variance component cannot be negative, an absorption rate constant must be positive, and a fraction such as bioavailability is often constrained to lie between zero and one. Near such a limit, the bootstrap distribution can be skewed or can contain a substantial number of estimates exactly at the bound.
That pattern is not automatically an error. A genuine boundary pattern can arise because the data provide weak evidence for a parameter larger than its smallest allowed value. But repeated estimates at a limit can also be caused by optimizer failure, poor parameter scaling, or inadequate identification of the parameter. The interpretation depends on both the parameter’s statistical behavior and the quality of the refits that generated the bootstrap estimates.
In a bootstrap analysis, each replicate produces a new data set according to the chosen resampling scheme, and the model is refitted to that data set. The resulting estimates approximate how the fitting procedure behaves across data sets similar to the observed one, subject to the model, design, and resampling assumptions. Near a boundary, that behavior may not be well summarized by a mean and standard deviation or by a symmetric interval around the original estimate.
Why a Point Mass Can Appear
A simple local calculation shows why estimates of a nonnegative parameter can pile up at zero. Suppose a scalar parameter \(\theta\) is restricted to \(\theta \geq 0\), and a locally quadratic log likelihood would, without that restriction, be maximized at \(\tilde{\theta}\). Under a regular approximation, let \(\tilde{\theta}\) have an approximately normal sampling distribution with mean \(\theta\) and standard error \(s_\theta\). In this simplified quadratic setting, imposing the constraint gives the estimate \(\hat{\theta}=\max(0,\tilde{\theta})\).
The calculation follows because, when the true value is zero, a centered normal \(\tilde{\theta}\) is nonpositive with probability one-half. Every such unconstrained estimate is mapped to zero by the constraint; positive values remain positive. Thus even this elementary approximation produces both a point mass at zero and a positive, asymmetric part of the distribution. A variance-component estimator in a nonlinear mixed-effects model does not generally equal this exact formula, but the calculation explains why a boundary pileup can be a genuine consequence of constrained estimation rather than a software defect.
This approximation also has limits. It assumes a single parameter, a locally quadratic objective, and an approximately normal unconstrained estimate. Correlations with other parameters, sparse data, weak identification, or a model with several interacting constraints can produce more complicated behavior. The bootstrap distribution should therefore be inspected rather than assumed to have this particular shape.
Reading Bootstrap Quantiles at a Boundary
For \(B\) bootstrap estimates, sort them as \(\hat{\theta}^{*}_{(1)}\leq \hat{\theta}^{*}_{(2)}\leq\cdots\leq \hat{\theta}^{*}_{(B)}\). Under the nearest-rank convention, the empirical \(p\) quantile is the observation at rank \(\lceil Bp\rceil\), where \(\lceil\cdot\rceil\) denotes rounding up to the next integer. This convention matters at the tails: the correct order statistic depends on the stated replicate count and probability.
Worked Example: A Variance Component With a Zero Pileup
Suppose a hypothetical subject-level bootstrap produces \(B=200\) estimates of an interindividual variance component on the log-concentration scale. The allowed values are nonnegative, and 34 of the 200 refits return exactly zero. The fifth ordered estimate is \(0\), the 100th is \(0.012\), and the 195th is \(0.085\), all in \((\text{log-concentration units})^2\). Assume the refits have been checked and these are valid converged fits.
For the 2.5th percentile, the nearest-rank index is \(\lceil 200(0.025)\rceil=\lceil5\rceil=5\). The percentile is therefore the fifth ordered estimate, \(0\). For the 97.5th percentile, the index is \(\lceil 200(0.975)\rceil=\lceil195\rceil=195\), so the percentile is \(0.085\), not the 196th estimate.
The interval is highly asymmetric around the bootstrap median of \(0.012\): its lower endpoint is at the constraint, while its upper endpoint is appreciably larger. This is compatible with uncertainty that is skewed toward positive values. It does not show that the variance is zero, nor does the fraction \(34/200=0.17\) prove that the true variance is zero. It shows that 17% of these refits landed at the bound; the scientific interpretation requires considering fit diagnostics and the information in the data.
A percentile interval reports quantiles of the bootstrap estimates. Its frequentist coverage, when the procedure is used repeatedly under the relevant sampling conditions, is the property one hopes to achieve; it is not a statement that a fixed parameter has a specified probability of lying in the interval. Near a boundary, standard bootstrap intervals may not achieve their nominal coverage, particularly when the sampling distribution changes abruptly at the constraint. A boundary endpoint is therefore a reason for caution, not a guarantee that the interval is reliable.
Positive Absorption Parameters and Other Limits
An absorption rate constant such as \(k_a\), measured in \(\text{h}^{-1}\), is positive. In practice, fitting software may impose a finite lower and upper limit, for example \(0.05\leq k_a\leq 3.0\;\text{h}^{-1}\). If many bootstrap estimates equal the lower limit, the distribution is truncated by the fitting constraint. A percentile interval can then have a lower endpoint equal to that limit even if the scientifically meaningful parameter is not believed to be exactly at the limit.
Worked Example: A Lower-Bounded Absorption Rate Constant
Consider a hypothetical bootstrap with 200 valid estimates of \(k_a\), each fitted under \(0.05\leq k_a\leq 3.0\;\text{h}^{-1}\). Eighteen estimates equal \(0.05\;\text{h}^{-1}\); the fifth ordered estimate is \(0.05\;\text{h}^{-1}\), the 100th is \(0.46\;\text{h}^{-1}\), and the 195th is \(1.24\;\text{h}^{-1}\). Suppose the original-data estimate is \(0.38\;\text{h}^{-1}\), and the bootstrap standard deviation is \(0.31\;\text{h}^{-1}\).
Using the nearest-rank convention, the 2.5th and 97.5th percentile ranks are \(\lceil200(0.025)\rceil=5\) and \(\lceil200(0.975)\rceil=195\). The percentile interval is therefore \([0.05,\;1.24]\;\text{h}^{-1}\). Its lower endpoint is the imposed fitting limit, and 18 of 200 estimates, or \(18/200=0.09\), are at that limit.
For comparison, a symmetric normal approximation using the stated bootstrap standard deviation gives \(0.38\pm1.96(0.31)\;\text{h}^{-1}\). The margin is \(0.6076\;\text{h}^{-1}\), so the approximate interval is \([-0.228,\;0.988]\;\text{h}^{-1}\). The negative lower endpoint is incompatible with a positive rate constant. Simply replacing it with \(0.05\;\text{h}^{-1}\) does not restore the nominal coverage of the normal interval; it changes the procedure without establishing its coverage.
The bootstrap estimates suggest a long upper tail relative to the constrained lower side, but this example alone cannot decide whether that asymmetry reflects genuine sampling behavior or fitting problems. The original estimate is not the bootstrap median, and estimates accumulate at the lower limit, so the fit diagnostics and parameter identification need review before treating the interval as a complete uncertainty summary.
A monotone transformation can sometimes make an interval more compatible with a parameter’s range. For example, a log transformation is natural for a strictly positive parameter. However, transforming a bootstrap estimate and transforming its quantiles back does not eliminate an actual pileup at a finite bound or resolve weak identification. For a parameter constrained to \([0,1]\), a logit transformation is undefined at exactly zero and one unless an additional convention is introduced. Such conventions must be reported and justified; they cannot be used to hide boundary estimates.
Distinguishing Boundary Behavior From Fitting Artifacts
The central diagnostic question is whether boundary estimates are valid optima of the stated constrained model. A refit that stops at a bound because the optimizer failed is not equivalent to a valid boundary solution. For each suspicious replicate, assess convergence status, objective-function values, gradients or other optimizer diagnostics, and sensitivity to reasonable starting values. At a constraint, the relevant optimality condition is one-sided: a solution at the endpoint can be valid even though an unconstrained search would continue beyond the allowed range. Conversely, a reported bound estimate is suspect if a different start yields a materially better objective value within the allowed parameter space.
Worked Example: Checking a Pileup at an Upper Limit
In a hypothetical analysis, \(F\), a fraction constrained to \(0\leq F\leq1\), is estimated from 200 bootstrap refits. The first run reports 29 estimates at \(F=1\). Inspection shows that, for 24 of these replicates, rerunning from alternative starting values finds an interior solution between \(0.82\) and \(0.96\) with a lower objective function value. Those original boundary results were not the best fits and should not be interpreted as evidence of a genuine boundary pileup. They require correction through valid refitting under the same model and constraints.
After correcting the refits, suppose all 200 estimates converge, seven remain at \(F=1\), the fifth ordered estimate is \(0.62\), the 100th is \(0.91\), and the 195th is \(1.00\). The nearest-rank 95% percentile interval is \([0.62,\;1.00]\), since the ranks are again 5 and 195. The upper endpoint is consistent with the observed upper-bound estimates; it does not imply that \(F\) is known to equal one. The valid boundary fits should be retained, while the 24 suboptimal results from the first run should not be treated as valid bootstrap estimates.
A useful interpretation is therefore conditional: after checking the refits, the corrected bootstrap distribution has a modest upper-bound pileup and an interval reaching the bound. This indicates uncertainty near the upper limit under the fitted model and design. It does not establish that the bound is scientifically correct or that the model identifies \(F\) precisely near one.
Evidence supporting genuine boundary behavior includes stable estimates at the same endpoint across alternative starts, acceptable convergence diagnostics, and no better feasible solution. Evidence of an artifact includes repeated failures, sensitivity to starts, abrupt changes in objective values, or a cluster of estimates exactly at a software limit while nearby solutions fit better. Numerical precision also matters: a reported value rounded to the bound may be merely close to it. Inspect unrounded estimates and the optimizer’s constraint status when possible.
A boundary pileup can also indicate limited information. In a population PK model, parameters may be correlated or only weakly informed by the sampling design. For example, the data may not distinguish a small \(k_a\) from other model features. Even if all estimates are valid optima, a pileup may reveal that the parameter is poorly identified under the available data and model. The bootstrap describes the instability; it does not by itself explain its scientific source.
Choosing How to Summarize the Distribution
| Summary | Potential use near a boundary | Main caution |
|---|---|---|
| Bootstrap histogram, quantiles, and boundary frequency | Shows skew, discreteness, and the proportion at a constraint directly | Shape depends on the resampling scheme and valid refits |
| Normal or Wald interval | Can be a rough comparison when the distribution is nearly symmetric and far from limits | May cross an impossible value or conceal a point mass |
| Percentile or basic bootstrap interval | Uses bootstrap estimates to reflect non-symmetric behavior | Nominal coverage is not assured at a nonregular boundary |
| Transformed-scale interval | Can respect positivity or a bounded parameter range when the transformation is suitable | Does not repair weak identification, boundary mass, or an inappropriate transformation |
| Profile likelihood or other model-based analysis | Can examine support for values near and at a constraint using the fitted likelihood | Depends on model adequacy and its own regularity and computation assumptions |
These summaries answer related but different questions. A histogram and boundary frequency are essential descriptive checks, while an interval compresses the distribution into endpoints. A normal interval is least persuasive when the estimate is near a limit and the bootstrap distribution is visibly skewed. Percentile, basic, and BCa intervals may also behave irregularly when many estimates coincide at a bound; methods that rely on smooth estimation behavior should not be presumed to solve a nonregular problem. A model-based profile can provide a useful comparison, but it is not a substitute for checking the bootstrap refits.
Report the constraint, the number and proportion of estimates at or near it, the interval method and quantile convention, and whether the boundary refits passed diagnostics. If a substantial pileup remains, explain how it affects interpretation. A single standard error or interval endpoint without this context can make a constrained distribution look more regular and informative than it is.
Common Mistake and Practical Conclusion
Wrong reasoning: “The bootstrap has many estimates at zero, so the true variance is zero,” or, conversely, “those zero estimates must be failed fits, so I will discard them.” Neither conclusion follows from the pileup alone. Correction: establish whether each boundary estimate is a valid constrained optimum, report how often it occurs, and interpret the distribution as uncertainty under the model and design. Retain valid boundary estimates; address invalid fits using the established convergence and refitting procedures rather than silently deleting them.
A boundary changes what an ordinary symmetric summary can convey. The most informative assessment combines the parameter’s allowed range and units, the full bootstrap shape, the frequency at the limit, the interval’s construction, and evidence that refits are valid. This approach helps separate a genuine consequence of constrained estimation from a computational artifact while keeping the limitations of bootstrap inference visible.
Check Your Understanding
Answer each question, then compare with the solutions below.
- With 200 bootstrap estimates, what order-statistic ranks give the nearest-rank 2.5th and 97.5th percentiles?
- A nonnegative variance estimate is zero in 40 of 200 valid refits. What proportion is at the boundary, and what does that proportion establish?
- Why can a normal interval be misleading for a positive absorption rate constant near its lower fitting limit?
- Name two diagnostics that help distinguish a valid boundary solution from an optimizer artifact.
- Does a percentile interval ending at a parameter’s upper bound prove that the true parameter equals that bound? Explain.
Solutions. (1) The ranks are \(\lceil200(0.025)\rceil=5\) and \(\lceil200(0.975)\rceil=195\). (2) The proportion is \(40/200=0.20\), or 20%; it establishes that this fraction of the valid refits reached the constraint, not that the true variance is zero. (3) A symmetric interval may extend below the allowed range and fails to represent a skewed distribution or point mass at the limit. (4) Check convergence and objective-function values, and rerun from alternative reasonable starting values; gradient or constraint diagnostics are also useful. (5) No. It says the empirical upper percentile reaches the constraint; it does not establish equality of the fixed parameter to that endpoint.