Basic Bootstrap Confidence Intervals in NLME Models

Why Reflect Bootstrap Quantiles?

A bootstrap distribution of an NLME estimate can be asymmetric, especially for weakly identified parameters, variance components, and nonlinear functions of model parameters. The percentile interval uses the lower and upper quantiles of that distribution directly. The basic bootstrap interval instead treats bootstrap estimates as a way to approximate the estimation error, then reverses that error around the original estimate. This reflection changes the interval's location and can produce endpoints outside the parameter space.

The construction is simple once its direction is clear: the upper bootstrap quantile determines the lower basic endpoint, and the lower bootstrap quantile determines the upper basic endpoint. This tutorial derives that reversal, calculates intervals in hypothetical NLME examples, and explains what the differences from percentile intervals do—and do not—tell us.

Definition: Let $\hat{\theta}$ be the estimate of a scalar parameter $\theta$ from the observed data. Let $\hat{\theta}^{*}$ denote estimates from valid bootstrap data sets generated according to the chosen bootstrap scheme. Write $q_L$ and $q_U$ for the empirical $\alpha/2$ and $1-\alpha/2$ quantiles of the bootstrap estimates, respectively. The two-sided $100(1-\alpha)\%$ basic bootstrap confidence interval is $$ \left[2\hat{\theta}-q_U,\;2\hat{\theta}-q_L\right]. $$ The corresponding percentile bootstrap confidence interval is $[q_L,q_U]$.

Deriving the Basic Endpoints

The derivation uses a bootstrap approximation to the sampling error of the estimator. Define the bootstrap error as $E^{*}=\hat{\theta}^{*}-\hat{\theta}$. The basic method approximates the distribution of the estimation error $\hat{\theta}-\theta$ by the distribution of $E^{*}$. This is an approximation, not an identity: it is the central assumption behind the method.

The $\alpha/2$ and $1-\alpha/2$ quantiles of $E^{*}$ are $q_L-\hat{\theta}$ and $q_U-\hat{\theta}$. A central interval for the estimation error therefore gives the approximate coverage statement

P\left(q_L-\hat{\theta}\leq\hat{\theta}-\theta\leq q_U-\hat{\theta}\right)\approx 1-\alpha.

Solving both inequalities for $\theta$ reverses their order: subtracting an inequality from a constant changes its direction. Thus

\begin{aligned} q_L-\hat{\theta}\leq\hat{\theta}-\theta &\quad\Longrightarrow\quad \theta\leq 2\hat{\theta}-q_L,\\ \hat{\theta}-\theta\leq q_U-\hat{\theta} &\quad\Longrightarrow\quad \theta\geq 2\hat{\theta}-q_U. \end{aligned}

Combining the bounds gives $[2\hat{\theta}-q_U,\;2\hat{\theta}-q_L]$. This also explains why the quantiles exchange roles: reflecting the bootstrap distribution around $\hat{\theta}$ sends a value above $\hat{\theta}$ to one equally far below it, and vice versa.

Interpretation: A basic interval is obtained by reflecting the endpoints of the percentile interval around the observed-data estimate. Reflection does not generally make the basic interval centered at that estimate. If $m_P=(q_L+q_U)/2$ is the percentile interval's midpoint, the basic interval's midpoint is $2\hat{\theta}-m_P$. Their widths are equal, $q_U-q_L$, but their centers are generally different.

Calculating and Comparing the Intervals

In practice, the bootstrap quantiles are calculated from the estimates in the valid replicates. The confidence level, parameter scale, and bootstrap design should be fixed before interpreting the resulting endpoints. The resampling scheme must represent the sampling process relevant to the target inference; the earlier material on choosing the resampling unit and constructing bootstrap replicates addresses that design. The calculation below does not repair a poorly chosen bootstrap scheme.

Worked Example: Clearance With Unequal Bootstrap Tails

Suppose a hypothetical NLME analysis estimates typical clearance as $\hat{\theta}=12.0$ L/h. From valid subject-level bootstrap replicates, the empirical 2.5th and 97.5th percentiles are $q_L=10.8$ L/h and $q_U=14.1$ L/h. Calculate and compare the nominal 95% percentile and basic intervals.

The percentile interval is the pair of bootstrap quantiles: $$ [10.8,\;14.1]\ \text{L/h}. $$ Its width is $14.1-10.8=3.3$ L/h and its midpoint is $(10.8+14.1)/2=12.45$ L/h.

For the basic interval, reflect the upper quantile to obtain the lower endpoint, and reflect the lower quantile to obtain the upper endpoint: $$ [2(12.0)-14.1,\;2(12.0)-10.8]=[9.9,\;13.2]\ \text{L/h}. $$ The width is $13.2-9.9=3.3$ L/h, and the midpoint is $(9.9+13.2)/2=11.55$ L/h.

Both intervals have the same width, but their centers differ: the basic interval's midpoint is 11.55 L/h, while the percentile interval's midpoint is 12.45 L/h. The basic endpoints reflect the percentile endpoints around 12.0 L/h; they do not make 12.0 L/h the basic interval's midpoint. These are approximate 95% confidence intervals for typical clearance under the stated bootstrap procedure, not statements that there is a 95% probability the fixed parameter lies in either realized interval.

Assumptions and Practical Checks

A basic interval is meaningful only when the bootstrap estimates approximate the relevant sampling behavior. The original estimator and bootstrap fits should target the same parameter under the same model specification, and replicates should preserve the aspects of the design needed for the intended inference. In an NLME analysis, this includes using the appropriate resampling unit and retaining relevant dosing and sampling features. Estimates from failed or invalid fits cannot be treated as ordinary draws from the bootstrap distribution; selective removal of difficult fits can distort its quantiles.

The error-distribution approximation is particularly important. The method relies on the distribution of $\hat{\theta}^{*}-\hat{\theta}$ being a useful approximation to that of $\hat{\theta}-\theta$. If the estimator has substantial bias, a strongly changing standard error, or behavior near a boundary that the bootstrap does not reproduce adequately, the reflected interval may have poor coverage. The formula alone offers no guarantee of nominal coverage.

Before reporting a basic interval, inspect the fitted replicates and the resulting distribution. Check whether the quantiles are based on the intended set of valid fits, whether a small number of extreme estimates dominate the tails, and whether the endpoints are scientifically and mathematically possible. A basic interval can extend beyond a parameter's allowed range because the reflection is performed on the chosen numerical scale without enforcing constraints.

Confirm the target and scale.
State which NLME parameter is being estimated and its units or scale. Compute the interval on that same scale.

Obtain the bootstrap quantiles.
From the bootstrap estimates, determine $q_L$ and $q_U$ at the specified tail probabilities, using a documented quantile convention.

Reflect and check.
Calculate $2\hat{\theta}-q_U$ and $2\hat{\theta}-q_L$, in that order. Check parameter constraints, units, and whether the interval is consistent with the observed bootstrap behavior.

When Reflection Conflicts With a Parameter Constraint

Some NLME parameters are restricted, such as a variance component constrained to be nonnegative or a correlation restricted to $[-1,1]$. The basic formula does not impose those restrictions on its endpoints. A result outside the permitted range is a warning that reflection on the current scale may not provide a usable interval; it is not permission to interpret an impossible value as a possible parameter value.

Worked Example: A Between-Subject Variance Near Zero

Suppose a hypothetical model estimates a between-subject variance as $\hat{\theta}=0.20$ (log-scale units)$^2$. The bootstrap quantiles are $q_L=0.08$ and $q_U=0.55$ (log-scale units)$^2$. Calculate both intervals and assess the basic interval.

The percentile interval is $[0.08,\;0.55]$ (log-scale units)$^2$, which stays nonnegative. The basic interval is $$ [2(0.20)-0.55,\;2(0.20)-0.08]=[-0.15,\;0.32]\ \text{(log-scale units)}^2. $$ The arithmetic is $0.40-0.55=-0.15$ and $0.40-0.08=0.32$.

A variance cannot be negative, so the lower basic endpoint is outside the parameter space. The calculated interval faithfully applies the basic formula, but it is not a plausible interval on the variance scale. The result calls for caution and consideration of an approach that respects the parameter's constraints; simply replacing $-0.15$ with zero changes the stated interval and does not, by itself, establish its coverage.

What the Two Methods Share—and What They Do Not

The percentile and basic methods use the same two empirical bootstrap quantiles, so their widths are exactly equal before rounding. Their difference is location: the percentile interval reports the quantiles themselves, whereas the basic interval reflects them around the original estimate. A right-skewed bootstrap distribution can therefore shift the intervals in opposite directions around the original estimate, but neither construction should be described as automatically centered there.

The percentile method is direct and respects the observed range of bootstrap estimates, although that range may not reflect the estimator's error in the way required for good coverage. The basic method explicitly uses an approximation to the estimation-error distribution, but reflection can yield implausible endpoints and depends on the quality of that approximation. Neither is universally preferable. Compare the interval with model constraints, the shape of the bootstrap distribution, and the scientific question.

A Wald interval, discussed in the earlier material on Wald confidence intervals, uses an estimate and an estimated standard error with a normal approximation. It can be convenient but can behave poorly when that approximation or the parameter scale is unsuitable. The basic interval does not require a normal-shaped bootstrap distribution, but it is not a general cure for bias or weak identification. Other approaches address different features of uncertainty; the basic interval should be reported as the particular bootstrap construction it is, not as a guarantee that bootstrap-based inference is reliable.

Worked Example: A Positive Rate Parameter

Suppose a hypothetical NLME model gives an absorption-rate estimate $\hat{\theta}=1.20$ h$^{-1}$, with bootstrap quantiles $q_L=0.70$ h$^{-1}$ and $q_U=1.50$ h$^{-1}$. Calculate both nominal 95% intervals and compare their centers.

The percentile interval is $[0.70,\;1.50]$ h$^{-1}$. Its width is $1.50-0.70=0.80$ h$^{-1}$, and its midpoint is $(0.70+1.50)/2=1.10$ h$^{-1}$.

Reflecting the quantiles around the estimate gives $$ [2(1.20)-1.50,\;2(1.20)-0.70]=[0.90,\;1.70]\ \text{h}^{-1}. $$ The basic interval also has width $1.70-0.90=0.80$ h$^{-1}$, while its midpoint is $(0.90+1.70)/2=1.30$ h$^{-1}$. Thus the widths agree, but the centers are 1.10 h$^{-1}$ for the percentile interval and 1.30 h$^{-1}$ for the basic interval. Both intervals are positive in this example, but that fact alone does not demonstrate adequate coverage.

Common Mistakes and Limitations

Reflecting in the wrong direction. Using $q_L$ for the lower basic endpoint gives the wrong order. The lower endpoint is $2\hat{\theta}-q_U$; the upper endpoint is $2\hat{\theta}-q_L$.
Claiming the basic interval is centered at the estimate. Reflection of each endpoint around the estimate does not generally center the interval there. Its midpoint is $2\hat{\theta}-(q_L+q_U)/2$.
Assuming equal widths mean equal intervals. Both methods have width $q_U-q_L$, but their endpoints and midpoints can differ.
Ignoring the parameter space. A negative variance endpoint or a correlation endpoint outside $[-1,1]$ signals a constraint problem for the interval on that scale.
Treating a nominal level as guaranteed coverage. The stated confidence level is an approximation whose performance depends on the bootstrap scheme and the error-distribution approximation.

The basic interval is most useful when the bootstrap distribution provides a credible approximation to estimation error and its reflected endpoints remain interpretable. It has limitations: it does not correct every source of bias, enforce parameter constraints, or eliminate uncertainty due to finite bootstrap samples and model assumptions.

Check Your Understanding

Work each problem, then compare your reasoning with the solutions given in the same item.

For $\hat{\theta}=5.0$ mg/L, $q_L=4.2$ mg/L, and $q_U=6.1$ mg/L, calculate the basic interval. Solution: $[2(5.0)-6.1,\;2(5.0)-4.2]=[3.9,\;5.8]$ mg/L.
For that same example, calculate the midpoints of the basic and percentile intervals. Is either necessarily the original estimate? Solution: The basic midpoint is $(3.9+5.8)/2=4.85$ mg/L. The percentile midpoint is $(4.2+6.1)/2=5.15$ mg/L. Neither equals the original estimate of 5.0 mg/L.
A variance estimate is $0.10$, and its bootstrap quantiles are $0.02$ and $0.30$, all in squared units. What is the basic interval, and what issue should be reported? Solution: The basic interval is $[0.20-0.30,\;0.20-0.02]=[-0.10,\;0.18]$. Its negative lower endpoint is outside the nonnegative variance parameter space.
Why does the upper bootstrap quantile become the lower basic endpoint? Solution: The error inequality is rearranged to solve for the parameter; subtracting the upper error bound reverses the inequality. Equivalently, reflection around $\hat{\theta}$ maps a high bootstrap estimate to a low endpoint.