BCa Intervals for Nonlinear Mixed-Effects Estimates

Why Adjust Bootstrap Quantiles?

A percentile bootstrap interval takes quantiles of the bootstrap estimates directly. But if the bootstrap distribution is shifted relative to the original estimate, or if its spread changes unevenly across the parameter scale, those nominal quantiles may not give the intended tail balance. The bias-corrected and accelerated (BCa) interval adjusts the quantile probabilities to account for these two features.

In an NLME analysis, the estimate may be a fixed effect, a variance parameter, or a derived quantity such as clearance. BCa changes which bootstrap quantiles are used; it does not change the bootstrap resampling scheme or make an unsuitable model fit reliable. The resampling units and design must still represent the sampling process and target of inference, as established in the earlier material on choosing the resampling unit and constructing NLME bootstraps.

Definition: Let $\hat{\theta}$ be the estimate from the original data and $\hat{\theta}^{*}_1,\ldots,\hat{\theta}^{*}_B$ the corresponding estimates from $B$ bootstrap replicates. The two BCa quantities are the bias-correction $z_0$, which measures the bootstrap rank of $\hat{\theta}$, and the acceleration $a$, which measures how the estimate's variability changes across the data. For a nominal tail probability $\alpha$, the adjusted probability is $$ \alpha_{\mathrm{BCa}}(\alpha) =\Phi\left[ z_0+\frac{z_0+z_{\alpha}}{1-a(z_0+z_{\alpha})} \right], \qquad z_\alpha=\Phi^{-1}(\alpha). $$ The BCa interval uses the bootstrap quantiles at the adjusted lower and upper probabilities, $\alpha_{\mathrm{BCa}}(\alpha/2)$ and $\alpha_{\mathrm{BCa}}(1-\alpha/2)$, for a nominal two-sided $100(1-\alpha)\%$ interval.

Bias Correction: The Rank of the Original Estimate

A common definition, with a specified treatment of ties, is

\hat{p}_0= \frac{\#(\hat{\theta}^{*}_b<\hat{\theta})+ \frac{1}{2}\#(\hat{\theta}^{*}_b=\hat{\theta})}{B}, \qquad z_0=\Phi^{-1}(\hat{p}_0).

Here $\hat{p}_0$ is the empirical percentile rank of the original estimate in the bootstrap distribution, using half the tied estimates. If there are no ties, it is simply the proportion of bootstrap estimates below $\hat{\theta}$. A positive $z_0$ means that the original estimate has a percentile rank above 50%—it is above the bootstrap median. A negative $z_0$ means its rank is below 50%. The value $z_0$ is on the standard normal scale; it is not a bias measured in the parameter's units.

This rank correction is useful when the bootstrap distribution is not centered around the original estimate. It does not establish that the original estimate is “biased” in a general scientific sense: it describes a specific comparison between the original estimate and its bootstrap replicates.

Worked Example: Calculating the Bias-Correction Term

Suppose a hypothetical subject-level bootstrap has $B=40$ successful estimates of clearance, in L/h. Of these, 26 are strictly below the original-data estimate $\hat{\theta}=5.00$ L/h, none equals it, and 14 are above it. Calculate $z_0$ and interpret its sign.

The empirical percentile rank is $$ \hat{p}_0=\frac{26}{40}=0.6500. $$ Thus $$ z_0=\Phi^{-1}(0.6500)\approx 0.3853. $$ The positive sign says that the original estimate has an empirical percentile rank of approximately 65% in the bootstrap distribution. It is above the bootstrap median, not below it. The value $0.3853$ is dimensionless: it is not a clearance difference of 0.3853 L/h.

Acceleration: Change in Estimator Variability

The acceleration $a$ is usually estimated by a delete-one-independent-unit jackknife. In a subject-level analysis, fit the model repeatedly, each time omitting one subject and retaining the rest of the design and data structure. Let $\hat{\theta}_{(i)}$ be the estimate when subject $i$ is omitted, for $i=1,\ldots,m$, and let $\bar{\theta}_{(.)}$ be the mean of these $m$ jackknife estimates. Define $d_i=\bar{\theta}_{(.)}-\hat{\theta}_{(i)}$. Then

\hat{a}= \frac{\sum_{i=1}^{m}d_i^3} {6\left(\sum_{i=1}^{m}d_i^2\right)^{3/2}}.

The signed cubic deviations distinguish asymmetric influence: large positive and negative deviations do not cancel in the same way as they do in the squared sum. The denominator scales the result, so $a$ is dimensionless. Acceleration is not simply the skewness of the bootstrap estimates. It is a jackknife estimate of how the estimator's standard error changes as the underlying parameter changes.

Worked Example: Computing Acceleration From Jackknife Fits

For illustration, suppose five hypothetical delete-one-subject fits give clearance estimates of 11, 11, 11, 9, and 8 L/h. Their mean is $$ \bar{\theta}_{(.)}=\frac{11+11+11+9+8}{5}=10.00\text{ L/h}. $$ The deviations $d_i=\bar{\theta}_{(.)}-\hat{\theta}_{(i)}$, in L/h, are $-1,-1,-1,1,2$. Therefore $$ \sum d_i^3=-1-1-1+1+8=6,\qquad \sum d_i^2=1+1+1+1+4=8. $$ Substitution gives $$ \hat{a}=\frac{6}{6(8)^{3/2}} =\frac{1}{22.627}\approx 0.04419. $$ This positive acceleration reflects the asymmetry of these jackknife deviations. It is not a statement that clearance increases by 0.04419 L/h.

How the Probability Adjustment Works

For a nominal probability $\alpha$, start with its standard normal score $z_\alpha=\Phi^{-1}(\alpha)$. BCa maps that score to $$ w=z_0+\frac{z_0+z_\alpha}{1-a(z_0+z_\alpha)} $$ and uses $\Phi(w)$ as the adjusted probability. The bias correction shifts the normal score, while acceleration modifies the adjustment differently in the two tails. If $a=0$, the expression simplifies to $w=2z_0+z_\alpha$. If both $z_0=0$ and $a=0$, the adjusted probability is $\Phi(z_\alpha)=\alpha$, so BCa reduces to the percentile interval.

This formula is the BCa adjustment; it is not a new estimate of the parameter. After calculating the two adjusted probabilities, sort the $B$ bootstrap estimates and take the corresponding quantiles. The exact finite-sample endpoint depends on the quantile interpolation convention, which should be stated and used consistently.

Worked Example: A 95% BCa Interval for Clearance

Suppose a hypothetical analysis has $B=100$ sorted bootstrap estimates of clearance in L/h. The original estimate has 55 bootstrap estimates strictly below it and no ties, so $\hat{p}_0=0.55$ and $z_0=\Phi^{-1}(0.55)\approx0.1257$. Suppose the delete-one-subject calculation gives $a=0.0573$. Calculate the adjusted probabilities for a 95% interval.

The nominal lower and upper probabilities are 0.025 and 0.975, with normal scores approximately $-1.9600$ and $1.9600$. For the lower tail, $$ \alpha_{\mathrm{BCa}}(0.025) =\Phi\left(0.1257+ \frac{-1.9600+0.1257}{1-0.0573(-1.9600+0.1257)}\right). $$ The denominator is $1.1051$; the normal argument is approximately $-1.5341$, giving an adjusted probability of approximately $0.0625$. For the upper tail, $$ \alpha_{\mathrm{BCa}}(0.975) =\Phi\left(0.1257+ \frac{1.9600+0.1257}{1-0.0573(1.9600+0.1257)}\right). $$ The denominator is $0.8805$; the normal argument is approximately $2.4945$, giving an adjusted probability of approximately $0.9937$.

Use linear interpolation with the type-7 quantile convention, whose position is $h=1+(B-1)p$ for probability $p$. The lower position is $1+99(0.0625)=7.1875$. If the 7th and 8th sorted estimates are 8.400 and 8.800 L/h, the endpoint is $$ 8.400+0.1875(8.800-8.400)=8.475\text{ L/h}. $$ The upper position is $1+99(0.9937)=99.3763$. If the 99th and 100th estimates are 11.900 and 12.006 L/h, the endpoint is $$ 11.900+0.3763(12.006-11.900)\approx11.940\text{ L/h}. $$ The resulting BCa interval is approximately $[8.475,\ 11.940]$ L/h. The positive $z_0$ means the original estimate is above the bootstrap median and has empirical percentile rank about 55%; it does not mean that clearance is biased upward by 0.1257 L/h.

Assumptions, Diagnostics, and Alternatives

BCa intervals rely on more than the arithmetic of the adjustment. The bootstrap replicates must reflect the sampling design and inferential target; the independent units must be resampled appropriately; and the estimator must be sufficiently regular for the bootstrap distribution and jackknife approximation to be informative. The acceleration calculation also requires usable delete-one fits. For an NLME model, numerical convergence and adequate parameter identification matter: a set of apparently successful bootstrap fits can still be unrepresentative if difficult fits were selectively excluded.

Before reporting an interval, check the sorted bootstrap estimates for extreme values, clusters, or a long tail; compare the original estimate with their median and empirical rank; inspect whether deleting a single subject produces an exceptionally large change; and verify that each adjusted probability lies between 0 and 1 and can be evaluated from the available replicates. A denominator $1-a(z_0+z_\alpha)$ near zero signals an unstable adjustment. Very extreme adjusted probabilities are also a warning that the interval depends heavily on a small number of tail replicates. These are reasons to investigate, not automatic grounds for silently changing the endpoints.

BCa and the main bootstrap interval alternatives differ in what they do with the same replicates:

Method	Endpoint construction	Main distinction
Percentile	Uses the nominal lower and upper bootstrap quantiles	Simple; does not adjust for the original estimate's rank or jackknife acceleration
Basic	Reflects the percentile endpoints around the original estimate	Uses the original estimate as a center for reflection, rather than adjusting probabilities with $z_0$ and $a$
BCa	Uses quantiles at BCa-adjusted probabilities	Accounts for empirical rank bias and acceleration, but requires a stable jackknife calculation and adequate tail replicates

BCa is not guaranteed to outperform the alternatives in every NLME problem. It can be unreliable for parameters at or near a boundary, highly discrete or nonsmooth estimators, severe practical nonidentifiability, or a bootstrap distribution with sparse tails. It cannot repair model misspecification, poor design, or a resampling scheme that targets the wrong population. When a parameter is constrained, calculate estimates on the scale and under the constraints that define the target, and scrutinize whether the jackknife and bootstrap are meaningful there.

Common Mistakes and Practical Use

Reversing the interpretation of $z_0$. If 55% of replicates are below the original estimate, the original estimate has rank about 55% and is above the bootstrap median.
Giving $z_0$ or $a$ parameter units. Both are dimensionless quantities used to adjust probabilities; the interval endpoints retain the parameter's units.
Using the wrong acceleration inputs. The standard jackknife formula uses delete-one-unit estimates and their centered deviations, not the bootstrap estimates substituted in their place.
Mixing endpoint probabilities and quantiles. Calculate both adjusted probabilities first, then take the corresponding quantiles of the sorted bootstrap estimates.
Ignoring finite-replicate uncertainty. An endpoint in a far tail may be determined by very few replicates. Increasing $B$ can reduce this simulation instability, but does not correct a flawed model or resampling design.

The practical value of BCa is its targeted adjustment of bootstrap quantiles: $z_0$ locates the original estimate within the bootstrap distribution, and $a$ captures asymmetry in the estimator's jackknife behavior. The interval is most useful when both quantities are stable and the underlying bootstrap itself is credible.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

If 60 of 100 bootstrap estimates are strictly below the original estimate, what is the empirical percentile rank, and what does a positive $z_0$ indicate?
Why is the acceleration term dimensionless even when the parameter is measured in L/h?
If $z_0=0$ and $a=0$, what happens to the adjusted probabilities?
For a nominal 90% interval, which two probabilities are adjusted to obtain the BCa endpoints?
Name one diagnostic warning that the BCa adjustment or its tail quantiles may be unstable.

Solutions. (1) The rank is $60/100=60\%$; the original estimate is above the bootstrap median and has an empirical percentile rank of approximately 60%. (2) The numerator scales as the cube of the parameter units and the denominator also scales as the cube, so the units cancel. (3) The adjusted probability equals the nominal probability because the normal argument reduces to $z_\alpha$. (4) Adjust the 0.05 and 0.95 probabilities. (5) Examples include a denominator near zero, an extreme adjusted probability with very few tail replicates, or a delete-one-subject fit that changes the estimate exceptionally strongly.