Bootstrap Distributions of Effective Doses

Why Bootstrap an Effective Dose?

An effective dose is obtained by inverting a dose-response model: it is the dose at which the modeled response reaches a specified target. Even when the model parameters are estimated reliably, the inverse can be highly uncertain. Small changes in a fitted curve can produce substantial changes in the dose that reaches the target, especially when the target is near the curve’s upper plateau. A bootstrap makes this sampling variability visible by repeatedly generating data sets that resemble the observed study and recalculating the target-attaining dose.

The bootstrap distribution is an empirical approximation to the sampling distribution of the dose estimator under the resampling scheme and model used. It is not a distribution of possible doses for an individual patient, and it is not a posterior distribution for the unknown dose. Its interpretation depends on how the data are resampled and whether the fitted model adequately represents the data-generating process.

For example, consider an increasing Emax curve \(m(d)=E_0+E_{\max}d/(ED_{50}+d)\). If the target is an increase \(r\) above baseline, then solving \(r=E_{\max}d/(ED_{50}+d)\) gives \(ED_r=ED_{50}r/(E_{\max}-r)\). This inversion requires \(0<r<E_{\max}\), positive \(ED_{50}\), and the increasing branch of the model. If a bootstrap fit has \(E_{\max}\leq r\), no finite dose reaches the target under that fitted curve; that result is a substantive failure to attain the target, not a value to replace silently with a large dose.

Constructing Bootstrap Distributions

A bootstrap replicate repeats the entire estimation procedure on a resampled or simulated data set. For replicate \(b\), fit the dose-response model to that data set, solve for its target-attaining dose, and record \(\widehat{ED}_r^{*(b)}\). Repeating this \(B\) times produces \(B\) estimates whose empirical distribution approximates sampling variability. The same fitting constraints, target definition, dose restrictions, and convergence checks should be used for every replicate as for the original analysis.

In a parametric bootstrap, simulate new outcomes from the fitted model while retaining the original design, such as the dose assignments and number of observations at each dose. For binary outcomes, for example, simulate the number of responses at each dose from the fitted binomial probabilities. This approach relies on the specified response distribution, the mean model, and the assumption that the fitted model is a reasonable approximation to the data-generating process.

In a nonparametric bootstrap, resample observed experimental units with replacement and refit the model. For a fixed-dose study, resampling separately within each dose group generally preserves the design and the group sample sizes. If observations are clustered, resample independent clusters rather than treating correlated observations as independent. This method relies on the observed units being representative of the relevant sampling population; with few units or sparse dose groups, the empirical data may provide a poor representation of sampling variability.

The justification is direct. If the original estimator is a function \(T\) of the sample, its sampling distribution describes \(T\) across repeated samples generated under the study’s sampling assumptions. A bootstrap substitutes the empirical distribution of observed units, or a fitted parametric distribution, for the unknown data-generating distribution. Applying the same function \(T\)—including model fitting and dose inversion—to each bootstrap data set approximates how the estimator would vary under repeated sampling. This approximation is useful only to the extent that the resampling distribution captures the features that drive variation in the original study.

Percentile and Bias-Corrected Intervals

A two-sided percentile interval with nominal confidence level \(1-\alpha\) uses the empirical \(\alpha/2\) and \(1-\alpha/2\) quantiles of the bootstrap estimates. For a 95% interval, these are the 2.5th and 97.5th percentiles. The method is simple and preserves the observed skewness of the bootstrap distribution. Its coverage can nevertheless be poor when the estimator is biased, highly skewed, constrained by a boundary, or unstable under resampling.

The bias-corrected (BC) interval adjusts the percentile levels to account for the bootstrap distribution’s median position relative to the original estimate. Let \(\Phi\) be the standard normal cumulative distribution function and define

where \(P^*\) is the proportion of bootstrap replicates and \(\widehat{ED}_r\) is the estimate from the original data. For nominal tail probability \(a\), the adjusted percentile level is

Thus, the BC interval endpoints are the bootstrap quantiles at \(\Phi(2z_0+\Phi^{-1}(\alpha/2))\) and \(\Phi(2z_0+\Phi^{-1}(1-\alpha/2))\). If fewer than half the bootstrap estimates fall below the original estimate, then the proportion in the definition of \(z_0\) is less than 0.5, so \(z_0<0\) and the adjusted quantile levels move downward. If more than half fall below the original estimate, \(z_0>0\) and the adjusted levels move upward. Ties can be handled with a stated convention, such as counting half of the tied estimates, to avoid probabilities of exactly zero or one in finite bootstrap samples.

Diagnostics, Comparisons, and Limitations

A bootstrap interval is not a substitute for checking whether the dose estimate is well defined. Inspect the replicate estimates in a histogram or empirical quantile plot, and report the number and proportion of replicates that fail to converge, produce inadmissible parameter estimates, or cannot attain the target. Also check whether estimates accumulate at dose boundaries, whether a long upper tail is driven by a few replicates, and whether the interval changes materially when the number of replicates increases. Increasing \(B\) reduces Monte Carlo error in the estimated quantiles, but it does not repair model misspecification or an unrepresentative sample.

The parametric and nonparametric approaches answer the same broad question using different approximations. The parametric bootstrap can be more stable in small samples when the response model and distributional assumptions are credible; it can also reproduce the model’s constraints and dose design exactly. Its weakness is dependence on that model. The nonparametric bootstrap avoids specifying a full outcome distribution, but may be unstable in small samples, cannot generate response patterns absent from the observed data, and must respect dependence and design structure. When only a few observations occur at a dose, resampling may create many replicates with sparse or separated data and unstable inversions.

These methods are complementary rather than interchangeable. The earlier material on delta-method uncertainty and profile likelihood provides alternatives based on local approximation and likelihood shape. A bootstrap is especially useful for displaying asymmetry and the frequency of problematic estimates, but it does not automatically deliver reliable coverage. The BC adjustment corrects median bias in a particular sense; it does not correct model misspecification, boundary problems, or all forms of skewness. The BCa interval adds an acceleration adjustment, but it is not the interval calculated here.

A large fraction of unattainable targets is itself an important result: the data and fitted model do not support a stable finite dose estimate under the chosen target definition. Reporting an interval only among successful replicates can conceal this problem and change the question being answered. Report the failure fraction alongside the interval, investigate why failures occur, and consider whether the target, dose range, or model needs reconsideration. If the target is outside the supported response range, a numerical interval may give a false impression of precision.