Comparing Wald, Bootstrap, and Profile Intervals

Why Three Intervals Can Differ

A clearance estimate is often positive and can have an asymmetric sampling distribution, especially when the data provide limited information about the lower or upper part of its plausible range. A single estimate and standard error do not show that full shape. Wald, bootstrap, and profile-likelihood intervals summarize uncertainty in different ways: the Wald method uses local curvature near the estimate, the bootstrap uses the distribution of estimates from resampled data, and the profile method finds where the likelihood falls to a specified threshold.

These methods are not interchangeable calculations of an identical quantity. Each relies on assumptions, and each can be misleading when those assumptions or its numerical requirements fail. The goal is to understand what each interval uses, how to calculate it, and what evidence would make one more informative than another for a particular clearance estimate.

Wald Intervals: Local Curvature

Let \(\widehat{CL}\) be the fitted clearance estimate and \(SE(\widehat{CL})\) its standard error. The usual two-sided 95% Wald interval is

The standard error is obtained from the estimated covariance matrix of the model parameters, commonly based on the inverse observed information matrix. The interval approximates the sampling distribution of the estimate by a normal distribution centered at the estimate. In likelihood terms, it uses the curvature close to the maximum; it does not trace how the likelihood behaves far from that maximum.

This approximation is most defensible when the model is identifiable, the estimate is away from parameter boundaries, the likelihood is sufficiently smooth and locally close to quadratic, and the estimator is approximately normal in the relevant sample size. In an NLME model, these conditions also depend on the model being adequately specified and the numerical optimization having found the intended maximum. A Wald interval for clearance on its original scale can extend below zero, even though clearance must be positive. Such an endpoint is a warning about the approximation or parameter scale, not a plausible negative clearance.

Bootstrap and Profile Intervals

A percentile-bootstrap interval approximates the sampling distribution by repeatedly resampling the independent sampling units and refitting the same model. In a typical subject-level NLME design, the unit is the subject: resample whole subjects with their within-subject records, rather than treating those records as independent observations. If \(B\) successful refits produce estimates \(\widehat{CL}^{*(1)},\ldots,\widehat{CL}^{*(B)}\), the percentile interval uses the empirical 2.5th and 97.5th percentiles for a 95% interval. Its endpoints can be asymmetric around the original estimate. This method relies on the resampling scheme representing the data-generating sampling structure and on the refits being sufficiently stable. A percentile interval is one bootstrap construction; other constructions, such as BCa or normal-based bootstrap intervals, need not have the same endpoints.

The profile-likelihood interval starts from the likelihood with the clearance fixed at a candidate value, while nuisance parameters are reoptimized at that value. Write \(\ell_p(CL)\) for this maximized constrained log likelihood and \(\widehat{CL}\) for the value maximizing the full likelihood. Under regular large-sample likelihood conditions, a 95% interval for one scalar, interior parameter uses the likelihood-ratio set

Here \(\ell(\widehat{CL})\) denotes the maximized full log likelihood, including optimization over nuisance parameters. The interval endpoints are the values on either side of the maximum where the likelihood-ratio statistic reaches 3.841. Unlike the Wald method, this procedure follows the profile's shape away from the maximum. It still depends on a well-defined likelihood, reliable constrained optimization, and an appropriate likelihood-ratio approximation. Boundary parameters or weak identification can undermine the usual cutoff.

Comparing the Endpoints and Their Meaning

In this hypothetical example, the intervals are:

The Wald interval is widest here, but that is not a general ranking. Its symmetric endpoints arise from the estimate and standard error, while the other two methods can reflect asymmetry. The bootstrap and profile intervals are closer to one another in this example, but they are not identical: resampling describes variation across constructed data sets, whereas profiling evaluates relative likelihood for fixed clearance values in the observed data. Different endpoints do not mean that one calculation is necessarily erroneous.

A useful justification for expecting the methods to be similar in a well-behaved large sample comes from a local quadratic approximation. If the profile log likelihood near \(\widehat{CL}\) is approximately

then substituting this approximation into the likelihood-ratio inequality gives

Taking square roots yields \(|CL-\widehat{CL}|\leq\sqrt{3.841}\,SE(\widehat{CL})\), and \(\sqrt{3.841}\) is approximately 1.96. Thus, the profile interval is approximately the Wald interval when the local quadratic description is good. Marked disagreement is a prompt to inspect the profile shape, the bootstrap distribution, and the model fit—not a proof that the Wald calculation alone is defective.

Diagnostics, Limitations, and a Common Pitfall

For a clearance interval, check whether the endpoint and distribution behavior make scientific and statistical sense. A negative Wald lower limit, strong skew in bootstrap estimates, a long flat segment in the profile, or very different interval widths each call for examination. For bootstrap results, inspect failed or boundary-converged fits as well as the distribution of successful estimates; silently dropping a substantial fraction can distort the reported quantiles. For profile results, verify that nuisance parameters were reoptimized, that the curve crosses the cutoff on both sides, and that the apparent crossings are not artifacts of a coarse grid or optimization failure. These checks assess numerical and inferential reliability; they do not guarantee that the model itself is correctly specified.

The methods also have practical limits. Wald intervals are fast but can be unreliable under skewness, weak information, or boundary constraints. Bootstrap intervals can represent asymmetry, but they require many stable refits and can be sensitive to a small or unrepresentative subject sample. Profile intervals can show likelihood asymmetry directly, but constrained optimization can be expensive or unstable, and the usual cutoff is an approximation whose regularity conditions may fail. If the clearance parameter is transformed, intervals should be interpreted on the intended scale: a symmetric interval for log clearance, for example, becomes asymmetric after exponentiation.

Common mistake: “The Wald interval uses a standard error, so it must be the most reliable; if the bootstrap or profile endpoints differ, they must be wrong.” This reasoning treats a local normal approximation as if it described the entire likelihood and sampling distribution. The correction is to identify what each method actually uses, check the assumptions and diagnostics relevant to each, and report the interval construction clearly. A wider or asymmetric interval is not automatically better, and endpoint agreement alone does not validate a model.

When the three intervals differ materially, do not choose one merely because its endpoints look most convenient. Examine the fitted model's identifiability and convergence, the bootstrap resampling unit and replicate failures, and the profile's curvature and crossings. Then explain which interval is primary and why, and use the others as informative sensitivity checks when their assumptions are credible.