Why Joint Bootstrap Results Matter
A set of marginal bootstrap intervals describes uncertainty in each parameter separately. It does not show how estimates of two parameters move together across refits. That joint behavior can matter in population pharmacokinetic models: two parameters may be individually uncertain, yet their estimates may be strongly coupled. Reporting the marginal intervals alone hides that feature.
Bootstrap covariance and correlation summarize the joint distribution of parameter estimates across bootstrap replicates. They describe how the estimates vary together under the chosen resampling and refitting procedure. They do not, by themselves, describe a biological correlation between individual patients’ parameters or establish that changing one parameter causes another to change.
The covariance has the product of the parameters’ units. For example, if one parameter is clearance in L/h and another is volume in L, their covariance is in L²/h. Correlation has no units and lies between \(-1\) and \(1\), provided both parameters have nonzero bootstrap variability. A positive covariance or correlation means larger-than-average replicate estimates of one parameter tend to occur with larger-than-average estimates of the other. A negative value means they tend to move in opposite directions. Neither sign alone explains why the estimates are associated.
How to Calculate and Interpret the Summary
The estimates must come from paired refits: row \(b\) contains all estimates obtained from the same bootstrap resample and model refit. The bootstrap covariance matrix applies the covariance formula to every pair of parameter columns. Its diagonal entries are the bootstrap sample variances; its off-diagonal entries are the covariances. Standardizing each entry by the corresponding standard deviations gives the correlation matrix.
For independently generated bootstrap replicates, the replicate parameter vectors are draws from the bootstrap procedure conditional on the observed data and fitted analysis. Their sample covariance, with denominator \(B-1\), estimates the covariance of that procedure’s replicate distribution. This is a finite-replicate estimate: with too few replicates, Monte Carlo variation can make the estimated covariance and correlation unstable. It does not turn the bootstrap distribution into a population distribution or remove uncertainty due to the original sample.
The denominator \(B-1\) follows from centering on the replicate means. For independent draws \(X_1,\ldots,X_B\) with common mean \(\mu_X\), write \(\bar{X}=B^{-1}\sum_b X_b\). The identity $$ \sum_{b=1}^{B}(X_b-\bar{X})(Y_b-\bar{Y}) = \sum_{b=1}^{B}(X_b-\mu_X)(Y_b-\mu_Y) - B(\bar{X}-\mu_X)(\bar{Y}-\mu_Y) $$ shows why centering by sample means reduces the expected sum of cross-products by a factor of \(B-1\), rather than \(B\), when \(X\) and \(Y\) have common covariance. Dividing by \(B-1\) therefore gives the usual unbiased sample covariance for independent draws from that distribution. In practice, the bootstrap distribution is not fully known; this result justifies the standard finite-replicate calculation, not a claim that the bootstrap covariance is an unbiased estimate of every possible target uncertainty.
Worked Example: Covariance and Correlation for Clearance and Volume
Suppose four hypothetical, successfully refitted bootstrap replicates give clearance estimates of 28, 29, 31, and 32 L/h and volume estimates of 17, 16, 16, and 15 L. The replicate rows remain paired as listed. Calculate the sample bootstrap covariance and correlation.
The means are \(\bar{\theta}_{CL}^*=(28+29+31+32)/4=30\) L/h and \(\bar{\theta}_{V}^*=(17+16+16+15)/4=16\) L. The centered deviations are \((-2,-1,1,2)\) L/h for clearance and \((1,0,0,-1)\) L for volume.
The sum of paired cross-products is \((-2)(1)+(-1)(0)+(1)(0)+(2)(-1)=-4\). Thus $$ \widehat{\operatorname{Cov}}_{\mathrm{boot}}(CL,V) = \frac{-4}{4-1} = -1.333\ \mathrm{L^2/h}. $$
The clearance sum of squared deviations is \(4+1+1+4=10\), giving \(s_{\mathrm{boot},CL}^2=10/3=3.333\ (\mathrm{L/h})^2\). The volume sum is \(1+0+0+1=2\), giving \(s_{\mathrm{boot},V}^2=2/3=0.6667\ \mathrm{L}^2\). Therefore, $$ \widehat{\operatorname{Corr}}_{\mathrm{boot}}(CL,V) = \frac{-1.333}{\sqrt{3.333}\sqrt{0.6667}} = -0.8944. $$
In these four hypothetical replicates, higher clearance estimates tend to coincide with lower volume estimates. The correlation is strongly negative, but four replicates are far too few for a stable practical estimate. The calculation illustrates the method; an analysis should use a replicate count adequate for the intended precision and inspect Monte Carlo stability.
Pairing, Scale, and What Marginal Summaries Conceal
A covariance uses the paired deviations from each replicate. If the columns are independently shuffled, each parameter’s marginal estimates and marginal intervals remain unchanged, but their covariance and correlation may change. Pairing is not a data-management detail: it is the information that defines the joint bootstrap distribution. Never sort each parameter column separately, pair values from unrelated replicates, or independently resample columns to calculate the covariance.
The parameter scale also matters. Multiplying a parameter by a positive constant rescales its covariance with another parameter but does not change their correlation. Nonlinear transformations, such as taking logarithms, can change both the correlation and the shape of the joint distribution. State whether the calculation uses natural-scale estimates or transformed estimates, and use the same scale consistently for all replicates. A correlation on a log scale describes linear association on that scale, not necessarily on the natural scale.
Worked Example: Same Marginals, Different Pairing
Consider five hypothetical bootstrap replicates for two log-scale quantities, \(x=\log(CL/(1\ \mathrm{L/h}))\) and \(y=\log(V/(1\ \mathrm{L}))\). The paired values are \(x=(2.8,2.9,3.0,3.1,3.2)\) and \(y=(3.6,3.9,4.0,4.1,4.4)\). Their means are 3.0 and 4.0, so the centered deviations are \((-0.2,-0.1,0,0.1,0.2)\) and \((-0.4,-0.1,0,0.1,0.4)\).
The cross-product sum is \(0.08+0.01+0+0.01+0.08=0.18\). The squared-deviation sums are \(0.10\) for \(x\) and \(0.34\) for \(y\). With \(B-1=4\), $$ \widehat{\operatorname{Cov}}_{\mathrm{boot}}(x,y)=0.18/4=0.045, \qquad \widehat{\operatorname{Corr}}_{\mathrm{boot}}(x,y) = \frac{0.045}{\sqrt{0.10/4}\sqrt{0.34/4}} = \frac{0.18}{\sqrt{0.10(0.34)}} = 0.9762. $$
Now reverse the order of the \(y\) values while leaving the \(x\) values unchanged. The marginal values, and therefore the separate marginal summaries, are identical. But the cross-product sum becomes \(-0.18\), giving correlation \(-0.9762\). This reversal is not a valid way to analyze the original paired replicates; it demonstrates why marginal intervals cannot recover the joint dependence.
A Joint Covariance Matrix
A matrix is more informative than a list of isolated pairwise correlations when several parameters are being examined. Each diagonal entry is a variance, each off-diagonal entry is a covariance, and corresponding standardized entries give the correlation matrix. Covariance entries have different units when the parameters have different units, so correlations can be easier to compare across parameter pairs. A matrix computed from the same complete set of replicate vectors is positive semidefinite, apart from small numerical rounding errors.
Worked Example: Finding Which Pair Is Strongly Coupled
Suppose five hypothetical replicates have the following centered deviations from their respective bootstrap means. Clearance deviations are in L/h, volume deviations in L, and absorption-rate deviations in h\(^{-1}\).
| Replicate | Clearance deviation | Volume deviation | Absorption-rate deviation |
|---|---|---|---|
| 1 | -2 | -4 | 0.2 |
| 2 | -1 | -1 | -0.2 |
| 3 | 0 | 0 | 0 |
| 4 | 1 | 1 | -0.2 |
| 5 | 2 | 4 | 0.2 |
There are four degrees in the sample-covariance denominator, \(B-1=4\). For clearance and volume, the cross-product sum is \(8+1+0+1+8=18\), so their covariance is \(18/4=4.5\ \mathrm{L^2/h}\). The sums of squares for clearance and volume are 10 and 34, giving variances 2.5 and 8.5. Their correlation is \(18/\sqrt{10(34)}=0.9762\).
The absorption-rate sum of squares is \(0.04+0.04+0+0.04+0.04=0.16\), so its variance is \(0.16/4=0.04\ \mathrm{h^{-2}}\). Its cross-product sums with clearance and volume are both zero: for clearance they are \(-0.4+0.2+0-0.2+0.4=0\); for volume they are \(-0.8+0.2+0-0.2+0.8=0\). The covariance matrix, in the parameter order clearance, volume, absorption rate, is therefore $$ \widehat{\Sigma}_{\mathrm{boot}} = \begin{pmatrix} 2.5 & 4.5 & 0\\ 4.5 & 8.5 & 0\\ 0 & 0 & 0.04 \end{pmatrix}. $$
The clearance-volume correlation is 0.9762, while the other two correlations are zero in this constructed example. The result identifies a strongly coupled pair in this replicate set; it does not establish a biological relationship between these parameters in individuals.
Assumptions, Diagnostics, and Alternatives
The interpretation depends on the bootstrap design. Replicates should be generated using a resampling scheme appropriate to the sampling units and design, and each retained vector should come from the intended refitting procedure. For example, resampling subjects and refitting is not interchangeable with holding parameters fixed during resampling. The replicate vectors must use a consistent parameter definition and scale. The estimates also need sufficient finite variability for the requested covariance or correlation to be meaningful.
Inspect the joint scatterplots and the marginal distributions of the parameters, not just the numerical matrix. A single correlation summarizes linear association and can be misleading when the point cloud is curved, clustered, or dominated by outliers. Look for boundary estimates, nonconvergence, extreme values, separate clusters, and a small effective number of successful refits. If a parameter is constant across retained replicates, its bootstrap standard deviation is zero and its correlation with any other parameter is undefined, not zero. A near-zero standard deviation can make the computed correlation highly unstable.
Failed replicates require particular care. Excluding failures can change the joint distribution if success is related to the resampled data or to parameter values. Report the number and handling of failures, and follow the prespecified convergence strategy described in the earlier material on bootstrap convergence and failed replicates. Do not silently replace failures, pair columns from different successful refits, or interpret a covariance matrix from a selected subset as though it necessarily represented the full bootstrap procedure.
Bootstrap covariance complements rather than replaces other summaries. Marginal percentile or BCa intervals describe each parameter’s uncertainty separately; they do not encode dependence. A model-based covariance estimate from local likelihood curvature can be useful as a computationally efficient approximation near a well-behaved optimum, but it relies on local approximation and model assumptions. Bootstrap covariance reflects the chosen resampling-and-refitting procedure, including some finite-sample behavior, but can be noisy and sensitive to the resampling design. Strong disagreement between methods is a diagnostic prompt to examine model geometry, parameter boundaries, weak identification, and bootstrap stability—not a reason to choose whichever estimate is more convenient.
The practical value of the joint summary is that it reveals estimation trade-offs that marginal intervals conceal. A strongly coupled pair may indicate that the data and model distinguish only certain combinations of parameters well. The covariance matrix can describe the direction and scale of local joint variation, while plots and model diagnostics help determine whether that description is adequate. It is not, on its own, a remedy for weak identification or a complete uncertainty analysis.
Check Your Understanding
Answer each question, then compare with the solutions.
- What data must be paired to calculate bootstrap covariance between two PK parameters?
- Can a bootstrap correlation be calculated if one parameter has exactly zero variability across the retained replicates? Why?
- Suppose clearance and volume have bootstrap covariance \(-6\ \mathrm{L^2/h}\). What does the negative sign indicate, and what does it not establish?
- Why can marginal intervals remain unchanged while bootstrap correlation changes?
- How does a positive rescaling of one parameter affect covariance and correlation with another parameter?
Solutions. (1) The two estimates from each same bootstrap resample and refit must stay together as one replicate row; the paired deviations determine the cross-products. (2) No. Its bootstrap standard deviation is zero, so the correlation denominator is zero and the correlation is undefined. (3) It indicates that, across paired replicates, deviations of the two estimates tend to have opposite signs. It does not establish a biological correlation among patients or a causal relationship. (4) Marginal intervals depend on each parameter’s values separately, whereas correlation depends on how values are paired across replicates. (5) Multiplying one parameter by a positive constant multiplies the covariance by that constant, but leaves correlation unchanged because its standard deviation is multiplied by the same constant.