What Rejection Sampling Selects
A candidate population can satisfy physiological constraints and still miss important targets for the population it is intended to represent. Rejection sampling provides a direct rule for screening candidates: retain a candidate only when its specified summaries fall within prespecified target ranges. In virtual patient generation, the candidate may be an entire cohort of simulated patients, rather than one patient at a time. That distinction matters because a cohort can pass or fail based on its aggregate behavior even when each patient is individually feasible.
The method is simple to state but sensitive to its design choices. The summaries, target ranges, and rule for combining them must be defined before screening. Otherwise, repeated adjustments to the rule can make the retained cohorts appear to fit targets chosen after seeing the results.
This rule is a conjunction: every required condition must pass. For example, a cohort might need to fall within a target interval for both mean drug exposure and mean biomarker suppression. Passing one does not compensate for failing the other unless a different rule was explicitly specified. Target intervals should be meaningful in context, use the same units and definitions as the simulated summaries, and be set before evaluating the candidate cohorts.
Assumptions and the Acceptance Rate
A valid application requires a well-defined candidate-generation procedure. It should generate cohorts from the intended virtual-patient model, including its parameter distributions, dependence structure, and feasibility constraints. The summaries and target intervals must also be specified in advance. If model simulations contain random variability, the procedure must state how that variability is handled—for example, whether outcomes are simulated once or averaged across a fixed number of replicates.
For \(M\) candidate cohorts, define \(I_i=1\) if cohort \(i\) passes every condition and \(I_i=0\) otherwise. The observed acceptance rate is the fraction of candidates retained. It describes the performance of this screening rule under the candidate-generation procedure used; it is not a measure of the probability that the targets are correct.
Here \(A\) is the number accepted. If candidate cohorts are independently generated by the same procedure, each has the same probability \(r\) of acceptance, and \(\hat{r}\) estimates \(r\). Under those assumptions, the standard error of the estimated rate is \(\sqrt{\hat{r}(1-\hat{r})/M}\). This standard error describes Monte Carlo variation in the estimated acceptance rate; it does not capture uncertainty in the target ranges or misspecification of the virtual-patient model.
Worked Example: Screening Cohorts Against Two Targets
Suppose the prespecified targets are a cohort mean clearance from 4.0 to 5.0 L/h and a cohort mean potency parameter from 1.8 to 2.2 nM. Eight candidate cohorts have the following summaries. The illustrative numbers are hypothetical.
| Cohort | Mean clearance (L/h) | Mean potency parameter (nM) | Decision |
|---|---|---|---|
| A | 4.5 | 2.0 | Accept |
| B | 3.9 | 2.0 | Reject |
| C | 4.8 | 2.3 | Reject |
| D | 4.0 | 1.8 | Accept |
| E | 5.1 | 2.1 | Reject |
| F | 4.2 | 2.2 | Accept |
| G | 4.9 | 1.7 | Reject |
| H | 4.6 | 1.9 | Accept |
The acceptance rule is \(4.0 \leq S_{\mathrm{clearance}}(C) \leq 5.0\) and \(1.8 \leq S_{\mathrm{potency}}(C) \leq 2.2\). Cohort D passes both, including the interval endpoints. Cohort B fails the clearance condition; C fails the potency condition. Applying the same rule to all eight gives four accepted cohorts, so
Thus, half of these candidate cohorts met both specified targets. The result does not mean that half of individual patients were acceptable: the decision was made once per cohort, using cohort-level summaries.
Joint Criteria and Why the Rate Can Fall Quickly
With multiple targets, acceptance is the probability that all required events occur together. If \(A_j\) is the event that target \(j\) is met, then acceptance is the intersection of those events. The joint probability is not generally the product of the separate probabilities: that product is valid only when the events are independent. In practice, cohort summaries may be related, so the observed joint acceptance rate is usually the safest direct calculation.
The relationship can be seen with a simple bound. A cohort accepted on every target must pass target 1, so the joint acceptance probability cannot exceed the probability of passing target 1. The same argument holds for every target. Therefore, \(P(\text{accept}) \leq P(A_j)\) for each \(j\). Adding required conditions cannot increase the acceptance probability. The rate may become small even when each separate condition is often met.
Worked Example: A Multivariate Target and Empirical Acceptance Rate
Ten hypothetical candidate cohorts are evaluated against three prespecified ranges: mean exposure from 80 to 100 mg·h/L, the standard deviation of exposure from 15 to 25 mg·h/L, and mean biomarker suppression from 35% to 45%. All endpoints are included.
| Cohort | Mean exposure (mg·h/L) | Exposure standard deviation (mg·h/L) | Mean suppression (%) | Decision |
|---|---|---|---|---|
| 1 | 90 | 20 | 40 | Accept |
| 2 | 78 | 19 | 41 | Reject |
| 3 | 96 | 26 | 39 | Reject |
| 4 | 84 | 18 | 36 | Accept |
| 5 | 101 | 20 | 42 | Reject |
| 6 | 88 | 24 | 46 | Reject |
| 7 | 80 | 15 | 35 | Accept |
| 8 | 94 | 22 | 44 | Accept |
| 9 | 86 | 25 | 40 | Accept |
| 10 | 92 | 17 | 34 | Reject |
Cohort 2 fails the mean exposure range; cohort 3 fails the exposure standard deviation range; cohort 6 fails the suppression range. Cohorts 5 and 10 also fail one target each. Cohorts 1, 4, 7, 8, and 9 pass all three, including cohorts on interval boundaries. Hence \(A=5\), \(M=10\), and the empirical joint acceptance rate is \(5/10=0.500\).
If these ten cohorts were independently generated by the same procedure, an estimated Monte Carlo standard error is
The relatively large standard error indicates that ten candidates give a rough estimate of the procedure’s acceptance probability. It does not change the observed count of five accepted cohorts, and it says nothing about whether the target intervals themselves are scientifically appropriate.
Deriving the Sampling Interpretation
The acceptance-rate formula follows directly from the indicator representation. Suppose the candidate cohorts are independent and identically generated, with probability \(r\) of meeting the complete rule. Each \(I_i\) is then a Bernoulli random variable with mean \(r\) and variance \(r(1-r)\). Since \(\hat{r}\) is the average of \(M\) such indicators, its expectation is \(r\), and independence gives its variance.
\(E(\hat{r})=E(\sum_i I_i/M)=\sum_i E(I_i)/M=Mr/M=r\). Thus, the rate estimate is unbiased under the stated sampling assumptions.
\(\operatorname{Var}(\hat{r})=\operatorname{Var}(\sum_i I_i/M)=\sum_i\operatorname{Var}(I_i)/M^2=Mr(1-r)/M^2=r(1-r)/M\). The standard error formula substitutes \(\hat{r}\) for the unknown \(r\).
If candidate cohorts are dependent, generated under different procedures, or selected adaptively based on earlier outcomes, this simple standard-error formula need not apply. In particular, the formula should not be presented as a general uncertainty measure for a set of candidates that was not independently sampled from one stable procedure.
Worked Example: Planning Candidate Generation From a Pilot Rate
In a hypothetical pilot, 6 of 24 independently generated candidate cohorts pass a fixed set of target ranges. The estimated acceptance rate is
If the same generation and screening procedure is continued, the expected number of accepted cohorts from \(M\) new candidates is \(M\hat{r}\), using the pilot rate as a planning estimate. To obtain an expected 30 accepted cohorts, the corresponding candidate count is
Thus, 120 new candidates would yield 30 accepted cohorts on average under the pilot estimate. It is not a guarantee of exactly 30: the number accepted varies from run to run, and the true rate may differ from the pilot estimate. The pilot-rate standard error is \(\sqrt{0.25(0.75)/24}=0.0884\), so planning based on this small pilot has substantial uncertainty.
Diagnostics, Alternatives, and Limitations
A low acceptance rate is a diagnostic, not automatically a reason to loosen the ranges. First identify which conditions are most often failed and whether failures are concentrated near a boundary or far outside it. Confirm that the simulated summaries were computed with the specified definitions and units, and that the candidate generator actually produces the intended virtual-patient distributions and dependence. If candidates are produced using stratified sampling, as in earlier material on candidate populations with stratified sampling, check that the strata and within-stratum allocation are consistent across the cohorts being compared.
Rejection sampling is transparent and retains complete candidates that satisfy a clear rule. Its main alternative at this stage is to revise the candidate-generation design or the prespecified acceptance criteria, then evaluate that revised procedure explicitly. Unlike a method that changes candidates’ relative contribution rather than applying an all-or-none cohort rule, rejection produces a straightforward accepted set—but can require generating many candidates when the joint acceptance probability is small. Any changed target range should be justified by the scientific question, not merely by a desire to increase the yield.
Acceptance is not proof of overall distributional agreement. It certifies only that the selected summaries fell within the stated ranges. Features that were not included in the rule may still differ from the intended population. Also, conditioning on passing several criteria changes which candidate cohorts are represented among the retained set; acceptance is a selection step, not a guarantee that all other properties of the original candidate-generating process remain unchanged.
A common mistake is to calculate the separate pass rate for each target and multiply those rates to obtain the joint rate. That multiplication is justified only if passing the targets is independent. The correction is to count cohorts that pass all conditions together, or to establish independence before using a product. Another mistake is to call a cohort “acceptable” because most of its target summaries pass. Under an all-target rule, even one failed condition means rejection; a different rule must be specified in advance if partial compliance is intended.
Check Your Understanding
Answer each question using the all-targets-must-pass rule unless otherwise stated.
- A cohort’s mean exposure is within its target range, but its mean biomarker response is outside its range. Is it accepted?
- Four of 16 independently generated candidate cohorts pass every target. Calculate the empirical acceptance rate.
- Two target pass events have probabilities 0.8 and 0.7. Is their joint probability necessarily \(0.8 \times 0.7\)?
- Explain why a cohort-level acceptance decision is not the same as deciding whether each patient is acceptable.
- Using an estimated rate of 0.20, how many candidate cohorts would yield an expected 15 accepted cohorts if the procedure remains unchanged?
Solutions: 1. No. The cohort fails at least one required condition. 2. \(\hat{r}=4/16=0.250\), or 25.0%. 3. No; the product is valid only if the pass events are independent. The joint rate should otherwise be calculated from cohorts that pass both. 4. The rule evaluates aggregate summaries of the cohort, so individual patients are not separately accepted or rejected by that cohort-level decision. 5. \(M=15/0.20=75\) candidates, giving 15 accepted on average, not with certainty.