Why Between-Patient Variation Changes Dose Selection
A dose that produces a promising response at the typical exposure may not work well for many patients. Some patients have lower exposure and may miss an efficacy target; others have higher exposure and may face greater toxicity risk. Dose selection therefore requires more than evaluating a dose-response curve at one representative exposure. It requires estimating what happens across the distribution of exposures in the intended population.
The relevant quantities are population probabilities: the fraction expected to attain an efficacy target, the fraction expected to experience toxicity, and the fraction expected to attain efficacy without toxicity. These probabilities combine the exposure distribution with the response model. The calculation below uses a small set of exposure strata to make the averaging explicit. The numbers are hypothetical and illustrate the method, not a real drug or study.
Model and Assumptions
Suppose exposure is dose proportional over the candidate range. At 50 mg, assume the population has three exposure strata: 25% have exposure index 0.6, 50% have index 1.0, and 25% have index 1.5. At dose \(D\), each stratum’s index is multiplied by \(D/50\). The index is dimensionless; it represents exposure relative to a reference exposure, not a measured concentration with physical units.
For a patient with exposure index \(C\), model the probability of attaining the efficacy target and the probability of toxicity as:
Here \(p_E(C)=P(E\mid C,D)\) and \(p_T(C)=P(T\mid C,D)\); dose affects these probabilities through the exposure it produces. For the worked calculations, assume the exposure-stratum proportions apply to the intended population at every candidate dose, the exposure distribution is adequately represented by these three strata, and the conditional response curves are correctly specified. Also assume that efficacy and toxicity events are conditionally independent given exposure. That last assumption allows the conditional probability of efficacy without toxicity to be calculated as \(p_E(C)\{1-p_T(C)\}\).
The model is intentionally simple. In an application, the exposure distribution and response curves would need to be estimated from suitable data, and uncertainty in those estimates would matter. The calculations here treat them as known so that the effect of integrating over patient variability is clear.
From Conditional Probabilities to Population Probabilities
For exposure strata \(j=1,\ldots,J\), let \(w_j\) be the proportion of patients in stratum \(j\), and let \(C_j(D)\) be its exposure index at dose \(D\). The weights satisfy \(w_j\geq 0\) and \(\sum_j w_j=1\). The law of total probability gives the population probability of each event by weighting its conditional probability in each stratum.
Under conditional independence of efficacy and toxicity given exposure, the joint target probability is:
The joint probability must be averaged within strata before averaging across the population. Multiplying the marginal population efficacy probability by one minus the marginal toxicity probability would generally be wrong: patients with higher exposure tend to have both a greater efficacy probability and a greater toxicity probability. The two events are therefore associated in the population even when they are independent conditional on exposure.
Worked Example: Calculating Probabilities at 75 mg
At 75 mg, the three exposure indices are \(0.6(75/50)=0.90\), \(1.0(75/50)=1.50\), and \(1.5(75/50)=2.25\). For example, at \(C=1.50\), the efficacy probability is:
Applying the same formulas to all three exposure strata gives the following approximate conditional probabilities:
| Exposure index \(C\) | Population weight | \(p_E(C)\) | \(p_T(C)\) | \(p_E(C)\{1-p_T(C)\}\) |
|---|---|---|---|---|
| 0.90 | 0.25 | 0.500 | 0.0835 | 0.458 |
| 1.50 | 0.50 | 0.822 | 0.297 | 0.578 |
| 2.25 | 0.25 | 0.940 | 0.587 | 0.388 |
Weighting the efficacy probabilities gives:
Thus, under the stated model, about 77.1% of the intended population is predicted to attain the efficacy target at 75 mg. The toxicity probability is:
For simultaneous efficacy and no toxicity, use the conditional joint probabilities in the final column:
The predicted probability of attaining efficacy without the toxicity event is therefore about 50.1%. It is not obtained by multiplying 0.771 by \(1-0.316\), which would ignore the association induced by exposure variation.
Comparing Candidate Doses
The same calculation can be repeated for each candidate dose. The approximate results are shown below. Minor differences from hand calculations can arise from rounding the stratum-specific probabilities.
| Dose | Exposure indices (low, typical, high) | \(P(E\mid D)\) | \(P(T\mid D)\) | \(P(E\cap T^c\mid D)\) |
|---|---|---|---|---|
| 50 mg | 0.60, 1.00, 1.50 | 0.552 | 0.136 | 0.457 |
| 75 mg | 0.90, 1.50, 2.25 | 0.771 | 0.316 | 0.501 |
| 100 mg | 1.20, 2.00, 3.00 | 0.878 | 0.487 | 0.429 |
For instance, the 100 mg joint probability is approximately \(0.25(0.578)+0.50(0.458)+0.25(0.223)=0.429\). The higher dose has the largest marginal efficacy probability, but also the largest toxicity probability. Under this model, 75 mg has the largest probability of attaining efficacy without toxicity among these three candidates.
Worked Example: Applying a Decision Constraint
Suppose a decision rule requires the population toxicity probability to be below 0.30, and among doses that meet this constraint the aim is to select the dose with the greatest population efficacy probability. Applying the rule to the table:
At 50 mg, \(P(T\mid D)=0.136<0.30\), so the dose meets the toxicity constraint. At 75 mg, \(0.316>0.30\), and at 100 mg, \(0.487>0.30\), so both fail it. Of the candidate doses, 50 mg is the only one meeting the stated constraint; its efficacy probability is 0.552.
This differs from selecting 75 mg to maximize \(P(E\cap T^c\mid D)\) without a separate toxicity cap. The difference is not a contradiction: the two rules encode different decision criteria. A population probability is an input to a decision, not a complete decision rule on its own.
Why the Mean Exposure Is Not Enough
A tempting shortcut is to calculate the mean exposure and evaluate the response model only at that value. This does not generally recover the population probability because the logistic response curves are nonlinear: the average of the probabilities at individual exposures need not equal the probability at average exposure.
Worked Example: Averaging Before Versus After the Response Model
At 75 mg, the mean exposure index in the three-stratum population is:
Evaluating the efficacy curve at this mean gives:
But the population efficacy probability, averaging the stratum-specific probabilities, is 0.771. The shortcut overstates the model-based population probability by about \(0.833-0.771=0.062\), or 6.2 percentage points. The discrepancy arises because the shortcut first averages exposures and then applies a nonlinear function, whereas the target quantity applies the function to each exposure and then averages.
Justification, Diagnostics, and Limitations
The averaging formula follows directly by partitioning the population into mutually exclusive exposure strata. Let \(S_j\) denote membership in stratum \(j\). Since exactly one stratum applies to each patient, the events \(E\cap S_j\) are disjoint and their union is \(E\). Consequently:
The toxicity expression follows by the same partition. Within each stratum, conditional independence gives \(P(E\cap T^c\mid S_j,D)=p_E\{C_j(D)\}[1-p_T\{C_j(D)\}]\); summing these stratum-specific joint probabilities yields the joint formula. If conditional independence is not defensible, replace the product with a fitted or otherwise justified conditional joint probability \(P(E\cap T^c\mid C_j(D),D)\).
Before using these results, check that exposure is defined for the relevant dosing regimen and that the exposure distribution represents the intended population. In observed data, compare the model-predicted exposure distribution with the empirical distribution, inspect whether the response curves capture observed efficacy and toxicity across exposure, and check calibration of predicted probabilities. Examine the tails: a small high-exposure subgroup can contribute substantially to toxicity even if the mean exposure is moderate. Also check that predicted probabilities remain between 0 and 1 and that the dose-proportional exposure assumption is plausible over the candidate range.
The main failure modes are an unrepresentative exposure distribution, misspecified exposure-response curves, omitted dependence between efficacy and toxicity after conditioning on exposure, and treating estimated model quantities as if they were known. The discrete strata are a numerical approximation: more strata or integration over a continuous exposure distribution may be needed if response changes sharply across exposure. Parameter uncertainty and uncertainty in the population distribution should be propagated when the goal is to quantify uncertainty in the population probabilities. The present calculations are conditional predictions under fixed assumptions, not confidence intervals or guarantees for individual patients.
A single-population dose comparison integrates over one common exposure distribution. A different strategy would tailor doses using patient characteristics, but that requires a separate comparison of stratified dosing with a single population dose. Here, the essential contrast is between evaluating a typical patient and averaging patient-specific predictions. The latter is necessary whenever the decision criterion concerns a population fraction or probability.
Common Mistakes
- Evaluating only at typical exposure. That gives a conditional prediction for one exposure, not the population probability. Average conditional probabilities over the exposure distribution.
- Multiplying marginal probabilities to get joint attainment. Shared exposure can associate efficacy and toxicity in the population. Calculate the joint probability within exposure strata and then average, unless a justified joint model says otherwise.
- Choosing a dose from efficacy alone. The highest efficacy probability may also have an unacceptable toxicity probability. State the decision criterion and any constraints explicitly.
- Ignoring uncertainty. A precise-looking probability calculated from estimated curves and weights is not known without error. Distinguish a model-based point prediction from uncertainty about that prediction.
Check Your Understanding
Use the assumptions and approximate 75 mg stratum probabilities in this tutorial. Answers are included for checking the calculations.
- Why is the mean-exposure efficacy estimate at 75 mg not the same as the population efficacy probability?
- If efficacy and toxicity are conditionally independent given exposure, write the formula for the population probability of efficacy without toxicity.
- At 75 mg, suppose the high-exposure stratum weight were 0.20, the typical-stratum weight 0.40, and the low-stratum weight 0.40. Calculate the efficacy probability.
- Which candidate dose has the greatest efficacy probability in the comparison, and which has the greatest joint efficacy-without-toxicity probability?
- What change is needed if efficacy and toxicity are not conditionally independent given exposure?
Solutions. (1) The efficacy curve is nonlinear, so averaging its values across patients generally differs from evaluating it at the mean exposure. Here the mean-exposure estimate is about 0.833, whereas the population average is 0.771. (2) The formula is \(\sum_jw_jp_E(C_j)[1-p_T(C_j)]\). (3) The result is \(0.40(0.500)+0.40(0.822)+0.20(0.940)=0.200+0.3288+0.188=0.717\). (4) The 100 mg dose has the greatest efficacy probability, about 0.878; 75 mg has the greatest joint probability, about 0.501. (5) Use a conditional joint probability model for efficacy and no toxicity in each exposure stratum, then average those probabilities using the stratum weights.