Why Vary the Dependence Assumption?
At a dose level, efficacy and toxicity can be related even when their marginal probabilities are known. For example, a dose may have a high efficacy rate and a high toxicity rate, but those events could occur largely in the same patients or in different patients. Those patterns have the same marginal rates but imply different probabilities of a patient experiencing efficacy without toxicity.
Sensitivity analysis makes this uncertainty visible. It asks how dose-level joint probabilities—and decisions based on them—change over a set of defensible dependence assumptions. The aim is not to select an association parameter by inspecting which one produces the preferred dose. Rather, the analysis shows which conclusions follow from the marginal models alone and which depend on an uncertain association.
This tutorial uses the Pearson correlation \(\rho_d=\operatorname{Corr}(E,T\mid d)\) of the two binary indicators as a transparent sensitivity parameter. A scenario with one common \(\rho\) across doses is convenient for comparing dose choices, but it is an assumption about the association pattern, not a consequence of the marginal dose-response models. If dose-specific associations are plausible, vary \(\rho_d\) separately at each dose or examine structured scenarios for the vector of associations.
From Marginal Probabilities to a Joint Table
For binary outcomes, the joint distribution is determined by the two marginal probabilities and the joint probability \(q_d\). At any dose, the four cell probabilities are
Each cell must be nonnegative. Equivalently, \(q_d\) must satisfy the Fréchet bounds
The bounds matter: not every correlation between \(-1\) and \(1\) is possible for Bernoulli outcomes with given margins. Using the covariance identity for a Bernoulli pair, \(\operatorname{Cov}(E,T\mid d)=q_d-p_{E,d}p_{T,d}\), and the standard deviations of the two indicators gives
Thus, once a feasible \(\rho_d\) is specified, all four cell probabilities follow. Independence is the special case \(\rho_d=0\), for which \(q_d=p_{E,d}p_{T,d}\). Positive correlation increases the probability of joint efficacy and toxicity relative to independence while reducing the two discordant cells; negative correlation does the reverse.
Worked Example: Reconstructing a Joint Table
Suppose a hypothetical dose has efficacy probability \(p_E=0.50\) and toxicity probability \(p_T=0.20\). Consider a dependence scenario with \(\rho=0.30\). First calculate the scale factor:
\(\sqrt{0.50(0.50)0.20(0.80)}=\sqrt{0.0400}=0.200.\) Therefore, \(q=0.50(0.20)+0.30(0.200)=0.100+0.060=0.160.\)
The four probabilities are \(P(E=1,T=1)=0.160\), \(P(E=1,T=0)=0.50-0.160=0.340\), \(P(E=0,T=1)=0.20-0.160=0.040\), and \(P(E=0,T=0)=1-0.50-0.20+0.160=0.460\). They are all nonnegative and sum to \(1.000\), so this is a valid joint distribution. The margins check: \(0.160+0.340=0.500\) for efficacy and \(0.160+0.040=0.200\) for toxicity.
Under independence, \(q=0.100\), giving cell probabilities \(0.100, 0.400, 0.100,\) and \(0.400\), respectively. The efficacy and toxicity rates are unchanged; only how outcomes co-occur differs.
Feasibility and the Sensitivity Range
The marginal probabilities determine which correlations are possible. Substituting the Fréchet bounds into the correlation formula yields the dose-specific interval
This expression assumes both marginal probabilities are strictly between zero and one, so the denominator is positive. If a margin is exactly zero or one, that outcome has no variation and its correlation with the other outcome is undefined; work directly with the joint probabilities instead. When using a common \(\rho\) across doses, it must lie in the feasible interval at every dose included in the scenario.
Worked Example: Checking Whether a Scenario Is Feasible
For the margins \(p_E=0.50\) and \(p_T=0.20\), the Fréchet bounds are \(0\leq q\leq0.20\). The correlation denominator is \(0.200\), and the independence value of \(q\) is \(0.100\). Therefore the feasible correlation range is
\((0-0.100)/0.200\leq\rho\leq(0.200-0.100)/0.200\), or \(-0.500\leq\rho\leq0.500\).
A proposed \(\rho=0.30\) is feasible, as the preceding joint table confirms. A proposed \(\rho=0.70\) is not: it would give \(q=0.100+0.70(0.200)=0.240\), which exceeds the maximum possible joint probability \(0.200\). Equivalently, \(P(E=0,T=1)=0.20-0.24=-0.04\), an impossible negative probability. The appropriate correction is to restrict the scenario to the feasible range, not to truncate a negative cell while leaving the other probabilities unchanged.
How Dependence Can Change a Dose Choice
Consider a hypothetical decision rule: among doses with marginal toxicity probability no greater than \(0.40\), select the dose with the largest probability of efficacy without toxicity, \(P(E=1,T=0\mid d)\). This rule is only an illustration; a trial’s clinical priorities and decision criteria must be specified separately. It is useful here because the target probability depends directly on \(q_d\):
The following hypothetical dose-level margins will be used. Dose C exceeds the stated toxicity limit and is therefore ineligible under this example’s rule.
| Dose | \(p_{E,d}\) | \(p_{T,d}\) | Eligible when \(p_T\leq0.40\)? |
|---|---|---|---|
| A | 0.50 | 0.20 | Yes |
| B | 0.65 | 0.38 | Yes |
| C | 0.75 | 0.55 | No |
Worked Example: Recomputing Dose Choices Across Scenarios
Compare independence (\(\rho=0\)) with a positive association scenario (\(\rho=0.30\)). For Dose A, the correlation scale factor is \(\sqrt{0.50(0.50)0.20(0.80)}=0.200\). Thus \(q_A=0.100+0.200\rho\), and \(P(E=1,T=0\mid A)=0.400-0.200\rho\).
For Dose B, the scale factor is \(\sqrt{0.65(0.35)0.38(0.62)}=\sqrt{0.053599}=0.2315\) (rounded). Thus \(q_B=0.247+0.2315\rho\), and \(P(E=1,T=0\mid B)=0.403-0.2315\rho\).
Under independence, Dose A has \(q_A=0.100\) and probability \(0.500-0.100=0.400\) of efficacy without toxicity. Dose B has \(q_B=0.247\) and probability \(0.650-0.247=0.403\). Both satisfy the toxicity limit, so Dose B is selected by the stated rule.
At \(\rho=0.30\), Dose A has \(q_A=0.100+0.200(0.30)=0.160\), so its target probability is \(0.500-0.160=0.340\). Dose B has \(q_B=0.247+0.2315(0.30)=0.3165\), so its target probability is \(0.650-0.3165=0.3335\). Both remain feasible under the toxicity limit, but Dose A now has the larger target probability. The preferred dose changes across these two feasible association scenarios.
For completeness, Dose C is excluded because its marginal toxicity probability is \(0.55>0.40\), regardless of the dependence parameter. This illustrates that a constraint on a marginal toxicity probability and a criterion based on a joint probability play different roles.
Locating the Decision Threshold
A sensitivity analysis can identify where the decision changes, rather than reporting only two selected scenarios. For the two eligible doses above, the difference in their target probabilities is
The two doses tie when \(0.003-0.0315\rho=0\), giving \(\rho=0.003/0.0315=0.0952\). Within the common feasible range for these margins, Dose B has the larger target probability below this threshold and Dose A above it. This crossover is a property of this decision rule and these hypothetical margins; it is not a universal threshold for clinical decisions.
Assumptions, Diagnostics, and Alternatives
The calculation rests on several explicit assumptions. Efficacy and toxicity are binary indicators defined over clear assessment windows, and the marginal probabilities refer to the same dose and target population. The margins are treated as fixed for the sensitivity calculation; this isolates uncertainty about dependence rather than propagating uncertainty in the margins. If the margins are estimated, their uncertainty should also be represented in a fuller analysis. A common correlation across dose levels is an additional simplifying assumption, not a requirement of the method.
Several checks help detect problems:
- Check every dose’s bounds. Each selected \(\rho_d\) must produce four nonnegative cell probabilities. A common value must be feasible for all doses to which it is applied.
- Check the margins after reconstruction. The two efficacy cells must sum to \(p_{E,d}\), and the two toxicity cells must sum to \(p_{T,d}\).
- Inspect whether the decision criterion uses dependence. A criterion depending only on \(p_E\) and \(p_T\), such as a fixed weighted difference of the two margins, will not change when only dependence varies. A criterion based on a joint category can change.
- Use clinically defensible scenarios. Sensitivity values should be motivated by clinical knowledge, previous evidence, or a stated range of uncertainty—not chosen after seeing which dose each value favors.
- Check the association across dose levels. If available data suggest different associations at different doses, a shared value can conceal that pattern. Report dose-specific scenarios or explain why the shared-association simplification is reasonable.
Independence is a useful reference scenario but does not resolve uncertainty when the outcomes may be associated. Copula and shared latent-variable models provide structured ways to connect marginal distributions, and can represent more detailed assumptions about how dependence behaves. A correlation grid is simpler and makes the change in a binary joint table easy to interpret, but it does not itself explain the clinical mechanism generating association. Nor does it account for uncertainty in estimated margins unless that uncertainty is added.
A further limitation is that the Pearson correlation scale is constrained by the margins, with feasible ranges that can differ across doses. The same numerical correlation can therefore represent different joint-probability changes at different doses. For reporting, give the association scenarios, the corresponding joint probabilities of decision-relevant outcomes, the dose rule, and whether the selected dose changes over the examined range. If it does change, the decision is sensitive to dependence assumptions; if it does not, the result is robust only over the range and assumptions actually examined.
Common Pitfall
A frequent error is to compute the joint probability as \(p_Ep_T\) at every dose and then describe the result as a general joint analysis. That multiplication assumes independence. It is appropriate as a reference case, but it is not justified merely because the marginal efficacy and toxicity rates are known. Another error is to vary \(\rho\) without checking its dose-specific bounds, which can generate negative cell probabilities. The correction is to state the dependence assumption, calculate the implied joint table, verify its margins and nonnegativity, and repeat the dose comparison over a prespecified feasible range.
Check Your Understanding
Answer each question using the definitions and calculations above. Brief solutions follow the questions.
- At a dose, \(p_E=0.40\), \(p_T=0.25\), and \(q=0.15\). Find the other three joint probabilities and verify the margins.
- For \(p_E=0.40\) and \(p_T=0.25\), calculate the independence value of \(q\). Is a proposed \(q=0.30\) feasible?
- Why can a dose ranking change for the probability of efficacy without toxicity, but not for a decision criterion that uses only fixed marginal probabilities?
- Suppose a common dependence value is feasible at one dose but outside the Fréchet-implied correlation range at another. What should the analysis do?
Solutions. 1. \(P(1,0)=0.40-0.15=0.25\), \(P(0,1)=0.25-0.15=0.10\), and \(P(0,0)=1-0.40-0.25+0.15=0.50\). They sum to \(1\); the efficacy margin is \(0.15+0.25=0.40\) and the toxicity margin is \(0.15+0.10=0.25\). 2. Under independence, \(q=0.40(0.25)=0.10\). The upper Fréchet bound is \(\min(0.40,0.25)=0.25\), so \(q=0.30\) is not feasible. 3. The probability of efficacy without toxicity is \(p_E-q\), so changing dependence changes it when margins are fixed. A criterion determined only by the fixed margins has no \(q\) term and therefore does not change. 4. Do not use that common value for both doses. Restrict the shared scenario to the intersection of all dose-specific feasible ranges, or use explicitly stated dose-specific dependence scenarios that remain feasible.