Worked Example With Binary Efficacy And Toxicity

Why Analyze Efficacy and Toxicity Together?

In dose finding, a dose can produce efficacy, toxicity, both, or neither. Looking at efficacy and toxicity separately may conceal how often benefit occurs without toxicity, or how often the two outcomes occur together. A joint analysis keeps these possibilities explicit and allows a dose rule to use them directly.

This tutorial follows a small hypothetical phase 1/2 trial through the complete calculation. The endpoints are binary and assessed within a prespecified follow-up window. The analysis uses a separate four-category model at each dose, then applies a stated efficacy and safety rule. The numerical thresholds are illustrative: a real protocol should justify them clinically and specify them before outcomes are reviewed.

Model, Assumptions, and Updating

For this worked analysis, each patient contributes one jointly classified outcome at the assigned dose. Within a dose, outcomes are assumed independent and to share the same four category probabilities. The endpoint definitions and assessment windows are consistent across doses, and there are no missing or unevaluable outcomes in the example. These assumptions make the category counts a multinomial sample. The model does not assume that efficacy and toxicity are independent, nor does it impose a monotone dose-response curve.

For each dose, use a Dirichlet prior with one unit of prior weight in each category. This is a simple symmetric prior used here to make the calculation transparent; it is not a default clinical choice. If \(n_{et,d}\) patients have outcome \((e,t)\) at dose \(d\), the posterior is Dirichlet with category parameters \(1+n_{et,d}\). The posterior mean for a category is its updated parameter divided by the sum of all four updated parameters.

This update follows directly from the likelihood. For category probabilities that sum to one, the multinomial likelihood is proportional to \(\prod_{e,t}\pi_{et}^{n_{et}}\). The symmetric Dirichlet prior has constant density over the probability simplex, so multiplying prior and likelihood gives powers \(\pi_{et}^{n_{et}}\), which are the Dirichlet posterior kernel with parameters \(1+n_{et}\). Summing parameters gives \(n+4\), where \(n\) is the number treated at that dose.

The marginal posterior for efficacy is also easy to obtain: \(p_E(d)\) has a beta distribution whose first parameter is the sum of the updated efficacy-category counts and whose second parameter is the sum for the non-efficacy categories. The corresponding statement holds for toxicity. This lets the decision rule use both posterior means and posterior tail probabilities.

Calculate Joint and Marginal Probabilities

At dose 2, the observed category counts are \((4,0,3,1)\), in the order \((1,0),(1,1),(0,0),(0,1)\). Adding the prior parameters gives \((5,1,4,2)\), with total \(12\). Therefore the posterior mean joint probabilities are obtained by dividing each updated count by \(12\). The posterior mean efficacy probability is the sum of the first two category means, and the posterior mean toxicity probability is the sum of the second and fourth.

The joint calculation can differ from a calculation that assumes independence. For dose 2, multiplying the posterior mean efficacy probability by one minus the posterior mean toxicity probability gives \(0.5000(1-0.2500)=0.3750\). That product is not the posterior mean of \(\pi_{10}(2)\), which is \(0.4167\). The product imposes a factorization using marginal summaries; it does not use the posterior distribution of the joint categories. With only eight observations, neither estimate is precise, but they answer different calculations.

Apply the Dose Rule

Suppose the protocol specifies the following illustrative rule for the final recommendation among the three evaluated doses. A dose is eligible only if its posterior mean efficacy probability is at least \(0.48\) and its posterior probability that toxicity exceeds \(0.35\) is no more than \(0.25\). Among eligible doses, recommend the one with the largest posterior mean probability of efficacy without toxicity, \(E[\pi_{10}(d)\mid\text{data}]\). If no dose is eligible, recommend no dose under this rule.

The safety criterion is a posterior tail probability, not a comparison of the observed toxicity rate with \(0.35\). At doses 1 and 2 there was one toxicity in eight patients, so the toxicity marginal posterior is \(\operatorname{Beta}(3,9)\). Its tail probability above \(0.35\) is approximately \(0.2001\). For example, using the integer-parameter beta-binomial identity, this tail is

At dose 3, four of eight patients experienced toxicity, so the marginal posterior is \(\operatorname{Beta}(6,6)\). Its probability of exceeding \(0.35\) is approximately \(0.8513\). Thus dose 3 does not satisfy the safety condition. The following calculations apply every part of the rule rather than selecting a dose from efficacy alone.

Diagnostics, Alternatives, and Limitations

Before applying a rule, check that each patient appears in exactly one joint category, that category counts sum to the dose total, and that marginal counts are derived consistently. For example, toxicity counts must include patients with both efficacy and toxicity; omitting them understates toxicity. Also verify that endpoint definitions, observation windows, and handling of unevaluable outcomes were fixed consistently. If outcomes are missing or assessed at different times, the complete-outcome calculations above do not directly apply.

The saturated model estimates four probabilities separately at every dose. It makes few dose-response assumptions, but small samples can leave categories empty and produce substantial uncertainty. The symmetric prior contributes one count to each category, so prior choice matters more when sample sizes are small. A dose-response model could borrow information across doses by imposing a smooth or monotone pattern, but its conclusions would then depend on those structural assumptions. Separate efficacy and toxicity models can be simpler when decisions use only marginal endpoints; they do not, without an additional dependence model, directly estimate probabilities such as efficacy without toxicity.

The rule itself also has limits. Posterior cutoffs do not automatically reflect the consequences of a false-safe or false-ineffective recommendation; those should inform the protocol thresholds. The example ranks eligible doses by one joint quantity, not by a complete utility function, and it ignores uncertainty in the ranking beyond the eligibility checks. A different clinical objective could rationally select a different eligible dose. In a real analysis, prior sensitivity and the consequences of alternative thresholds should be examined rather than treating one numerical rule as self-justifying.