Why subgroup benefit–risk needs a joint model
A dose can have different efficacy and toxicity profiles in different patient subgroups. Moreover, the association between the two outcomes may differ: for example, efficacy might occur alongside toxicity more often in one subgroup than another. A model that compares only subgroup efficacy rates, or only toxicity rates, can miss this joint pattern. The decision is not simply whether two subgroups have different observed rates; it is whether the evidence supports different benefit–risk recommendations after accounting for uncertainty.
The earlier material on patient covariates in joint dose-response models provides the general idea of including patient characteristics in a dose model. Here the focus is a categorical subgroup and the decision implications of allowing its joint efficacy–toxicity profile to vary.
The four probabilities distinguish efficacy without toxicity, efficacy with toxicity, neither outcome, and toxicity without efficacy. This distinction matters whenever clinical value depends on which outcomes occur together. A standard measure of within-subgroup association is the odds ratio \(\theta_g(d)=\pi_{11,g}(d)\pi_{00,g}(d)/[\pi_{10,g}(d)\pi_{01,g}(d)]\), when all four probabilities are positive. An odds ratio of one corresponds to independence between \(E\) and \(T\) at that subgroup and dose; it does not mean that either outcome is rare or clinically unimportant.
Representing subgroup variation
A direct model treats the four outcomes as a multinomial response for each patient. For patients in subgroup \(g\) assigned dose \(d\), the cell counts have a multinomial likelihood with probabilities \(\pi_{00,g}(d), \pi_{10,g}(d), \pi_{01,g}(d), \pi_{11,g}(d)\). A model can let these probabilities change with dose and subgroup. For example, separate logistic models for the marginal efficacy and toxicity probabilities can include dose, subgroup, and dose-by-subgroup terms; a further parameter or model component must then specify their association. The resulting four cell probabilities must be valid probabilities, nonnegative and summing to one.
The dose-by-subgroup terms represent differences in dose-response pattern, not merely a difference in average outcome between subgroups. Subgroup-specific association can also be modeled, but is difficult to estimate when joint outcome cells are sparse. A parsimonious model may instead assume a common association across subgroups or doses. That is a substantive assumption to examine, not a consequence of using a joint model.
The analysis requires clearly defined efficacy and toxicity endpoints and assessment times, consistent outcome ascertainment, and subgroup membership defined in a way that is available for decision-making. Conditional on subgroup and dose, patients are commonly assumed independent, and the chosen dose-response form must adequately represent the probabilities over the studied dose range. The allocation must also provide information at relevant doses in each subgroup; if a subgroup receives almost no patients at a dose, its response there is mostly model extrapolation. Randomization, if used, supports dose comparisons within the design, but does not by itself guarantee enough information in every subgroup.
Worked Example: Different joint profiles at one dose
Suppose a hypothetical study has 40 evaluable patients in each of two prespecified subgroups at the same dose. The table gives joint outcome counts. The benefit–risk scores used here are protocol-defined illustrative values: efficacy without toxicity scores \(+1\), efficacy with toxicity scores \(+0.2\), neither scores \(0\), and toxicity without efficacy scores \(-1\).
| Subgroup | Efficacy, no toxicity | Efficacy and toxicity | Neither | Toxicity, no efficacy |
|---|---|---|---|---|
| A | 18 | 6 | 12 | 4 |
| B | 10 | 8 | 10 | 12 |
For subgroup A, \(\hat p_E=(18+6)/40=0.600\) and \(\hat p_T=(6+4)/40=0.250\). The estimated odds ratio is \((6\times12)/(18\times4)=1.00\). For subgroup B, \(\hat p_E=(10+8)/40=0.450\), \(\hat p_T=(8+12)/40=0.500\), and the odds ratio is \((8\times10)/(10\times12)=0.667\). Thus, these data show different marginal rates and different estimated associations; the estimates alone do not establish that the population profiles differ.
The subgroup A estimated mean score is \(\hat U_A=(18\times1+6\times0.2+12\times0+4\times(-1))/40=(18+1.2-4)/40=0.380\) score units per patient. For subgroup B it is \(\hat U_B=(10+8\times0.2-12)/40=(10+1.6-12)/40=-0.0100\) score units per patient. These are descriptive estimates for the hypothetical data, not dose recommendations. Their uncertainty and the clinical acceptability of each profile must also be considered.
Why the association can change the decision
A joint benefit–risk summary should reflect how the protocol or clinical stakeholders value each outcome combination. The following derivation shows why marginal rates alone may not determine that summary. Write \(q=\pi_{11,g}(d)\), \(p_E=p_{E,g}(d)\), and \(p_T=p_{T,g}(d)\). Then \(\pi_{10}=p_E-q\) and \(\pi_{01}=p_T-q\). For the illustrative scores above, expected score per patient is
The expected score depends on \(q\), the probability of both efficacy and toxicity, as well as on the two marginal probabilities. If the score for efficacy with toxicity were different, the coefficient of \(q\) would change. Consequently, the utility scale must be justified clinically; it is not a universal statistical default. If utility were strictly additive in \(E\) and \(T\), association would not affect its expectation, though it could still matter for other clinical decisions.
A subgroup-specific recommendation should combine such a benefit–risk criterion with explicit constraints. For instance, a protocol might require toxicity probability no greater than \(0.30\) and efficacy probability at least \(0.35\), then choose the eligible dose with the largest expected score. Thresholds should be specified for the clinical context rather than selected after viewing subgroup results.
Worked Example: Applying constraints by subgroup
Suppose a hypothetical fitted joint model gives the following plug-in probabilities at two doses. The score is the same as in the previous example. For this illustration, a dose is eligible only if its estimated toxicity probability is at most \(0.30\) and its estimated efficacy probability is at least \(0.35\).
| Subgroup and dose | \(\pi_{10}\) | \(\pi_{11}\) | \(\pi_{00}\) | \(\pi_{01}\) |
|---|---|---|---|---|
| A, dose 1 | 0.40 | 0.10 | 0.40 | 0.10 |
| A, dose 2 | 0.50 | 0.15 | 0.25 | 0.10 |
| B, dose 1 | 0.32 | 0.08 | 0.45 | 0.15 |
| B, dose 2 | 0.34 | 0.16 | 0.30 | 0.20 |
For subgroup A at dose 1, efficacy is \(0.40+0.10=0.50\), toxicity is \(0.10+0.10=0.20\), and score is \(0.40+0.2(0.10)-0.10=0.320\). At dose 2, efficacy is \(0.50+0.15=0.65\), toxicity is \(0.15+0.10=0.25\), and score is \(0.50+0.2(0.15)-0.10=0.430\). Both doses meet the stated point-estimate constraints, so dose 2 has the higher estimated score.
For subgroup B at dose 1, efficacy is \(0.32+0.08=0.40\), toxicity is \(0.08+0.15=0.23\), and score is \(0.32+0.2(0.08)-0.15=0.186\). At dose 2, efficacy is \(0.34+0.16=0.50\), toxicity is \(0.16+0.20=0.36\), and score is \(0.34+0.2(0.16)-0.20=0.172\). Dose 2 violates the toxicity constraint. On point estimates alone, this rule selects dose 2 for A and dose 1 for B.
That arithmetic does not yet show that separate recommendations are supported. In an actual analysis, the estimated probabilities and decision criterion are uncertain. For example, a Bayesian protocol could require the posterior probability that both constraints hold to exceed \(0.80\) before declaring a dose eligible. The model, data, and posterior would have to establish that probability; the point estimates in this example do not supply it.
Assessing whether subgroup recommendations are supported
A difference between subgroup point estimates can arise from sampling variation, especially when one subgroup is small or joint outcome cells are rare. A useful assessment focuses on the quantities relevant to the decision: subgroup-specific probabilities, differences between them, and the probability that each dose satisfies the clinical criteria. In a Bayesian analysis, for instance, one can examine posterior draws of \(p_{E,g}(d)\), \(p_{T,g}(d)\), and the joint probabilities, and calculate the posterior probability that a clinically meaningful contrast or decision condition holds. This is not the probability that a null hypothesis is true; it is a probability under the specified model and prior, conditional on the observed data.
A frequentist analysis can instead report uncertainty intervals for subgroup-specific risks and risk differences, or use an interaction test as one component of the assessment. Failure to reject a no-interaction hypothesis does not prove that subgroups respond identically, and a small p-value alone does not show that different doses are warranted. The scale matters: differences on a probability scale, odds-ratio scale, and utility scale answer different questions.
Worked Example: A noisy subgroup contrast
Consider hypothetical efficacy data at one dose: 8 of 12 patients respond in subgroup A, compared with 24 of 48 in subgroup B. The estimated risk difference is \(8/12-24/48=0.667-0.500=0.167\), or 16.7 percentage points. Using the usual independent-binomial large-sample standard error,
An approximate 95% interval for the risk difference is \(0.167\pm1.96(0.154)\), or \((-0.135,0.469)\). The estimate is compatible with no difference and also with a substantial positive difference for subgroup A. Because subgroup A is small, the normal approximation is rough; a more suitable binomial interval or a model-based interval may be preferred. Either way, these data alone are weak support for a distinct dose recommendation. Toxicity, joint outcome patterns, dose-response information, and the decision thresholds would also be needed.
Diagnostics, alternatives, and limitations
Several checks help reveal when subgroup conclusions are fragile. Tabulate the four joint outcome cells by subgroup and dose before interpreting a fitted model. Zero or very small cells can make association estimates unstable; a saturated estimate of the odds ratio may be zero, infinite, or extremely imprecise. Check whether each subgroup has observations across the dose range that informs its recommendation. If not, a smooth dose model may produce precise-looking predictions that depend strongly on its form. Also check that the chosen efficacy and toxicity definitions, assessment windows, and subgroup classifications are consistent.
The main alternatives answer narrower questions. Separate efficacy and toxicity models can be easier to fit and interpret, but do not characterize their association unless an additional joint component is supplied. A pooled model with no subgroup terms is more parsimonious, but assumes away subgroup differences. A fully separate model allows maximal variation, but can be unstable with limited data. Partial pooling—allowing subgroup effects to differ while shrinking estimates toward a common distribution—offers a compromise, but depends on its distributional assumptions and does not create information where subgroup data are absent.
Subgroup-specific recommendations are most credible when the subgroups are clinically defined in advance, enough data inform the relevant doses and joint cells, and the recommendation remains stable under reasonable modeling choices. Searching many subgroups after seeing the outcomes increases the chance of finding an apparently favorable pattern by chance. Even a prespecified subgroup may have limited support if its sample is small. Model-based evidence can guide decisions; it does not establish that subgroup membership itself causes different responses.
A common mistake is to compare subgroup efficacy rates, observe a numerical difference, and immediately select different doses. The correction is to estimate the joint benefit–risk profile, quantify uncertainty in the subgroup contrasts, apply the prespecified clinical constraints, and require adequate support at the candidate doses. Another mistake is to treat a common efficacy–toxicity association as harmless simplification without checking its consequences. If association affects the clinical score or a safety constraint, that assumption can change the recommendation and should be justified.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- For a subgroup at one dose, suppose \(\pi_{10}=0.30\), \(\pi_{11}=0.20\), \(\pi_{00}=0.35\), and \(\pi_{01}=0.15\). Find the efficacy and toxicity probabilities.
- Using the illustrative scores in this tutorial, calculate the expected score for the probabilities in Question 1.
- Why can a subgroup with the higher estimated efficacy rate still have the less attractive benefit–risk profile?
- What does an odds ratio of one for efficacy and toxicity mean at a given subgroup and dose?
- Name two reasons a subgroup-specific recommendation may be unreliable even when subgroup point estimates differ.
Solutions. (1) The probabilities sum to one. Efficacy is \(\pi_{10}+\pi_{11}=0.30+0.20=0.50\); toxicity is \(\pi_{01}+\pi_{11}=0.15+0.20=0.35\). (2) The expected score is \(\pi_{10}+0.2\pi_{11}-\pi_{01}=0.30+0.2(0.20)-0.15=0.190\) score units per patient. (3) The subgroup may also have more toxicity, or efficacy may more often occur alongside toxicity; the joint distribution and clinical score determine benefit–risk. (4) It means efficacy and toxicity are independent within that subgroup at that dose, assuming all four probabilities are positive. (5) Examples include a small subgroup, sparse joint outcome cells, little dose coverage, or a result sensitive to model assumptions and exploratory subgroup selection.