Endpoint Dependence And Joint Estimands

Why Endpoint Dependence Must Be Defined

A biomarker and a clinical response measured in the same participant may be related, but “the endpoints are associated” does not say exactly what quantity an analysis targets. It could refer to the population-level relationship between the outcomes, the relationship after accounting for baseline characteristics, or the probability that both outcomes occur. Those are different questions and can lead to different numerical answers.

For example, a PK/PD analysis might ask whether treatment changes the probability of a biomarker response, whether clinical response differs between participants with and without that biomarker response, or whether treatment changes the probability of achieving both responses. The first is marginal over the other endpoint; the second is conditional on it; the third is joint. Defining the question first prevents a fitted model’s parameters from being mistaken for the scientific quantity of interest.

An estimand is not just a model coefficient. Its interpretation depends on the population, treatment or exposure condition, endpoint definitions, measurement times, and summary being targeted. For paired pharmacodynamic outcomes, “paired” means that both endpoint values belong to the same participant under the defined analysis conditions. Outcomes measured at different, clinically mismatched times may not represent the intended pairing.

Marginal, Conditional, and Joint Quantities

Suppose both endpoints are binary, with \(B=1\) denoting biomarker response and \(C=1\) denoting clinical response. Under a treatment condition \(a\), the marginal response probabilities are \(P(B=1\mid A=a)\) and \(P(C=1\mid A=a)\). A treatment contrast might be a difference or ratio between these probabilities under two treatment conditions. Each marginal summary averages over the observed distribution of the other endpoint.

A conditional probability answers a different question. For instance, \(P(C=1\mid B=1,A=a)\) is the clinical-response probability among participants with biomarker response under condition \(a\). The conditioning event must be named: conditioning on biomarker response is not equivalent to conditioning on baseline severity, and neither is equivalent to a population-average probability.

A joint probability specifies a combination of outcomes, such as \(P(B=1,C=1\mid A=a)\). The full joint distribution for two binary endpoints consists of four cell probabilities. The marginals can be obtained by summing appropriate cells, and the conditional probabilities by dividing a joint cell probability by the probability of its conditioning event.

This conditional probability is defined only when \(P(B=1\mid A=a)>0\). More generally, all conditional summaries require a conditioning event with positive probability in the target population. In a finite sample, sparse or empty conditioning groups can also make estimates unstable or unavailable.

What Cross-Endpoint Association Should a Joint Analysis Represent?

A joint analysis needs an explicit target for dependence. For binary endpoints, possible targets include the joint probability of both responses, the conditional response probability across biomarker groups, or an association measure such as an odds ratio. These quantities answer related but nonidentical questions. A joint probability is directly interpretable as the chance of a specified outcome combination; an association measure compares the endpoint distributions across groups or relative to independence.

For two binary outcomes, independence under a specified population and condition means \(P(B=1,C=1)=P(B=1)P(C=1)\). A convenient signed measure of departure from independence is the covariance. Its value is determined by the joint probability and the two marginal probabilities:

For the hypothetical counts above, the product of the marginal probabilities is \(0.300\times0.350=0.105\). The observed joint probability is \(0.200\), so the covariance is \(0.200-0.105=0.095\). The positive value indicates that the two binary response indicators tend to occur together more often than would be expected under independence with those same marginals. It is not a causal effect, and its size depends on the outcome probabilities as well as their association.

The association target should also match the scientific interpretation. If the question concerns whether biomarker and clinical responses tend to co-occur across the target population, a marginal association may be relevant. If the aim is to describe the relationship at fixed baseline characteristics, a conditional association may be more appropriate. These need not agree: averaging across covariate groups can change the apparent association, even when each group has its own well-defined relationship.

Conditioning, Averaging, and the Target Population

A conditional quantity should not be averaged over covariates by taking an unweighted mean of conditional probabilities unless that weighting is the intended target. Standardizing to a target population means averaging the relevant stratum-specific quantities using the target population’s stratum proportions. For a joint estimand, the same principle applies to the joint probabilities, not merely to one endpoint at a time.

For example, suppose a target population is divided equally between low- and high-severity baseline groups. In a hypothetical paired-outcome distribution, the joint probability of both responses is 0.50 in the low-severity group and 0.15 in the high-severity group. The population-standardized joint probability is

The two strata also have different conditional clinical-response probabilities among biomarker responders. In the low-severity group, let \(P(B=1,C=1)=0.50\) and \(P(B=1)=0.60\), giving \(P(C=1\mid B=1)=0.50/0.60=0.833\). In the high-severity group, let those probabilities be 0.15 and 0.40, giving \(0.15/0.40=0.375\). The conditional probability for the target population is not the simple average \((0.833+0.375)/2\). The total joint probability is 0.325 and the total biomarker-response probability is \(0.50(0.60)+0.50(0.40)=0.50\), so the population conditional probability is \(0.325/0.50=0.650\). Participants with biomarker response contribute in different proportions from the two severity groups, which determines the correct weighting.

Diagnostics, Alternatives, and Limitations

Before interpreting any dependence summary, verify that the paired records correspond to the same participants and the intended time window. Tabulate the four outcome combinations and confirm that the cell counts add to the analyzed sample size. Compare the joint table’s row and column totals with the reported endpoint marginals. Inconsistencies can reveal mismatched records, different analysis populations, or an incorrect denominator.

Sparse cells are a warning that conditional probabilities and odds ratios may be unstable; a zero cell can make an unadjusted odds ratio zero, infinite, or undefined. Also examine whether missingness or exclusion differs by endpoint or treatment. A complete-case paired table describes participants with both endpoints observed; it does not automatically describe the full target population. The intended handling of missing outcomes must be part of the estimand and analysis plan.

A marginal analysis is often the clearest choice when the scientific question concerns one endpoint’s population-average response. Separate marginal summaries can describe each endpoint, but they do not themselves supply a joint probability or cross-endpoint association. A conditional summary is useful when the question explicitly concerns one outcome given another or given covariates. A joint distribution summary is necessary when the target concerns co-occurrence or dependence. These are alternatives in the question being answered, not interchangeable labels for the same result.

A limitation is that association summaries depend on the chosen scale and population. For example, an odds ratio and a covariance summarize different aspects of a binary joint distribution, and conditional and marginal associations can differ. Results can also be distorted by selection into the analyzed paired sample. Finally, statistical association between a biomarker and a clinical endpoint does not establish mediation, mechanism, or causal surrogacy.

Common Mistakes

• Calling a marginal probability a joint probability. The probability of clinical response alone is not the probability of both clinical and biomarker response.
• Leaving the conditioning event unstated. “Clinical response given biomarker response” and “clinical response given baseline severity” are different conditional quantities.
• Interpreting association causally. A higher clinical-response rate among biomarker responders does not show that biomarker response caused the clinical response.
• Averaging conditional probabilities with the wrong weights. The correct population average uses the target population’s composition; conditioning on an endpoint can change the subgroup weights.
• Assuming separate endpoint summaries define dependence. Marginal distributions alone do not determine the joint distribution; many different patterns of co-occurrence can have the same two marginals.