Dependence Between Efficacy And Toxicity

Why the Two Kinds of Association Matter

In a dose-finding trial, efficacy and toxicity are both recorded for each participant, while their marginal rates are also summarized across dose levels. These summaries answer different questions. At a fixed dose, patient-level association asks whether participants with efficacy tend to be more or less likely to experience toxicity. Dose-level correlation asks whether doses with higher efficacy rates also tend to have higher toxicity rates. One does not determine the other.

The distinction matters whenever a decision depends on a joint event, such as achieving efficacy without toxicity. Marginal efficacy and toxicity probabilities alone do not determine that event's probability. Multiplying the two marginals supplies an answer only under an independence assumption; if the outcomes are dependent, the product can overstate or understate clinically relevant probabilities.

The binary outcomes must be defined consistently: for example, efficacy by a specified visit and dose-limiting toxicity within a specified window. The association is descriptive, not evidence that efficacy causes toxicity or that toxicity causes efficacy. It may reflect biological mechanisms, patient characteristics, or other shared factors.

Joint Probabilities at One Dose

At a particular dose, the four possible outcome probabilities can be written in terms of the two marginals and their joint probability. The first entry in each pair refers to efficacy and the second to toxicity.

Every cell must be nonnegative. In particular, \(q(d)\) cannot exceed either marginal, and the probability of neither outcome must also be nonnegative. These requirements give the Fréchet bounds:

This bound is useful even when the association is not precisely estimated: it shows how much the joint probability could vary while the marginals remain fixed. Under independence at dose \(d\), and only under that assumption, \(q(d)=p_E(d)p_T(d)\). The probability of efficacy without the specified toxicity is \(p_E(d)-q(d)\), so it too depends on the association.

Measuring Association and Checking the Assumptions

For binary outcomes with nonzero marginal variances, a standard measure is the phi correlation. It is the covariance of the two binary indicators divided by the product of their standard deviations. Because \(E\) and \(T\) are zero-one indicators, their covariance is \(q(d)-p_E(d)p_T(d)\).

A positive value means that efficacy and toxicity co-occur more often than the independence calculation predicts; a negative value means they co-occur less often. A value of zero corresponds to independence for two non-degenerate binary variables. If either outcome has probability zero or one, the denominator is zero and phi is undefined, even though the joint probability can still be described.

For direct clinical interpretation, conditional probabilities can be more informative than a correlation coefficient. For example, compare \(P(T=1\mid E=1,d)=q(d)/p_E(d)\) with \(P(T=1\mid E=0,d)=\{p_T(d)-q(d)\}/\{1-p_E(d)\}\), when their denominators are positive. These compare toxicity rates between outcome groups at the same dose; they are not causal effects.

Simple estimates from outcome counts require clear outcome definitions, consistent follow-up, and complete or appropriately handled ascertainment. Within a dose, the frequency in each of the four cells estimates its corresponding probability when the participants provide an appropriate sample for the target population. Sparse cells make the association estimate unstable. If patient characteristics differ substantially, a marginal within-dose association can also conceal different associations in subgroups. Missing efficacy or toxicity outcomes cannot simply be classified as absent: excluding such participants may bias the joint probabilities if missingness is related to either outcome.

Why Dose-Level Correlation Is Not Patient-Level Association

A dose-level correlation is calculated from a set of pairs \(\{p_E(d),p_T(d)\}\), one pair per dose. It describes how two dose-specific rates move together as dose changes. Patient-level association instead uses paired outcomes from participants within a dose. Correlating dose-level rates does not supply the within-dose joint probability \(q(d)\): many different joint tables can have the same two marginals.

A Short Justification for the Independence Calculation

The product rule gives \(P(E=1,T=1\mid d)=P(E=1\mid d)P(T=1\mid E=1,d)\). Independence at dose \(d\) means that \(P(T=1\mid E=1,d)=P(T=1\mid d)\) whenever the conditioning event has positive probability. Substitution yields the product of the two marginals. If this conditional probability differs from the marginal toxicity probability, the product is not the joint probability.

Equivalently, the joint probability can be written as the independence product plus a covariance term:

The bracketed term is the covariance of the binary outcome indicators. Ignoring dependence discards precisely this term. The error in a calculated joint probability is not guaranteed to have one direction: positive association makes the product too small for the “both” event, while negative association makes it too large.

Practical Checks, Alternatives, and Limitations

Before using a joint probability in dose selection or utility calculations, check that the two outcomes refer to the same participants, dose, and clinically relevant observation period. Then verify that the four cell counts reconcile with the reported marginals. The estimated \(q(d)\) must satisfy the Fréchet bounds; a value outside them signals inconsistent definitions, denominators, or calculations. At small sample sizes, report the counts as well as rates, since a large apparent association may be driven by only a few participants.

The main alternative to assuming independence is to estimate the joint outcome distribution directly from paired outcomes at each dose. This preserves the observed pairing, but can be imprecise when dose-specific samples are small or some cells are rare. A product of marginals is simpler and uses fewer quantities, but it imposes independence and may distort probabilities of clinically important combinations. A third practical approach is to report the marginals together with feasible bounds for \(q(d)\) when the joint association is poorly estimated; those bounds are informative but may be wide.

Pooling participants across doses to estimate a single association is generally not a substitute for dose-specific assessment. Dose changes both marginal risks, and the mix of participants at each dose can affect pooled summaries. Likewise, a correlation computed from a small number of dose-level rates is a summary of those dose levels, not an estimate of patient-level association. Neither quantity alone establishes the joint distribution needed for a particular dose-level decision.

These calculations describe binary outcomes at a chosen assessment horizon. They do not by themselves handle event timing, ordinal severity, repeated outcomes, or covariate adjustment. They also do not establish a causal mechanism. When the patient-level dependence is uncertain, state the assumption being used and examine how conclusions change across plausible joint probabilities rather than presenting an independence product as an observed fact.