Joint Inclusion Probabilities For Adaptive Assignments

Why Pairwise Assignment Probabilities Matter

In a sequential trial, a participant’s probability of receiving treatment may depend on the assignments and responses observed earlier. Marginal inclusion probabilities summarize each participant’s chance of treatment, but they do not say whether two participants tend to receive treatment together. That dependence is part of the assignment design: ignoring it can misrepresent the uncertainty of estimates based on the randomization.

The key calculation is to trace the assignment histories that can lead to both participants receiving treatment. The calculation must respect the order of assignment and the probability used at each step. When the rule adapts to observed responses, those responses belong in the history too.

The joint probability is not generally the product of the marginal probabilities. That product would equal \(\pi_{ij}\) if the two assignment indicators were independent, but sequential adaptation can make them dependent. For example, an early treatment assignment may change the allocation probability for later participants.

Calculating Probabilities From Assignment Histories

Suppose there are \(N\) participants assigned in a fixed order. Let \(H_{t-1}\) denote the information available to the assignment algorithm just before participant \(t\) is assigned. It can include earlier assignments, observed responses, and other information used by the prespecified rule. Write \(q_t(H_{t-1})=P(Z_t=1\mid H_{t-1})\). For a binary assignment, the conditional probability of control is \(1-q_t(H_{t-1})\).

An assignment path \(z=(z_1,\ldots,z_N)\) has a probability found by multiplying the appropriate conditional probabilities along that path. If responses enter the rule, the history and response observed on each path must be consistent with the assignments on that path. For randomization-based calculations with fixed potential outcomes, this consistency is determined by the assignment and the potential outcomes; the algorithm is then applied along each possible path.

To obtain \(\pi_{ij}\), sum the probabilities of all complete paths on which \(z_i=z_j=1\). Equivalently, sum over histories up to the later of the two assignments and include the conditional probability of the later treatment assignment. If the participants are not consecutive, paths for the intervening assignments must be included or integrated out. Multiplying two marginal probabilities skips this shared-history calculation.

Worked Calculations

Connection to Design-Based Uncertainty

The role of a joint inclusion probability can be seen in a Horvitz–Thompson estimator. Suppose \(Y_i(1)\) is a fixed potential outcome under treatment for participant \(i\), and the target is the finite-population treatment total \(\sum_i Y_i(1)\). With positive treatment inclusion probabilities, the estimator is \(\widehat{T}_1=\sum_i Z_iY_i(1)/\pi_i\). Its randomization covariance between the contributions from participants \(i\) and \(j\) depends on \(\pi_{ij}\):

This identity follows because \(E(Z_iZ_j)=\pi_{ij}\) and \(E(Z_i)E(Z_j)=\pi_i\pi_j\). Therefore \(\operatorname{Cov}(Z_i,Z_j)=\pi_{ij}-\pi_i\pi_j\); scaling each indicator by its inclusion probability and its fixed potential outcome gives the displayed expression. The estimator’s overall design variance includes such pairwise covariance contributions as well as the individual contributions. The next calculation shows why using only the marginals can discard a nonzero covariance.

Checks, Alternatives, and Limitations

Before using calculated joint probabilities, check that they describe a coherent assignment design. For distinct participants, probabilities must satisfy \[ \max(0,\pi_i+\pi_j-1)\leq \pi_{ij}\leq \min(\pi_i,\pi_j), \] because the probability of both events cannot exceed either event, and the probability of their intersection cannot be smaller than \(P(Z_i=1)+P(Z_j=1)-1\). Also check symmetry, \(\pi_{ij}=\pi_{ji}\), and the diagonal identity \(\pi_{ii}=\pi_i\). These checks can reveal arithmetic errors, but passing them alone does not prove that all probabilities match the assignment algorithm.

For a small design, explicit enumeration is often clearest. For a large design, exact enumeration may be impractical; Monte Carlo simulation from the fully specified algorithm can approximate joint probabilities by counting the simulated paths on which both participants are treated. The estimate then has simulation error, so report simulation precision and use enough draws for the required accuracy. In either approach, the probabilities must be generated under the actual sequential rule, not a simplified rule that ignores adaptation.

A common mistake is to calculate \(\pi_{ij}\) as \(\pi_i\pi_j\) because the two participants have known marginal probabilities. This is correct only when their assignment indicators are independent under the design. In an adaptive design, they can share earlier assignments and responses, so the correction is to condition on histories, multiply the conditional probabilities along each compatible path, and sum those path probabilities.

Joint inclusion probabilities do not solve every inferential problem. They rely on a correctly specified assignment mechanism and, when response-dependent rules are involved, correct reconstruction of the histories that feed that rule. If some pair has \(\pi_{ij}=0\), that pair can never both be treated, which can limit variance estimation for estimators requiring that pair’s joint inclusion probability. Very small probabilities can also make inverse-probability contributions unstable. These are properties of the design, not defects repaired by multiplying marginal probabilities.