Positivity And Extreme Adaptive Assignment Probabilities

Why Assignment Probabilities Matter

In response-adaptive randomization, the chance of assigning a participant to treatment can depend on information accumulated earlier in the trial. This can direct more participants toward an arm that appears promising. But if an arm's assignment probability becomes very small, an inverse-probability-weighted analysis gives the rare assignments very large weights. The resulting statistic may be highly variable even when the assignment mechanism is correctly implemented. If the probability reaches zero, the issue is more fundamental: the design no longer provides randomization support for that treatment at that point.

Consider a finite set of \(n\) participants with fixed potential outcomes \(Y_i(1)\) and \(Y_i(0)\), the outcomes participant \(i\) would have under treatment and control. Assume consistency, so the observed outcome is \(Y_i=A_iY_i(1)+(1-A_i)Y_i(0)\), and no interference, so one participant's assignment does not change another participant's potential outcomes. Let \(H_{i-1}\) denote the assignment and outcome history available before assigning participant \(i\). The design uses the known, history-dependent probability \(e_i=P(A_i=1\mid H_{i-1})\). The treatment assignment is generated according to this probability conditional on that history; the potential outcomes are treated as fixed for randomization-based inference.

The relevant probabilities are the probabilities actually specified by the design, conditional on the information available when each assignment is made. Marginal inclusion probabilities can be useful for other design-based calculations, but a favorable marginal probability does not remove the instability caused by a very small conditional probability in a realized history.

What Extreme Probabilities Do to an IPW Score

For participant \(i\), the inverse-probability-weighted (IPW) score for the individual treatment effect is \(X_i=A_iY_i/e_i-(1-A_i)Y_i/(1-e_i)\). The observed outcome is weighted by the reciprocal of the probability of the treatment actually assigned. If \(e_i\) is known and strictly between zero and one, then, conditional on \(H_{i-1}\),

Thus the score is conditionally unbiased for that participant's treatment effect under the stated randomization assumptions. A small \(e_i\), however, means that a treated participant's outcome is multiplied by \(1/e_i\). Correct weighting does not guarantee stable estimation.

Here is the variance calculation in general. Conditional on the history, \(X_i\) equals \(Y_i(1)/e_i\) with probability \(e_i\), and equals \(-Y_i(0)/(1-e_i)\) with probability \(1-e_i\). Therefore

The cross term \(2Y_i(1)Y_i(0)\) is part of the variance formula. It results from subtracting the square of the mean, \(\{Y_i(1)-Y_i(0)\}^2\), from the second moment. For example, if \(e_i=0.5\) and both potential outcomes equal 1 unit, the score is 2 or \(-2\), each with probability one half. Its mean is zero and its variance is 4 squared units; omitting the cross term would incorrectly give 2.

How a Positivity Bound Limits Variance

Suppose the potential outcomes are bounded in magnitude, \(|Y_i(a)|\leq M\) for \(a\in\{0,1\}\), and the design has a uniform positivity bound \(\epsilon\leq e_i\leq 1-\epsilon\), with \(0<\epsilon\leq 0.5\). Since variance cannot exceed the second moment,

The last inequality holds because \(e_i(1-e_i)\geq\epsilon(1-\epsilon)\) throughout the interval \([\epsilon,1-\epsilon]\). This is a valid, generally conservative bound. It makes explicit that a uniform probability floor and bounded outcomes together prevent the conditional score variance from becoming arbitrarily large. Positivity by itself supplies no such uniform guarantee.

For the average of the IPW scores, \(\hat{\tau}=n^{-1}\sum_{i=1}^n X_i\), the centered scores \(X_i-\{Y_i(1)-Y_i(0)\}\) have conditional mean zero given \(H_{i-1}\). They are martingale differences under this sequential assignment setup, so different centered terms have zero covariance. Consequently, the randomization variance of \(\hat{\tau}\) is \(n^{-2}\sum_i E\{\operatorname{Var}(X_i\mid H_{i-1})\}\), when the finite-population effects are fixed and the assignment mechanism follows the stated conditional probabilities. A near-zero probability can make one or more terms dominate this sum. An observed estimate of variance can also be unstable when only a few assignments receive very large weights.

Diagnosing Instability and Comparing Responses

A practical review should examine the full sequence of assignment probabilities, not just the final allocation totals. Useful diagnostics include the minimum of \(e_i\) and \(1-e_i\), how often probabilities approach the design's allowed limits, the largest reciprocal weights, and whether a small number of observations dominate the weighted estimate. Review these summaries across prespecified subgroups or histories as well: an acceptable overall allocation can conceal almost deterministic assignment in a clinically important subgroup.

These checks identify potential instability; they do not establish that an estimator is biased or that a particular result is invalid. A very large weight may occur rarely, and the actual uncertainty depends on the outcomes and the assignment process. Conversely, a modest average weight does not rule out a near-zero probability for a relevant subgroup. The design probabilities should be recorded accurately so that diagnostics and randomization-based calculations use the probabilities the algorithm actually assigned.

A floor is a design choice: it protects future assignments if the rule is specified and implemented throughout the trial. Truncation is an analysis choice: if a weight \(1/e_i\) is replaced by a smaller cap, its contribution no longer has the same conditional expectation in general. For example, capping the treated weight at \(c<1/e_i\) changes its conditional expected contribution from \(Y_i(1)\) to \(e_i cY_i(1)\), which differs from \(Y_i(1)\) unless special conditions hold. Weight stabilization or model-based adjustment may be useful in particular analyses, but neither should be presented as a universal cure for a lack of design support.

When Support Is Lost

If \(e_i=0\), treatment is never assigned at that history, so \(Y_i(1)/e_i\) is undefined and the data provide no randomized observation of that treatment condition there. If \(e_i=1\), the analogous problem applies to control. This is not merely high variance: the design has removed one treatment option from its support. In a subgroup where the design always assigns treatment, for instance, the observed outcomes alone cannot supply a randomization-based comparison with control for that subgroup. A model could extrapolate a control outcome, but that conclusion would rely on additional assumptions.

A probability that is positive but extremely small differs logically from a probability of zero. With positive probabilities, the relevant assignment remains in the design's support, although estimates may be so variable that useful precision is unattainable in a finite trial. With zero probability, inverse weighting is not defined and the absent treatment condition cannot be recovered by taking a limit or by trimming other weights. Inference must be restricted to contrasts supported by the design, or rely explicitly on additional assumptions.

The same distinction matters for randomization tests. The reference distribution is generated from assignments allowed by the design. If a treatment cannot occur in a relevant history, the reference distribution cannot create that missing assignment. A test may still address a null hypothesis for the assignments that are possible, but it cannot manufacture evidence about an unsupported treatment comparison.

Common Mistake and Practical Conclusion

Common mistake: “All assignment probabilities are nonzero, so the IPW estimate is safe; if a few weights are huge, I can cap them without changing the inference.” Strict positivity ensures only that reciprocals exist. It does not bound them or guarantee tolerable variance. Capping a weight generally changes the estimator's expectation, while a zero probability leaves a treatment condition unsupported altogether.

A sound assessment reports the smallest relevant assignment probabilities, the corresponding largest weights, and the implications for precision and support. When the goal is stable randomization-based estimation, a prespecified probability floor can be preferable to relying on post hoc weight corrections. If support has already been lost, state clearly which contrasts remain design-supported and which require additional modeling assumptions.