Inverse-Probability Weighting With Adaptive Assignments

Why Weight Outcomes in an Adaptive Trial?

In a response-adaptive trial, the probability of assigning a treatment can change as earlier participants’ outcomes accumulate. Participants assigned to an arm are therefore not, in general, a simple random sample of the trial population. The unweighted mean of observed outcomes in an arm may not represent that arm’s mean outcome for the full population.

Inverse-probability weighting addresses this selection mechanism using the assignment probabilities specified by the randomization policy. An outcome observed under a treatment assigned with low probability receives more weight than an outcome observed under that treatment assigned with high probability. The result targets a fixed-population treatment mean, rather than the mean among only those who happened to receive the treatment.

Definition: Consider a fixed population of $N$ participants and a treatment $a$. Let $Y_i(a)$ be the potential outcome participant $i$ would have under treatment $a$. The fixed-population treatment mean is $$ \bar{Y}(a)=\frac{1}{N}\sum_{i=1}^{N}Y_i(a). $$ If participant $i$ is assigned treatment $A_i$, the observed outcome is $Y_i=Y_i(A_i)$. Let $H_i$ denote the information available just before assigning participant $i$, and let $\pi_{ia}=P(A_i=a\mid H_i)$ be the treatment’s assignment probability at that point. The inverse-probability weighted estimator is $$ \widehat{\bar{Y}}_{\mathrm{IPW}}(a)=\frac{1}{N}\sum_{i=1}^{N}\frac{\mathbb{1}(A_i=a)Y_i}{\pi_{ia}}, $$ where $\mathbb{1}(A_i=a)$ equals 1 if participant $i$ receives $a$, and 0 otherwise.

The denominator is the fixed population size $N$, not the number of participants assigned to treatment $a$. A participant not assigned to $a$ contributes zero. A participant assigned to $a$ contributes their observed outcome divided by the probability of that assignment. The assignment probabilities must be the probabilities in force immediately before each assignment, including their dependence on prior trial history.

Assumptions and the Unbiasedness Argument

The design-based result applies under a clearly defined target and assignment mechanism. The population and its potential outcomes are treated as fixed; randomness comes from the trial’s assignment process. Each participant receives one treatment, the observed outcome equals the potential outcome for the treatment received, and one participant’s treatment does not change another participant’s potential outcome. The randomization policy must assign treatment $a$ with the stated probability $\pi_{ia}$ given $H_i$, and that probability must be known and strictly positive for every participant in the target population.

In particular, $\pi_{ia}>0$ is a positivity condition. If the design makes treatment $a$ impossible for a participant, no observed outcome can directly represent that participant’s potential outcome under $a$. The weighting identity cannot recover that missing contribution. The design’s randomization, conditional on the information available before assignment, is what justifies using these probabilities; the same conclusion does not follow merely because probabilities can be calculated after seeing the data.

To see the key step, condition on the pre-assignment history $H_i$. The potential outcome $Y_i(a)$ is fixed under the stated finite-population framework, and assignment to $a$ has conditional probability $\pi_{ia}$. If participant $i$ is assigned to $a$, consistency gives $Y_i=Y_i(a)$. Thus:

E\left[\left.\frac{\mathbb{1}(A_i=a)Y_i}{\pi_{ia}}\right|H_i\right] = \frac{Y_i(a)}{\pi_{ia}}P(A_i=a\mid H_i) = \frac{Y_i(a)}{\pi_{ia}}\pi_{ia} = Y_i(a).

Taking expectations over the possible histories gives an expected weighted contribution of $Y_i(a)$ for each participant. Summing those contributions and dividing by $N$ yields:

E\left[\widehat{\bar{Y}}_{\mathrm{IPW}}(a)\right] = \frac{1}{N}\sum_{i=1}^{N}E\left[\frac{\mathbb{1}(A_i=a)Y_i}{\pi_{ia}}\right] = \frac{1}{N}\sum_{i=1}^{N}Y_i(a) = \bar{Y}(a).

This is an expectation over repetitions of the randomization procedure, not a claim that one realized weighted estimate must equal the fixed-population mean. The policy may adapt: $\pi_{ia}$ can depend on earlier assignments and outcomes. The identity still holds because the current assignment indicator has conditional expectation $\pi_{ia}$ given the history used to set that probability.

Worked Example: Calculating a Weighted Mean From One Realization

Suppose a hypothetical trial has $N=4$ participants. The recorded probabilities of assignment to treatment A and the observed assignments and outcomes are below. Outcomes are measured in points, with larger values indicating more benefit.

Participant	Probability of A, $\pi_{iA}$	Observed assignment	Observed outcome
1	0.50	A	8 points
2	0.40	B	6 points
3	0.25	A	12 points
4	0.20	B	5 points

Only participants 1 and 3 contribute to the estimate for A. Substituting the observed data and the known assignment probabilities gives:

\widehat{\bar{Y}}_{\mathrm{IPW}}(A) = \frac{1}{4}\left(\frac{1\times 8}{0.50} +\frac{0\times 6}{0.40} +\frac{1\times 12}{0.25} +\frac{0\times 5}{0.20}\right) = \frac{16+0+48+0}{4} = 16\text{ points}.

The weighted estimate is 16 points for this realization. The outcome of 12 points receives a larger contribution because participant 3 had only a 0.25 probability of receiving A. This single estimate need not equal the unknown mean of the four A potential outcomes; the unbiasedness result concerns its expectation over repeated assignments under the policy.

Adaptation Does Not Break the Identity

A concern is that assignment probabilities are themselves affected by earlier outcomes. The conditional calculation resolves it: when participant $i$ is assigned, the probability used in the weight is the probability conditional on the history available at that moment. It need not be the same for every participant, nor independent of earlier outcomes. It must, however, be the actual probability that governed the current randomization.

Worked Example: Checking the Expected Contribution Under Adaptation

Consider a hypothetical two-participant population. Their potential outcomes under treatment A are fixed at 4 and 10 points, respectively. Participant 1 has probability 0.50 of receiving A. The policy assigns participant 2 to A with probability 0.25 if participant 1 received A, and with probability 0.75 if participant 1 received B. The policy may use the first assignment as its history; any outcomes it uses would also have to be observed before the second assignment.

For participant 1, the weighted contribution to the sum for A is 8 points if assigned A, because $4/0.50=8$, and zero otherwise. Its expectation is:

0.50(8)+0.50(0)=4\text{ points}.

For participant 2, calculate the expectation conditional on each possible first assignment. If participant 1 received A, participant 2’s probability of A is 0.25. The expected weighted contribution in that branch is:

0.25\left(\frac{10}{0.25}\right)+0.75(0)=10\text{ points}.

If participant 1 received B, participant 2’s probability of A is 0.75, giving:

0.75\left(\frac{10}{0.75}\right)+0.25(0)=10\text{ points}.

Both histories produce the same conditional expectation, so participant 2’s unconditional expected contribution is also 10 points. Dividing the expected total by $N=2$ gives $(4+10)/2=7$ points, exactly the fixed-population mean under A. The example uses different probabilities across histories; it is the correct conditional probability in each history that makes the calculation work.

Positivity, Weight Size, and Diagnostics

A positive probability is not necessarily a practically adequate probability. When $\pi_{ia}$ is very small, an assignment to $a$ produces a large weight $1/\pi_{ia}$. For example, an outcome of 9 points observed with probability 0.10 contributes $9/0.10=90$ points to the weighted sum, whereas the same outcome observed with probability 0.90 contributes 10 points. The larger contribution is required by the expectation identity, but it also means that a small number of assignments can have substantial influence on a realized estimate.

Before calculating or interpreting an estimate, check that the recorded probabilities were available before each assignment, match the assigned treatment, and are strictly positive for the target arm. Also verify the target population size, the treatment and outcome coding, and that each participant contributes only the outcome observed under the assigned treatment. If probabilities were rounded, reconstructed incorrectly, or replaced by probabilities from a different policy, the algebra may describe a different estimator from the one actually computed.

Worked Example: A Very Small Probability and a Zero Probability

Suppose a hypothetical trial targets $N=3$ participants and treatment A. The assignment probabilities for A are 0.10, 0.50, and 0.00. The observed assignments and outcomes are: participant 1 receives A and has outcome 9 points; participant 2 receives B and has outcome 4 points; participant 3 receives B and has outcome 7 points.

For participants 1 and 2, the weighted contributions to A are $9/0.10=90$ points and zero. Participant 3 cannot contribute a weighted A outcome: the assignment probability for A is zero, so division by that probability is undefined. More fundamentally, the design provides no chance to observe this participant under A. The required positivity assumption fails, and the stated estimator cannot be used to recover the A mean for all three participants.

If the third probability had instead been 0.05 and participant 3 had received B, the observed-data estimate would be $90/3=30$ points. That value could be far from the true fixed-population mean in this realization; the small probability makes a large contribution possible on the occasions when A is assigned. Strict positivity permits the expectation identity, but does not guarantee a stable estimate from one trial.

How This Differs From Other Summaries

The ordinary adaptive arm mean divides the sum of outcomes observed under A by the number assigned to A. It describes the participants who received A in the realized trial. Because the allocation process can depend on earlier outcomes, those participants need not represent the full fixed population. This is the selection issue behind bias in adaptive arm means.

The IPW estimator instead uses every participant in the fixed-population denominator and corrects each observed A outcome using that participant’s known chance of receiving A. A design with the same assignment probability for every participant is a simpler special case of the same weighting principle. But weighting does not make the realized estimate identical to the target, and it does not by itself supply a valid standard error, confidence interval, or test. Those require additional inferential work.

Outcome-model approaches offer a different route: they predict missing treatment outcomes using a model, rather than correcting observed outcomes through assignment probabilities. Such predictions require their own modeling assumptions. For the design-based IPW identity derived here, the essential ingredients are the known randomization probabilities, positivity, and the potential-outcome setup—not a correct outcome regression model.

Common Mistakes and Limits

Using the final overall allocation fraction as every participant’s probability. In an adaptive trial, probabilities can change. Use $\pi_{ia}$ in force just before participant $i$ was assigned.
Dividing by the number assigned to the arm. The fixed-population estimator divides by $N$. Replacing that denominator changes the estimator and its target.
Treating a zero probability as an extreme but usable weight. A zero probability makes the weight undefined and violates positivity; it cannot be repaired by arithmetic.
Calling one weighted estimate unbiased because it is close to the target. Unbiasedness is a repeated-randomization expectation statement, not a property that can be verified from one realization.
Assuming unbiasedness guarantees precision. Small assignment probabilities can yield highly influential weights. Unbiasedness does not prevent substantial variation in realized estimates.

The target also matters. The estimator above targets the mean of the potential outcomes for the specified fixed population. It does not automatically target a population beyond the trial participants, and the no-interference and consistency assumptions may be inappropriate if treatment versions vary or participants’ outcomes affect one another. The weighting identity is exact for its stated target and assumptions; it is not a general cure for every source of bias.

Key takeaway: Under sequential randomization with known, positive assignment probabilities, each participant’s inverse-probability weighted contribution has expectation equal to that participant’s potential outcome under the treatment. Averaging these contributions over the fixed population therefore recovers its treatment mean in expectation, even when allocation adapts to earlier history.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A hypothetical study has $N=2$. For treatment A, participant 1 is assigned A with probability 0.50 and has observed outcome 6 points; participant 2 is assigned B with probability of A equal to 0.25 and has observed outcome 5 points. Calculate the IPW estimate of the fixed-population mean under A.
In one sentence, explain why the IPW expectation argument still works when the probability for the current participant depends on earlier trial outcomes.
What fails if one participant in the target population has probability zero of receiving treatment A?
Does an IPW estimate from one trial have to equal the fixed-population treatment mean? Explain what “unbiased” means here.

Solutions. (1) Participant 1 contributes $6/0.50=12$ points and participant 2 contributes zero, so $\widehat{\bar{Y}}_{\mathrm{IPW}}(A)=(12+0)/2=6$ points. (2) Conditional on the history used to set the probability, the assignment indicator has expectation equal to that probability, which cancels its inverse in the weighted contribution. (3) The weight is undefined, and the design never observes that participant under A; positivity fails for the target population mean. (4) No. Unbiasedness means that the estimator’s expectation over repeated assignments under the specified policy equals the fixed-population mean; an individual realization can differ from it.