Updating Model-Based Allocation Probabilities

Why Update Allocation Probabilities Sequentially?

In a covariate-adjusted response-adaptive design, a participant’s baseline profile can inform the probability of assigning that participant to each treatment. As responses accumulate, the outcome model used to make those predictions can also change. Sequential updating means that, after each newly observed response, the model is refitted using the data then available, and the allocation probabilities for the next participant are recalculated.

This is a procedure, not a single universal allocation rule. A design must specify both the outcome model and how its predictions are converted into allocation probabilities. The procedure below uses a binary favorable response and one transparent conversion rule: allocate in proportion to the model-predicted probabilities of a favorable response. Other target rules are possible, but they should be chosen before using the accumulating outcomes to guide assignments.

A Model and an Allocation Rule

For a binary response, one possible working model is logistic regression:

Here \(\mu(z,x)=P(Y_i=1\mid Z_i=z,X_i=x)\), and \(I(\cdot)\) is an indicator. The interaction allows the treatment difference to vary with the covariate. For the data available after response \(t\), the fitted coefficients are denoted \(\hat{\beta}_t\), and the fitted response probability is \(\hat{\mu}_t(z,x)\). The model can instead use several covariates or a different prespecified structure; the choice must reflect the design and the available data.

For the illustrative proportional-response rule, a participant with profile \(x\) is assigned to A with probability

These probabilities are calculated using the model fitted through participant \(t\); they govern the next assignment, not assignments already made. This rule normalizes the two fitted response probabilities so they sum to one. It is a specified allocation policy, not a theorem that this is the uniquely optimal policy. In particular, it does not by itself guarantee covariate balance, adequate estimation, or a desired treatment comparison.

The procedure relies on important conditions. Baseline covariates must be measured before randomization, and the treatment assignment must follow the recorded probabilities. The outcome model should adequately represent the conditional response probabilities relevant to allocation. Responses should be observed and recorded consistently, and one participant’s assignment should not change another participant’s response in a way the model ignores. In a randomized trial, each treatment whose response is to be estimated must have a positive chance of assignment for relevant profiles. If the fitted model cannot be estimated or gives unreliable predictions, its allocation probabilities should not be treated as trustworthy merely because they can be calculated.

The Sequential Update

The operational cycle is simple, but the order matters: a response cannot be used to determine the assignment that produced it. It can affect assignments only after it has been observed and incorporated into the next model fit.

Worked Examples

Why the Update Has This Form

The calculation has two distinct parts: estimate conditional response probabilities, then apply an allocation policy. In the saturated example, the model for one treatment-profile cell is Bernoulli with response probability \(\mu\). If that cell has \(r\) favorable responses among \(n\) participants, its likelihood is proportional to \(\mu^r(1-\mu)^{n-r}\). For \(0<r<n\), maximizing its log likelihood gives

Thus the cell response proportions in the examples are the fitted probabilities under the stated model. The allocation step then divides each fitted probability by their sum. If both fitted probabilities are positive, this produces valid probabilities between 0 and 1 that sum to 1. A new response changes the likelihood, can change the fitted probabilities, and therefore can change the next allocation probabilities.

This justification does not establish that proportional allocation is best for every scientific aim. It establishes how the chosen rule operates once the model and policy are specified. A different objective requires a different prespecified conversion from fitted outcomes to allocation probabilities.

Diagnostics, Alternatives, and Limitations

A sequential procedure should be monitored as it runs. Useful checks include whether the model fit converges after each update; whether coefficients or fitted probabilities change sharply after a single response; whether important treatment-profile combinations have little or no data; and whether probabilities recorded by the system match the probabilities actually used for randomization. A probability calculation based on a failed fit, stale data, or a profile coded incorrectly is not a valid update.

In a saturated logistic model, a cell with all favorable responses or all unfavorable responses has no finite maximum-likelihood estimate for its log odds. A logistic model with more covariates can also encounter separation or unstable estimates. These are substantive operational problems: the model may fail to return the probabilities the rule requires, or its predictions may be highly sensitive to a small number of responses. The design should specify what to do if a fit fails, rather than improvising a replacement rule after seeing outcomes.

Sequential refitting differs from two common alternatives. Fixed allocation probabilities do not change as responses arrive, so they avoid feedback from the current outcome model but do not adapt to accumulating responses. Fitting a model once and using its probabilities for all subsequent participants allows adaptation only at the initial update; it does not incorporate later responses. Sequential refitting uses more of the accumulating information but also makes assignments depend on a series of estimated models. Randomized minimization, by contrast, uses covariate imbalance to guide assignment rather than refitting an outcome model after each response.

The main limitations follow from this dependence on a working model and a chosen policy. Early estimates can be noisy; misspecified response relationships can produce misleading fitted probabilities; sparse profiles can make some predictions unreliable; and an adaptive allocation probability does not guarantee balanced treatment groups. The rule described here can also assign very low probabilities when one fitted response probability is much smaller than the other. These limitations are reasons to prespecify the model, allocation mapping, recording procedures, and response to estimation failures. They are not repaired simply by recalculating the probabilities more often.