Sequentially Updating Response Estimates

From One Observed Response to the Next Assignment

A response-adaptive design changes its allocation as information accumulates. The operational unit is therefore not just an estimate or an allocation rule, but a sequence: record an assignment, observe its response, update the treatment-specific estimates, recalculate the target shares, and then calculate the probability for the next assignment. Keeping those steps in the correct order prevents information from a future response from influencing a probability that should already have been set.

This tutorial works with two treatment arms. The treatment-specific response estimator and the target map must be specified by the design; the update mechanics do not determine which response measure is scientifically appropriate. The examples use running means of numerical response scores and a simple normalized-estimate target solely to make each calculation explicit. They are hypothetical illustrations, not a recommendation to use that target map in a trial.

For two arms, the target shares and next-patient probabilities have different roles. The target share \(\rho_i(n)\) is the allocation proportion the design is currently tracking. The probability \(q_i(n+1)\) is the chance of the next assignment, adjusted using the current allocation counts. With tuning parameter \(\gamma\geq 0\), the two-arm DBCD rule is

The rule requires positive allocation counts and target shares so its ratios and weights are defined. Initialization addresses the early stage before estimates and counts are usable. At each later step, the calculation uses the current state; it does not assume the estimates or target shares remain fixed after a response.

The Update Order and the Information Set

The probability \(q_i(n+1)\) is calculated before the \((n+1)\)st assignment. Let \(A_{n+1}\) denote the arm assigned and \(Y_{n+1}\) its response once observed. The assignment updates the allocation count immediately: \(N_{A_{n+1}}(n+1)=N_{A_{n+1}}(n)+1\). The response estimator is updated when the response becomes available. Then the target map is applied to the updated estimates, and the DBCD formula gives the probability for assignment \(n+2\).

This order is a form of sequential bookkeeping. A probability must depend only on information available before its assignment. Thus, the response to assignment \(n+1\) cannot be used in \(q_i(n+1)\), but it can be used in \(q_i(n+2)\) if observed in time. If outcomes are delayed, the allocation count may already include a participant whose response has not yet updated the estimate. The design must state how such pending outcomes are handled; a response that has not been observed is not silently treated as a known response.

Updating a Running Mean

One common estimator update for a numerical response is the running sample mean. If arm \(i\) has \(m_i\) observed responses with mean \(\bar{x}_i\), and its next observed response is \(y\), the updated mean is

The first expression adds the new response to the previous total and divides by the new number of observed responses. The second shows the direction and size of the change: the estimate moves toward \(y\), with a smaller change when more responses have already contributed to the mean. This update is valid for a running mean of the observed numerical responses; other estimators may require different stored information and update equations.

A Second Sequential Update

The calculation repeats after every observed response. Counts change because of assignments; estimates change because of observed outcomes; target shares are recalculated from those estimates. These state changes need not move the target share and the next-patient probability in the same direction.

Pending Responses and Sequential Validity

When responses are not immediate, the design needs a precise rule for distinguishing participants assigned from responses observed. For example, suppose an arm has five observed responses with mean 6 points and a sixth participant has been assigned to that arm but has not yet responded. The count used to describe assignments is six; the running mean still summarizes five observed responses. When the sixth response arrives, the estimator can be updated using all six observed values.

A response-adaptive procedure should specify whether it waits for an outcome before making another assignment, permits multiple pending outcomes, or uses a prespecified estimator that accommodates incomplete follow-up. The relevant requirement for the sequential probability is that it be determined from the information actually available at that point. Updating an estimate with an unobserved value, or retrospectively changing a probability using a response that arrived later, breaks that ordering.

Why the Update Rule Is Coherent

The DBCD formula produces valid probabilities at each step when its inputs are valid. To see this, define \(w_i(n)=\rho_i(n)\{\rho_i(n)/N_i(n)\}^{\gamma}\). If \(N_i(n)>0\), \(\rho_i(n)>0\), and \(\gamma\geq0\), then \(w_i(n)>0\). Dividing each weight by the sum of the two positive weights gives positive probabilities whose sum is one:

The update sequence also preserves the intended timing: the current state determines the next randomization, and newly observed data can affect later randomizations. This is not a claim that the point estimates are unbiased, stable, or scientifically adequate. Those properties depend on the estimator, the response process, and the target map. The mechanics ensure only that the specified estimator and target are carried forward in the correct sequence.

Diagnostics, Alternatives, and Limitations

A practical implementation should retain an audit trail for each assignment: the counts, observed-response information, estimates, target shares, tuning parameter, computed probabilities, randomization, and later outcome update. Recomputing a few steps independently is a useful check. At every step, verify that the target shares sum to one, the assignment probabilities sum to one, and the probability used was calculated before the corresponding assignment.

• Using a target share as the randomization probability. The target is not generally the next-patient probability. The count correction in the DBCD rule changes the probability when current allocations differ from the target.
• Updating an estimate at assignment rather than observation. An assignment supplies no response value. Keep the estimate unchanged until the response is observed or use a clearly specified method for pending data.
• Mixing counts and estimator sample sizes. Allocation counts and numbers of observed responses may differ with delayed outcomes. Use each quantity for its intended role.
• Changing the target map midstream without specification. The response estimates alone do not define target shares. A map must be prespecified or any adaptation to it must itself be explicitly defined.

A fixed allocation probability does not respond to the estimates. Directly randomizing according to the target shares also differs from DBCD because it omits the correction for current allocation imbalance. Batch updating can be appropriate in designs that deliberately randomize groups of participants, but it is not the same sequence as recalculating after each available response. Each alternative changes the design’s allocation behavior and should not be substituted silently.

Finally, an online update is not a guarantee of good performance. Early estimates can be unstable; an extreme estimate can shift a target sharply if the target map is sensitive; and delayed or missing responses can make the current estimates less current than the assignment counts. The method described here specifies how to propagate estimates through a DBCD rule. It does not by itself solve estimator selection, uncertainty quantification, missing outcomes, or the scientific choice of target.