From an Outcome Model to a Sequential Design
A covariate-adjusted response-adaptive design uses observed outcomes, as well as baseline covariates, to guide later treatment assignments. The practical challenge is to make that idea precise: specify the outcome model before the trial, define how its predictions become allocation targets, and retain enough randomization that the algorithm does not deterministically assign participants based on early, noisy results.
This tutorial builds one possible policy for a two-arm trial with a binary success outcome. The policy gives greater allocation probability to the arm with the higher model-predicted success probability within a participant's covariate profile. The target formula is a design choice, not a universally optimal rule. Its purpose here is to make each step explicit and reproducible.
For a simple illustration, suppose \(X_i\) is a binary profile indicator: \(X_i=0\) is the lower-risk profile and \(X_i=1\) the higher-risk profile. A prespecified logistic regression is
where \(\operatorname{logit}(p)=\log\{p/(1-p)\}\). The interaction permits the treatment-control difference in predicted success to vary by profile. At each planned update, the fitted coefficients give profile-specific predicted probabilities \(\hat p_0(x)\) and \(\hat p_1(x)\) for control and treatment.
Specify the Target and the Bounds
For this worked policy, define the raw treatment target within profile \(x\) as the treatment share of the two fitted success probabilities:
When both predictions are positive, this target is above one half if the model predicts greater success under treatment, and below one half if it predicts greater success under control. This ratio is an explicit rule for favoring the arm with higher predicted success; it is not, by itself, a proof of optimality for power, precision, or patient welfare.
Before the trial, choose bounds \(q_{\min}\) and \(q_{\max}\), with \(0<q_{\min}<q_{\max}<1\). The actual treatment assignment probability is the raw target clipped to these bounds:
Thus, conditional on the available history and the next participant's profile, assignment is randomized with probability \(q(x)\) to treatment and \(1-q(x)\) to control. The bounds prevent the model from producing near-certain assignments and preserve a positive chance of either assignment for every profile. They should be selected and documented in advance, not altered in response to inconvenient interim results.
Worked Example: Fit Predictions and Calculate Profile Targets
Worked Example: Fit Predictions and Calculate Profile Targets
Suppose a hypothetical trial has completed an initial randomized stage. Its observed data at the first planned update are:
| Profile | Arm | Successes | Participants | Observed success proportion |
|---|---|---|---|---|
| Lower-risk | Control | 2 | 10 | 0.200 |
| Lower-risk | Treatment | 4 | 10 | 0.400 |
| Higher-risk | Control | 5 | 10 | 0.500 |
| Higher-risk | Treatment | 7 | 10 | 0.700 |
For this illustration, the specified model is saturated across the four arm-by-profile cells. Each cell has both successes and failures, so its fitted success probability equals the observed proportion and the corresponding cell logit is finite. The control, lower-risk cell gives \(\hat\beta_0=\log(0.2/0.8)=-1.3863\). The control, higher-risk cell has logit \(0\), so \(\hat\beta_X=0-(-1.3863)=1.3863\). For treatment in the lower-risk profile, the logit is \(\log(0.4/0.6)=-0.4055\), giving \(\hat\beta_A=-0.4055-(-1.3863)=0.9808\). The treatment, higher-risk cell has logit \(\log(0.7/0.3)=0.8473\), so \(\hat\beta_{AX}=0.8473-(-1.3863)-1.3863-0.9808=-0.1335\), up to rounding.
The fitted probabilities are therefore \(\hat p_0(0)=0.200\), \(\hat p_1(0)=0.400\), \(\hat p_0(1)=0.500\), and \(\hat p_1(1)=0.700\). The raw treatment targets are
With prespecified bounds \(q_{\min}=0.25\) and \(q_{\max}=0.75\), neither target is changed. The design assigns a lower-risk participant to treatment with probability 0.6667 and a higher-risk participant with probability 0.5833. These are conditional assignment probabilities, not guarantees about the final treatment proportions.
Run the Algorithm One Participant at a Time
A usable implementation needs an initialization stage, a rule for updating the model, and an assignment step. The following sequence makes those dependencies clear.
Record the endpoint, covariates and their coding, logistic model, target formula, update schedule, probability bounds, and a rule for initialization or model-fitting failure.
Use a prespecified randomized stage to collect outcomes across arms and relevant profiles. Do not use a model that cannot yet be estimated reliably to make adaptive assignments.
At a planned look, fit the logistic model using eligible observed outcomes and the assignments and baseline covariates recorded for those participants.
For the incoming participant's profile, calculate \(\hat p_0(x)\) and \(\hat p_1(x)\), form \(q^*(x)\), and clip it to \([q_{\min},q_{\max}]\).
Generate the assignment using the bounded probability, then retain that probability, the model version, profile, and trial history needed to reproduce the decision.
If outcomes arrive with delay, the update must use only outcomes actually available at that time; otherwise the design implicitly uses information it did not have. The update schedule also matters. Updating after every outcome and updating in prespecified batches are distinct policies, even if they use the same model and target equation.
Worked Example: Assign the Next Lower-Risk Participant
Use the first-update estimates above and suppose the next participant has \(X=0\). Their raw target is \(0.6667\), within the bounds \([0.25,0.75]\), so the treatment assignment probability remains \(q(0)=0.6667\). If the random-number generator produces \(U=0.63\), and the implementation assigns treatment when \(U\le q(0)\), this participant is assigned to treatment because \(0.63\le0.6667\). If \(U=0.81\), the participant would instead be assigned to control because \(0.81>0.6667\). The model determines the probability, while the random draw determines this participant's assignment.
The assignment itself does not change the fitted model. If the participant's binary outcome later becomes available, it can enter the next scheduled update. In particular, the design must not use this participant's unobserved outcome to choose their own assignment.
Worked Example: Update Targets After More Outcomes
Worked Example: Update Targets After More Outcomes
Suppose, hypothetically, that by the next planned update each of the four cells has received five additional participants. In the lower-risk profile, control has 1 success among 5 new participants and treatment has 3 among 5. In the higher-risk profile, control has 0 among 5 and treatment has 4 among 5. Adding these to the first-stage counts gives cell totals of 3/15, 7/15, 5/15, and 11/15 for lower-risk control, lower-risk treatment, higher-risk control, and higher-risk treatment, respectively. Each cell still has successes and failures, so the saturated logistic fit has finite estimates.
The updated predictions are \(3/15=0.2000\) and \(7/15=0.4667\) for lower-risk control and treatment, and \(5/15=0.3333\) and \(11/15=0.7333\) for higher-risk control and treatment. The new raw targets are
Both lie between 0.25 and 0.75, so both remain unchanged. The higher-risk target has risen from 0.5833 to 0.6875 because the newly observed cell proportions changed the fitted predictions. This does not establish that treatment truly benefits this profile more; it records how this particular model-based rule reacts to the observed data.
Worked Example: Why the Bounds Matter
Worked Example: Why the Bounds Matter
Suppose a later fit for a particular profile predicts \(\hat p_1(x)=0.90\) and \(\hat p_0(x)=0.10\). The raw target is
With bounds \([0.25,0.75]\), the actual assignment probability is \(q(x)=\min(0.75,\max(0.25,0.90))=0.75\). Treatment is favored, but control still has probability \(1-0.75=0.25\). Reporting 0.90 as the assignment probability would be incorrect: it is the unbounded target, not the probability used by the randomization procedure.
Diagnostics, Alternatives, and Limitations
The design depends on both a model and a randomization mechanism. Useful checks include confirming that fitted probabilities are finite and within \([0,1]\), that all assignments use probabilities inside the prespecified bounds, that the model converges at each update, and that the recorded probability matches the calculation from the recorded model version and profile. Review cell counts and prediction calibration by arm and profile; sparse data, separation, or persistent prediction errors can make adaptive targets unstable or misleading.
If a model fails to converge or a profile has insufficient information, the algorithm needs a prespecified fallback, such as retaining the previous valid model or using a fixed randomized probability until fitting is possible. Choosing a fallback after seeing which arm looks favorable can change the design in an outcome-dependent way. Likewise, unplanned covariate recoding or changing the interaction structure during the trial makes the allocation rule difficult to interpret and reproduce.
| Approach | How assignments are guided | Key distinction |
|---|---|---|
| Fixed randomization | Uses a prespecified probability, often one half, regardless of observed outcomes. | Does not adapt to fitted response probabilities. |
| Minimization | Uses covariate information and existing assignments to reduce imbalance on selected factors. | Targets covariate balance rather than predicted response. |
| Covariate-adjusted response adaptation | Uses a fitted outcome model to set profile-specific assignment probabilities. | Responds to observed outcomes through model predictions; bounded probabilities retain randomization. |
The policy does not guarantee exact covariate balance, nor does a favorable model prediction guarantee a better outcome for a future participant. Early estimates can be noisy, and the bounded randomization deliberately sacrifices some degree of response-based allocation to retain assignment to both arms. The chosen target is one defensible rule among alternatives; a design aimed at a different objective, such as allocation for a specified variance criterion, would require a different target formula.
Check Your Understanding
Answer each question, then compare with the solutions below.
- For a profile with \(\hat p_1=0.60\) and \(\hat p_0=0.40\), calculate the raw target and the bounded probability under bounds \([0.25,0.75]\).
- For a raw target of 0.10 and the same bounds, what are the actual treatment and control assignment probabilities?
- Why must the next participant's covariates be known before calculating their assignment probability, and why must their outcome not be used for that assignment?
- Name two diagnostics or safeguards that address unstable model-based targets.
Solutions. 1. \(q^*=0.60/(0.60+0.40)=0.6000\), which is already within the bounds, so \(q=0.6000\). 2. The raw target is clipped to \(q=0.25\); treatment probability is 0.25 and control probability is \(1-0.25=0.75\). 3. The covariates identify the participant's profile-specific predictions, while the outcome is not yet observed at assignment and cannot legitimately inform that participant's own assignment. 4. Examples include checking convergence and cell counts, checking calibration by profile and arm, and specifying in advance a fixed-randomization or previous-valid-fit fallback when estimation fails.