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Handling Covariates in the Design of Clinical Trials

William F. Rosenberger, Oleksandr Sverdlov

arXiv:1102.3773v1stat.ME

TL;DR

Clinical-trial design faces a split over whether known covariates should be incorporated into randomization to improve balance. The paper reviews covariate-adaptive and model-based approaches, shows that balance can conflict with efficiency and ethical allocation, and advocates randomized CARA procedures as a way to address those competing considerations.

  • Problem

    The paper addresses disagreement over how clinical-trial designs should handle important baseline covariates when balance, efficiency, and ethical allocation may diverge.

  • Method

    The authors review stratified, covariate-adaptive, model-based, and CARA randomization procedures, alongside theoretical discussion and simulation studies.

  • Results

    The paper demonstrates by example that balancing known covariates does not necessarily produce the most efficient or ethically attractive design, and vice versa.

  • Takeaways & Limitations

    CARA randomization is advocated as a randomized design class intended to address efficiency and ethical considerations while incorporating covariate information.

  • Takeaways & Limitations

    The literature reviewed contains very little theoretical work, with no theoretical justification for minimization methods and unresolved questions about their properties.

Abstract

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There has been a split in the statistics community about the need for taking covariates into account in the design phase of a clinical trial. There are many advocates of using stratification and covariate-adaptive randomization to promote balance on certain known covariates. However, balance does not always promote efficiency or ensure more patients are assigned to the better treatment. We describe these procedures, including model-based procedures, for incorporating covariates into the design of clinical trials, and give examples where balance, efficiency and ethical considerations may be in conflict. We advocate a new class of procedures, covariate-adjusted response-adaptive (CARA) randomization procedures that attempt to optimize both efficiency and ethical considerations, while maintaining randomization. We review all these procedures, present a few new simulation studies, and conclude with our philosophy.

1. INTRODUCTION

Clinical-trial design must address important covariates, but the field disagrees about whether and how to incorporate them during randomization. This paper examines trade-offs among balance, efficiency, and ethical allocation, advocating CARA randomization as a randomized approach to these competing goals.

  • Motivation: Design flaws, including inadequate power, incorrect planning assumptions, covariate imbalances, and patient heterogeneity, can undermine clinical-trial conclusions.A breast-cancer trial was difficult to interpret after imbalance in prognostic risk factors and limited prospective collection of survival factors.
  • Background: Traditional practice prestratifies a small number of important covariates, analyzes by strata, and relies on randomization for other covariates.This approach becomes difficult when many covariates are considered important.
  • Controversy: The statistics and clinical-trials communities lack consensus about covariate-adaptive procedures, despite their growing use and regulatory skepticism.The debate includes disagreement about minimization, dynamic allocation, and the theoretical basis for these procedures.
  • Paper scope: The paper compares balance, efficiency, and ethical considerations, showing that balancing known covariates need not produce the most efficient or ethically attractive design.The authors also state that balanced and efficient designs do not necessarily assign more patients to the better treatment.
  • Contribution: The authors review covariate-adaptive and model-based procedures and advocate covariate-adjusted response-adaptive randomization procedures that retain randomization while targeting efficiency and ethical considerations.The paper also reports simulation studies comparing CARA procedures.

2. COVARIATE-ADAPTIVE RANDOMIZATION

Covariate-adaptive randomization makes treatment assignment depend on observed patient covariates, with procedures designed to balance covariates across treatment groups. The section covers prestratification, marginal and urn-based methods, distance-based allocation, and the continuing debate over retaining randomization.

  • Definition: Covariate-adaptive randomization assigns the next patient using prior treatment assignments and observed covariates, often targeting balance across treatment groups.Its assignment probability can depend on the full history of assignments and covariates.
  • Prestratification: Prestratification uses separate restricted randomization within each combination of levels of a small set of known discrete covariates.For K covariates with k_i levels, this creates a separate procedure within the product of the numbers of levels.
  • Marginal procedures: Marginal procedures adjust assignment probabilities according to weighted imbalances in the observed covariate levels, with p = 1 yielding nonrandomized minimization.Pocock and Simon investigated p = 3/4 in their biased-coin formulation.
  • Urn procedures: Wei’s urn procedure selects the covariate-level urn with the greatest imbalance, assigns treatment through a drawn ball, and updates observed urns.Under a standard linear model without covariate interactions or treatment-effect interactions, marginal balance is sufficient for an unbiased treatment-difference estimate.
  • Randomization debate: Randomization remains contested: proponents cite reduced selection and accidental bias and a basis for inference, while some authors argue covariate randomness makes it unnecessary.The disagreement is especially visible in debates over minimization without an additional random component.
  • Other procedures: Covariate-adaptive methods can target balance within margins and strata, while distance-based procedures assign patients according to covariate-profile distances from existing treatment groups.Several proposed methods emphasize balancing interactions when interactions exist.

3. RANDOMIZATION-BASED INFERENCE

The section contrasts likelihood-based analysis, which treats the randomization as ancillary, with randomization tests that derive the null distribution from treatment assignments allowed by the design. Covariate-adaptive randomization complicates the choice and implementation of these tests.

  • Likelihood analysis: Likelihood-based analysis can condition on responses, treatments, and covariates while treating treatment assignments as independent of model parameters.Under the stated population-model formulation, the randomization is ancillary to the likelihood.
  • Randomization tests: A randomization test computes the null distribution by considering treatment-assignment sequences permitted by the randomization procedure.Under no treatment effect, responses are treated as a deterministic sequence unaffected by treatment assignment.
  • Open issue: Researchers disagree about how randomization tests should handle covariate-adaptive assignments, including whether covariate and response sequences should be treated as fixed.Different choices can produce different results in reported examples.

4. WHAT WE KNOW ABOUT COVARIATE-ADAPTIVE RANDOMIZATION PROCEDURES

Covariate-adaptive randomization is widely studied and often improves balance with many covariates, but its theoretical basis and inferential benefits remain contested. The literature includes contradictory simulations and unresolved questions about whether marginal balance improves power or efficiency.

  • Evidence base: Theoretical work on covariate-adaptive randomization is sparse relative to the large simulation literature.Original papers often provide little justification for biasing probabilities, covariate weights, inference, or the behavior of imbalances.
  • Theory: Aickin’s analysis shows that balancing a measured covariate can improve balance in its linear projection onto an unmeasured covariate.The residual component remains balanced as under restricted or complete randomization.
  • Interpretation: A published claim that Smith established minimization’s theoretical validity is contradicted by the cited paper’s actual focus on a model-based optimal design.The paper states that Smith provided no theoretical justification for minimization methods.
  • Balance: Covariate-adaptive randomization generally improves balance when many covariates are involved.
  • Inference: Simulation findings on inference are mixed: minimization can be inferior to stratification, while other studies report little difference in power or error rates.
  • Open questions: The literature leaves open whether marginal balance improves power and efficiency or is merely cosmetic.

5. MODEL-BASED OPTIMAL DESIGN APPROACHES

Model-based optimal designs target treatment-effect precision rather than balance alone. They use regression models and sequential allocation criteria to select assignments that optimize a contrast-variance objective, with randomization introduced through biased probabilities.

  • Design objective: Model-based designs minimize treatment-effect variance in the presence of covariates, but the resulting optimal designs are initially deterministic.
  • Model: The model includes treatment indicators and selected covariates, with estimator variance determined by the inverse dispersion matrix.
  • Sequential allocation: Atkinson’s sequential design assigns each incoming patient to the treatment maximizing the directional design criterion for that patient’s covariates.
  • Randomization: A biased coin converts the deterministic allocation into randomized assignment probabilities through a monotone increasing function.
  • Tuning: The function ψ controls a compromise between randomness and efficiency, but its choice has not been adequately explored.
  • Evaluation: Atkinson’s simulations evaluated information loss and selection bias for covariate-adaptive procedures in linear-model trials of up to n = 200 patients.

6. WHAT WE KNOW ABOUT ATKINSON’S CLASS OF PROCEDURES

Atkinson’s procedure belongs to a broader class of randomized, model-based allocation rules with substantial theoretical development. The framework balances covariate information, treatment allocation, and randomization through a function ψ and its tuning parameter ρ.

  • General class: Smith generalized Atkinson’s allocation rule using a nonincreasing function ψ satisfying symmetry and smoothness conditions.
  • Assumptions: The procedure assumes independent identically distributed covariates with nonsingular covariance matrix Q and finite third moments.Implementation also requires the covariate distribution to be known at trial initiation.
  • Tuning: Smith’s class commonly uses a proportional biased coin raised to a power ρ, where ρ is determined by the slope of ψ at zero.
  • Theory: Smith derived the asymptotic variance of a conditional randomization test but did not otherwise analyze the procedure or draw substantive conclusions.
  • Trade-offs: Smith recommended choosing ρ large enough to balance covariate balance, accidental bias, and selection bias.

7. BALANCE, EFFICIENCY OR ETHICS?

Balance, efficiency, and ethics can conflict in clinical-trial design. The examples show that probability-informed unbalanced allocations may preserve efficiency while reducing treatment failures, whereas balance is not generally equivalent to efficiency outside specialized models.

  • Competing objectives: Clinical trials pursue balance, efficiency, randomization, and ethical treatment allocation, which can sometimes conflict.
  • Efficiency: Balance is equivalent to minimizing treatment-effect variance in homoscedastic linear models but not generally in logistic or traditional survival models.
  • Conclusion: The authors reject cosmetic balance as a goal when forcing it sacrifices power or efficiency.
  • Allocation rules: In the binary-response example, Rules 2 and 3 use stratum-specific success probabilities and are generally unbalanced.
  • Example: For n0 = n1 = 100, Rule 2 allocates treatment A at πA0 = 0.68 and πA1 = 0.32 across the two strata.
  • Ethics: Rule 3 yields 16 fewer expected failures and Rule 2 yields 8 fewer than balanced allocation in the example.
  • Response adaptation: Response-adaptive designs had similar average power to the PBD and fewer treatment failures, especially when success probabilities were 0.8–0.9 and treatment differences were large.

8. CARA RANDOMIZATION

CARA randomization assigns treatments using previous responses and covariates together with the current patient’s covariates, aiming to incorporate treatment effects into allocation. The section reviews proposed designs, their theoretical properties, and limitations for inference.

  • CARA probabilities use treatment assignments, responses, and covariates from previous patients, plus the current patient’s covariate vector.This extends response-adaptive randomization by adjusting allocation for covariates.
  • A linear-regression allocation rule is not CARA because it omits the current patient’s covariates, and small scaling constants can cause severe treatment imbalance and power losses.Its allocation probability depends on the estimated covariate-adjusted treatment-mean difference and a scaling constant.
  • Atkinson and Biswas combined efficiency and ethical considerations through a weighted DA-optimal criterion, but did not derive asymptotic properties needed to assess inference validity.Their operating characteristics were investigated through simulation.
  • For larger treatment effects, a logistic CARA procedure had similar power to complete randomization but a smaller expected proportion of treatment failures.The comparison used simulations with delayed responses.
  • Bandyopadhyay and Biswas proposed a two-stage logistic-regression design that estimates model parameters before adapting allocation to treatment effects and current covariates.The first stage randomizes 2m patients equally; the second stage uses estimated effects for allocation.
  • Later theory established strong consistency and asymptotic normality for maximum likelihood estimators and allocation proportions in CARA procedures.The framework covered a broad class of models, including generalized linear models.

9. COMPARING DIFFERENT RANDOMIZATION PROCEDURES WHICH ACCOUNT FOR COVARIATES

The simulations compare covariate-adaptive and CARA procedures using balance, power, allocation variability, and treatment failures. CARA procedures generally retain good balance and power while reducing failures relative to balanced designs.

  • Simulation design: The simulations evaluate balance, efficiency, and ethics under logistic models with gender, age, cholesterol, and treatment-by-covariate interactions.Balance uses allocation proportions and covariate-distribution distances; efficiency uses average power, while ethics uses treatment failures.
  • Procedures: CARA procedures sequentially estimate treatment-specific logistic models and randomize each patient using covariate-adjusted allocation targets.The first patients are allocated with Pocock and Simon’s method so model parameters can be estimated.
  • Simulation results: Under the null, all procedures produce balanced allocations; CARA 1, CARA 3, and CARA 4 have type I error 0.06, whereas CARA 2 and CARA 5 preserve 0.05.Pocock and Simon’s procedure is the least variable among the eight rules considered.
  • Simulation results: Balanced designs achieve average power of 90% for stratified blocks and Pocock–Simon randomization versus 89% for complete randomization.These procedures also show lower allocation variability than complete randomization.
  • Simulation results: CARA 2, CARA 3, and CARA 5 achieve 81% average power and two fewer failures than balanced designs, while CARA 4 has 80% power and four fewer failures.CARA procedures are more variable than balanced methods but less variable than complete randomization; CARA 4 matches complete-randomization power.
  • Conclusion: CARA procedures may be an alternative to balanced covariate-adaptive designs for nonlinear responses, despite added allocation variability that may reduce power.The simulations indicate that this potential power impact is not dramatic.
  • Implementation constraint: CARA implementation requires initial covariate-adaptive or stratified-block allocations so enough responses accrue to estimate model parameters accurately.The authors’ numerical experiments found that at least 80 patients must be randomized before maximum-likelihood estimates can be computed.

10. DISCUSSION

The discussion argues that trial design deserves the same care as analysis because covariate handling can affect efficiency, ethics, and interpretability. It favors designs chosen according to outcome severity and supported by theory and simulation.

  • Design practice: Clinical-trial design is often treated as a rote, regulator-driven exercise, even though randomization details are rarely used as a basis for inference.The paper criticizes protocol language that specifies randomization without explaining its design properties.
  • Balance and efficiency: Balanced allocation is most efficient under normally distributed outcomes with similar treatment variances, but this relationship breaks down for heterogeneous variances and binary or survival responses.Covariate-adaptive randomization can therefore create balance without guaranteeing efficiency in nonlinear settings.
  • Alternative designs: New design-stage randomization techniques can improve efficiency and ethical attractiveness while assigning more patients to the better treatment.The discussion links these procedures to optimal experimental design.
  • Design philosophy: Efficiency and ethical considerations should guide randomization, with careful design potentially saving time, money, and, in some cases, patients’ lives.The authors recommend combining theoretical analysis with simulations spanning standard through worst-case models.
  • Design philosophy: Standard balanced designs may remain acceptable when trade-offs are modest, but careful design is essential when outcomes are grave and imbalance could cause severe inefficiency or excess inferior-treatment assignments.The appropriate choice depends on the relative gravity of the outcome and the design’s operating characteristics.
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