Covariate-Adaptive Randomization

• Covariate-adaptive randomization is a method that partitions subjects into strata based on baseline covariates to achieve pre-specified treatment balance.
• It employs mechanisms like stratified block and biased-coin randomization to control imbalances and improve estimation efficiency.
• Design-aware inference under CAR requires adjustments to standard methods to account for induced dependency and achieve semiparametric efficiency.

Covariate-adaptive randomization (CAR) refers to treatment assignment procedures in randomized controlled trials and experiments that use observed baseline covariates to improve covariate balance across treatment arms. In the canonical formulation, units are first partitioned into strata defined by baseline covariates and are then randomized within strata so that realized treatment proportions track prespecified targets. Because assignment is conditioned on strata and often constrained within strata, CAR generally induces dependence across assignments, so its design-based properties, semiparametric efficiency, and valid inference differ in systematic ways from simple randomization (Rafi, 2023).

1. Design structure and randomization mechanisms

CAR experiments can be written through a measurable stratification map $s:\mathbb{R}^k \to \{1,\dots,\mathcal{S}\}$ applied to baseline covariates $Z$, with stratum labels $S=s(Z)$. The experimenter fixes target proportions $\pi(s)\in(0,1)$ for each stratum and assigns treatment using a known randomization mechanism $\alpha_n(a_1,\dots,a_n\mid s_1,\dots,s_n)$ that is exogenous given strata, known to the experimenter, independent of the population distribution, and balancing in the sense that $N_n(1,s)/N_n(s)\to_p \pi(s)$ within each stratum. In this formulation, CAR allows dependent assignments within strata, but the asymptotic theory depends on the target proportions rather than on the particular mechanism, provided the balance and exogeneity conditions hold (Rafi, 2023).

Two benchmark mechanisms recur throughout the literature. Simple stratified random assignment uses independent Bernoulli draws within each stratum with $P(A=1\mid s)=\pi(s)$ and achieves weak balance at the $n^{-1/2}$ rate. Stratified permuted block randomization assigns exactly $\lfloor \pi(s)N_n(s)\rfloor$ treated units within stratum $s$ by uniformly sampling a block and achieves strong balance at the $Z$0 rate. Clinical-trial and RCT frameworks discussed in recent work also include stratified block randomization, stratified biased-coin randomization, Pocock–Simon minimization, and simple randomization as a special case (Rafi, 2023, Gu et al., 23 Dec 2025).

Modern asymptotic theory distinguishes fixed-$Z$1 and diverging-$Z$2 regimes, where $Z$3 is the number of strata. With many strata, the design conditions connecting balance and non-emptiness become rate-sensitive. For equal-probability strata, simple randomization within strata satisfies the core balance condition when $Z$4 for $Z$5, whereas stratified block randomization satisfies it for $Z$6. This establishes a formal sense in which stratified block randomization tolerates much faster growth in the number of strata than simple randomization while preserving valid large-sample inference (Xin et al., 2024).

A parallel design literature studies CAR as a sequential imbalance process. In the two-arm setting with treatments coded as $Z$7 and $Z$8, Hu and Hu’s design uses a weighted sum of overall, marginal, and within-stratum imbalances and an Efron-type biased coin; under explicit weight conditions, the joint within-stratum imbalance process is a positive recurrent Markov chain, yielding $Z$9 control of imbalance components (Hu et al., 2012). Later work reframed the same design tension as one between asymptotic power and selection bias, emphasizing that many popular CAR procedures achieve strong balance but also generate nontrivial selection bias relative to complete randomization (Zhang, 2019).

2. Probabilistic structure, target parameters, and the role of the propensity score

For binary treatment, the standard superpopulation formulation takes $S=s(Z)$0 with finite second moments and observes $S=s(Z)$1, where $S=s(Z)$2. The average treatment effect is

$S=s(Z)$3

Under CAR, the treatment propensity conditional on all baseline covariates is design-determined:

$S=s(Z)$4

Because $S=s(Z)$5 is fixed by the design, it is known and does not need to be estimated; positivity follows from $S=s(Z)$6 (Rafi, 2023).

This design-based propensity interpretation is central. In CAR, the stratum-specific target proportions act as the propensity score conditional on all baseline covariates, not merely conditional on the coarse stratum labels. The conditional outcome regressions may therefore be written as $S=s(Z)$7, and the conditional variances as $S=s(Z)$8. The distinction between conditioning on $S=s(Z)$9 and conditioning only on $\pi(s)\in(0,1)$0 drives later efficiency comparisons (Rafi, 2023).

The same design logic extends to multi-arm experiments. With treatments $\pi(s)\in(0,1)$1 and stratum-specific targets $\pi(s)\in(0,1)$2 such that $\pi(s)\in(0,1)$3, the estimands of interest are typically arm-versus-control contrasts

$\pi(s)\in(0,1)$4

CAR then means that units are first stratified by $\pi(s)\in(0,1)$5 and then assigned so as to achieve the target shares $\pi(s)\in(0,1)$6 within each stratum (Bugni et al., 2018).

Recent work has also extended CAR beyond mean effects. For distributional treatment effects, the basic targets are

$\pi(s)\in(0,1)$7

with identification under CAR obtained from stratum-specific assignment probabilities $\pi(s)\in(0,1)$8 and conditional independence given strata (Byambadalai et al., 6 Jun 2025). Under imperfect compliance, CAR complicates the usual i.i.d. arguments, but sharp identified sets for the ATE and ATT can still be characterized in terms of $\pi(s)\in(0,1)$9, monotonicity, and bounded outcomes (Bugni et al., 2024).

3. Semiparametric efficiency and design-aware estimation

A central result for binary-treatment CAR is that the semiparametric efficiency bound for the ATE is Hahn’s bound specialized to the known propensity score $\alpha_n(a_1,\dots,a_n\mid s_1,\dots,s_n)$0 and conditioning on all baseline covariates:

$\alpha_n(a_1,\dots,a_n\mid s_1,\dots,s_n)$1

If one conditions only on stratum labels, the corresponding bound replaces $\alpha_n(a_1,\dots,a_n\mid s_1,\dots,s_n)$2 by $\alpha_n(a_1,\dots,a_n\mid s_1,\dots,s_n)$3. Thus, when $\alpha_n(a_1,\dots,a_n\mid s_1,\dots,s_n)$4 varies within strata, conditioning on $\alpha_n(a_1,\dots,a_n\mid s_1,\dots,s_n)$5 can strictly improve efficiency over a purely saturated analysis based only on strata (Rafi, 2023).

The efficient influence function has the familiar augmented inverse-probability form:

$\alpha_n(a_1,\dots,a_n\mid s_1,\dots,s_n)$6

Its variance equals the efficiency bound. This leads to a feasible estimator

$\alpha_n(a_1,\dots,a_n\mid s_1,\dots,s_n)$7

with $\alpha_n(a_1,\dots,a_n\mid s_1,\dots,s_n)$8 known by design. Under the stated CAR assumptions and mild first-stage conditions, a cross-fitted Nadaraya–Watson kernel estimator yields $\alpha_n(a_1,\dots,a_n\mid s_1,\dots,s_n)$9-consistency, asymptotic normality, and attainment of the efficiency bound (Rafi, 2023).

Cross-fitting plays the same role in more general AIPW theory under CAR. A unified analysis of AIPW estimators shows that machine learning methods with cross-fitting remain valid despite the dependence induced by CAR, because the nuisance estimation and score evaluation are separated fold by fold. Under a universal applicability condition,

$N_n(1,s)/N_n(s)\to_p \pi(s)$0

the asymptotic variance becomes invariant to the CAR scheme; a joint calibration strategy based on constructed covariates $N_n(1,s)/N_n(s)\to_p \pi(s)$1 guarantees both universality and efficiency improvement, and these procedures are implemented in the R package RobinCar (Bannick et al., 2023).

The same semiparametric program has now been extended from means to distributional effects. A distribution-regression AIPW estimator for $N_n(1,s)/N_n(s)\to_p \pi(s)$2 takes the form

$N_n(1,s)/N_n(s)\to_p \pi(s)$3

and yields an efficient estimator of the DTE $N_n(1,s)/N_n(s)\to_p \pi(s)$4 under CAR, with a valid influence-function-based inference procedure and a semiparametric efficiency bound specialized to stratum-specific assignment probabilities (Byambadalai et al., 6 Jun 2025).

4. Many strata, growing dimension, and assumption-lean adjustment

When the number of strata diverges, the basic stratified difference-in-means estimator

$N_n(1,s)/N_n(s)\to_p \pi(s)$5

remains asymptotically normal under moment, overlap, and balance conditions, with variance decomposition

$N_n(1,s)/N_n(s)\to_p \pi(s)$6

where the within-strata terms scale conditional outcome variances by inverse assignment probabilities and the between-strata term captures heterogeneity in conditional treatment effects. In this regime, weighted regression residualization is especially important when assignment probabilities vary across strata. The proposed weighted adjustment replaces unweighted residualization by stratum-arm weighted covariances and guarantees

$N_n(1,s)/N_n(s)\to_p \pi(s)$7

with equality only in special cases; practical algorithms based on complete-case or imputation approaches are provided for extremely large numbers of strata (Xin et al., 2024).

A distinct high-dimensional issue arises when the number of adjustment covariates grows with sample size. In the assumption-lean setting with $N_n(1,s)/N_n(s)\to_p \pi(s)$8 and no structural assumptions on the outcome model, ordinary least squares adjustment under CAR suffers from a V-statistic diagonal bias of order $N_n(1,s)/N_n(s)\to_p \pi(s)$9, which can dominate the sampling error when $P(A=1\mid s)=\pi(s)$0 is moderately large. To address this, a second-order $P(A=1\mid s)=\pi(s)$1-statistic estimator replaces the OLS diagonal terms by off-diagonal terms and yields almost unbiased estimation of the ATE with a guaranteed efficiency gain relative to the stratified difference-in-means when $P(A=1\mid s)=\pi(s)$2 (Gu et al., 23 Dec 2025).

The feasible $P(A=1\mid s)=\pi(s)$3-statistic estimator replaces the population Gram matrix $P(A=1\mid s)=\pi(s)$4 by the sample Gram matrix $P(A=1\mid s)=\pi(s)$5 and takes the form

$P(A=1\mid s)=\pi(s)$6

Its asymptotic variance decomposes as

$P(A=1\mid s)=\pi(s)$7

with $P(A=1\mid s)=\pi(s)$8 and $P(A=1\mid s)=\pi(s)$9. The estimator is $n^{-1/2}$0-asymptotically normal under CAR, and the accompanying variance estimator is consistent; implementation is available in the R package HOIFCar (Gu et al., 23 Dec 2025).

These results sharpen a recurring design-analysis tradeoff. Finer stratification can reduce within-stratum heterogeneity and potentially improve saturated analyses, but it reduces effective sample size per stratum, stresses nonparametric or high-dimensional adjustment, and can destabilize fold construction or Gram-matrix inversion. The many-strata and $n^{-1/2}$1 literatures make that tradeoff explicit rather than treating it as a purely finite-sample concern (Xin et al., 2024, Gu et al., 23 Dec 2025).

5. Specialized inferential problems

CAR changes the validity of familiar inferential procedures in ways that are now well understood. With multiple treatments, fully saturated regression of the outcome on all treatment-by-strata interactions yields consistent effect estimators, but the usual heteroskedasticity-consistent variance estimator is invalid because it omits a design-based heterogeneity term. Exact inference instead uses

$n^{-1/2}$2

where $n^{-1/2}$3. When target treatment shares are constant across strata and strong balance holds, a strata fixed-effects regression can also be used, but if target shares vary across strata, that regression is generally inconsistent for the superpopulation contrast (Bugni et al., 2018).

In survival analysis, CAR similarly invalidates simple reuse of tests derived under simple randomization. The log-rank test is conservative under CAR, and the robust score test developed under simple randomization is no longer robust under CAR. Calibration restores validity: the paper proposes calibrated log-rank and score tests whose denominators estimate

$n^{-1/2}$4

with $n^{-1/2}$5 under stratified permuted blocks and $n^{-1/2}$6 under simple randomization, yielding tests that are valid and robust under both simple randomization and a large family of CAR schemes (Ye et al., 2018).

Interaction testing under CAR exhibits the same pattern. Usual treatment–covariate interaction tests are conservative: their limiting rejection probabilities under the null do not exceed the nominal level and are typically strictly lower. Modified tests based on CAR-aware variance estimators are valid, and a distinct class of stratified-adjusted interaction tests is simpler and more powerful than the usual and modified tests. This applies both when the interacting covariate is itself a stratification covariate and when it is an additional covariate not used for randomization (Zhang et al., 2023).

Imperfect compliance introduces a different complication. Under binary treatment CAR with monotonicity and bounded outcomes, the ATE and ATT are generally partially identified. The ATE identified set is an interval $n^{-1/2}$7, and the ATT identified set is $n^{-1/2}$8. For the ATE bounds, using sample analog assignment frequencies $n^{-1/2}$9 is weakly more efficient than using target probabilities $\lfloor \pi(s)N_n(s)\rfloor$0; for the ATT bounds, the most efficient approach uses the target in the numerator weights and the sample analog in the denominators. Uniformly valid confidence intervals are then constructed using Stoye-type shortest valid intervals (Bugni et al., 2024).

CAR has also been extended to two-stage experiments with partial interference. In cluster-randomized designs with first-stage cluster-level CAR and second-stage unit-level randomization inside treated clusters, difference-in-“average of averages” estimators are consistent and asymptotically normal for primary and spillover effects. Ignoring covariate information at the design stage can result in efficiency loss, and commonly used inference methods that ignore or improperly use covariate information can be conservative or invalid. Within the large-sample class studied there, a generalized matched-pair design minimizes asymptotic variance (Liu, 2023).

6. Design tensions, shift, and recent innovations

A major recent design issue is the “shift problem” under unequal targeted allocation ratios. In a general non-Markovian CAR framework with imbalance vector

$\lfloor \pi(s)N_n(s)\rfloor$1

the shift problem is equivalent to a mismatch between the conditional average allocation ratio and the target among units sharing specific covariate values. For classical minimization-style procedures under $\lfloor \pi(s)N_n(s)\rfloor$2, this mismatch can persist even when balancing covariates are well controlled, so the average imbalance of additional covariates does not center at zero. A new class of allocation functions

$\lfloor \pi(s)N_n(s)\rfloor$3

was constructed precisely to enforce the no-shift condition $\lfloor \pi(s)N_n(s)\rfloor$4 while retaining $\lfloor \pi(s)N_n(s)\rfloor$5 balance; the resulting feasible non-Markovian randomization procedure updates its parameters from accumulated covariate information and achieves balance without shift (Fang et al., 26 Feb 2026).

A closely related 2026 development aims at CAR “without inflated variances.” The proposed class uses

$\lfloor \pi(s)N_n(s)\rfloor$6

with $\lfloor \pi(s)N_n(s)\rfloor$7. Under this design, the specified covariate imbalance satisfies $\lfloor \pi(s)N_n(s)\rfloor$8, the shift problem does not appear, and for any unspecified covariate $\lfloor \pi(s)N_n(s)\rfloor$9,

$s$0

where $s$1. The corresponding asymptotic variance does not exceed that under simple randomization (Li-Xin, 11 Feb 2026).

These innovations sharpen a broader misconception. CAR does not merely “improve balance”; it changes the stochastic structure of assignment and therefore the null distribution, variance decomposition, and efficiency calculations of downstream estimators. Across regression, survival analysis, interaction testing, multi-arm inference, high-dimensional adjustment, and partially identified compliance problems, the consistent theme is that design-aware procedures are required. The recent literature therefore treats CAR not as a minor perturbation of simple randomization, but as a distinct asymptotic regime whose inferential consequences can be characterized precisely (Ma et al., 2020).