## 1 Introduction

A theory for estimation in a linear model, specifically accounting for the randomization used in a clinical trial will be developed. A particular goal is to compare the inferences from the usual population-based linear model with those from the randomization model. A further goal is to compare randomization model inference that follows from permuted-blocks randomization (PBR) with those from dynamic randomization methods. Here, ‘dynamic randomization’ refers to methodology in which the probability of assignment of a given patient to experimental treatment is a function of the patient's stratification variables and the stratification variables and treatment assignments of previously randomized patients. Other terminology has been used, such as adaptive randomization and minimization.

Randomization of treatments to patients in clinical trials has perhaps most commonly been performed with PBR, often within strata defined by crossings of prognostic factors and/or investigational sites. Permuted blocks are used instead of ‘complete randomization’, which assigns the next patient to experimental treatment essentially by a fair coin toss, to ensure that the number of patients assigned to each treatment group remains similar within strata, regardless of the number of patients ultimately enrolled in the strata [1]. Efron [2] introduced biased-coin randomization, which assigns the next patient to experimental treatment with a biased coin if the counts of patients previously assigned to the two treatments are imbalanced. Pocock and Simon [3] generalized the method to include stratification variables, where the probability of assignment to experimental treatment depends on the balance of treatment counts across the margins of the stratification variables. There have subsequently been many proposals for other dynamic randomization procedures [4-7].

A theory for estimation and testing exists for permutation-based randomizations paired with a linear model, where the linear model includes the randomization's stratification variables in addition to a treatment term, and leads to unbiased estimators of treatment effects and unbiased estimators of the variance of the treatment effect estimator [8-11]. The unbiasedness is over the distribution of the observations induced by repeated randomizations of treatments to fixed experimental units, under the assumption of unit-treatment additivity with fixed unit and treatment effects.

There have been many theoretical developments for randomizations that are not of the permutation type, but these have mostly focused on randomization tests rather than estimation.

- Smythe and Wei [12] derived the asymptotic distribution of a linear rank test of the null hypothesis of equality of two treatments when the assignment of treatments to patients is according to an urn-based randomization scheme.
- Smith [13] summarized treatment allocation methods developed to date. For treatment allocation probabilities that are functions of (
*n*_{1}−*n*_{0}) / (*n*_{0}+*n*_{1}), where the*n*_{j}'s are the numbers of patients previously assigned to the two treatments, Smith developed an approximation to a randomization test based on the difference in sample means. Smith also developed similar approximations when treatment assignment probabilities are a particular function of stratification variables of the current and previous patients and developed asymptotic approximations to randomization tests. - Wei
*et al.*[14] proposed a biased-coin scheme for the allocation of*k*treatments in proportion ξ_{1}, … ,ξ_{k}, where ξ_{1}+ … + ξ_{k}= 1, and generalized the work of Smythe and Wei [12] to*k*> 2. - Shao
*et al.*[15] proposed a theory for clinical trials with dynamic randomization, but the results were developed on an assumption of a model that relates the dependent variable to the variables used in constructing hypothesis tests. - Rosenkranz [16] explored the sampling properties of a difference in sample means under various randomization schemes and noted that a
*t*-test can be very conservative with biased- coin randomization.

Regardless of the randomization method used, it is common in clinical trials data analysis to apply a statistical test that is based on assumptions of random sampling from a distribution. In general, these tests behave as follows for biased-coin and urn-based randomizations [2, 13, 16-18].

- If there is low-frequency variation in the responses of patients over time of entry into the trial, then a population-based test will tend to result in a larger
*p*-value than the corresponding randomization test result. With this pattern of variation, patients enrolled closer together in time tend to respond more similarly than patients enrolled farther apart in time. - If there is high-frequency variation in responses over enrollment time, then a population-based test will tend to yield a smaller
*p*-value than the corresponding randomization test result. In this case, patients enrolled closer together in time tend to respond less similarly than patients enrolled farther apart in time. This is generally noted to be less likely than the scenario above. - Finally, when responses appear to come from a homogeneous, time-independent model, then the two test results tend to be similar.

These same qualitative comparisons apply for PBR when the blocking in time is ignored in the analysis [1], in which case the variation in patient responses is quantified by the intrablock correlation.

Motivation for the present work on estimation and testing under a randomization model comes from a ‘points to consider’ document on adjustment for baseline covariates [19] that states ‘… techniques of dynamic allocation such as minimization are sometimes used to achieve balance across several factors simultaneously. Even if deterministic schemes are avoided, such methods remain highly controversial. Thus, applicants are strongly advised to avoid such methods.’ Furthermore, Halpern and Brown [18] noted that dynamic randomization schemes ‘can lead to complex randomization distributions and hence to complex analyses about which little is currently known, and which, in fact, may present intractable difficulties.’ With a randomization-based theory for estimation and testing applicable across all randomization methods, perhaps regulatory authorities will be more accepting of dynamic randomization methods.

Section 2 contains the theory for randomization-based estimation in a linear model with unit-treatment additivity. Section 3 evaluates the estimation of the variance of the treatment effect estimator through a population-based linear model, in comparison with the randomization-based variance estimator. Section 4 compares the estimation results with randomization test results. Sections 5 through 7 explore randomization-based inference when there is planned unbalanced treatment allocation, like two experimental to one control. A key requirement, often not met with proposed dynamic randomization methods for such unbalanced allocation, is that the probability that a patient receives experimental treatment should be constant across patients. The paper finishes with a discussion and conclusion.