Estimation after adaptive allocation

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Abstract

In recent years a vast number of adaptive designs have been proposed, often in the context of clinical trials or industrial applications. The adjective adaptive refers to the feature that, in such designs, the current allocation may depend on data already collected. When analyzing such designs, proving consistency and asymptotic normality of estimators is of fundamental importance. For some designs, this has not yet been done. When it has, the proofs have been tailored to the particular design being analyzed, often utilizing martingale arguments. In this paper independence properties of the allocated sequence are proved and then used to provide a simple method for proving consistency and asymptotic normality of estimators for a wide class of designs.

Introduction

In clinical trials and in industrial work, adaptively designing an experiment is often desirable. Such experiments allocate observations sequentially, where past data can influence present design decisions. Often these designs are constructed in an attempt to achieve ethical allocation (in the context of clinical trials) or optimal allocation (in the industrial context). In this paper, we are concerned with the effect that the adaptive design has on the independence structure of the observations, and the large sample properties of the resulting estimators. From independence properties of the sequence of allocated observations, we give a direct approach for proving strong consistency and an easy, general, non-martingale approach for proving asymptotic normality of estimators based on allocated observations. The method is applied in a variety of settings in Section 4.

Let (X1,Y1),(X2,Y2),… be a sequence of independent and identically distributed random vectors. Let FX denote the distribution of X1 and let FY denote the distribution of Y1. Note that there may be dependence within pairs, i.e., Xk and Yk may not be independent. These random vectors are the (potential) observations from which the adaptive designs are constructed.

In the setting of clinical trials, suppose that there are two competing treatments, X and Y; an arriving patient must be allocated to either treatment X or treatment Y. In this case, Xk and Yk represent the kth patient's (potential) responses to treatments X and Y, respectively, of which only one will actually be observed. In this setting it is not reasonable to assume that Xk and Yk are independent.

In the industrial setting, suppose that there are two populations, X and Y. At each time k, an observation may be taken from Population X or Population Y. In this context, it may be reasonable to assume that Xk and Yk are independent.

Note that the adjective adaptive refers to the fact that the allocation decision for the current patient or population may depend on the previous allocations and the previous outcomes.

At stage k, exactly one of Xk,Yk is observed. A sequential allocation procedure is represented by a sequence of random variables, δ12,…, where δk is 1 if Xk is observed and δk is 0 if Yk is observed. So the observation at time k can be represented by δkXk+(1−δk)Yk. In this setting, an adaptive sequential design allows δk+1 to depend on the history {(δiiXi+(1−δi)Yi):i=1,…,k}, and possibly on auxiliary randomization.

For example, in the adaptive biased-coin design of Wei (1978), δk+1 depends on the previous allocations i:i=1,…,k} and auxiliary randomization, but not on the observations iXi+(1−δi)Yi:i=1,…,k}. In the design of Robbins et al. (1967), δk+1 depends on both the previous allocations and the previous observations, but not on auxiliary randomization. In the doubly adaptive biased-coin design of Eisele (1994), however, δk+1 depends on all three: the previous allocations, the previous observations, and auxiliary randomization. The Eisele design is described more fully in Section 4.

In this paper it is shown that a simple and unified treatment of independence and large-sample properties is possible for adaptive designs, assuming the sampling model above holds. Section 2 concentrates on independence properties that are inherited by the allocated sequence; Section 3 uses these results to give simple, general proofs of consistency and asymptotic normality of estimators based on the allocated sequence; and Section 4 applies these results to some common designs.

Much effort has gone into using martingale methods to prove asymptotic normality for estimators arising from adaptive designs. For some examples of this approach, see Wei et al. (1990) and Rosenberger et al. (1997).

Remark 1.1

It should be noted that the results below are not restricted to maximum likelihood estimators or moment estimators or any other particular class of estimators. Rather, the results proceed from an assumption on the behavior of the estimator under a non-adaptive design (for example, consistency) to a result about the behavior of the estimator when the design is adaptive.

Section snippets

Independence properties

This section begins with a general result (Theorem 2.1) about random selections of elements from two i.i.d. sequences. Then the theorem is applied to allocated sequences in Corollary 2.1.

Throughout this section, (X1,Y1),(X2,Y2),… is assumed to be a sequence of independent and identically distributed random vectors. The marginal distribution of X1 is denoted by FX and the marginal distribution of Y1 is denoted by FY. In addition, {Fn:n⩾1} is an increasing sequence of sigma-algebras such that (Xn,

Strong consistency and asymptotic normality

From Section 2.2, sequentially allocated observations are randomly truncated i.i.d. sequences. Thus, we can model the allocated sequence of k observations as X1,X2,…,XM(k) and Y1,Y2,…,YN(k) where M(k) and N(k) are the number of the first k observations taken on X and Y, respectively. These (dependent) random variables randomly truncate the sequences X1,X2,… and Y1,Y2,….

Applications

In this section applications of the strong consistency (Theorem 3.1) and asymptotic normality (Corollary 3.1) results of the previous sections are given for four adaptive designs which have been proposed. At this point it is useful to recall the model under which these results are applicable.

Comments and conclusions

The importance of the results is two-fold. First, the independence results in Section 2 provide a picture of how the adaptive allocation affects the probability structure of the allocated observations. In particular, see the discussion at the end of Section 2. Second, these independence results lead to a simple, unified treatment of asymptotic properties of estimators arising from adaptive designs. Once the convergence of the allocation proportions is known, consistency and asymptotic normality

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