Response-adaptive treatment allocation for clinical studies with ordinal responses

Abstract

Ordinal responses are common in clinical studies. Although the proportional odds model is a popular option for analyzing ordered-categorical data, it cannot control the type I error rate when the proportional odds assumption fails to hold. The latent Weibull model was recently shown to be a superior candidate for modeling ordinal data, with remarkably better performance than the latent normal model when the data are highly skewed. In clinical trials with ordinal responses, a balanced design is common, with equal sample allocation for each treatment. However, a more ethical approach is to adopt a response-adaptive allocation scheme in which more patients receive the better treatment. In this paper, we propose the use of the doubly adaptive biased coin design to generate treatment allocations that benefit the trial participants. The proposed treatment allocation scheme not only allows more patients to receive the better treatment, it also maintains compatible test power for the comparison of treatment efficiencies. A clinical example is used to illustrate the proposed procedure.

Keywords

Adaptive treatment allocation ordered-categorical responses doubly adaptive biased coin design latent Weibull model allocation function

1 Introduction

In clinical trials, ordered categorical response is a familiar type of patient outcome. For example, a study of various anti-malarial drugs by Youdom et al.¹ used the classification of clinical outcomes suggested by the World Health Organization.² The measure is an ordinal scale, including adequate clinical and parasitological response, late parasitological failure, late clinical failure, and early treatment failure. Another study that assessed various treatments for depression distinguished patients according to the severity of their depression based on a patient health questionnaire.³ The possible categories are mild depression, moderate depression, moderately severe depression, and severe depression.

For comparisons of treatments with ordinal outcomes, the proportional odds model has long been a popular choice for testing the equality of treatment efficacy.^4,5 However, as demonstrated by Lu et al.,⁶ the proportional odds model fails to control the type I error rate when the proportional odds assumption is invalid. A better alternative is to use the latent variable model, which conceptualizes ordinal responses as manifestations of underlying continuous variables. The latent variable model has long been an attractive technique for analyzing ordinal responses.^7–9 Poon¹⁰ proposed the use of the latent normal model to compare two treatments. Her method was extended to pairwise comparisons¹¹ and multiple comparisons with a control^6,12 when more than two treatments are available. Although the latent normal model shows satisfactory performance and can control the type I error rate in ordinary cases, it also suffers from the type I error rate inflation problem when the degree of skewness differs greatly for data from various treatments. Lu et al.^13,14 recently found that the latent Weibull model is a superb candidate to compare treatments with highly skewed ordinal outcomes. Testing procedures have been developed with very competitive test power even when the data are not skewed. The latent Weibull model appears to be superior to the aforementioned procedures in comparing treatments with ordered categorical outcomes.

In clinical studies, the traditional option for treatment allocation is a balanced design by which equal numbers of patients are assigned to each treatment, and trials with ordinal outcomes are no exception.^15,16 However, during the patient recruitment period, incoming patients almost always arrive sequentially, which permits the use of a response-adaptive treatment allocation scheme. Response-adaptive treatment allocation procedures are able to incorporate information about the responses of previous patients amassed during the trial thus far and skew the probability of treatment assignment to favor the better treatment.¹⁷ Hence, a response-adaptive design provides a desirable alternative to the popular balanced-design norm by reducing the number of patients who undergo the less effective treatment in clinical trials.¹⁸

The primary objective of this paper is to introduce a response-adaptive treatment allocation scheme that can be applied to compare treatments with ordinal responses. Of the various types of available response-adaptive designs, the doubly adaptive biased coin design (DBCD) proposed by Hu and Zhang¹⁹ has two important features. First, it enables the inclusion of pre-specified targets, such as the treatment of fewer patients with the inferior treatment. Second, the allocation scheme yields very competitive test power.

The layout of this paper is as follows. In Section 2, the latent Weibull model is outlined and the testing procedure to compare treatment efficacy is explained. In Section 3, the basic framework of response-adaptive design is given, including a discussion of the DBCD design. Under the DBCD framework, various allocation rules are given based on different objectives. In Section 4, a simulation study is used to demonstrate the benefits of the suggested design compared with the balanced design. In Section 5, a clinical study is redesigned to demonstrate the advantage of the DBCD design in practice. Finally, a few concluding remarks are given in Section 6.

2 Latent Weibull model

2.1 Clinical example

To facilitate understanding of the data structure and the proposed model, a clinical study to determine whether intensive or moderate alveolar recruitment strategies added to lung-protective ventilation will affect postoperative pulmonary complications²⁰ is presented here for illustrated purposes. The intended allocation ratio for the two treatments was 1:1 as stated in their study protocol. Eventually, this randomized clinical trial recruited 320 patients, of whom 157 were assigned to the intensive recruitment strategy and 163 to the moderate recruitment strategy. The severity of postoperative pulmonary complications was the primary study outcome. An ordinal scale was used to measure the outcome, ranging from grade 0 (no symptoms) to grade 4 or above (reintubation or death), thus giving five outcome categories (K = 5). The data are tabulated in Table 1, and the study concludes that the severity of pulmonary complications can be reduced with the use of an intensive alveolar recruitment strategy.

Table 1.

Severity of postoperative pulmonary complications with two recruitment strategies.

	Treatment
Modified score of pulmonary complications	Intensive recruitment strategy	Moderate recruitment strategy
Grade 4 or above (k = 1)	4	8
Grade 3 (k = 2)	20	35
Grade 2 (k = 3)	75	79
Grade 1 (k = 4)	55	41
Grade 0 (no symptoms) (k = 5)	3	0

2.2 Model and parameter estimation

Assume the existence of two treatments in a clinical study with ordinal outcomes, similar to the aforementioned example. Let Y_i be the ordinal response of treatment i, i = 1, 2, with possible values k, $k = 1, \dots, K$ . For simplicity, let a larger value of k represent a preferred outcome. With reference to the above clinical example, the coded outcomes are grade 4 or above (k = 1), grade 3 (k = 2), grade 2 (k = 3), grade 1 (k = 4), and grade 0 (k = 5). As stated in the Introduction, the latent variable model is a useful tool to compare treatments with ordinal outcomes. In particular, it avoids the likelihood of an undesirable inflated type I error rate. The observable ordinal response Y_i is assumed to be the manifestation of an underlying continuous variable Z_i via a threshold model

Y_{i} = k iff α_{k - 1} < Z_{i} \leq α_{k}, k = 1, \dots, K, i = 1, 2

(1)

where the thresholds

α_{1} < \dots < α_{K - 1}

are assumed to be the same across treatments with

α_{0} = - \infty

and

α_{K} = \infty

. Let the cumulative distribution function of Z_i be

G (z | ω_{i})

, where

ω_{i}

contains all unknown parameters of the distribution. In general, any continuous distribution can be chosen as

G (z | ω_{i})

. Popular choices, including the normal distribution and the logistic distribution, have been proposed.^5,21 Lu et al.²² gave a unified framework for the comparison of treatments with ordinal responses, with G being any member of the location-scale distribution family. Estimation and testing procedures have also been developed under this general framework. When G is selected to be the normal distribution, previous studies have indicated that it is superior to the proportional odds model and is able to eliminate the inflated type I error rate problem in general.^6,11,12 However, a recent study by Lu et al.¹³ has reported that although the performance of the latent normal model is satisfactory in most situations, it also suffers from a lack of control of the type I error rate in certain extreme cases when treatment efficacies are compared, specifically, when the degrees of skewness of the treatments differ greatly.

To provide a better model to cope with the problem of the inflated type I error rate for a wider spectrum of data environments, Lu et al.^13,14 suggested the latent Weibull model. The Weibull distribution is a member of the log-location-scale distribution family, and its corresponding distribution in the location-scale family is the extreme value distribution.²³ Hence, the estimation procedures can easily be obtained by generalizing the estimation procedures that were previously derived from a model using the corresponding location-scale distribution.¹³ Unlike the normal distribution, which is symmetric, the Weibull distribution has a very flexible distribution that can be applied to model data that are highly skewed. With the latent Weibull model, the distribution function of Z_i is

\begin{matrix} G (z | ω_{i}) = 1 - exp {- (\frac{z}{η_{i}}) β_{i}}, z > 0 \end{matrix}

with

ω_{i} = (η_{i}, β_{i})'

, i = 1, 2. The thresholds in equation (1) are set to

0 = α_{0} < α_{1} < \dots < α_{K - 1} < α_{K} = \infty

for the Weibull distribution.

Let n_i patients undergo treatment i, i = 1, 2, and the total sample size, n, is therefore $n_{1} + n_{2}$ . Let n_ik be the number of patients who undergo treatment i with outcomes classified into category k, $k = 1, \dots, K$ . The log-likelihood function for the two treatments is then

\begin{matrix} L (θ) = \sum_{i = 1}^{2} \sum_{k = 1}^{K} n_{ik} log {π_{ik} (θ)} \end{matrix}

(2)

where

θ = (α', ω_{1}', ω_{2}')'

is a vector that contains all unknown parameters. Here,

α' = (α_{1}, \dots, α_{K - 1})

is the vector of thresholds. Furthermore,

π_{ik} (θ)

represents the cell probability for treatment i, category k. That is

\begin{matrix} π_{ik} (θ) = G (α_{k} | ω_{i}) - G (α_{k - 1} | ω_{i}) \end{matrix}

Let $\hat{θ} = (\hat{α}', {\hat{ω}}_{1}', {\hat{ω}}_{2}')'$ be the estimate of $θ$ . To tackle the model identifiability problem which is linked to the likelihood function (2), a two-step estimation procedure was given by Lu et al.¹³ The details for the derivation, together with the formulas, of $\hat{θ}$ are provided in their paper. They also gave the following asymptotic result regarding ${\hat{ω}}_{i}, i = 1, 2$

\begin{matrix} \sqrt{n_{i}} ({\hat{ω}}_{i} - ω_{i}) \to^{d} N (0, Ω_{i}) \end{matrix}

(3)

where

\begin{matrix} Ω_{i} = (\begin{matrix} a_{1, i} & a_{2, i} \\ a_{2, i} & a_{3, i} \end{matrix}) \end{matrix}

The entries $a_{1, i}, a_{2, i}, a_{2, i}$ , and $a_{3, i}$ of the 2 by 2 covariance matrix $Ω_{i}$ are functions of the parameters of the Weibull distribution, and their expressions are derived using the suggestions given by Lu et al.¹³ and can be found in Appendix 1.

2.3 Comparison of treatments

For the latent Weibull model, the median of the latent Weibull distribution is a reasonable measure of the mean treatment efficacy because the Weibull distribution could be highly skewed. Let M_i be the median of the underlying Weibull distribution $G (z | ω_{i})$ . In addition, let $γ_{i}^{2} / n_{i}$ be the variance of the sample median ${\hat{M}}_{i}$ . Following the arguments of Lu et al.,¹³ $M_{i} = η_{i} (\ln (2))^{1 / β_{i}} .$ Asymptotically, they also showed that

\begin{matrix} \sqrt{n_{i}} ({\hat{M}}_{i} - M_{i}) \to^{d} N (0, γ_{i}^{2}) \end{matrix}

where

\begin{matrix} γ_{i}^{2} = (\ln (2)) 2 / β_{i} a_{1, i} - 2 \frac{η_{i}}{β_{i}^{2}} (\ln (2)) 2 / β_{i} \ln (\ln (2)) a_{2, i} + \frac{η_{i}^{2}}{β_{i}^{4}} (\ln (2)) 2 / β_{i} (\ln (\ln (2))) 2 a_{3, i} \end{matrix}

(4)

To compare these two treatments, we test the null hypothesis

H_{0} : M_{1} = M_{2} against H_{1} : M_{1} = M_{2}

with the pivotal test statistic

\begin{matrix} W_{1} = \frac{{\hat{M}}_{2} - {\hat{M}}_{1}}{\sqrt{{\hat{γ}}_{1}^{2} / n_{1} + {\hat{γ}}_{2}^{2} / n_{2}}} \to^{d} N (0, 1) \end{matrix}

3 Response-adaptive design

3.1 Response-adaptive treatment allocation versus balanced design

Balanced design has turned out to be the most popular option to avoid subjective judgment and possible bias in treatment allocation in clinical studies, that is, an equal number of patients are assigned to each treatment regardless of differences in efficacy among the treatments. For trials with ordinal outcomes, balanced treatment allocation schemes also predominate in design protocols.^20,24

Even though treatment randomization using the balanced design is widely adopted in clinical trials, response-adaptive randomization is becoming more acceptable.²⁵ The attractive feature of response-adaptive randomization is that the probabilities of treatment assignment can be shifted according to pre-determined objectives, such as lowering the number of patients who undergo the inferior treatment. In a response-adaptive randomization scheme, the patients are required to accrue sequentially, which enables the assessment of treatment efficacies with the data amassed at any given point in the trial. The usefulness of response-adaptive designs has recently been recognized by clinical researchers. For example, Berry et al.²⁶ proposed the use of response-adaptive randomization in a study of the Ebola virus, and Das and Lo²⁷ discussed the advantages of adaptive randomization in breast cancer trials.

Among the various classes of response-adaptive design, DBCD, which was first proposed by Eisele,²⁸ is considered to be superior to many other procedures.^19,29,30 Desirable optimal criteria can be incorporated into the DBCD, and treatment allocations converge to the pre-specified targets. The mechanism to implement treatment randomization with DBCD is given in the following subsection. Note that the use of a response-adaptive design with ordinal responses is more difficult than its counterpart with continuous outcomes. This mainly happens because the response-adaptive design requires continuous estimation of the parameters, even during the early stages of a clinical study when the number of individuals recruited for the trial is still small and, therefore, the ordinal data may be quite skewed. Traditional estimation procedures do not work satisfactorily, but the latent Weibull model performs much better in terms of estimation and testing, even when the data are very skewed. Hence, with the use of the latent Weibull model, response-adaptive treatment randomization is a feasible treatment allocation scheme.

3.2 Treatment allocation scheme of DBCD

Assume that n patients are to be assigned to two treatments in a clinical study. At time t, q $(q = 1, \dots, n)$ patients have been assigned treatments, and their responses up to time t have been observed. Among these q patients, let $n_{(q) i}$ be the number of patients who underwent treatment $i, i = 1, 2$ . Denote $n_{(q)} = (n_{(q) 1}, n_{(q) 2})$ as the sample size vector until time t. The unknown parameter vector is denoted by $θ$ , and the corresponding estimate at time t is ${\hat{θ}}_{(q)}$ . The subscript (q) is attached to a statistic obtained based on the responses of q patients observed up to time t. An attractive characteristic of DBCD is that it can incorporate appealing allocation targets (allocation rule) in the treatment randomization scheme. Let ρ and $1 - ρ$ be the desired allocation rules, a function of $θ$ , for treatments 1 and 2, respectively. The allocation rule can be constructed according to certain pre-specified targets. Different allocation rules will be examined later, but first, let us state the DBCD treatment randomization scheme.

DBCD treatment randomization scheme

The first n₀ patients are allocated to the two treatments using restricted randomization to obtain an initial estimate of the parameters, ${\hat{θ}}_{(n_{0})}$ .³¹

For $q \geq n_{0}$ , patient $(q + 1)$ is assigned to treatment 1 with a probability $g (\frac{n_{(q)}}{q}, {\hat{ρ}}_{(q)})$ and treatment 2 with a probability $1 - g (\frac{n_{(q)}}{q}, {\hat{ρ}}_{(q)})$ , where ${\hat{ρ}}_{(q)} = ρ ({\hat{θ}}_{(q)}) .$

In Stage 1, the parameters $α', ω_{1}'$ , and $ω_{2}'$ are estimated by ${\hat{θ}}_{(n_{0})}$ . In Stage 2, the thresholds estimated in Stage 1 are fixed without further updating to avoid unnecessary additional variations produced by obtaining the allocation proportion. The rationale for this approach is based on the argument of Lu et al.²² that the test statistics that are used to compare the efficacy of treatments are invariant to the choice of thresholds. Hence, according to the need to achieve model identifiability, the estimated thresholds are first treated as fixed known values for the subsequent process of estimation and testing.

Using the treatment allocation procedure in Hu and Zhang,¹⁹ the allocation function for treatment 1 in the second step is

\begin{matrix} g (\frac{n_{(q)}}{q}, {\hat{ρ}}_{(q)}) = \frac{{\hat{ρ}}_{(q)} {(q {\hat{ρ}}_{(q)} / n_{(q) 1})}^{τ}}{{\hat{ρ}}_{(q)} {(q {\hat{ρ}}_{(q)} / n_{(q) 1})}^{τ} + (1 - {\hat{ρ}}_{(q)}) {(q (1 - {\hat{ρ}}_{(q)}) / n_{(q) 2})}^{τ}} \end{matrix}

(5)

where

τ \geq 0

is a tuning parameter that has a negative association with the level of randomness of the treatment allocation process, with a maximum degree of randomness when τ = 0. The recommended range for τ is between 2 and 4, striking a balance between too much variability (

τ < 2

), which slows the speed for the allocation proportion to converge to the desire allocation target, and too little variability (

τ > 4

), which increases predictability, which could in turn induce an undesirable selection bias.²⁹

3.3 Allocation rules

With reference to the DBCD treatment randomization scheme, we must construct the target allocation rule ρ, proportion allocated to treatment 1, according to certain desired allocation objectives. Here, we examine several viable candidates.

Rule 1: Balanced design (ρ₁). This is the most popular allocation rule in clinical studies. Under the DBCD framework

\begin{matrix} ρ_{1} = 1 / 2 \end{matrix}

Rule 2: Maximize power (ρ₂). As explained by Hu and Rosenberger,²⁹ the DBCD allocation rule to maximize test power for comparison of treatments 1 and 2 is equivalent to searching for a ρ function such that the variance $γ_{1}^{2} / n_{1} + γ_{2}^{2} / n_{2}$ is a minimum. It is then straightforward to show that

ρ_{2} = \frac{γ_{1}}{γ_{1} + γ_{2}}

Rule 3: Minimize failure (ρ₃). An ethical allocation target is to reduce the number of patients who undergo the inferior intervention. Hence, with fixed test power, or equivalently with fixed $γ_{1}^{2} / n_{1} + γ_{2}^{2} / n_{2}$ , a reasonable rule is one that minimizes $n_{1} / M_{1} + n_{2} / M_{2}$ . (Note that a larger median implies that the treatment is better, a convention that was included in Section 2 for convenience.) It is then easy to obtain the corresponding allocation rule ρ₃, where

ρ_{3} = \frac{\sqrt{M_{1}} γ_{1}}{\sqrt{M_{1}} γ_{1} + \sqrt{M_{2}} γ_{2}}

Rule 4: Modified Rule 3 (ρ₄). Rule 4 is a modification of Rule 3, with the median replaced by the standardized median to reduce the effects of extreme heterogeneity between the two treatments. Hence, the required minimization function is $\frac{n_{1}}{M_{1} / γ_{1}} + \frac{n_{2}}{M_{2} / γ_{2}}$ . The resulting rule becomes

ρ_{4} = \frac{\sqrt{M_{1} γ_{1}}}{\sqrt{M_{1} γ_{1}} + \sqrt{M_{2} γ_{2}}}

4 Simulation study

4.1 Comparisons of allocation rules under the latent Weibull model

A simulation study was conducted to study the performance of the allocation rules given in Section 3. Data were generated from a Weibull distribution and classified into five ordered categories (K = 5). Other values of K were also used, although these results are not presented here because the major findings are similar.

Without loss of generality, the parameters of the first treatment, η₁ and β₁, were fixed at 1. A variety of combinations of η₂ and β₂ were selected. The number of simulated replications was 10,000 for each parameter configuration. The total sample sizes n were selected to be 200 and 500. For Stage 1 of the DBCD procedure, we adopted restricted randomization, with $n_{0} = 60$ and $n_{0} = 100$ for n = 200 and n = 500, respectively. The value of τ is set at 2, and the thresholds for Weibull distribution are fixed at (0.5, 1.0, 1.5, 2.5). The hypothesis test to compare the two treatments was conducted at level $α = 0.05$ in each replication. The simulated results are shown in Tables 2 to 4. To compare the allocation rules, three major measures are used:

the proportion of patients who undergo the better treatment (i.e. the treatment with a larger median),

the simulated test power, and

the average proportion of patients in the worst category, $(n_{11} + n_{21}) / n$ .

Table 2.

Simulated treatment allocation under H₀ with $(η_{1}, β_{1}) = (1.0, 1.0)$ and $M_{1} = M_{2} = 0.693$ .

			Type I
$(n, n_{0})$	$(η_{2}, β_{2}, γ_{1}, γ_{2})$	Rule	Error Rate^a	$n_{1} / n$ (SD, $σ_{n_{1} / n}$ )	$\hat{ρ}$	ρ
(200, 60)	(1.0, 1.0, 0.911, 0.911)	1	0.049	0.500 (0.016, 0.016)	0.500	0.500
		2	0.050	0.503 (0.042, 0.043)	0.502	0.500
		3	0.052	0.505 (0.055, 0.061)	0.503	0.500
		4	0.052	0.502 (0.040, 0.042)	0.502	0.500
	(0.9827, 1.05, 0.911, 0.865)	1	0.049	0.500 (0.016, 0.016)	0.500	0.500
		2	0.050	0.514 (0.041, 0.042)	0.514	0.513
		3	0.051	0.516 (0.055, 0.059)	0.515	0.513
		4	0.050	0.508 (0.040, 0.041)	0.507	0.506
	(1.0195, 0.95, 0.911, 0.961)	1	0.050	0.500 (0.016, 0.016)	0.500	0.500
		2	0.046	0.490 (0.042, 0.044)	0.488	0.487
		3	0.052	0.492 (0.055, 0.062)	0.490	0.487
		4	0.051	0.496 (0.040, 0.043)	0.495	0.493
(500, 100)	(1.0, 1.0, 0.911, 0.911)	1	0.047	0.500 (0.010, 0.010)	0.500	0.500
		2	0.050	0.500 (0.027, 0.027)	0.500	0.500
		3	0.052	0.500 (0.034, 0.038)	0.500	0.500
		4	0.052	0.500 (0.026, 0.027)	0.500	0.500
	(0.9827, 1.05, 0.911, 0.865)	1	0.050	0.500 (0.010, 0.010)	0.500	0.500
		2	0.050	0.512 (0.027, 0.027)	0.512	0.513
		3	0.050	0.511 (0.036, 0.038)	0.512	0.513
		4	0.052	0.505 (0.026, 0.026)	0.506	0.506
	(1.0195, 0.95, 0.911, 0.961)	1	0.048	0.500 (0.010, 0.010)	0.500	0.500
		2	0.049	0.486 (0.027, 0.028)	0.486	0.487
		3	0.052	0.487 (0.036, 0.039)	0.486	0.487
		4	0.049	0.493 (0.026, 0.027)	0.493	0.493

SD: empirical standard deviation, $σ_{n_{1} / n}$ : asymptotic standard deviation (Appendix 2).

Simulated type I error rate.

The major findings are summarized below:

Proportion who underwent the better treatment. As demonstrated in Tables 3 and 4, Rules 3 and 4 are in general more capable of sending more patients to the better treatment, reflecting the inherent construction criteria of these two rules. Nevertheless, Rule 2, a function of γ, is constructed to maximize the test power. Hence, under Rule 2, it is possible that more patients receive the better or the worse treatment. For instance, in Table 3, when n = 200, $η_{2} = 2$ , and $β_{2} = 1$ (i.e. treatment 2 is better), treatment 1 allocation proportions $n_{1} / n$ for Rules 2, 3, and 4 are 0.364, 0.298, and 0.356, respectively; whereas for n = 200, $η_{2} = 0.8$ , and $β_{2} = 0.6$ (i.e. treatment 1 is better), the treatment allocation proportions $n_{1} / n$ for Rules 2, 3, and 4 are 0.468, 0.532, and 0.547, respectively. The contrast in Rules 3 and 4 is subtler. Although Rules 3 and 4 both assign more patients to the superior treatment in general, Rule 4 is less sensitive to the value of γ because the standardized median is being used. This specific feature can be revealed in cases in which γ₁ differs greatly from γ₂, where Rule 3 could possibly assign more patients to the inferior treatment. An example can be found in Table 3 with n = 200, $η_{2} = 0.9$ , and $β_{2} = 0.7, (M_{1}, γ_{1}) = (0.693, 0.911)$ and $(M_{2}, γ_{2}) = (0.533, 1.069)$ , the treatment allocation proportions $n_{1} / n$ for Rules 2, 3, and 4 are 0.457, 0.494, and 0.513, respectively. With this parameter configuration, Rules 2 and 3 both allocate more patients to the inferior treatment. Therefore, the performance of Rule 4 is superior if the target to allow fewer patients to undergo the inferior treatment is deemed to be the prime objective.

Table 3.

Simulated treatment allocation under H₁ ( $η_{1} = β_{1} = 1.0, M_{1} = 0.693, (n, n_{0}) = (200, 60)$ ) under latent Weibull model.

			Simulated
M ₂	$(η_{2}, β_{2}, γ_{1}, γ_{2})$	Rule	Power	$n_{1} / n$ (SD)	$\hat{ρ}$	ρ	$\frac{n_{11} + n_{21}}{n}$
0.442^a	(0.6, 1.2, 0.911, 0.544)	1	0.682	0.500 (0.016)	0.500	0.500	0.473
		2	0.700	0.619 (0.031)	0.623	0.626	0.454
		3	0.689	0.667 (0.037)	0.675	0.677	0.446
		4	0.698	0.613 (0.032)	0.617	0.618	0.455
0.434^a	(0.8, 0.6, 0.911, 1.067)	1	0.449	0.500 (0.016)	0.500	0.500	0.461
		2	0.458	0.468 (0.051)	0.464	0.460	0.469
		3	0.454	0.532 (0.086)	0.530	0.519	0.459
		4	0.455	0.547 (0.067)	0.546	0.538	0.457
0.533^a	(0.9, 0.7, 0.911, 1.069)	1	0.216	0.500 (0.016)	0.500	0.500	0.439
		2	0.218	0.457 (0.044)	0.457	0.460	0.444
		3	0.223	0.494 (0.062)	0.491	0.493	0.439
		4	0.219	0.513 (0.045)	0.511	0.513	0.439
0.924	(1.2, 1.4, 0.911, 0.824)	1	0.452	0.500 (0.016)	0.500	0.500	0.323
		2	0.452	0.515 (0.044)	0.514	0.525	0.326
		3	0.453	0.470 (0.061)	0.469	0.489	0.319
		4	0.454	0.465 (0.043)	0.463	0.477	0.319
0.883	(1.2, 1.2, 0.911, 0.928)	1	0.304	0.500(0.016)	0.500	0.500	0.344
		2	0.302	0.493 (0.043)	0.492	0.495	0.343
		3	0.302	0.461 (0.058)	0.458	0.465	0.339
		4	0.306	0.464 (0.041)	0.462	0.467	0.340
1.386	(2.0, 1.0, 0.911, 1.724)	1	0.956	0.500 (0.016)	0.500	0.500	0.307
		2	0.964	0.364 (0.042)	0.358	0.346	0.284
		3	0.960	0.298 (0.046)	0.285	0.272	0.271
		4	0.964	0.356 (0.036)	0.349	0.339	0.282
2.498	(3.0, 2.0, 0.911, 1.795)	1	≈1	0.500 (0.016)	0.500	0.500	0.212
		2	≈1	0.327 (0.052)	0.318	0.337	0.148
		3	≈1	0.223 (0.040)	0.197	0.211	0.110
		4	≈1	0.274 (0.034)	0.260	0.273	0.129

$M_{1} > M_{2}$ (treatment 1 is better).

Test power. As mentioned previously, the DBCD allocation scheme yields very competitive test power and is consistent with our simulation results. As shown in Tables 3 and 4, the test powers of all four rules are very similar. Although Rule 2 always attains the greatest power, the test powers of Rules 3 and 4 are only slightly lower than that of Rule 2. There is no significant gain or loss among the various choices of allocation rules.

Equal efficacy (under the null). When the two treatments have equivalent medians (refer to Table 2), all allocation rules can send roughly half the patients to each treatment, as expected. Furthermore, the simulated type I error rates are all close to the pre-specified level, 0.05.

Sample size n. First, for the simulated power, when the sample size is 500 (Table 4), the simulated test power is much greater than that when sample size is 200 (Table 3). Furthermore, the allocation proportion $n_{1} / n$ and the estimated allocation function $\hat{ρ}$ are much closer to the target ρ when the sample size is larger. Second, the empirical standard deviations of the allocation proportions are provided in the tables. As expected, Tables 3 and 4 show that as the sample size increases, the standard deviations grow smaller. Note that the standard deviations for Rules 2, 3, and 4 are all larger than that of Rule 1. Intuitively, this is reasonable because Rule 1 is constructed to target a balanced treatment allocation and is not a function of the estimated response distribution parameters. Hence, its resulting treatment allocation shows less variation attributed to the variation in the response outcomes.

Standard deviation of allocation proportion ( $σ_{n_{1} / n}$ ). The findings in Tables 2 to 4 indicate that Rule 3, which is related to the variance of the estimators of $\sqrt{M_{i}}$ and $γ_{i}, i = 1, 2$ , always produces higher standard deviation for the allocation proportion. For Rules 2 and 4, they are related to the variances of the estimators of γ_i and $\sqrt{M_{i} γ_{i}}$ , respectively. The formula to obtain the asymptotic standard deviation of the allocation proportion for treatment 1 is given in Appendix 2. In Table 2, these theoretical standard deviations are tabulated next to the empirical standard deviations for illustrative purposes. The simulated values are very close to the theoretical values.

Average proportion of patients in the worst category, $(n_{11} + n_{21}) / n$ . In clinical studies with categorical outcomes, a response-adaptive design should be considered more ethical if fewer patients eventually fall into the worst category, especially when the worst category implies a serious undesirable clinical consequence. The preferred adaptive allocation rule should be one that on average sends fewer patients to the worst category. In Tables 3 and 4, the empirical proportions of patients in the worst category, $(n_{11} + n_{21}) / n$ are tabulated. The reduction of this proportion is substantial when one of the treatments is much better in terms of efficacy. For instance, in Table 3, when n = 200, $η_{2} = 3$ , and $β_{2} = 2, (M_{1}, γ_{1}) = (0.693, 0.911)$ and $(M_{2}, γ_{2}) = (2.498, 1.795)$ (i.e. treatment 2 is significantly better in terms of median), the average proportions of patients in the worst category for Rules 1, 2, 3, and 4 are 0.212, 0.148, 0.110, and 0.129, respectively. Therefore, for Rules 3 and 4, there are respectively 48% and 39% reduction in the proportions of patients in the worst category when compared to Rule 1.

Tuning parameter τ. For the tabulated results, the value of τ is set to 2. However, other values of τ, such as 3 and 4, were also adopted. However, with no substantial differences in terms of the general features of the findings, the results are omitted.

In summary, although a balanced design is a popular approach in clinical studies, it is not a desirable treatment randomization scheme because it allows too many patients to undergo the inferior treatment when the efficacy of one of the treatments is much higher. Rule 4 is the better choice among the four allocation schemes because it has four advantages. First, it can allocate more patients to the better treatment and reduce the number of patients in the worst category. Second, the allocation of Rule 4 is more stable when the variances of two treatments differ substantially (i.e. very different values of γ₁ and γ₂). Third, as compared to Rule 3, Rule 4 consistently has a smaller standard deviation for the allocation proportion. Fourth, it attains a level of test power that is very competitive when compared to the other rules.

4.2 Comparisons of latent Weibull and latent normal models

In addition to the latent Weibull model, the latent normal model is a viable option for treatment comparisons when the responses are categorical.⁶ As indicated by Lu et al.,¹³ the latent normal model is able to serve as an appropriate inference tool when the data are not highly skewed. Hence, the rest of this section is devoted to comparing the latent normal model and the latent Weibull model with reference to their comparative performance in treatment allocation.

According to Model (1), assume that Z_i has a normal distribution with distribution function $G (z | ω_{i})$ where $ω_{i} = (μ_{i}, σ_{i}^{2})'$ . The comparison of treatment efficacy is conducted by testing the null hypothesis $H_{0} : μ_{1} = μ_{2}$ with test statistic

\begin{matrix} W_{2} = \frac{{\hat{μ}}_{1} - {\hat{μ}}_{2}}{\sqrt{\frac{v_{1}^{2}}{n_{1}} + \frac{v_{2}^{2}}{n_{2}}}} \to^{d} N (0, 1) \end{matrix}

where

{\hat{μ}}_{i}, \frac{v_{i}^{2}}{n_{i}}

are the corresponding maximum likelihood estimates of μ_i and variance of

{\hat{μ}}_{i}, i = 1, 2

. The formulas of these estimates are given in Lu et al.⁶

Parallel to the development of the allocation rules using the latent Weibull model, the allocation rules under the latent normal model are given in the following:

Rule 1: Balanced design ( $ρ_{1}'$ )

ρ_{1}' = ρ_{1} = 1 / 2

Rule 2: Maximize power ( $ρ_{2}'$ ). Similar to ρ₂, the allocation function is one that the variance $v_{1}^{2} / n_{1} + v_{2}^{2} / n_{2}$ is a minimum, leading to Rule 2, where

ρ_{2}' = \frac{v_{1}}{v_{1} + v_{2}}

Rule 3: Minimize failure ( $ρ_{3}'$ ). With fixed test power, to minimize failure, Zhang and Rosenberger³² proposed a failure measure $n_{1} / μ_{1} + n_{2} / μ_{2}$ , with the condition that both μ₁ and μ₂ are positive. However, the requirement that $μ_{i}, i = 1, 2$ , are both positive is too restrictive. Hence, Xu et al.³³ revised the failure measure to $\frac{n_{1}}{e^{μ_{1} / σ_{1}}} + \frac{n_{2}}{e^{μ_{2} / σ_{2}}}$ . The minimization of the above produces the following allocation rule

ρ_{3}' = \frac{v_{1} \sqrt{e^{μ_{1} / σ_{1}}}}{v_{1} \sqrt{e^{μ_{1} / σ_{1}}} + v_{2} \sqrt{e^{μ_{2} / σ_{2}}}}

For the simulation study, data are generated from normal distributions with threshold fixed at $(- 1.0, - 0.4, 0.4, 1.0)$ . We do not explore the situation where the data are generated from Weibull because Lu et al.¹³ demonstrated that the latent normal model is unable to control the type I error rate. In fact, when degrees of skewness of the two treatments are very different, the type I error rate of the latent normal model could be much inflated.

Three configurations of

μ_{1}, σ_{1}, μ_{2}, σ_{2}

under the alternative hypothesis are studied. The first,

(μ_{1}, σ_{1}, μ_{2}, σ_{2}) = (0, 1, 0.2, 1)

, represents a case where treatment 2 is tangentially better than treatment 1. In the second configuration,

(μ_{1}, σ_{1}, μ_{2}, σ_{2}) = (0, 1, - 0.5, 1.5)

, treatment 1 is moderately better. In the final configuration,

(μ_{1}, σ_{1}, μ_{2}, σ_{2}) = (- 0.5, 1.5, 0.5, 1)

, treatment 2 is substantively better. In addition, except for the first parameter configuration, the treatment variances are different. The findings are tabulated in Table 5.

Table 4.

Simulated treatment allocation under H₁ ( $η_{1} = β_{1} = 1.0, M_{1} = 0.693, (n, n_{0}) = (500, 100)$ ).

			Simulated
M ₂	$(η_{2}, β_{2}, γ_{1}, γ_{2})$	Rule	Power	$n_{1} / n$ (SD)	$\hat{ρ}$	ρ	$\frac{n_{11} + n_{21}}{n}$
0.442^a	(0.6, 1.2, 0.911, 0.544)	1	0.967	0.500 (0.010)	0.500	0.500	0.473
		2	0.973	0.622 (0.020)	0.623	0.626	0.454
		3	0.972	0.671 (0.025)	0.674	0.677	0.446
		4	0.972	0.615 (0.021)	0.616	0.618	0.455
0.434^a	(0.8, 0.6, 0.911, 1.067)	1	0.804	0.500 (0.010)	0.500	0.500	0.461
		2	0.814	0.453 (0.033)	0.452	0.460	0.469
		3	0.808	0.510 (0.053)	0.508	0.519	0.460
		4	0.802	0.533 (0.038)	0.532	0.538	0.458
0.533^a	(0.9, 0.7, 0.911, 1.069)	1	0.440	0.500 (0.010)	0.500	0.500	0.440
		2	0.441	0.450 (0.028)	0.449	0.460	0.444
		3	0.435	0.479 (0.042)	0.478	0.493	0.442
		4	0.436	0.504 (0.029)	0.504	0.513	0.440
0.924	(1.2, 1.4, 0.911, 0.824)	1	0.834	0.500 (0.010)	0.500	0.500	0.325
		2	0.844	0.513 (0.028)	0.513	0.525	0.327
		3	0.840	0.467 (0.041)	0.467	0.489	0.321
		4	0.841	0.462 (0.028)	0.462	0.477	0.320
0.883	(1.2, 1.2, 0.911, 0.928)	1	0.643	0.500 (0.010)	0.500	0.500	0.345
		2	0.650	0.490 (0.028)	0.490	0.495	0.344
		3	0.645	0.456 (0.038)	0.455	0.465	0.340
		4	0.648	0.461 (0.027)	0.461	0.467	0.341
1.386	(2.0, 1.0, 0.911, 1.724)	1	≈1	0.500 (0.010)	0.500	0.500	0.308
		2	≈1	0.358 (0.028)	0.356	0.346	0.284
		3	≈1	0.286 (0.032)	0.282	0.272	0.272
		4	≈1	0.350 (0.024)	0.348	0.339	0.282
2.498	(3.0, 2.0, 0.911, 1.795)	1	≈1	0.500 (0.010)	0.500	0.500	0.211
		2	≈1	0.319 (0.042)	0.316	0.337	0.145
		3	≈1	0.202 (0.036)	0.193	0.211	0.102
		4	$\approx 1$	0.263 (0.029)	0.259	0.273	0.125

$M_{1} > M_{2}$ (treatment 1 is better).

Table 5.

Comparison between latent normal and latent Weibull models in treatment allocation ( $(n, n_{0}) = (200, 60)$ , data are generated from two normal distributions with unequal means).

	Latent Weibull				Latent normal
$(μ_{1}, σ_{1}, μ_{2}, σ_{2})$	R^a	Power	$n_{1} / n$ (SD)	$\frac{n_{11} + n_{21}}{n}$	R^a	Power	$n_{1} / n$ (SD)	$\frac{n_{11} + n_{21}}{n}$
(0, 1, 0.2, 1)	1	0.263	0.50 (0.016)	0.141	1	0.262	0.50 (0.016)	0.141
	2	0.275	0.46 (0.050)	0.139	2	0.267	0.50 (0.041)	0.141
	3	0.278	0.43 (0.065)	0.138	3	0.267	0.47 (0.046)	0.140
	4	0.274	0.45 (0.045)	0.139
(0, 1, −0.5, 1.5)^b	1	0.776	0.50 (0.016)	0.263	1	0.712	0.50 (0.016)	0.263
	2	0.782	0.49 (0.050)	0.266	2	0.728	0.40 (0.044)	0.285
	3	0.784	0.55 (0.067)	0.253	3	0.723	0.44 (0.047)	0.275
	4	0.780	0.55 (0.046)	0.252
(−0.5, 1.5, 0.5, 1.0)	1	≈1	0.50 (0.016)	0.216	1	≈1	0.50 (0.016)	0.216
	2	≈1	0.45 (0.054)	0.200	2	$\approx 1$	0.59 (0.046)	0.243
	3	$\approx 1$	0.34 (0.064)	0.170	3	≈1	0.50 (0.053)	0.215
	4	≈1	0.37 (0.044)	0.177

Allocation rule.

$μ_{1} > μ_{2}$ (treatment 1 is better).

The major findings are summarized in the following:

When treatment means are similar, $μ_{1} = 0, μ_{2} = 0.2$ , and variances are identical, the adaptive procedures derived using the latent Weibull model or the latent normal model produce similar results. Treatment 2 is slightly better, and all response-adaptive allocation rules using Weibull and Rule 3 of the latent normal model are able to assign more patients to the better treatment. Because the treatment efficiencies are similar, the proportions of patients in the worst category are quite close for different choices of allocation rules.

When the variances of the treatments are different, the findings reveal the superiority of the latent Weibull model, especially when the treatment means are very different. With reference to the latent normal model, the response-adaptive rules are highly influenced by the variance of the data and undesirably sending more patients to the inferior treatments in most cases. Furthermore, as compared to the allocation rules using the latent Weibull model, the proportion of patients in the worst category is much higher. For example, $(n_{11} + n_{21}) / n$ with Rule 4 under the latent Weibull model when $(μ_{1}, σ_{1}, μ_{2}, σ_{2}) = (- 0.5, 1.5, 0.5, 1)$ is 0.177 while that with Rule 3 under the latent normal model is 0.215. This demonstrates an increase of about 21% of patients in the worst category. Note that this corresponds to the case where treatment 2 is much better and the test power to differentiate treatment efficacies is almost 1. The findings here convey an important message. That is, when treatment efficacies are very different, under the latent Weibull model and the proposed allocation rules, there will be fewer patients in the worst response category.

5 Clinical example

To demonstrate the advantages of response adaptive treatment allocation procedures, the clinical trial given in Section 2.1 has been redesigned to demonstrate the practical advantages of the proposed treatment randomization scheme. A simulation study is carried out using the response-adaptive treatment allocation scheme. The parameters for the simulation are estimated using the data collected in a clinical trial. In this example, the estimates of the parameters M₁ and γ₁ for treatment 1 (intensive recruitment strategy) are 0.857 and 1.031, respectively. For treatment 2 (moderate recruitment strategy), the estimates of the parameters M₂ and γ₂ are 0.562 and 0.778, respectively. The corresponding threshold estimates are 0.038, 0.235, 1.173, and 4.670, respectively. Moreover, the statistic for testing equal treatment efficacies is −2.960, and the corresponding p-value is 0.003, which is strong evidence against the null hypothesis, which indicates that intensive recruitment strategy can effectively reduce postoperative pulmonary complications.

Referring to the simulation results in Table 6, more patients would have undergone the better treatment if Rules 2, 3, and 4 of DBCD were used. Following the aforementioned arguments, Rule 4 is recommended. Under this scheme, 9% more patients will be allocated to the intensive recruitment strategy; therefore, 29 patients will undergo the better treatment if the proposed method is used. In terms of the proportion of patients in the worst category, the response-adaptive procedures are able to lower that by 5 to 7%.

Table 6.

Comparison of allocation rules for the clinical example ( $n = 320, n_{0} = 60$ ).

Allocation				Simulated
Rule	$n_{1} / n$ (SD)	$\hat{ρ}$	ρ	Power	$\frac{n_{11} + n_{21}}{n}$
1	0.500 (0.012)	0.500	0.500	0.841	0.039
2	0.588 (0.033)	0.586	0.589	0.855	0.037
3	0.633 (0.044)	0.633	0.639	0.846	0.036
4	0.592 (0.032)	0.591	0.597	0.852	0.037

6 Conclusions

The purpose of this study is to generate useful and pragmatic treatment allocation randomization schemes that outperform the conventional balanced design option for ordinal outcomes. For the comparison of the efficacies of two treatments, we show that a response-adaptive treatment allocation scheme is more ethical because it allows more patients to undergo the better treatment. In addition, each of the discussed allocation procedures has very similar test power. Among the various allocation rules introduced in this paper, Rule 4 appears to be the best candidate because it not only reduces the number of patients who receive the inferior treatment but it is also more robust to the differences between the standard errors of the median estimators of the two treatments. Hence, Rule 4 is the recommended treatment allocation scheme.

The latent Weibull model is an important component in the formulation of the proposed treatment allocation scheme. This is the first attempt in the literature to have applied the latent Weibull model for the derivation of response-adaptive treatment allocation schemes for ordinal responses. The allocation rules are derived using very general basic principles in DBCD. The asymptotic variance of the allocation proportion is also provided. The comparative study between the latent Weibull model and the latent normal model has illustrated the ethical advantage of the former with respect to proportion of patients being treated by the better treatment and the proportion of patients in the worst category. The latent Weibull model provides a much better estimation procedure than other comparable models, especially during the early stages of the treatment randomization process, when the number of observed outcomes is still small, thus increasing the likelihood of obtaining a very skewed distribution. In this paper, the randomization procedure is restricted to two treatments, and an exploration of response-adaptive randomization schemes for clinical trials with multiple treatments would be an interesting project.

Footnotes

Acknowledgements

We are grateful to the referees for many valuable suggestions.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Lu’s research was supported by the project of the National Social Science Foundation of China (no. 15BTJ016). The research of Poon and Cheung was supported by the Research Grants Council of the Hong Kong Special Administrative Region (CUHK14301118) and a direct grant from The Chinese University of Hong Kong.

References

Youdom

Samson

Basco

, et al. Multiple treatment comparisons in a series of anti-malarial trials with an ordinal primary outcome and repeated treatment evaluations. Malaria J 2012; 11: 147.

World Health Organization. Assessment and monitoring of antimalarial drug efficacy for the treatment of uncomplicated falciparum malaria. Geneva: World Health Organization, 2003.

Rossom

Solberg

Vazquez-Benitez

, et al. Predictors of poor response to depression treatment in primary care. Psychiatr Serv 2016; 67: 1362–1367.

Whitehead

. Sample size calculation for ordered categorical data. Stat Med 1993; 12: 2257–2271.

McCullagh

. Regression models for ordinal data. J R Statist Soc B 1980; 42: 109–142.

Poon

Cheung

. Multiple comparisons with a control for a latent variable model with ordered categorical responses. Stat Methods Med Res 2015; 24: 949–967.

Bekele

Thall

. Dose-finding based on multiple toxicities in a soft tissue sarcoma trial. J Am Stat Assoc 2004; 99: 26–35.

Leon-Novelo

Zhou

Bekele

, et al. Assessing toxicities in a clinical trial: Bayesian inference for ordinal data nested within categories. Biometrics 2010; 66: 966–974.

Poon

Wang

. Bayesian analysis of multivariate probit models with surrogate outcome data. Psychometrika 2010; 75: 498–520.

10.

Poon

. A latent normal distribution model for analysing ordinal responses with applications in meta-analysis. Stat Med 2004; 23: 2155–2172.

11.

Lin

Cheung

Poon

, et al. Pairwise comparisons with ordered categorical data. Stat Med 2013; 32: 3192–3205.

12.

Lin

Kwong

Cheung

, et al. Step-up testing procedure for multiple comparisons with a control for a latent variable model with ordered categorical responses. Stat Med 2014; 33: 3629–3638.

13.

Poon

Cheung

. Comparison of two treatments with skewed ordinal responses. Stat Med 2016; 35: 189–201.

14.

Poon

Cheung

. Multiple comparison of treatments with skewed ordinal responses. Comput Stat Data Anal 2016; 104: 223–232.

15.

Honarmand

Safavi

. Magnesium sulphate pretreatment to alleviate pain on propofol injection: a comparison with ketamine or lidocaine. Acute Pain 2008; 10: 23–29.

16.

L'Hermite

Dubout

Bouvet

, et al. Sore throat following three adult supraglottic airway devices: a randomised controlled trial. J Anaesthesiol 2017; 7: 417–424.

17.

Rosenberger

. The theory of response-adaptive randomization in clinical trials, New Jersey: Wiley, 2006.

18.

Montgomery

. From standardization to adaptation: clinical trials and the moral economy of anticipation. Sci Cult 2017; 2: 232–254.

19.

Zhang

. Asymptotic properties of doubly adaptive biased coin designs for multitreatment clinical trials. Ann Stat 2004; 32: 268–301.

20.

Leme

Hajjar

Volpe

. Effect of intensive vs moderate alveolar recruitment strategies added to lung-protective ventilation on postoperative pulmonary complications: a randomized clinical trial. J Am Med Assoc 2017; 14: 1422–1432.

21.

Anderson

Philips

. Regression, discrimination and measurement models for ordered categorical variables. Appl Stat 1981; 30: 22–31.

22.

Poon

Cheung

. A unified framework for the comparison of treatments with ordinal responses. Psychometrika 2014; 79: 605–520.

23.

Lawless

. Statistical models and methods for lifetime data., 2nd ed. Wiley: New Jersey, 2003.

24.

Vera

Ochoa

Romero

, et al. Intracanal cryotherapy reduces postoperative pain in teeth with symptomatic apical periodontitis: a randomized multicenter clinical trial. J Endod 2018; 44: 4–8.

25.

Park

JJH

Thorlund

Mills

. Critical concepts in adaptive clinical trials. Clin Epidemiol 2018; 10: 343–351.

26.

Berry

Petzold

Dull

, et al. A response adaptive randomization platform trial for efficient evaluation of Ebola virus treatments: a model for pandemic response. Clin Trials 2016; 13: 22–30.

27.

Das

. Re-inventing drug development: a case study of the I-SPY 2 breast cancer clinical trials program. Contemp Clin Trials 2017; 62: 168–174.

28.

Eisele

. The doubly adaptive biased coin design for sequential clinical trials. J Stat Plan Inference 1994; 38: 249–261.

29.

Rosenberger

. Optimality, variability, power: evaluating response-adaptive randomization procedures for treatment comparisons. J Am Stat Assoc 2003; 98: 671–678.

30.

Zhang

. Efficient randomized-adaptive designs. Ann Stat 2009; 37: 2543–2560.

31.

Rosenberger

Lachin

. Randomization in clinical trials: theory and practice, New York: Wiley, 2002.

32.

Zhang

Rosenberger

. Response-adaptive randomization for clinical trials with continuous outcomes. Biometrics 2006; 62: 562–596.

33.

Gao

, et al. Response-adaptive treatment allocation for non-inferiority trials with heterogeneous variances. Comput Stat Data Anal 2018; 124: 168–179.