Computational and Mathematical Methods in Medicine

Computational and Mathematical Methods in Medicine / 2018 / Article

Research Article | Open Access

Volume 2018 |Article ID 8025827 | 15 pages | https://doi.org/10.1155/2018/8025827

Maximin Efficiencies under Treatment-Dependent Costs and Outcome Variances for Parallel, AA/BB, and AB/BA Designs

Academic Editor: Reinoud Maex
Received26 Jan 2018
Accepted19 Jul 2018
Published01 Oct 2018

Abstract

If there are no carryover effects, AB/BA crossover designs are more efficient than parallel (A/B) and extended parallel (AA/BB) group designs. This study extends these results in that (a) optimal instead of equal treatment allocation is examined, (b) allowance for treatment-dependent outcome variances is made, and (c) next to treatment effects, also treatment by period interaction effects are examined. Starting from a linear mixed model analysis, the optimal allocation requires knowledge on intraclass correlations in A and B, which typically is rather vague. To solve this, maximin versions of the designs are derived, which guarantee a power level across plausible ranges of the intraclass correlations at the lowest research costs. For the treatment effect, an extensive numerical evaluation shows that if the treatment costs of A and B are equal, or if the sum of the costs of one treatment and measurement per person is less than the remaining subject-specific costs (e.g., recruitment costs), the maximin crossover design is most efficient for ranges of intraclass correlations starting at 0.15 or higher. For other cost scenarios, the maximin parallel or extended parallel design can also become most efficient. For the treatment by period interaction, the maximin AA/BB design can be proven to be the most efficient. A simulation study supports these asymptotic results for small samples.

1. Introduction

The standard design of a randomized clinical trial is the parallel group design: subjects are randomly assigned to one of two treatments, say A or B. An alternative, well-known design is the AB/BA crossover trial in which subjects receive both treatments, A and B, but the sequencing of the treatments is opposite for two randomly allocated groups [1, 2]. An AB/BA crossover trial is considered most suited when examining treatments for chronic or ongoing diseases, such as rheumatism, chronic obstructive pulmonary disease, or (frequent) heartburn. In these cases, there is no real possibility that the disease gets cured, and the aim is to moderate the effects of the disease [2]. A third design that we will consider involves treatment sequences AA and BB. This design extends the parallel design across two treatment periods, allows for testing treatment by time interaction effects, and is a realistic alternative for the AB/BA design in case the treatment regime should continue.

If the outcome variable is continuous and (approximately) normally distributed, the data can be analyzed by mixed effects regression [3]. Of primary interest is testing the treatment effect of, for instance, a new medication for chronic obstructive pulmonary disease. A relevant issue then is which design is the most efficient in estimating the treatment effect, thereby yielding maximum power for testing this effect. Such optimality has already been examined when comparing crossover and parallel designs [2] and when comparing all three designs introduced before [4, 5]. If there are no carryover effects and no dropouts, the sample sizes are equal and equally allocated to the treatments, an AB/BA design yields more efficient estimates of the treatment effect than a parallel and extended parallel design and consequently, will yield more power to test this effect.

The present study extends results on the relative efficiencies of these designs in that (a) optimal instead of equal treatment allocation is examined, (b) allowance is made for treatment-dependent outcome variances, and (c) next to treatment effects, also treatment by period interaction effects are examined. Outcome variances may differ between treatments [6, 7]. This also is to be expected if treatments differ in terms of their effectiveness. Furthermore, since research costs and outcome variances may differ between treatments, equal allocation to treatments may not be the most efficient. The issue then is how to allocate subjects to treatments such that a design’s efficiency is optimized, and how different designs relate in terms of efficiency under such optimal allocation. Optimal allocation requires a priori knowledge on parameters of the analysis model, that is, intraclass correlations for the mixed effects model that we consider. Since this knowledge typically is rather vague, optimal allocations and corresponding efficiencies for maximin versions of the (extended) parallel design and crossover design will be derived. These maximin designs guarantee a power level across plausible ranges of the intraclass correlations at the lowest research costs.

In designs where treatments are successively given to the same group of subjects, carryover may occur. For the AB/BA trial, it may be that, in the AB sequence treatment, A still has an effect on the outcome, when B has been given and the second measurement is done. When in the BA sequence, the effect of B is present, once A has been administered and this effect differs from the carryover effect for the AB sequence, differential carryover occurs. The present study assumes that differential carryover can be safely excluded or is negligible and that this effect does not need to be estimated in analyzing the data.

The paper is structured as follows. Section 2 will present the linear mixed model for analyzing data from each of the three designs. Section 3 will introduce the efficiency criterion and will provide asymptotic expressions for this criterion in the case of maximum likelihood estimation of the treatment effect. Starting from a flexible cost function, optimal allocations to treatments will be derived as well as resulting design efficiencies. Since the efficiencies depend on the intraclass correlations and knowledge on these parameters is often limited, in Section 4, we will derive maximin designs. Section 5 will show to what extent the asymptotic efficiencies translate into desired power levels for small sample sizes. Section 6 will give an application of the results, and Section 7 will discuss some issues for further research.

2. Linear Mixed Effects Models

In the case of a parallel design, an extended parallel design, and a crossover design, the subjects are randomly allocated to one of the two arms. In a parallel design to treatment A or treatment B, in an extended parallel design, they are allocated to treatment sequence AA or BB, and in a crossover trial to treatment sequence AB or BA. We consider a quantitative outcome variable, denoted as yij for person j () at measurement occasion i, and assume yij is (approximately) normally distributed.

For a parallel design and outcome variances that differ between treatments A and B, simple linear regression with heterogeneous variances may be an adequate tool for data analysis:where treatment is coded 0 for persons having treatment A and coded 1 for persons having treatment B, and and are normally distributed, with mean 0 and variances and , respectively. The random terms and can be thought of as consisting of a random person (between-subject) effect, , and a treatment-dependent random error (within-subject) effect, and . In formula, , and . These two sources of random variation cannot be separated in a single-period parallel trial.

For a crossover AB/BA design and an extended, two-period, parallel AA/BB design, however, the variances of and of and can be identified. The linear regression model can then be extended with a random intercept as well as a fixed effect of time, yielding the following mixed effects model:

In (2), time is coded 0 for observations at the first measurement and coded 1 for observations at the second measurement. The random terms , , and are independently normally distributed, with mean 0 and variances , , and , respectively. Their relation with the variances in (1) is for treatment A and for treatment B.

In the case we want to examine whether there is an interaction between treatment and period, the model in (2) is extended as follows:where represents the treatment by period interaction effect. The parameters in (1)–(3) can be estimated through maximum likelihood (ML). In what follows, we are interested in optimally estimating in (1) and (2), which will be denoted as , and in optimally estimating in (3), which will be denoted as . A relevant concept is the intraclass correlation, which is between-subject variation on the outcome as compared to the total outcome variation. For the models in (2) and (3),this can be expressed as and for treatments A and B, respectively. The larger the person (between-subject) variance as compared to the error (within-subject) variance, the larger the intraclass correlations. Note that we assume a common between-subject variance, but allow for treatment-dependent within-subject variances, leading to treatment-dependent within-subject correlations. We also define a variance ratio , which can be expressed as a ratio of the intraclass correlations, .

3. Optimal Allocations and Corresponding Design Efficiencies

Let denote the variance of the estimator of the treatment effect in (1) or (2) or the variance of the treatment by the period interaction effect in (3), given a design . The efficiency of an estimator of is defined as the inverse of its variance, that is, . In the sequel, we will consider the efficiency of one design, , versus another design, , which is defined as and denoted as the relative efficiency. Since no closed-form expressions are available for the variances of the maximum likelihood (ML) estimator, asymptotic variances of the ML estimator were derived (Appendices A.1 and A.2).

The optimal allocation to treatments minimizes the variance of the estimator of in (1) or (2) and of in (3), given a fixed research budget. Note that changing the coding of the treatment factor or the time factor in (1)–(3), for instance into 1 versus −1 instead of 1 versus 0, will not affect the optimal allocation. Such a change of coding leads to a linear transformation of or , and this will change the variance of their estimators only by a multiplicative constant. This implies that allocations that minimize the variance of the estimators do not depend on the coding of treatment and time.

To derive the optimal allocations under a budget restriction, we need to define a budget function. Let the costs involved with each subject in the parallel design be csp euros, in an extended parallel design be csep euros, and in a crossover design be csc euros. These costs may represent financial rewards given to subjects for participating in the trial but also the (average) costs of recruiting a subject. Furthermore, for treatments A and B there are, for each subject, costs cA and cB, respectively, and each measurement may involve ct euros. Finally, attached to each treatment sequence, there may be administration costs cts.

In the case of allocation proportions pA for treatment A and pB = 1 − pA for treatment B in a parallel design having subjects, the following budget is required:

For the designs that we consider, this budget function can be reparametrized such that it is the same as the cost function given by Yuan and Zhou [8], thereby generalizing the cost function proposed by Brown [9] and Berger and Wong [4].

For an AB/BA crossover design, involving subjects and allocation proportions pAB for treatment sequence AB and pBA = 1 − pAB for treatment sequence BA, noting that each subject receives both treatment A and B and is measured twice, the following budget is required:

Finally, the required budget for an AA/BB design, involving subjects, with allocation proportions pAA and pBB = 1 − pAA for the treatment sequences AA and BB, respectively, is as follows:

Note that, for the functions in (4)–(6), the budget may simply be the total number of observations involved in a study, by setting ct = 1 and the other costs to 0. It can also represent the total number of subjects involved, by setting csp = csep = csc = 1 and the remaining costs to 0.

In what follows, we will assume that the subject-specific costs of the two-period designs are the same; that is, csep = csc = cs_2p. Since subjects in these designs receive two treatments and a washout period may be involved, these costs are very likely larger than those of a parallel design. We also assume that the subject-specific costs for the two-period designs will not exceed 2 times the subject-specific costs for the parallel design, so that csp ≤ cs_2p ≤ 2csp. Finally, since each design involves two treatment sequences, the administration costs are the same for each of the three designs considered, and thus, the budgets that are available for remaining costs are identical; that is, the budget C =  − 2cst is the same for each design.

3.1. Treatment Effect

For treatment effect estimation, the optimal allocations to the treatment sequences are derived in Appendix B. The optimal allocations and corresponding (asymptotic) variances of the treatment effect estimators are shown in the second and third column of Table 1, respectively. The optimal allocation ratios of the parallel and the extended parallel design depend on the costs and intraclass correlations: the more the expensive treatment A (or the cheaper treatment B) and the larger the intraclass correlation in treatment A (or the smaller the intraclass correlation in treatment B), the more the subjects have to be assigned to treatment B. The optimal allocation ratio for a crossover design is 1, which may be expected, since both groups receive both treatment A and B.


Treatment effect
DesignAllocation ratio (n1/n2)

Crossover design1
Parallel design
Extended parallel design

Treatment by period interaction
DesignAllocation ratio (n1/n2)

Crossover design1
Extended parallel design

Note. n1: sample size for A (parallel design), AB (crossover design), or AA (extended parallel design) sequence; n2: sample size for B (parallel design), BA (crossover design), or BB (extended parallel design) sequence;
3.2. Treatment by Period Interaction Effect

In the case the treatment by period interaction effect is of primary interest, the optimal allocations can be derived along lines similar to the derivations for the treatment effect (Appendix B). The allocations and corresponding optimal variances are displayed in Table 1. Note that, similar to treatment effect estimation, the allocation ratio for a crossover design is 1, whereas the allocation ratio for an extended parallel design depends on the treatment costs and intraclass correlations, such that more persons are allocated to treatment sequence AA if the intraclass correlation of A decreases, the intraclass correlation of B increases, the costs of treatment A decrease, or the costs of treatment B increase.

4. Maximin Designs

Choosing the optimal allocation requires knowledge on the intraclass correlations and (remember that the variance ratio is fixed if and are given). Commonly, there is only limited knowledge on these parameters. A possible solution is the maximin strategy [4], consisting of 2 steps: (1) for each design determine the minimum efficiency of the effect estimator across the plausible ranges for the intraclass correlations and and (2) choose that design which maximizes this minimum efficiency. Such a design optimizes a worst case scenario and is called a maximin design. The maximin strategy implies choosing the design that minimizes the maximum variance of the estimator of the effect of interest. In determining sample sizes, choosing values for the intraclass correlations and within their plausible ranges (and thus a variance ratio within its plausible range) for which the variance is maximum will guarantee the desired power level also for all other values of these parameters. Moreover, the maximin design guarantees this power level at the lowest research costs. In what follows, we will refer to ranges of and that have lower bounds and and upper bounds and , respectively.

4.1. Treatment Effect

From the asymptotic variances in Table 1, one can derive for which values of and (and thus for which value of the variance ratio ), the variance of the treatment effect estimator is maximized. These derivations are given in Appendix C. The maximin parameter values and corresponding variances for the treatment effect estimator under optimal allocation to the treatments are shown in Table 2. The corresponding optimal allocations for the maximin designs are obtained by substituting the maximin parameter values of Table 2 into the allocation ratios as given in Table 1.


DesignMaximin parameter values for the maximin design

Crossover design

Parallel design

Extended parallel design

Note. .

If for a parallel design the maximin value for the variance ratio  =  is within the plausible range for , that is, [] and cs_2p ≤ 2csp, then a parallel design is always less efficient than a maximin crossover design. If for an extended parallel design the maximin value for one of the intraclass correlations is within the plausible range for the corresponding intraclass correlation, then also this design is less efficient than a maximin crossover design. For other scenarios, the relations between the maximin designs are more complicated, depending on the ranges for and , the costs of treatments, subject recruitment, and measurement.

A systematic numerical evaluation was done to examine under what conditions the crossover design is the best choice in terms of efficiency. For and , we consider ranges of width 0.10 (small), 0.30 (medium), and 0.60 (large). The lower bounds were {}, where the largest possible lower bound was determined by the width of the range under consideration. For instance, if the range is 0.30 (medium), the largest lower bound for the intraclass correlation is 0.70. All combinations of small, medium, and large ranges for and were considered. The values of the variance ratio thus considered vary from 1/100 to 100. Since in most crossover trials, the intraclass correlation exceeds 0.30 [13, 10, 11], ranges with lower bounds of 0.30 or higher are empirically most relevant. The empirical evidence on the costs cA, cB, ct, csp, and cs_2p is scarce, and we thus choose costs covering a wide range of scenarios. Let CRA = (cA + ct)/csp, CRB = (cB + ct)/csp, and CRp = cs_2p/csp (note that the relative efficiencies of the maximin designs depend only on these cost ratios). CRA and CRB take on the values 100, 20, 10, 1, 0.1, 0.05, and 0.01. For CRp, we consider 1 and 2.

If the costs of treatments are identical between the treatment arms, that is, CRA = CRB, for most scenarios examined, the crossover maximin design turns out to be most efficient. For CRp = 1 and CRA = CRB ≤ 1, the crossover design always is the most efficient. For CRp = 1 and CRA = CRB > 1, or CRp = 2, only if the lower bound of one of the intraclass correlations is 0.05 or lower and the ranges of the intraclass correlations do not overlap, the parallel design can become most efficient. Since in most empirical studies the intraclass correlations will exceed 0.05, this implies that, for equal costs of treatments, the crossover maximin design will almost always be the most efficient design.

In the case the treatment costs differ and CRA ≤ 1 and CRB ≤ 1, only in the case the lower bound of one of the intraclass correlations is 0.10 or smaller, the parallel or the extended parallel maximin design can become most efficient. The extended parallel design can only become most efficient if CRp = 1. Hence, in all scenarios with unequal treatment costs and CRA ≤ 1 and CRB ≤ 1, for intraclass correlations of 0.15 or higher, the maximin crossover design is most efficient.

In the case the treatment costs differ and CRA > 1 or CRB > 1, the maximin crossover design is less often most efficient. For these cost scenarios, also for ranges of intraclass correlations exceeding 0.15, the maximin parallel and extended parallel design may become more efficient. This especially occurs if the costs of treatment A and lower bound of the range of are both larger (or smaller) than the costs of treatment B and lower bound of , respectively. The efficiency improvement is large if treatment A is much more expensive than treatment B and if the costs of treatments and measurements are large compared to the subject-related costs. This is illustrated in Figure 1. The top row shows that if the costs of treatment A are larger than the costs of treatment B and the lower bound of is larger than the lower bound of , a parallel design is most efficient, even up to an upper bound 1 of if CRA = 100. As can also be seen, the upper bound of is not very relevant in terms of the relative efficiencies. The left plot of the middle row of Figure 1 shows that if the lower bounds of and are equal, then for almost all upper bounds of , the crossover design is most efficient. Again, as can be seen in the rightmost plot of the middle row, if the lower bound of is higher than the lower bound of , then for higher upper bounds of , the parallel design is most efficient but to a lesser extent as compared to a smaller lower bound of . As is evident from the four subplots in the top and middle row, when increasing the ratio CRA/CRB, the crossover design becomes less efficient as compared to the other two designs. The subplots of the bottom row furthermore show that the crossover design also becomes less efficient compared to the other designs if CRA and CRB increase while the ratio CRA/CRB remains constant. This illustrates that the efficiency of the other designs relative to the crossover design becomes larger if the costs of treatments and measurements are large compared to the subject-related costs. However, to summarize, if the treatment costs differ and CRA > 1 or CRB > 1, no simple rules of the thumb emerge and the most solid way to choose the most efficient design is just to calculate the maximin variances as given in Table 2.

Finally, if CRp = 2, the maximin parallel design is consistently more efficient than the maximin extended parallel design (as is illustrated in Figure 1). If CRp = 1, the maximin extended parallel design can also become more efficient than the maximin parallel design.

4.2. Treatment by Period Interaction Effect

The maximin parameter values and corresponding variances of the estimator of the treatment by period interaction effect are shown in Table 3. The derivations of these results can be done along lines similar to the derivations for the treatment effect estimator (Appendix C). The optimal allocation for the extended parallel design is obtained by substituting the maximin parameter values in the expression for the allocation ratio in Table 1. For a crossover design, the allocation ratio is 1. The maximin efficiency of an extended parallel design is always higher than that of a crossover design if the maximin value is within the plausible range for . This follows fromwhere the right-hand side of the inequality in turn is smaller than the variance of a maximin crossover design (Table 3). The higher maximin efficiency of the extended parallel design can also be shown to hold if the maximin value is within the plausible range for . Furthermore, if the variance maximizing values and are outside the plausible ranges for and , respectively, then values for and that coincide with one of the borders of their corresponding ranges should be chosen as values that maximize the variance. But in that case, even smaller variances result for the extended parallel design.


DesignMaximin parameter values

Crossover design

Extended parallel design

Note.