Designing dose-finding studies with an active control for exponential families

Abstract

Optimal design of dose-finding studies with an active control has only been considered in the literature for regression models with normally distributed errors and known variances, where the focus is on estimating the smallest dose that achieves the same treatment effect as the active control. This paper discusses such dose-finding studies from a broader perspective. We consider a general class of optimality criteria and models arising from an exponential family. Optimal designs are constructed for several situations and their efficiency is illustrated with examples.

1. Introduction

Dose-finding studies are important for investigating the effect of a compound on a response and have applications in various fields. They are of particular importance in drug development, because marketed doses have to be safe and provide clinically relevant efficacy (Ting, 2006). In most of the statistical literature on dose-response studies, a placebo is included as a control group (Bretz et al., 2008), and numerous authors have worked on optimal designs in such applications because the use of efficient designs can substantially increase the accuracy of statistical analysis (Dragalin et al., 2007; Miller et al., 2007; Bornkamp et al., 2011). Dose-response studies that include a marketed drug as an active control are becoming more popular, especially in preparation for active-controlled confirmatory noninferiority trials, where the use of a placebo may be unethical. Hence, there is growing interest in active-controlled studies. For example, Helms et al. (2015) investigated the finite-sample properties of maximum likelihood estimators of the target dose in an active-controlled study which achieves the same treatment effect as the active control. However, optimal design problems for such studies have only been considered in one paper (Dette et al., 2014). These authors investigated optimal designs for estimating the target dose under the assumption of a normal distribution with known variances. In particular, they demonstrated the superiority of the optimal designs over standard designs used in pharmaceutical practice.

In this paper we investigate optimal design problems for dose-finding studies with an active control from a more general perspective. First, we consider a general class of optimality criteria. Second, we study exponential families for modelling the distribution of the responses of the new drug and the active control, as in practice the assumption of normally distributed responses is often hard to justify. This enables the design of experiments for active-controlled studies with discrete data, as motivated by the consulting projects described in the next paragraph. Third, we demonstrate that even when the assumption of a normal distribution is justifiable, the estimation of the variances has a nontrivial effect on the optimal designs for an active-controlled study.

The research in the present paper is motivated by two examples where the assumption of normally distributed responses made by Dette et al. (2014) is hard to justify. The first example involves a 24-week dose-ranging Phase II study in patients with gouty arthritis to determine the target dose of a compound in preventing signs and symptoms of flares in chronic gout patients starting allopurinol therapy. The primary endpoint is the number of flares occurring per subject within 16 weeks of randomization, which is modelled using a negative binomial distribution for all treatment arms. The second example is a Phase IIb multicentre, randomized, double-blind, active-controlled dose-finding study in the treatment of acute migraine, as measured by the percentage of patients reporting freedom from pain at two hours post-dose.

For brevity, in this paper we restrict our attention to locally optimal designs (Chernoff, 1953). Following Chaloner and Verdinelli (1995) and Dette (1997), the methodology introduced in the present paper can be further developed to address uncertainty in the preliminary information for the unknown parameters.

2. Modelling active-controlled studies using exponential families

Consider a clinical trial in which patients are treated either with an active control, i.e., a standard treatment administered at a fixed dose level, or with a new drug using different dose levels in order to investigate a dose-response relationship. Let and denote the numbers of patients randomized to the new drug and to the active control, respectively. We determine the optimal number of different dose levels for the new drug, the dose levels themselves, and the optimal number of patients allocated to each dose level.

More formally, we assume that different dose levels, , are chosen in a dose range, , for the new drug, and that at each dose level the experimenter can investigate patients (), where . The optimal numbers , or more precisely the optimal proportions , will be determined by the design. The responses are modelled as realizations of independent real-valued random variables (; ). Similarly, the responses of patients treated with the active control are modelled as realizations of independent real-valued random variables , where all responses are assumed to be independent. We further assume that the random variables and have distributions from an exponential family, with the densities of the variables depending on the dose levels :

(1)

(2)

Here, and are unknown parameters, and we use common terminology for exponential families (Brown, 1986). In particular, the functions , , and are assumed to be twice continuously differentiable, with , and and denote - and -dimensional statistics defined on the corresponding sample spaces. The functions and are assumed to be positive and measurable.

Throughout the paper, let denote a variable indicating whether a patient receives the new drug, , or the active control, . Further, let

(3)

denote the design space of the experiment, where is the dose range for the new drug, is the dose level of the active control, and the second component of an experimental condition determines the treatment, or 1. The Fisher information matrix at the point is

(4)

where denotes a matrix of appropriate dimension with all entries equal to 0, is the vector of all parameters, is the indicator function of the event , and the matrices and are the Fisher information matrices of the two models (1) and (2), that is,

(5)

where the random variables and have densities (1) and (2). The Fisher information matrix in (4) is block-diagonal because of the independence of the samples, given that a patient is randomized to either the new drug or the active control.

Example 1 —

To demonstrate the different structures of the Fisher information matrix arising from different distributions of the exponential family, we consider several examples. Here we restrict our attention to the normal and negative binomial distributions. Further examples can be found in the Supplementary Material.

Dette et al. (2014) investigated normally distributed responses with known variances for the new drug and for the active control. For the expectation of the response of the new drug at dose level they assumed a nonlinear regression model, where , while the expectation of the response of the active control is assumed to equal for the active control. If the variances must be estimated from the data, we have and for the parameters in models (1) and (2), respectively. The Fisher information matrix at a point is given by (4) with

Next, assume a negative binomial distribution with parameter and a function for the probability of a success of the new drug at dose level , and parameters and for the active control. Then we have , and the Fisher information matrix is given by (4) with

Here the parameters are assumed to be known.

Throughout this paper we consider approximate designs in the sense of Kiefer (1974), which are defined as probability measures with finite support on the design space in (3). Therefore, an experimental design is given by

(6)

where are positive weights such that . Here, denotes the relative proportion of patients treated at dose level () or, for , the active control. If observations are taken, a rounding procedure is applied to obtain integers () and from the possibly real-valued quantities (). The information matrix of an approximate design of the form (6) is defined by the matrix

(7)

Here, the matrix and the matrix are given by

and (5), respectively, where

(8)

denotes the design on the design space for the new drug, which is induced by the design in (6) defining the weights (). If observations are taken according to an approximate design, then the covariance of the maximum likelihood estimators and in models (1) and (2) is given approximately by , and consequently optimal designs maximize an appropriate functional of the information matrix defined in (7).

In order to discriminate between competing designs, we consider Kiefer's -criteria (Kiefer, 1974). To be precise, let and let denote a matrix of full column rank . Then a design is said to be locally -optimal for estimating the linear combination in a dose-response model with an active control if is estimable by the design , i.e., , and maximizes the functional

(9)

among all designs for which is estimable, where and denote the trace and a generalized inverse of the matrix , respectively. The cases and correspond to the D- and E-optimality criteria, i.e., and , where denotes the smallest eigenvalue of the matrix . An application of the general equivalence theorem (Pukelsheim, 2006, §§7.19 and 7.21) yields the following result.

Lemma 1 —

If , a design with is locally -optimal for estimating the linear combination in a dose-response model with an active control if and only if there exists a generalized inverse of the information matrix such that the inequality

(10)

holds for all .

If , a design with is locally -optimal for estimating the linear combination if and only if there exist a generalized inverse of the information matrix and a nonnegative-definite matrix with such that the inequality

(11)

holds for all .

Moreover, there is equality in and for all support points of the design .

Below we assume that either or the matrix has the form

(12)

with elements and , such that . Roughly speaking, the choice or a block-diagonal structure of the matrix in (12) leads to a separation of the parameters from models (1) and (2) in the corresponding optimality criterion. Hence, optimal designs for dose-finding studies with an active control can be obtained from optimal designs for dose-finding studies that include a placebo group, which maximize the criterion

(13)

in the class of all designs for which is estimable, i.e., . We call these designs -optimal for estimating the parameter in the dose-response model (1). The proof can be found in the Appendix.

Theorem 1 —

Suppose that , the matrix is given by , and

(14)

is a locally -optimal design for estimating in . Then the design

is locally -optimal for estimating in the dose-response model with an active control, where the weights are

(15)

with

(16)

The case is interpreted as the corresponding limit.

In the case a more general statement is available, without the restriction to block matrices of the form . The proof is obtained using similar arguments to those in the proof of Theorem 1; we therefore omit it.

Theorem 2 —

Assume that with and , and let denote the -optimal design for estimating the parameter . Then the design defined in Theorem 1 is locally -optimal for estimating in the dose-response model with an active control.

The final result of this section concerns the special case of . The result is a direct consequence of Theorem 1, upon considering a corresponding limit and observing that the quantity defined in satisfies .

Corollary 1 —

Assume that the matrix is given by , and let denote the locally D-optimal design of the form for estimating the parameter in the dose-response model , which maximizes in the class of all designs for which is estimable. Then the design with masses and at the points and , respectively, is locally D-optimal for estimating the parameter in the dose-response model with an active control.

Remark 1 —

The assumption of a block matrix in Theorem 1 and Corollary 1 cannot be omitted. A counterexample is given in the Supplementary Material.

3. D-optimal designs for the MichaelisMenten and models

In this section we determine some D-optimal designs under different distributional assumptions for the Michaelis–Menten model and the model , where the dose range is the interval . Both models are widely used in practice and are therefore treated in detail. If the dose-response function describes a probability, some restrictions on the parameters are required. For example, if denotes the success probability for the negative binomial distribution in Example 1, then we implicitly assume that in the following discussion. In other models similar assumptions have to be made, but for brevity we do not mention them explicitly. In the following, denotes the maximum of .

Theorem 3 (Michaelis–Menten model) —

1. If the distributions of the responses corresponding to the new drug and active control are normal with parameters and , respectively, then the locally D-optimal design for the dose-response model with an active control allocates of patients to each of the dose levels and of the new drug and of patients to the active control.

2. If negative binomial distributions with probabilities and are used, then the locally D-optimal design for the dose-response model with an active control allocates of patients to each of the dose levels and of the new drug and of patients to the active control.

3. If binomial distributions with probabilities and are used, then the locally D-optimal design for the dose-response model with an active control allocates of patients to each of the dose levels and of the new drug and of patients to the active control.

4. If Poisson distributions with parameters and are used in and , then the locally D-optimal design for the dose-response model with an active control allocates of patients to each of the dose levels and of the new drug and of patients to the active control.

The proof of Theorem 3 follows directly from Corollary 1, if the locally D-optimal designs for model (1) are known. For example, in the case of a normal distribution, it follows from Dette et al. (2010) that the D-optimal design for the Michaelis–Menten model has equal masses at the points and , which yields statement (i) of Theorem 3. In the other cases, the D-optimal designs for model (1) are not known and the proof can be found in the Appendix.

The differences in the D-optimal designs derived under different distributional assumptions can be substantial. For example, if the design space is with a large right boundary , the nontrivial dose level for the new drug is approximately under the assumption of a normal distribution and under a Poisson distribution.

We now give the corresponding results for the model. The proof uses similar arguments and is therefore omitted.

Theorem 4 ( model) —

1. If the distributions of responses corresponding to the new drug and active control are normal with parameters and , respectively, then the locally D-optimal design for the dose-response model with an active control allocates of patients to each of the dose levels , and of the new drug and of patients to the active control.

2. If negative binomial distributions with probabilities and are used, then the locally D-optimal design for the dose-response model with an active control allocates of patients to each of the dose levels , and of the new drug and of patients to the active control, where is the solution of the equation

   
3. If binomial distributions with probabilities and are used, then the locally D-optimal design is of the same form as in (ii), where is the solution of the equation

   
4. If Poisson distributions with parameters and are used in and , respectively, then the locally D-optimal design is of the same form as in (ii), where

   

   with and .

Example 2 —

We now discuss D-optimal designs for the two clinical trials described in §1.

First let us consider the gouty arthritis example. The primary endpoint is modelled by a negative binomial distribution with parameters and for the new drug and parameters and for the comparator. The dose range is  mg, and we obtained from the clinical team the following preliminary information on the unknown parameters: , , , , and . In addition, are fixed. The D-optimal design is obtained from Theorem 4 and summarized in Table 1. The standard design actually used in this study allocates of patients to each of the dose levels 25, 50, 100, 200 and 300 mg of the new drug and of patients to the active control. To compare these designs, we also show in Table 1 the D-efficiency , where is the locally D-optimal design. We observe that, in this example, the optimal design is substantially more efficient than the standard design.

Next, consider the acute migraine example, which measured the percentage of patients reported to be free of pain two hours post-dose. We assume a binomial distribution for this case. The probabilities of success are for the new compound, where the dose level varies in the interval  mg, and for the active control. The sample sizes are and , and the preliminary information obtained from the clinical team consists of , and . The locally D-optimal designs under normal and binomial distributional assumptions are listed in Table 1. The design actually used for this study allocated 21, 5, 7, 10, 10, 11, 10 and 10% of the patients to dose levels 0, , 5, 10, 20, 50, 100 and 200 mg of the new drug, respectively, and 16% of the patients to the active control. Again, a substantial improvement can be observed under both distributions.

Table 1.

D-optimal designs in the two examples under different distributional assumptions, together with the efficiencies of the designs which were actually used in the study

	Gouty arthritis example					Acute migraine example
Distribution	D-optimal design					D-optimal design
Normal	(0, 0)	(981, 0)	(300, 0)		025	(0, 0)	(1095, 0)	(200, 0)		084
nB/Bin	(0, 0)	(823, 0)	(300, 0)		011	(0, 0)	(905, 0)	(200, 0)		086
	25%	25%	25%	25%		25%	25%	25%	25%

nB, negative binomial; Bin, binomial.

4. Optimal designs for estimating the target dose

4.1. AC-optimal designs

In this section we investigate the construction of locally optimal designs for estimating the smallest dose of the new compound that achieves the same treatment effect as the active control. We consider a dose range of the form and write

(17)

(18)

for the expected values of responses corresponding to the new drug for dose level and the active control, respectively. We assume that the function in (17) is strictly increasing with respect to and that is an element of the dose range for the new drug. The expectation in (18) is a function of the -dimensional parameter , say . Consequently, a natural estimator of is given by , where and denotes the vector of the maximum likelihood estimators of the parameters and in models (1) and (2), respectively. Standard calculations show that the variance of this estimator is where

(19)

Here, denotes the design for the new drug induced by the design , see (8), and and are generalized inverses of the information matrices and , respectively. Following Dette et al. (2014), we say that a design is locally AC-optimal if , , and minimizes the function among all designs satisfying the above estimability conditions. Criterion (19) corresponds to a -optimal design for estimating the parameter in a dose-response model with an active control, where . In particular, Theorem 2 is applicable and locally AC-optimal designs can be derived from the corresponding optimal designs for model (1). The following result provides an alternative representation of the criterion (19) in the case where . As a consequence, the design required in Theorem 2 is a locally -optimal design in model (1) for a specific vector , i.e., the design minimizing where . The proof can be obtained using arguments similar to those in Dette et al. (2014) and is given in the Supplementary Material.

Theorem 5 —

In the case where , the function in can be expressed as

4.2. Some explicit results for two-dimensional models

In this subsection we present some examples illustrating different structures of locally AC-optimal designs. We suppose that the Fisher information matrix defined in (5) is of the form where denotes a two-dimensional vector and a matrix, which does not depend on the dose level. By Theorem 2, the locally AC-optimal design can be determined from the design that minimizes the expression in the class of all designs defined on the dose range for the new drug, where the vector is given by . Because of the block structure of the Fisher information matrix , with a lower block not depending on the dose level, we may assume without loss of generality that , i.e.,

(20)

By Elfving's theorem (Elfving, 1952) a design with weights at the points () minimizes if and only if there exist a constant and such that the point is a boundary point of the Elfving set

(21)

and the representation is valid. Note that , where the curve is defined by . The structure of the Elfving set depends sensitively on the distributional assumptions, and we now consider several examples in the context of the Michaelis–Menten model.

Example 3 —

If the dependence on the dose in model (1) is described by the Michaelis–Menten model, then the vector in (20) has the form where the function varies with the distributional assumption.

If the responses are normally distributed, we have , and it follows by an obvious generalization of Theorem 5 that we must consider a -optimal design problem in model (1), where the vector is now given by . From Fig. 1(a) we see that the line intersects the boundary of the Elfving set at some point whenever , where

A typical situation is shown for the vector in Fig. 1(a) for and . Consequently, Elfving's theorem shows that a one-point design minimizes . An application of Theorem 2 yields a locally AC-optimal design which allocates % of the patients to dose level for the new drug and the remaining patients to the active control. On the other hand, if , the line does not intersect the set at the boundary of the Elfving set , and the situation is more complicated. A typical situation in this case is shown in Fig. 1(a) for the vector . The locally AC-optimal design allocates and of patients to dose levels and of the new drug, respectively, where , and allocates the remaining patients to the active control; here

(22)

, and .

As a further example, consider the Michaelis–Menten model for the probability of a negative binomially distributed response. We have , , and . A corresponding Elfving set is depicted in Fig. 1(b) for , and ; the locally AC-optimal design is always supported at three points. A straightforward calculation shows that the locally AC-optimal design allocates and of the patients to dose levels and of the new drug, respectively, where , and allocates the remaining patients to the active control; in this case

(23)

, and .

Next, consider the Michaelis–Menten model for binomially distributed responses. In this case we have , , , and . The corresponding Elfving set is depicted in Fig. 2(a) for , and ; we have to distinguish three different situations. Observe that the line intersects the boundary of the Elfving set at some point if and only if , where

A typical situation is shown for the vector in Fig. 2(a). Consequently, the same arguments as in the previous examples show that in this case the locally AC-optimal design allocates of the patients to the dose level of the new drug, where with and , and allocates the remaining patients to the active control. If , the locally AC-optimal design allocates and of patients to dose levels and of the new drug, respectively, and allocates the remaining patients to the active control, where is of the form (22) with . A typical situation is shown for the vector in Fig. 2(a). The case corresponds to the vector . Here the locally AC-optimal design allocates and of patients to dose levels and of the new drug and the remaining patients to the active control, where, with , is of the form (23) and .

Finally, we consider the case of Poisson-distributed responses. We have , , and ; by Theorem 5, we have to solve a -optimal design problem with . It is easy to see that the line intersects the boundary of the Elfving set at some point if and only if , where ; see Fig. 2(b) for illustration of the case with , , and the vector . Consequently, the same arguments as in the previous examples show that in this case the locally AC-optimal design allocates of patients to dose level of the new drug, where with and , and allocates the remaining patients to the active control. If , the locally AC-optimal design allocates and of patients to dose levels and of the new drug and allocates the remaining patients to the active control, where is of the form (22) with . A typical situation is shown for the vector in Fig. 2(b).

Fig. 1.

The Elfving set (21) in model (1), where the expected response is given by the Michaelis–Menten model: (a) normal distribution; (b) negative binomial distribution.

Fig. 2.

The Elfving set (21) in model (1), where the expected response is given by the Michaelis–Menten model: (a) binomial distribution; (b) Poisson distribution.

4.3. Locally AC-optimal designs in the model

Explicit expressions for the AC-optimal designs in the model are complicated, so for brevity we conclude this paper by discussing AC-optimal designs for the two examples from §1. All designs presented in this subsection were calculated numerically using particle swarm optimization (Clerc, 2006), and the optimality has been checked by applying Lemma 1.

We begin with the gouty arthritis clinical trial, for which we use the same prior information as in Example 2. AC-optimal designs under the assumption of normal and negative binomial distributions are shown in Table 2. For instance, under the assumption of normally distributed endpoints, the AC-optimal design allocates almost half of the patients to the dose level  mg and the rest to the active control. In order to compare these results with the standard design introduced in Example 2, we also report values of the efficiency where is defined in (19) and is the locally AC-optimal design. For example, the efficiency of the standard design for estimating the target dose under the assumption of a normal or negative binomial distribution is or , respectively.

Table 2.

AC-optimal designs in the two examples under different distributional assumptions, together with the efficiencies of the designs used in the study

	Gouty arthritis example				Acute migraine example
	(target dose  mg)				(target dose  mg)
Distribution	AC-optimal design				AC-optimal design
Normal
nB/Bin

nB, negative binomial; Bin, binomial.

For the acute migraine clinical trial, we again use the prior information from Example 2. AC-optimal designs for normally and binomially distributed responses are summarized in Table 2. The efficiencies of the standard designs are and under the assumptions of normal and binomial distributions, respectively.

Supplementary material

Supplementary material available at Biometrika online includes further examples and the proof of Theorem 5.

Appendix

Proof of Theorem 1 —

If the matrix is of the form (12), we obtain from (7) and (12) that , which gives, for the criterion (9), when , where and the function is as defined in (13). It is easy to see that the function is an increasing function of . Consequently, the locally -optimal design problem for the dose-response model with an active control can be solved by determining a design which maximizes the criterion (13) in a first step. If denotes the optimal value for this criterion, it remains to maximize the function with respect to the weight assigned to the active control, which gives the expression (15) and proves the assertion for the case where . The remaining cases of and are proved similarly.

Proof of Theorem 3 —

The proof of (i) was given in §3. For the remaining cases we restrict ourselves to the Poisson distribution, for which the Fisher information matrix in model (1) is . All other cases are treated similarly. By Corollary 1, the D-optimal design can be obtained from the D-optimal design in a regression model with Fisher information matrix . If denotes an information matrix of a design in this model, then is D-optimal if and only if the inequality holds for all ; see Lemma 1. Moreover, there must be equality at the support points of the design . This inequality is equivalent to an inequality of the form where is a polynomial of degree 3 with . A straightforward argument now shows that has exactly two support points and . Consequently, the D-optimal design for the regression model with information matrix has equal masses at the points and , where maximizes the function in the interval , that is, . The assertion now follows from an application of Corollary 1, upon observing that and in the case under consideration.