Optimal designs for dose finding studies with an active control

Authors


Summary

Dose finding studies often compare several doses of a new compound with a marketed standard treatment as an active control. In the past, however, research has focused mostly on experimental designs for placebo controlled dose finding studies. To the best of our knowledge, optimal designs for dose finding studies with an active control have not been considered so far. As the statistical analysis for an active controlled dose finding study can be formulated in terms of a mixture of two regression models, the related design problem is different from what has been investigated before in the literature. We present a rigorous approach to the problem of determining optimal designs for estimating the smallest dose achieving the same treatment effect as the active control. We determine explicitly the locally optimal designs for a broad class of models employed in such studies. We also discuss robust design strategies and determine related Bayesian and standardized minimax optimal designs. We illustrate the results by investigating alternative designs for a clinical trial which has recently appeared in a consulting project of one of the authors.

1. Introduction

One of the critical steps in developing a medicinal drug is a proper understanding and characterization of its dose response relationship. Failing to characterize well the dose response relationship may have severe consequences once the drug is available to patients: selecting too high doses may lead to unacceptable safety problems, whereas selecting too low doses may lead to insufficient efficacy. Further applications, where dose response modelling is of particular importance, include the investigation of a new herbicide or fertilizer, a molecular entity, an environmental toxin or an industrial chemical.

Much literature is available on dose response studies including a placebo group (see Ruberg (1995), Ting (2006) and Bretz et al. (2008) among many others). However, in some drug development programmes the dose-dependent efficacy relative to a standard treatment is of major interest, especially in preparation for an active controlled confirmatory non-inferiority trial. In addition to regulatory requirements related to drug approval, health technology assessments for national reimbursement decisions may be improved by dose finding studies that evaluate the incremental dose effect as compared with the standard treatment. Furthermore, in many situations the use of placebo could be considered unethical or unfeasible, even in a phase II dose finding study. If no placebo is used, the extrapolation of the dose response from the lowest dose to the zero dose (i.e. placebo) becomes problematic and the use of an active control (AC) could facilitate the assessment of the overall level of efficacy of the dose response curve.

The considerable interest by regulatory agencies in active controlled studies becomes evident from several related guidelines. For example, the tripartite International Conference on Harmonisation E4 guideline on dose finding encourages the inclusion of an active comparator in a dose finding study to improve assay sensitivity of the trial as well as to generate better data on comparative effectiveness and safety (International Conference on Harmonisation, 1994). In addition, several international disease-specific regulatory guidelines recommend the use of an active comparator in pivotal phase III studies ( European Medicines Agency, 2006, 2011). In a more general context, the European Medicines Agency guideline on the choice of a non-inferiority margin states that a placebo controlled trial is usually not sufficient and that the comparison between test and reference will often be of importance in its own right (European Medicines Agency, 2005). It thus becomes evident that, because of the regulatory requirements on active-controlled phase III trials, dose finding studies with an AC contribute significantly to the proper choice of a dose to be used in phase III and lead to a better risk–benefit profile in comparison with a marketed drug.

The research in the present paper is motivated by an active controlled dose finding phase II study to determine the optimal dose of the new compound for the management of acute flare in gout adult patients who are refractory or contraindicated to standard therapies. The primary objective is to determine the target dose of the new compound, which is the dose that leads to the same efficacy as the AC. It will be identified by assessing the dose response relationship of various doses of the new compound with regard to pain intensity in the target joint at 72 h (day 4) post dose measured on a 0–100 mm visual analogue scale. Approximately 200 patients will be included in the study. Patients who meet the entry criteria will be randomized to receive either the AC or one dose of the new compound. An important problem consists in specifying the dose levels for the new compound as well as the allocation ratio of patients across all treatments arms in this study. Once the optimal dose of the new compound has been selected on the basis of this phase II trial, phase III studies will be conducted to evaluate further the efficacy and safety of the new compound in the respective patient population (either acute or chronic gout patients).

It is well known that optimal designs can substantially improve the efficiency of statistical analyses and numerous researchers have worked on the problem of constructing optimal designs for placebo controlled dose response studies (see Miller et al. (2007), Dragalin et al. (2007), Dette, Bretz, Pepelyshev and Pinheiro (2008) and Dette, Kiss, Bevanda and Bretz (2010) among others). However, to our best knowledge, optimal design problems for active controlled dose finding studies have not been considered in the literature so far. In this paper we propose a strategy to obtain efficient designs for such situations. In Section 'Statistical model for active controlled dose finding studies' we introduce the statistical model that includes an AC together with several dose levels of the compound under investigation. Locally optimal designs for estimating the target dose for the new compound are constructed explicitly. These results are obtained by a non-standard application of the implicit function theorem, which yields a substantial simplification of the asymptotic variance of the target dose estimate. Locally optimal designs require a priori information about the unknown model parameters (see Chernoff (1953), Ford et al. (1992) and Fang and Hedayat (2008)) and usually serve as benchmarks for commonly used designs (see Section 'Relative efficiencies'). In addition, locally optimal designs serve as a basis for constructing optimal designs with respect to more sophisticated optimality criteria, which are robust against a misspecification of the unknown parameters (see Pronzato and Walter (1985), Chaloner and Verdinelli (1995), Dette (1997) or Imhof (2001) among others). In Section 'Robust optimal active-control-optimal designs' we consider standardized minimax and Bayesian optimal designs, which minimize the maximal efficiency and average efficiency over a given range of the unknown parameters respectively. For several models, including a reparameterization of the widely used EMAX model, it is demonstrated that the robust design problems are related to interpolation optimal design problems for polynomial regression models as considered in Kiefer and Wolfowitz (1964ab). These results are used to show that the robust optimal designs are saturated (i.e. the number of different experimental conditions coincides with the number of parameters for the underlying model) and optimal designs with respect to these criteria are determined explicitly. As a by-product we also obtain explicit solutions of some of the design problems that were raised in Kiefer and Wolfowitz (1964a, b). Several examples illustrating the results are presented in Section 'Examples', where we also study the efficiency of commonly used designs for the case-study that was described above. Finally, some conclusions and directions for further research are presented in Section 'Discussion', whereas Appendix A contains the proofs of our main results. Motivated by the regulatory International Conference on Harmonisation E4 guidance (International Conference on Harmonisation, 1994) and the cross-industry Pharmaceutical Research and Manufacturers of America working group on ‘Adaptive dose-finding studies’ (Bornkamp et al., 2007), we focus on estimating the minimum efficient dose relative to an AC. However, the methodology that is presented in this paper can be extended to other quantities of interest (e.g. estimating the model parameters) and we discuss this briefly in Section 'Optimal designs for estimating model parameters'.

2. Statistical model for active controlled dose finding studies

We assume that patients are treated either with an AC (a standard treatment administered at a fixed dose level) or with the new drug for which the dose response relationship is unknown. For a given total sample size, say n, the goal of an experimental design is to allocate math formula and math formula patients to the new drug and the AC respectively, and to determine the dose levels under which the math formula patients are treated with the new drug.

For the statistical analysis we assume that the observations are realizations of independent random variables math formula, math formula according to the model

display math
display math

where math formula and math formula are independent and normally distributed with expectation 0 and variances math formula and math formula respectively. The random variable math formula corresponds to a patient receiving the new drug at dose level math formula and math formula corresponds to a patient receiving the AC (at a fixed single dose level). Furthermore, μ denotes the expected effect for the AC, math formula a vector of unknown parameters and f a given function which describes the average response of the new drug at a given dose. Let κ be an indicator, whether a patient receives the new drug (κ=0) or the AC (κ=1). The design space is therefore given by the set

display math

where math formula denotes the dose range for the new drug, C the dose level of the AC and the second component of an experimental condition (d,κ) determines the treatment (κ=0,1). Straightforward calculation shows that the Fisher information at (d,κ) is given by the matrix math formula where the function math formula is defined by

display math(2.1)

and

display math(2.2)

denotes the gradient of the regression function f(d,θ) with respect to the vector θ. Note that the corresponding Fisher information is block diagonal because of the independence of the samples math formula and math formula, as different patients are treated with the new drug and the AC. 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 math formula. Therefore, an experimental design is given by

display math(2.3)

where math formula are positive weights, such that math formula. The weight math formula denotes the relative proportion of total observations treated with the AC, math formula the different dose levels used for the new drug and math formula the relative proportion of patients treated at dose level math formula (j=1,…,k). In typical studies the number k of different dose levels is in the range of 5–10 and almost never larger than 10. Note that in this paper we shall also optimize the number k when determining the optimal designs. It turns out that for the models considered here the optimal designs usually allocate observations at fewer than five dose levels.

The information matrix of an approximate design is given by the (p+2)×(p+2) matrix

display math

where math formula, the (p+1)×(p+1) matrix math formula is defined by

display math(2.4)

and

display math(2.5)

denotes a design (on the design space math formula) for the new drug with weights

display math

Because commonly used optimality criteria are homogeneous (see Pukelsheim (2006)), locally optimal designs will depend only on the ratio math formula and we may assume without loss of generality that math formula. In what follows we consider models of the form

display math(2.6)

where math formula and the function math formula is assumed to be strictly increasing; see Bretz et al. (2008). The aim is to estimate the treatment effect μ of the AC and the target dose

display math

i.e. the smallest dose of the new compound achieving the same treatment effect as the AC. A natural estimate of math formula is given by math formula, where math formula is the maximum likelihood estimate of the parameter math formula.

Standard calculation shows that the variance of this estimator is approximately given by

display math(2.7)

where the function ψ is defined by

display math(2.8)

Here, ∇ denotes the gradient of the function math formula with respect to the parameter math formula, math formula the gradient with respect to θ, and math formula and math formula are generalized inverses of the information matrices M(ξ,θ) and math formula respectively. A design math formula is called a locally AC-optimal design if math formula and if math formula minimizes the function math formula among all designs satisfying this condition. Identity (2.8) holds, because for a design with math formula the variance is independent of the choice of the general inverse (see Pukelsheim (2006)). Therefore we choose a generalized inverse with the same block structure as the information matrix. Note that the optimality criterion (2.8) is a special case of the c-optimality criterion, which corresponds to minimizing the expression

display math(2.9)

for a given vector math formula in the class of all designs ξ, such that c is estimable by the given design ξ, i.e. range(c)⊂ range{M(ξ,θ)}.

The accuracy of approximation (2.7) has been investigated in Dette, Bretz, Pepelyshev and Pinheiro (2008) in the context of dose finding studies including a placebo group by means of a simulation study. They concluded that the approximation is satisfactory for total sample sizes larger than 25. We have performed a similar simulation study investigating the quality of approximation (2.7) and obtained very similar results (details are omitted here for brevity). As typical clinical dose finding trials have sample sizes in the range 200–300 (Bornkamp et al., 2007), approximation (2.7) is reasonably accurate for all practical purposes.

In the next result we determine locally AC-optimal designs for a broad class of non-linear regression models accounting for an AC by minimizing criterion (2.8). These designs serve as benchmarks for commonly used designs and are the basis for the construction of optimal designs with respect to more sophisticated optimality criteria, in particular standardized minimax and Bayesian optimal designs discussed in the following section. Since the seminal paper of Chernoff (1953) numerous researchers have worked on the problem of constructing locally optimal designs for many regression problems (see for example Ford et al. (1992), He et al. (1996), Fang and Hedayat (2008) or Yang (2010) among many others) but—to the best knowledge of the authors—optimal design problems for active controlled dose finding studies have not been considered in the literature. In the following result we show that for all models of the form (2.6) there are locally AC-optimal designs with two support points independently of the dimension of the parameter vector θ. The proof is based on a non-standard application of the implicit function theorem and is given in Appendix A.

Theorem 1.. For a model of the form (2.6) the optimality criterion defined in equation (2.8) has the representation

display math

where

display math

and the matrix math formula and the design math formula are defined by equations (2.4) and ((2.5) respectively. Moreover, for model (2.6) with math formula the design

display math(2.10)

is a locally AC-optimal design, where math formula denotes the ratio of the variances of the two samples. In particular, the minimum value of the criterion ψ defined in equation (2.8) is given by

display math

It is of interest to note that the locally optimal designs that are determined in theorem 1. are not necessarily unique. To see this we consider the case p=1, where we use the notation math formula because the function math formula then does not depend on the parameter math formula. In this case model (2.6) reduces to a linear model

display math(2.11)

and there also exist locally AC-optimal designs with three or more support points.

Theorem 2.. Let math formula and math formula such that

display math(2.12)

Then the design with masses math formula and 1/(r+1) at the points math formula and (C,1) respectively, is a locally AC-optimal design for model (2.11).

Note that theorem 2. generalizes a result of Herzberg and Cox (1972), who considered the special case math formula, r=1, and showed that designs with masses math formula, math formula and math formula at the points math formula, math formula and (C,1) are locally AC-optimal designs on the design space math formula whenever math formula.

To utilize locally AC-optimal designs in practice, initial values for the model parameter θ are needed before any data are available. Thus, one has to rely on prior information or expectation about the dose response relationship. Strategies for deriving initial parameter estimates based on prior dose response assumptions and before data have been collected are given in Pinheiro et al. (2006). For example, an initial estimate for math formula can be obtained from knowledge of the prior expected percentage of the maximum effect math formula associated with a given dose math formula and inverting model (2.6). If different math formula pairs are available, one can use the average of the corresponding math formula-values as an initial value, or use a range of parameter values; see Section 'Active-control-optimal designs' for examples.

3. Robust optimal active-control-optimal designs

Locally optimal designs are often sensitive with respect to misspecification of the initial parameters and several alternative design strategies have been developed to address this issue. The literature mainly differentiates between adaptive or sequential (see Chaudhuri and Mykland (1995) or Dragalin et al. (2010) among others) and Bayesian or minimax optimal designs (see for example Chaloner and Verdinelli (1995) and Dette (1997) among others). In this section we shall investigate two robust design strategies for active controlled dose finding studies, namely minimax and Bayesian AC-optimal designs. To be precise, let Θ and math formula denote sets for the possible values for θ and μ respectively. A design math formula is called a standardized minimax AC-optimal design for the AC model with respect to the set math formula if math formula for all math formula and if math formula minimizes the maximum efficiency

display math(3.1)

calculated over a given range math formula of the parameters. The efficiency here is defined by

display math(3.2)

where math formula denotes the criterion function that was introduced in equation (2.8) and math formula is the locally AC-optimal design. The range math formula denotes a given set specified by the experimenter which reflects the prior belief about the unknown vector of parameters. Therefore a design minimizing equation (3.1) is expected to yield reasonable efficiencies for all values in the set math formula.

Similarly, the Bayesian AC-optimal design minimizes an average of the efficiencies. To be precise, let π denote a prior distribution with support given by math formula. Then a design math formula is called a Bayesian AC-optimal design with respect to the prior π if math formula for all math formula and if math formula minimizes a weighted average of the efficiencies (3.1), i.e.

display math(3.3)

In the following discussion we shall derive some robust designs with respect to these criteria. From theorem 1. it follows that the efficiency in expressions (3.1) and (3.3) is given by

display math

where

display math

In general, the determination of standardized minimax and Bayesian optimal designs is a very difficult problem (see for example Imhof (2001) or Braess and Dette (2007)) and in most cases these designs must be found numerically. In the following discussion we shall describe some general properties of these designs and construct robust optimal designs for some models in explicit form. For this we consider again model (2.6) and assume that for each parameter math formula the quantity

display math

is well defined and an element of the design space, i.e.

display math(3.4)

In other words, for any set of parameters math formula there is a (unique) dose math formula, such that math formula. In the following discussion we consider criteria (3.1) and (3.3), where the parameter space is of the form

display math

for some fixed vector math formula. Throughout this section we assume that math formula and math formula contain at least two points (otherwise we have a locally optimal design problem and the results of the previous section are applicable). Similarly, if the prior π is supported on the set math formula we reflect this in our notation, i.e. math formula, and assume that math formula has at least two support points.

3.1. Standardized minimax optimal designs

For a fixed math formula we introduce for a design of the form (2.5) an induced design on the design space math formula by

display math

with math formula, i=1,…,k. It is easy to see that condition (3.4) is equivalent to

display math(3.5)

Observing that for model (2.6) we have

display math

with an obvious definition of math formula. It follows that the efficiency in equation (3.2) is given by

display math

where the vector h is defined by

display math(3.6)

and math formula. In model (2.6) we have math formula and math formula for j≥2 and consequently the expression math formula does not depend on the parameters math formula and μ. Because math formula is assumed to be fixed, we reflect this property by the notation

display math(3.7)

where the definitions of h(z) and math formula depend on the specific context. Therefore, the optimization problem for the standardized minimax AC-optimal design with respect to the set math formula reduces to

display math(3.8)

As a consequence, the solution of the standardized minimax optimal design can be found in two steps. First, we determine a minimax optimal interpolation design math formula in a regression model with Fisher information math formula and design space math formula, where the range of interpolation is given by math formula, i.e.

display math(3.9)

Secondly, one determines the optimal weight math formula, which specifies the proportion of patients treated with the control. Problems of the type (3.9) have been discussed by Kiefer and Wolfowitz (1964a,b) for polynomial regression models, who indicated that explicit solutions of equation (3.9) are extremely difficult to obtain. Dette and O'Brien (1999) presented further examples and proved the following equivalence theorem, which can be used to check the optimality of a given design.

Theorem 3.. For a design math formula on the induced design space math formula define

display math

and

display math

Then the following two properties are equivalent.

  1. math formula minimizes math formula.
  2. There is a probability measure π on the set math formula, such that for all math formula the in equality
    display math(3.10)
    holds with math formula. Moreover, in this case there is equality in expression (3.10) for each support point of the design math formula.

In Sections 'Linear model' and 'A reparameterized EMAX model' we shall use this result to identify standardized minimax AC-optimal designs for model (2.11) and a reparameterization of the common EMAX model.

3.2. Bayesian optimal designs

An analogue of equation (3.8) is obtained by similar arguments to those in the previous section for the Bayesian optimality criterion with respect to the prior math formula, i.e.

display math(3.11)

where math formula is the prior induced on the set math formula by the transformation math formula. Consequently, the Bayesian AC-optimal design problem can be solved, by first determining the design math formula minimizing

display math(3.12)

and then determining the optimal weight math formula, where the matrix A is given by

display math

Note that the optimality criterion (3.12) is a classical A-optimality criterion which has been well studied in the literature. For example, the equivalence theorem states that a design math formula is an A-optimal design if and only if the inequality

display math(3.13)

holds for all math formula. Moreover, if math formula is an A-optimal design, there is equality in expression (3.13) for all support points of the design math formula.

3.3. Optimal designs for estimating model parameters

In some cases the target dose is not the sole interest of the trial and optimal designs for estimating the parameter θ in a dose finding study with an AC are also of interest. In this section we illustrate how designs for this purpose can be constructed. Numerous optimality criteria can be used to find efficient designs for parameter estimation (see Pukelsheim (2006)). For brevity we restrict ourselves to Bayesian D-optimal designs, which maximize

display math(3.14)

in the class of all designs of the form (2.3). From the discussion in Section 'Statistical model for active controlled dose finding studies' we obtain by straightforward calculation

display math

and the following result.

Theorem 4.. Let

display math

denote the Bayesian D-optimal design maximizing math formula. Then the design

display math

is a Bayesian D-optimal design for the model Y=f(d,θ)+ɛ with an AC.

For a one-point prior we obtain locally D-optimal designs for a dose finding study with an AC. A comparison of the latter designs with the locally AC-optimal designs from theorems 1. and 2. shows that locally D-optimal designs put more weight on the dose levels corresponding to the new drug (in particular, if the number of the parameters in the dose response model is large). A similar comment applies to Bayesian D-optimal designs (see remark 5. at the end of Section 'A reparameterized EMAX model'). This result is intuitive, because the D-criterion reflects the goal of estimating the parameters of the regression model which is used to describe the dose response relationship for the new drug.

3.4. Linear model

It follows from the definition of model (2.11) that the vector h(z,θ) in equation (3.7) is given by math formula. The following results show that standardized minimax and Bayesian AC-optimal designs for model (2.11) are saturated. The proofs are given in Appendix A.

Theorem 5.. If assumption (3.4) is satisfied, then the standardized minimax AC-optimal design for model (2.11) has at most three support points. Moreover, if math formula is an interval, then the standardized minimax AC-optimal design with respect to the set math formula is given by

display math(3.15)

where

display math

math formula, math formula and the parameter ρ is given by

display math

3.4.1. Example 1

Consider the linear regression model math formula and assume that math formula (i.e. no placebo effect), μ ∈ [0.1,0.2] and math formula, which implies math formula. If math formula, it follows from theorem 5. that the standardized minimax AC-optimal design math formula on the dose range math formula allocates 71.9% and 8.7% of the patients to the new drug at dose levels 10 and 150 respectively, whereas 19.4% of the patients are treated with the AC.

It was pointed out by a referee that it might be of interest to investigate the efficiencies of optimal designs for different dose ranges. For this we calculated the standardized minimax AC-optimal designs math formula and math formula on the dose ranges math formula and math formula respectively, obtaining

display math
display math

If these designs were used on the dose range math formula their minimax efficiencies would be 29.7% and 29.1% for the designs math formula and math formula) respectively. This means that optimal designs are closely associated with the given dose range, and a substantial loss of efficiency is observed if these designs are used on different dose ranges.

Theorem 6.. If assumption (3.4) is satisfied, then the standardized Bayesian AC-optimal design with respect to the prior distribution π has at most three support points. Moreover, if math formula is an interval, then the standardized Bayesian AC-optimal design with respect to the prior π on the set math formula is given by

display math(3.16)

where

display math(3.17)
display math(3.18)

and math formula and math formula denote the first and second moment of the induced prior distribution math formula on the set math formula respectively.

Remark 1.. There is a simple interpretation of the quantities appearing in theorems 5. and 6.. The weights math formula and math formula give the weights of the standardized minimax and Bayesian optimal design respectively, for estimating the target dose in a dose finding trial that includes placebo but not an AC. These weights depend on the prior information about the unknown parameters. For example, if there is no information about the location of the target dose, then we have math formula and math formula which give math formula and

display math

However, these quantities change substantially if some preliminary knowledge is available. For example, if it is known that the target dose is in a neighbourhood of the right boundary of the design space, one could use math formula (for some ɛ>0) and math formula, which yield

display math

i.e. the standardized minimax AC-optimal design puts most of its weight at the design points (R,0) and (C,0). A similar (but slightly more complicated) remark also applies to the Bayesian AC-optimal designs.

The quantities math formula and math formula represent the contribution to the average variance of the target dose estimate from the Bayesian AC-optimal design for the new drug relative to the AC. This follows from the proofs of theorem 5 and 6 and the representation of the optimality criterion in equation (3.11), which gives

display math

Similarly, the standardized maximin optimal design for estimating the target dose for the new drug satisfies

display math

Remark 2.. A similar calculation to that used in the proof of theorems 5 and 6 shows that the Bayesian D-optimal design for model (2.11) with an AC is given by

display math

Thus, in contrast with the standardized minimax and Bayesian AC-optimal design, the Bayesian D-optimal design allocates math formula of the patients to the AC independently of the dose range and prior information.

Remark 3.. In applications there appear typically restrictions regarding the number of subjects allocated to the AC, because pharmaceutical companies usually want to use most resources to investigate the new drug. This situation corresponds to the determination of optimal designs with restricted weights (Pukelsheim (2006), page 277) and the methodology proposed can easily be extended to this case.

To be precise, consider the situation in theorem 6. where at most 100p% of the subjects should be treated with the AC, and recall the definition of math formula in equation (3.18).

If math formula the standardized Bayesian AC-optimal design is given by equation (3.16). Otherwise the standardized Bayesian AC-optimal design is given by

display math

where math formula is defined in equation (3.17). This statement follows directly from the proof of theorem 6 in Appendix A. For all other results presented in this paper similar statements can be derived, which have been omitted for brevity (a further example is given in remark 5. below).

3.5. A reparameterized EMAX model

In our second example we consider standardized minimax and Bayesian AC-optimal designs for the model

display math(3.19)

for which an explicit determination of standardized minimax AC-optimal designs is substantially more difficult. Note that model (3.19) corresponds to a reparameterization of the well-known EMAX model. A straightforward calculation shows that for model (3.19) the gradient of the regression function can be written in the form

display math

where math formula. Observing the definition of the induced design space math formula it follows that the vector h in equation (3.6) is given by math formula and equation (3.7) holds with math formula. The following results describe the standardized minimax and Bayesian AC-optimal designs with respect to the set math formula.

Theorem 7.. If assumption (3.4) is satisfied, then the standardized minimax AC-optimal design with respect to the set math formula for model (3.19) has at most four support points. Moreover, if additionally math formula and math formula are intervals and the set

display math

is symmetric with respect to the centre of the interval math formula the following statements hold. If

display math(3.20)

then the standardized minimax AC-optimal design is given by

display math(3.21)

where math formula and math formula. Otherwise, if math formula, the standardized minimax AC-optimal design is given by equation (3.21) where

display math

and

display math

Theorem 8.. If assumption (3.4) is satisfied, then the standardized Bayesian AC-optimal des ign for the EMAX-type model (3.19) with respect to the prior distribution math formula has at most four support points. Moreover, if math formula is an interval and the induced prior distribution math formula in equation (3.11) is symmetric with respect to the centre of the interval math formula, then the standardized Bayesian AC-optimal design with respect to the prior math formula is given by

display math(3.22)

where

display math

and math formula denotes the jth moment of the induced prior distribution math formula (j=2,4).

Remark 4.. The interpretation of the parameters math formula, math formula and math formula is the same as in remark 1.. There is also an interesting interpretation of the parameter a referring to the metric

display math

on math formula. With this notation, the parameter a in expression (3.20) can be rewritten as

display math

and is thus a relative measure of the size of the set math formula with respect to the dose range math formula. If math formula is small compared with math formula (with respect to the metric M), then the standardized minimax optimal design has a different structure compared with the case where these sets are of similar size.

Note also that the Bayesian AC-optimal designs depend sensitively on the fourth moment of the induced prior distribution. If the support of math formula is given by [−z,z] and z is large, the fourth moment can be substantially larger than the second moment and consequently we have math formula and math formula. In this case the Bayesian AC-optimal design allocates most of the patients to the new drug and only a small part to the AC.

However, for a given second moment math formula the fourth moment math formula of a symmetric measure on the interval cannot vary arbitrarily. More precisely, it follows from chapter 1 in Dette and Studden (1997) that math formula, which gives lower and upper bounds for the quantities math formula and math formula in theorem 8.. In particular an efficient Bayesian AC-optimal design depending only on the second moment can be obtained by using the mean of the range for math formula, i.e.

display math

Similar approximations can be obtained for non-symmetric prior distributions but have been omitted here for brevity.

Remark 5.. The Bayesian D-optimal design for model (3.19) is given by

display math

and allocates 80% of the patients to the new drug, independently of the prior information. This result follows from theorem 4. and similar arguments to those given in the proof of theorem 8. Note that the support points of the standardized minimax, Bayesian AC-optimal and Bayesian D-optimal design for model (3.19) coincide. This was also observed by Dette, Kiss, Bevanda and Bretz (2010) for locally optimal designs in the EMAX model without AC. If the experimenter does not want to allocate 20% of the subjects to the AC, this design must be modified according to remark 3.. For example, the Bayesian D-optimal design for model (3.19) is given by

display math

if at least 90% of the subjects must be treated with the new drug.

4. Examples

In this section we illustrate the methodology by calculating several AC-optimal designs and investigate the efficiency of two designs that were considered by the clinical team for the active controlled dose finding study that was described in Section 'Introduction'. For simplicity we focus on the case r=1, i.e. math formula, which corresponds to the case of equal uncertainty about the new drug and the AC. Similar results are obtained in the case of unequal variances and we briefly indicate this in Section 'Active-control-optimal designs'

4.1. Active-control-optimal designs

We determine various AC-optimal designs for model (3.19), where the dose range math formula is given by the interval [10,150]. Information that was available at the design stage of the dose finding study led to best guesses of the model parameters, namely math formula, math formula, math formula and μ=22.5. By theorem 1. the corresponding locally AC-optimal design is given by (32,0) and (C,1), i.e. 50% of the patients are treated with the new drug (at dose level 32) and the control respectively. Table 1 displays the locally AC-optimal design for other parameter constellations, as obtained from theorem 1.. For each configuration, the corresponding locally AC-optimal design advises the experimenter to treat 50% of the patients with the control and 50% with the new drug at the dose level that is listed in Table 1. This allocation reflects the fact that we consider the new drug and AC as equally uncertain with respect to the standard deviations of the errors (i.e. math formula. If more information is available about the AC it might be reasonable to specify a larger value of r. For example, if math formula, the AC-optimal design advises the experimenter to treat 90% of the patients with the new drug at the dose level that is specified in Table 1, whereas only 10% of the patients are treated with the AC.

Table 1. Locally AC-optimal designs for various parameter specifications
math formula math formula math formula μ Local
0.0251.1452.522.531.0
0.0251.122.523.2534.2
0.0251.0852.523.537.5
0.02831.1452.522.534.5
0.02831.122.523.2538.6
0.02831.0852.523.542.8

The locally AC-optimal designs that are considered in Table 1 were calculated under the assumption that the elicited parameter values would be the true values. To account for the uncertainty about the parameter values, we next determine several standardized minimax and Bayesian AC-optimal designs. We initially keep the parameter math formula fixed and allow some uncertainty for the other parameters. Table 2 displays the results for math formula and various example intervals for math formula and μ. We show only the dose levels and corresponding weights for the new drug, because the proportion of patients who are treated with the AC is easily calculated from these quantities. For example, if

display math(4.1)

and the standardized minimax optimal design is used, we obtain from the corresponding row in Table 2 that 12.5%, 32% and 12.5% of the patients should be treated with the new drug at dose levels 10, 39.2 and 150, whereas the remaining 43% of the patients are treated with the standard treatment.

Table 2. Bayesian and standardized minimax AC-optimal designs for various specifications for the parameters math formula, math formula and μ where the intercept satisfies math formula
math formula math formula μ Bayes design Minimax design
  1. †The set math formula defined in equation (3.5) is symmetric with respect to the centre of the induced space math formula. First row of each section: dose levels of the new compound. Second row of each section: proportion of patients treated at these dose levels.

{0.025}[0.86,1.43][19,26]1035.21501039.2150
0.090.410.050.160.280.16
[0.91,1.33][21.5,25]1036.51501039.2150
0.060.440.040.1250.320.125
[0.97,1.20][23,24]1038.01501039.2150
0.030.470.030.080.390.08
[0.023,0.027][0.86,1.43][19,26]1035.21501037.1150
0.090.410.060.150.260.19
[0.91,1.33][21.5,25]1036.61501039.7150
0.060.440.040.140.320.11
[0.97,1.20][23,24]1038.11501039.2150
0.030.470.030.080.380.10
[0.016,0.025][0.86,1.43][19,26]1033.91501042.9150
0.110.400.040.190.270.14
[0.91,1.33][21.5,25]1034.11501043.1150
0.080.430.030.170.320.09
[0.97,1.20][23,24]1034.61501041.1150
0.050.460.020.130.370.07

We discuss again briefly the case where more information regarding the AC is available. In this case more patients are treated with the new drug. For example, if math formula it follows that the standardized minimax optimal design allocates 17.6%, 44.9% and 17.6% of the patients to the dose levels 10, 39.2 and 150 of the new drug, whereas the remaining 19.9% patients are treated with the AC.

The AC-robust designs that are reported in Table 2 correspond to a set math formula which is symmetric with respect to the centre of the induced design space math formula. The Bayesian AC-optimal designs have been calculated with respect to the uniform distribution on math formula and were determined numerically, even when assuming a fixed parameter math formula, because the induced prior distribution is not symmetric. The corresponding standardized minimax AC-optimal designs in Table 2, however, were calculated by using theorem 2.. For example, in scenario (4.1) the set math formula defined in equation (3.4) is given by math formula which yields L=10 and R=150, and math formula and math formula. The induced design space is given by math formula and we obtain math formula. Therefore, the standardized AC-optimal design can be directly obtained by an application of theorem 7..

To obtain efficient designs which are robust against misspecification of math formula, Table 2 also displays the results under the assumption that an interval for the parameter math formula is specified, more specifically that math formula and math formula. In this case, the standardized minimax AC-optimal designs must be determined numerically as well. A comparison of the standardized minimax optimal designs shows only minor differences between the cases math formula and math formula. However, if math formula the smallest dose level 10 receives more weight whereas the highest dose level 150 receives less weight. Moreover, the dose level in the interior design space math formula is larger. It is interesting to note that the proportion of patients who are treated with the AC is essentially not changing if there is more uncertainty about the parameter math formula. Similar observations can also be made for the Bayesian AC-optimal designs.

If the set math formula is not symmetric all designs must be calculated numerically, even in the case where the parameter math formula is fixed. Table 3 displays results for some examples in this case. There are no major differences between the Bayesian AC-optimal designs for math formula and math formula. The differences between the standardized minimax AC-optimal designs are more pronounced. The lowest dose level receives approximately three times more weight if uncertainty about the parameter math formula is taken into account in the optimality criterion. In addition, the weight at the dose level in the interior design space is decreased by 25%. Again the proportion of patients who are treated with the control is essentially the same in all three scenarios, regardless of whether a Bayesian or standardized minimax approach is employed. Note that the intermediate dose level for the Bayesian AC-optimal design is smaller than for the standardized minimax AC-optimal design. Moreover, the weights at the dose levels 10 and 150 are mostly smaller for the Bayesian designs compared with the minimax designs.

Table 3. Bayesian and standardized minimax AC-optimal designs under the assumption that math formula, math formula and μ ∈ [20,23]†
math formula Bayes design Minimax design Rule-of-thumb design
  1. †The rightmost part of the table contains a Bayesian design obtained by a rule of thumb, where the support points have been fixed as 10, √(10×150)≈38.7 and 150 and only the weights are optimized. The range for the parameter math formula is shown in the first column. The induced space math formula defined in equation (3.5) is not symmetric with respect to the centre of the induced design space math formula. First row of each section: dose levels of the new compound. Second row of each section: proportion of patients treated at these dose levels.

{0.025} math formula 1034.0150 math formula 1039.2150 math formula 1038.7150
0.070.440.030.050.440.050.080.430.04
[0.023, 0.027] math formula 1033.9150 math formula 1039.9150 math formula 1038.7150
0.070.440.030.140.330.100.080.430.04
[0.016, 0.025] math formula 1031.9150 math formula 1043.4150 math formula 1038.7150
0.090.420.020.180.340.050.120.390.04

4.2. Relative efficiencies

In this section we investigate the relative efficiencies of the robust designs determined in Section 'Active-control-optimal designs' in comparison with two designs that were considered by the clinical team at the planning stage of the dose finding study. These two standard designs math formula and math formula reflect the current practice of allocating patients equally across several doses, which are often chosen to be equally distant in the original or in a logarithmic scale, i.e.

display math(4.2)
display math(4.3)

Some robust optimal designs are given in Table 3, which also contains a design that was obtained by a rule of thumb suggested by one reviewer of this paper. This rule of thumb uses the Bayesian AC-optimal design in the class of all designs supported at the points (L,0), (√(LR),0), (R,0) and (C,1). In other words, the rule of thumb uses a fixed set of design points and the Bayesian AC-optimality criterion is optimized only with respect to the weights. The design points for the new compound are motivated by the observation that for L=10 and R=150 the geometric mean √(LR)≈38.73 yields a support point that is very close to the interior support point of the robust AC-optimal design. In the following discussion we shall refer to this rule of thumb as a Bayesian AC-optimal design with fixed support.

It follows from the discussion in Section 'Robust optimal active-control-optimal designs' that the relative efficiencies depend on the unknown parameter math formula only through the target dose math formula. Table 4 exemplarily displays the relative efficiencies of various designs for estimating the target dose math formula. A design is better if its relative efficiency (which is always larger than or equal to 1) is closer to 1. As seen from Table 4 the robust designs achieve considerably better relative efficiencies than the two standard designs. The Bayesian AC-optimal designs usually yield better relative efficiencies for estimating the target dose than the minimax AC-optimal designs. However, this observation depends on the specific target dose (in our case math formula). This can be seen in Fig. 1, where we exemplarily show the relative efficiency of the standardized minimax AC-optimal design math formula and Bayesian AC-optimal design math formula for various values of the target dose math formula and the parameter math formula (these are determined by the specification of the set math formula). We observe that the relative efficiencies of the standardized minimax AC-optimal designs vary between 1.6 and 2.0, whereas the range of efficiencies that is obtained from the Bayesian AC-optimal design is considerably larger, namely [1.4,3.9]. If, for example, the target dose is math formula, the standardized minimax AC-optimal design performs better than the Bayesian AC-optimal design.

Table 4. Relative efficiencies of reference and robust designs for estimating the target dose math formula
  Results for Results for Results for Results for
  standard design Bayes design minimax design rule-of-thumb design
  math formula math formula math formula math formula math formula math formula math formula math formula math formula math formula math formula
math formula 2.942.401.581.581.661.681.891.951.671.671.74
Figure 1.

Relative efficiencies of the designs (a) math formula and (b) math formula with varying parameter math formula and target dose math formula

It is also remarkable that the Bayesian AC-optimal designs with fixed support have only slightly worse efficiency than the Bayesian AC-optimal designs for estimating the target dose math formula and are slightly better than the standardized minimax AC-optimal designs; see the right-hand part of Table 4. Thus, using the dose levels L, √(LR) and R for the new drug seems to be a reasonable rule of thumb. However, some care is necessary with this design strategy, because any statement about the efficiency of a particular design depends sensitively on the design space and the model parameters. To illustrate this we display in Table 5 Bayesian AC-optimal and Bayesian AC-optimal designs with fixed support (determined by the rule of thumb) with their corresponding efficiencies for estimating the target dose for a smaller (math formula) and a larger design (math formula) space. In these cases we observe a substantial better efficiency of the Bayesian AC-optimal design compared with the rule of thumb.

Table 5. Bayesian AC-optimal designs and Bayesian AC-optimal designs with fixed support under the assumption that math formula, math formula, μ ∈ [20,23] and math formula and math formula
math formula Bayes design Efficiency rule-of-thumb design Efficiency
  1. †The range for the parameter math formula is shown in the first column. The induced space math formula defined in equation (3.5) is not symmetric with respect to the centre of the induced design space math formula. First row of each section: dose levels of the new compound. Second row of each section: proportion of patients treated at these dose levels.

{0.025}1028.7601.651024.5601.90
0.040.400.110.050.390.12
[0.023,0.027]1028.7601.871024.5602.54
0.040.400.050.050.390.13
[0.016,0.025]1027.5601.911024.5602.42
0.050.420.070.060.420.08
{0.025}1035.83001.571054.83001.92
0.080.430.020.120.390.06
[0.023,0.027]1035.83001.571054.83001.92
0.080.430.020.120.390.06
[0.016,0.025]1033.73001.641054.83002.25
0.100.410.020.160.350.06

Finally, we compare the efficiencies with the corresponding relative efficiencies obtained from the two standard designs, as displayed in Fig. 2. Both standard designs perform uniformly worse than the standardized minimax AC-optimal design. The efficiency of the equidistant design math formula varies between 2.5 and 2.9, whereas the efficiency of the design math formula is smaller and varies between 2.1 and 2.5. In contrast the Bayesian AC-optimal design outperforms the two standard designs whenever the target dose is less than 45 and larger than 20. If math formula there are no substantial differences between the Bayesian AC-optimal and the two standard designs, whereas the latter have a better performance if math formula. On the basis of these calculations (and similar results for other parameter specifications, which have been omitted here for brevity), we recommend the use of a standardized minimax AC-optimal design.

Figure 2.

Relative efficiencies of the reference designs (a) math formula and (b) math formula with varying parameter math formula and target dose math formula

4.3. Model with two ‘non-linear’ parameters

In this section we demonstrate that the methodology can easily be extended to models of the form (2.6) with a multivariate parameter math formula. Exemplarily we consider a logistic model of the form (2.6), where math formula and

display math

This model was proposed by Bretz et al. (2005) as an alternative to the EMAX model and we assume as prior information for the unknown parameters μ=22.5, math formula, math formula, math formula and math formula, and the target dose is given by math formula. Various Bayesian AC-optimal designs are shown in Table 6 where the relative proportions and dose levels for the new drug are displayed under various a priori assumptions regarding the parameters math formula and math formula. It is remarkable that the robust designs do not put very much weight at the smallest and largest dose level. The corresponding efficiencies for estimating the target dose math formula are given in Table 7 and are similar to those of the Bayesian AC-optimal designs that were derived for the other models. In Table 7 we also display the efficiencies of the two standard designs which have been used so far. These designs show a rather poor performance. For example the standard design math formula yields a variance which is at least three times larger than the variance obtained by the Bayesian designs. The standard design math formula has an even worse performance for estimating the minimum effective dose.

Table 6. Bayesian AC-optimal designs for the logistic model with uniform prior under the assumption that math formula, math formula, μ ∈ [20,25] and math formula
math formula, math formula Bayes design  
  1. †The range for the parameters math formula and math formula is shown in the first column. First row of each section: dose levels of the new compound. Second row of each section: proportion of patients treated at these dose levels.

math formula, math formula1047.356.7150 math formula
0.010.230.300.01
math formula, math formula1046.158.0150 math formula
0.020.240.290.02
math formula, math formula1046.258.0150 math formula
0.020.250.290.01
math formula, math formula1046.657.5150 math formula
0.020.240.280.03
math formula, math formula1049.057.8150 math formula
0.010.290.240.02
Table 7. Relative efficiencies of reference and robust designs for estimating the target dose math formula
  math formula math formula math formula math formula math formula math formula math formula
math formula 5.817.531.591.681.701.621.53

5. Discussion

In this paper we present a rigorous approach for the construction of optimal designs for dose finding studies with an AC. Despite their practical importance, optimal design problems of this type have not been studied in the literature so far to our best knowledge. Locally optimal designs for estimating the target dose are derived explicitly. These designs are used for the construction of robust designs which require much less prior knowledge about the parameters of the model that is used for describing the dose response relationship. It is demonstrated that the new designs outperform several standard designs which are currently used in clinical practice. Interestingly, in many cases the dose levels of the proposed new designs for dose finding studies with an AC coincide with those obtained for placebo controlled dose finding studies without an AC, although the weights of the two designs differ. For example, the Bayesian D-optimal design for the reparameterized EMAX model (3.19) without AC on the design space [0,150] advises the experimenter to take 33.33% of the observations at three doses levels including placebo (d=0) and the maximium dose 150; see Dette, Kiss, Bevanda and Bretz (2010). The Bayesian D-optimal design for the reparameterized EMAX model (3.19) with an AC uses the same dose levels for the new drug with equal weights 26.66%, whereas the remaining 20% of the patients are treated with the AC. This is a nice property from a practical point of view, because experimenters could use the same design points for estimating the minimum effective dose relative to placebo and AC, thus resulting in efficient multiobjective designs.

However, this observation depends sensitively on the assumption of normally distributed residuals in the regression model and the optimality criterion under consideration. For example, consider the c-optimality criterion

display math

for the vector math formula. Then the locally c-optimal criterion for the parameters math formula and μ=20 is given by

display math

for a dose finding study with an AC (here we assume math formula), whereas the corresponding design for a dose finding study without AC is given by

display math

As a further example, consider the Bayesian A-optimality criterion

display math

where π denotes a prior distribution on math formula. If we use a uniform distribution on the set

display math

(note that the A-optimal designs do not depend on the parameters math formula and μ), the Bayesian A-optimal designs are given by

display math

and

display math

These examples indicate that in general optimal designs for dose finding studies with and without an AC do not have any similarities, and it is an open problem under which conditions similar properties to those derived in this paper can be observed for other error distributions or other optimality criteria.

For example, if the dose response curve is modelled by a linear spline, we conjecture that similar results can be obtained. By combining the arguments of this paper with the reasoning in Dette, Melas and Pepelyshev (2008) it can be shown that the optimal designs are supported at the boundary points of the dose range and the knots of the spline function.

A further challenging future research project in the context of designing experiments for dose finding studies with an AC is the important problem of model uncertainty which typically appears in this type of investigations, because in many applications it is very difficult to specify an adequate non-linear model for the description of the dose response relationship. In most cases there are several competing models (e.g. EMAX, log-linear and logistic) for this purpose. A typical strategy to obtain a good model for the description of the dose response relationship is to test the null hypothesis of a constant dose response curve at a level of significance of, say, 5% (adjusted for multiplicity), against the alternative hypothesis of a non-constant dose response curve for each candidate model. Among those models where the null hypothesis is rejected, the model with the highest value of the Akaike information criterion AIC will be selected for the estimation of the target dose; see for example Bretz et al. (2005). Recently Dette, Bretz, Pepelyshev and Pinheiro (2008) proposed model robust designs for minimum efficient dose estimation for a class of ‘classical’ dose finding models by maximizing multiple objective criteria (see for example Cook and Wong (1994)). In the future we plan to adapt this methodology to the problem of constructing AC-optimal designs for several competing AC models (see also Bornkamp et al. (2011) for a Bayesian approach to address model uncertainty in a ‘classical’ dose finding study).

An important problem in this context is the construction of efficient designs for model discrimination, which has been discussed for dose response models without AC in Dette et al. (2009) and Dette, Pepelyshev, Shpilev and Wong (2010). A promising direction consists in the construction of T-optimal designs (see Dette and Titoff (2009)) to discriminate efficiently between the linear and EMAX model that was discussed in Sections 'Linear model' and 'A reparameterized EMAX model'.

A different approach to address model uncertainty is semiparametric or non-parametric methods to enhance the robustness of the dose response estimation; see Müller and Schmitt (1988), Mukhopadhyay (2000), Dette et al. (2005), Bornkamp and Ickstadt (2009), Dette and Scheder (2010) and Yuan and Yin (2011) among many others. These methods allow model-independent descriptions of a dose response relationship. However, their applicability in dose response studies is limited because they require observations on a rather dense set of different dose levels, which are rarely available in practice. For logistic and ethical reasons the number of different dose levels is typically of the order of 5–10 (Bornkamp et al., 2007) and non-parametric methods may not yield reliable results in these cases.

We have focused on one possibility of estimating a target dose that takes the treatment effect of an AC into account. However, alternative target dose definitions might be used in special situations. For example, if the safety profile of the new compound is better than that of the AC, one might be interested in estimating the smallest dose that is not relevantly inferior, i.e.

display math

for a fixed non-inferiority margin δ>0. Yet a different alternative target dose arises in dose finding studies which include both an AC and a placebo. In such situations, one might be interested in a combined objective by estimating the smallest dose that is not worse than the AC and is still better than a placebo by a certain clinically relevant amount. A careful inspection of the proofs in Appendix A shows that the methodology can directly be applied to this problem by replacing μ with μδ; details are left to the reader.

A different line of research is to derive optimal designs for outcomes that do not follow a normal distribution. For example, in chronic gout studies (which is a different situation from the acute gout study that was considered in Section 'Introduction') the primary end point is often defined as the number of flares occurring per subject within 16 weeks of randomization. These flares can be modelled by using a negative binomial distribution and a common overdispersion (variance divided by expectation minus 1) for all treatment arms. The logarithm of the expectation of the number of flares during 16 weeks is then described by a regression model for the dose response relationship between the (single) dose groups of the new compound and a constant parameter for the comparator. Again, we leave the determination of optimal designs in such situations for future research.

This paper focused on optimal designs for a single dose finding study. As pointed out by one reviewer, one possibility for broadening the scope is to connect optimal designs with decision theory for the reasons that

  1. the costs of administering the new drug and the AC drugs are usually different (sometimes substantially) and
  2. the expected utility of high dosage may be dampened by side effects.

Such considerations are particularly important when transitioning from phase II to phase III in drug development. We refer to Gilbert (2010) and Antonijevic et al. (2010) for initial research in this context. Another line of research is to investigate whether additional data that are routinely collected in contemporary trials (such as biomarker data) can benefit the optimal design of experiments. Biomarker information is particular relevant in adaptive dose finding studies (see Chaudhuri and Mykland (1995) or Dragalin et al. (2010) among others), where clinical response may not be fully available during an interim analysis. Fast read-outs of the clinical end point or relevant biomarker information could be used to guide patient allocation for subsequent cohorts of patients. We leave the extension of the proposed methods to such situations for future research.

Acknowledgements

The authors thank Martina Stein, who typed parts of this manuscript with considerable technical expertise. This work has been supported in part by the Collaborative Research Center ‘Statistical modeling of nonlinear dynamic processes’ (Sonderforschungsbereich 823, Teilprojekt G2) of the Deutsche Forschungsgemeinschaft. The authors also thank two referees and the Associate Editor for very constructive comments on an earlier version of this paper.

The views expressed in this article are views of the Norbert Benda and do not necessarily reflect the views of the Federal Institute for Drugs and Medical Devices.

Appendix A: Technical details

A.1. Proof of theorem 1

We shall show at the end of this proof that for the models under consideration we have

display math(A.1)

where math formula denotes the gradient with respect to the parameter θ. Therefore we obtain the representation

display math

for the gradient with respect to math formula, and the optimality criterion defined in equation (2.8) is in fact a special case of the c-optimality criterion (2.9) with the vector

display math(A.2)

It can be shown (see Pukelsheim (2006)) that a design ξ minimizes math formula if and only if there is a generalized inverse G of math formula such that for all math formula the inequality math formula is satisfied, where there is equality for all support points of any locally c-optimal design. Therefore a design math formula is a locally optimal AC design if and only if there is a generalized inverse of the matrix math formula, such that the inequalities

display math(A.3)

hold for all math formula, where the vector c is given by equation (A.2). Note that the inequalities in expression (A.3) correspond to the case κ=0 and κ=1 in expression (2.9) respectively (note also that we assume without loss of generality math formula). The information matrix of the candidate design math formula defined in equation (2.10) is obtained as

display math

Observing that in model (2.6) the first co-ordinate of the vector g(d,θ) is given by 1 it now follows by a straightforward calculation that the matrix

display math

is a generalized inverse of the information matrix math formula, where math formula denotes the ith unit vector. Using this generalized inverse in the equivalence theorem with the vector math formula it follows that the design math formula fulfils inequalities (A.3) because both sides are equal to math formula.

We conclude the proof showing the relationship (A.1) which is essential for the argument above. For this we recall the definition of the vector math formula and consider the function math formula By assumption there is a solution, say math formula, of the equation math formula with respect to d and ∂f(d,θ)/∂d≠0. Therefore the implicit function theorem implies that the function math formula is differentiable with derivative

display math

This yields

display math

and

display math

which proves result (A.1) and completes the proof of theorem 1.

A.2. Proof of theorem 2

In model (2.11) the gradient in equation (2.2) is given by math formula and we obtain from equation (2.11) and Elfving's theorem (see Elfving (1952)) that the design math formula defined in equation (2.5) minimizes math formula where the minimum value is given by 1. Therefore it follows from theorem 1. that the design math formula defined in theorem 2. is locally AC optimal.

A.3. Proof of theorem 5

We first show that the optimal design math formula that is defined by equation (3.9) is supported at at most two points, which implies the statement regarding the number of support points. For this we apply theorem 3. and recall that for model (2.11) the vector h in this criterion is given by math formula . Consequently, it is easy to see that the left-hand side of inequality (3.10) is a polynomial of degree 2. Because equality holds in expression (3.10) for all support points of math formula this implies that the optimal design math formula has at most two support points. Consequently, the standardized minimax AC-optimal design has at most three support points.

We shall also use theorem 3. to show that the designmath formula minimizing equation (3.9) has masses math formula and math formula at the points math formula and math formula respectively, where the points math formula, math formula, math formula and math formula are defined by math formula, math formula, math formula and math formula respectively. For this we note that a straightforward but tedious calculation gives math formula in theorem 3.. Equivalently,

display math

where math formula and math formula This yields

display math

and

display math

If π is a prior distribution on the set math formula with weights p and 1−p at the points math formula and math formula respectively, we have

display math

If the design math formula was optimal, then it follows from theorem 3. that for math formula there must be equality in expression (3.10) which determines the weight p, i.e.

display math

math formula is a polynomial of degree 2, with leading coefficient math formula and minimum at the point math formula. Therefore it attains maxima in the set math formula at the points math formula and math formula with value math formula and by theorem 3. the design math formula is optimal, i.e. it minimizes equation (3.9). For the determination of the standardized minimax AC-optimal design it remains to determine the weight math formula. Inserting math formula into equation (3.8) leads to

display math

where

display math

With the notation math formula the function k is minimal for math formula and with definition (2.5) it follows that

display math
display math

Resubstitution of math formula and math formula shows that the design given by equation (3.15) is standardized minimax AC optimal.

A.4. Proof of theorem 6

It follows from the discussion in Section 'Bayesian optimal designs' that the Bayesian AC-optimal design can be found by solving the A-optimal design problem (3.12), where

display math

and math formula. The statement regarding the support points is obtained in the same way as given in the proof of theorem 5 by using inequality (3.13). A further application of equation (3.13) shows that the design math formula with masses p and 1−p at the points −1 and 1 minimizes equation (3.12), where the weight p is given by equation (3.17). The corresponding minimal value is given by math formula. Therefore the assertion follows by minimizing expression (3.11) with respect to the remaining weight math formula and transforming the design math formula onto the design space math formula.

A.5. Proof of theorem 7

The statement regarding the number of support points follows along the lines given in the proof of theorem 5, where in this case the vector h in equation (3.7) is given by math formula. To prove the second part of the result we note that the set math formula is an interval and consequently the design math formula minimizing equation (3.9) has a non-singular information matrix math formula and has therefore exactly three support points. Note also that the criterion math formula is invariant with respect to linear transformations of the form zαz+β of the set math formula and math formula and therefore we assume without loss of generality that math formula and math formula. From a standard convexity argument it follows that the design math formula minimizing

display math

is symmetric with masses p,1−2p and p at the points −1,0 and 1 and inverse information matrix

display math

Recalling the definition of the optimization problem (3.8) we obtain that the function

display math

is a symmetric polynomial of degree 4 with positive leading coefficient. Therefore the maximum in equation (3.9) is attained at most at the two boundary points of the set math formula and one interior point of math formula, and it follows by symmetry that we must distinguish three cases for the set math formula in theorem 3., i.e.

  1. math formula
  2. math formula and
  3. math formula

In the first case we note that the identity math formula determines the weight p, i.e. math formula with value

display math(A.4)

In the second case the condition

display math(A.5)

is satisfied for any symmetric design and cannot be used to determine p directly. However, the optimal design would minimize math formula, and this yields

display math(A.6)

with value

display math(A.7)

Finally, for the third case note that by equation (A.5) the inequality math formula is equivalent to the inequality math formula. Therefore minimizing math formula with respect to this condition gives math formula and a design satisfying condition (a), i.e. math formula. Consequently, the third case cannot occur and we must only compare the results corresponding to cases (a) and (b), for which the optimal values are given by equations ((A.4)) and ((A.7)) respectively. A simple calculation shows that the inequality math formula holds in the interval [0,1] if and only if the inequality math formula is satisfied. It follows that criterion (3.9) is minimized for the design math formula with masses p,1−2p and p at the points −1,0 and 1 respectively, where the weight p is given by math formula if math formula and equation (A.6) if math formula. The assertion of theorem 7. finally follows by transforming these results to the original design space (note that the transformation is non-linear).

A.6. Proof of theorem 8

We may assume without loss of generality that math formula. Now by assumption the induced prior distribution is symmetric and as a consequence the elements in the matrix math formula vanish, whenever i+j is odd. Therefore a standard argument shows that there is a symmetric design math formula minimizing equation (3.12) and in what follows we shall investigate whether a symmetric design supported at the points −1, 0 and 1 is optimal. For such a design the optimal weights can be computed by using a result from chapter 8 in Pukelsheim (2006), which gives for the corresponding weights

display math
display math

respectively, where math formula denotes the jth moment of the distribution math formula. Finally, a straightforward application of the equivalence theorem (3.13) shows that the design with weights math formula and math formula at the points −1,0 and 1 is in fact minimizing equation (3.12). The value of the corresponding criterion in equation (3.12) is given by math formula. Minimizing criterion (3.11) with respect to the remaining weight math formula and transforming the support points from the induced design space math formula to the given design space math formula shows that the Bayesian AC-optimal design with respect to the prior math formula is given by equation (3.22), which completes the proof of theorem 8.

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