Optimal Experimental Design Strategies for Detecting Hormesis

Abstract

Hormesis is a widely observed phenomenon in many branches of life sciences ranging from toxicology studies to agronomy with obvious public health and risk assessment implications. We address optimal experimental design strategies for determining the presence of hormesis in a controlled environment using the recently proposed Hunt-Bowman model. We propose alternative models that have an implicit hormetic threshold, discuss their advantages over current models, construct and study properties of optimal designs for (i) estimating model parameters, (ii) estimating the threshold dose, and (iii) testing for the presence of hormesis. We also determine maximin optimal designs that maximize the minimum of the design efficiencies when we have multiple design criteria or there is model uncertainty where we have a few plausible models of interest. We apply these optimal design strategies to a teratology study and show that the proposed designs outperform the implemented design by a wide margin for many situations.

1. INTRODUCTION

Hormesis is a dose-response relationship which is characterized by low dose stimulation (beneficial effect) and high dose inhibition (destructive intoxication). Hormetic effects mean that there might actually be a reduced risk of exhibiting toxic effects at low exposure levels. Hormetic effects are observed in pharmacology⁽¹⁴⁾, toxicology^(9,16) and radiation experiments⁽²⁾. Calabrese and Baldwin⁽³⁾ reported that hormetic effects are also observed in non-toxicological fields: experimental psychology, plant biology and chemotherapy. In such areas, hormetic effects may mean enhanced longevity or decreased disease incidence. The presence of hormesis was clearly shown in⁽¹⁵⁾, where a reduction in the background tumor incidence was observed in an analysis of 25 cancer studies in the National Toxicology Program.

Hormesis implies the existence of a threshold dose level which is defined as the maximum nonzero exposure level below which no adverse events above background response occur. In particular, the background response occurs at this threshold dose level. This definition is widely accepted, see for example,^(17,13,7). However, despite the wide spread existence of hormesis and discussion in scientific circles, the subject of hormesis is not without controversy, see^(29,5,6), for example.

Hunt and Bowman⁽¹⁷⁾ characterized the overall dose-response relationship by a piecewise function that consists of a quadratic U-shape curve at dose levels that are lower than the threshold and a shifted logistic curve at dose levels that are higher than the threshold. The Hunt-Bowman model has advantages over several threshold models but retains two peculiarities common to many threshold models; the derivative is not continuous at the hormesis threshold dose level and the U-shape is symmetric at low dose levels. To overcome these disadvantages, we propose smooth analytic models that do not possess the threshold dose level parameter explicitly. Specifically, we suggest modeling the overall dose-response relationship by a sum of an exponential decay and a sigmoidal curve that may be, for example, the logistic or Weibull curve.

There are many papers in various disciplines that discuss the existence and estimation issues for a threshold model. Some examples are^{(22,30,28,10,25)} to name a few. However, optimal experimental strategies for detecting hormesis have not been studied. We believe that this paper is the first serious attempt to address design issues for detecting hormesis in an experiment using a variety of techniques. We first focuss on locally optimal designs where we assume nominal values of the model parameters are available. These optimal designs typically require the least effort to find and usually represent a first step in constructing more flexible designs later on. Specifically, we construct locally optimally designs for (i) estimating model parameters, (2) estimating the threshold and (3) testing for the presence of hormesis. Because optimal designs can perform inefficiently under another criteria, it is desirable to have designs that are robust under a variation of criteria, see for example,⁽³²⁾ and⁽²¹⁾. To this end, we construct maximin optimal designs that maximize the minimal efficiency across the multiple criteria. The resulting maximin optimal design provides some assurance that the design can deliver reasonable efficiencies for several optimality criteria at the same time. We also provide similar applications of maximin optimal designs to the situation when there is uncertainty in the model assumptions and we wish to design for a few plausible models at the same time.

2. APPROXIMATE DESIGN AND DESIGN CRITERIA

A design ξ is a discrete probability measure defined on a pre-selected dose interval Ω = [0, d̄]. We denote such a design by ξ={d₀, d₁,…, d_k; w₀, w₁, …, w_k}, where k is the number of distinct doses and d_i ∈ Ω. The weight w_i represents the relative proportion of the total number of observations allocated to the i^th dose, i = 1, 2, …, k. In practice, if N is the pre-determined sample size, the number of animals allocated to dose d_i is N_wi, i = 1, …, k, subject to N_w0 + N_w1 ··· + N_wk = N. Such designs are frequently referred to as continuous designs. They are easier to find and study than exact designs⁽¹⁹⁾.

Given a nonlinear model with mean response function μ(d, θ), under certain regularity conditions^(18,23) the covariance matrix of the least squares estimate of parameters is asymptotically proportional to the inverse of the information matrix

$$M(\xi ,\theta )=\sum _{i}f(d_i,\theta )f^T(d_i,\theta )w_{\iota },f(d,\theta )=\frac{\partial }{\partial \theta }\mu (d,\theta ).$$

Following convention, we formulate the design optimality criterion in terms of the matrix M(ξ, θ)^(27,24). This function is then optimized by choosing the optimal number of dose levels (k), the location of the dose levels (d_i’s) and the proportions (w_i’s) of observations to take these doses.

In hormesis studies, the threshold dose level is the maximum nonzero exposure level below which no adverse events above the background response occur and there is the background response at this level, that is

$$\tau =\tau (\theta )=max\{d\in \mathrm{\Omega }:\mu (d,\theta )\le \mu (0,\theta )\}.$$

Let τ̂ = τ(θ̂) be the estimate for the parameter τ obtained from the least squares estimator (LSE) θ̂ of θ. The optimality criterion for the most precise estimation of the threshold level requires us to minimize Var(τ̂) over all continuous designs on Ω. By the δ-method⁽³¹⁾, see also §17.5 in⁽¹⁾, we have that Var(τ̂) is approximately proportional to b^T(θ)M⁻¹(ξ, θ)b(θ), where

$$b(\theta )=\frac{\partial }{\partial \theta }\tau (\theta ).$$

The existence of b(θ) follows from the implicit function theorem⁽¹¹⁾ and M(ξ, θ) is the information matrix for θ using design ξ in the model

$$y_i=\mu (d_j,\theta )+{\epsilon }_j.$$

Here the $y_j^{\prime }$s are outcomes and the errors ${\epsilon }_j^{\prime }$s are independent and identically distributed random values with zero mean and finite variance. Designs with a singular information matrix are called singular designs. Such designs do not permit all parameters in the models to be estimated but they can serve as a benchmark for comparing various designs in practice. In practice, designs with a singular information matrix are usually adjusted by adding extra points so that the modified designs remain efficient for their original purpose and at the same time allow the researcher to estimate all parameters in the model, see⁽⁸⁾.

For a nonlinear model, the information matrix depends on the model parameters that we want to estimate. Consequently, nominal values of the parameters are required to construct an optimal design. Typically nominal values come from experts’ opinion, or results from similar experiments or a pilot experiment. Once nominal values for the model parameters are available, we assume that they are true values of the model parameters so that the information matrix is now free of unknown parameters. Upon optimization, we obtain an optimal design, which we use to produce the next set of estimates for the model parameters. Usually, these estimates stabilize after a few iterations and the optimal design does not change anymore, see §7 in⁽²⁷⁾ and §4.4 in⁽¹²⁾ for details. These locally optimal designs are easier to find and they usually form the basis for constructing designs that are more robust to model assumptions or designs that can meet the multiple objectives in the study, see §3.2 and §4.

There are design criteria useful for studying hormesis. For estimating model parameters in the mean function, D-optimality is appropriate. This criterion is popular because least squares estimates for θ are asymptotically normally distributed and a D-optimal design minimizes the volume of the asymptotic confidence ellipsoid for the parameter θ. If we are interested in the precise estimation of the threshold dose level τ, an appropriate design is a locally τ-optimal design that minimizes b^T(θ)M⁻¹(ξ, θ)b(θ). This criterion is a particular case of the widely used c-optimality criterion discussed in design monographs^(27,24). Techniques for finding c- and D-optimal designs are are commonly described in design monographs mentioned above.

A more challenging design question is how to design a study specifically for detecting the existence of hormesis. Depending on the context, hormesis may exhibit a J-shaped, U-shaped or an inverted U-shaped dose response, see⁽²⁵⁾ for details. We assume that the mean response as a function of the dose is differentiable and to fix ideas, assume that when hormesis exists, its derivative is negative at the zero dose. Consequently, the hypothesis for the existence of hormesis is

$$H_0:{\mu }^{\prime }(0,\theta )\ge 0\text{vs}{H}_1:{\mu }^{\prime }(0,\theta )<0$$

where μ′(d, θ) = ∂μ(d, θ)/∂d. It follows that if μ′(0, θ̂) is used to test the hypothesis, then a design that maximizes the power of the test also minimizes Var(μ′(0, θ̂)), see §3 in⁽²⁴⁾. By the δ-method, Var(μ′(0, θ̂)) is asymptotically proportional to

$$h^T(0,\theta )M^{-1}(\xi ,\theta )h(0,\theta )$$

where h(d, θ) = ∂f(d, θ)/∂d. This implies that we want to find a locally optimal design that minimizes h^T(θ)M⁻¹(ξ, θ)h(θ) where h(θ) = h(0, θ). This criterion is also a special case of c-optimality; we call the criterion h-optimality and designs that minimize this criterion h-optimal designs. Because these designs minimize the variance of the estimate of the derivative of the mean response at 0, they remain optimal even for the situation when we reverse the null and alternative hypotheses. This will correspond to the situation when we have a hormetic effect with a reverse shape.

Throughout, we measure the worth of a design by its efficiency. This number is between 0 and 1 and is typically the ratio of the criterion values from the current design and the locally optimal design for a selected value of parameters and reported as a percentage after multiplying by 100%. A design with 50%-efficiency means that it has to be replicated twice to do as well as the optimal design. For D-efficiency, we work with the p-root of the ratio to maintain this interpretation, where p is the number of parameters of interest.

In the next section, we focus on the Hunt-Bowman model and construct a variety of locally optimal designs for this model. In §4, we propose alternative models that do not have explicit threshold parameter and present a variety of optimal designs for these models. Robust designs are discussed in §5. These designs ensure the constructed designs have the best possible efficiencies under various design criteria or different model assumptions. Justifications for all the optimal designs are quite similar and we sketch the key ideas in Lemma 1 and Lemma 2 in the appendix.

3. THE HUNT-BOWMAN MODEL

Hunt and Bowman⁽¹⁷⁾ proposed modeling the mean response μ(d) at dose d using the piecewise quadratic-logistic function

$$\mu (d)=\{\begin{array}{cc}{c}_1{d}^2+c_{2}d+\kappa & 0\le d\le \tau \\ \frac{1}{1+e^{{\beta }_0-{\beta }_1(d-\tau )}}& d>\tau \end{array}$$ (1)

with two restrictions on the six parameters c₁, c₂, κ, τ, β₀, β₁: μ(0) = μ(τ) and μ(τ−) = μ(τ+). The former follows from the definition of the hormesis threshold and the latter follows from the continuity of the dose-response curve. These restrictions imply $\kappa =\frac{1}{1+e^{{\beta }_0}}$ and c₂ = −c₁τ. The parameter τ is the threshold dose and the vector of model parameters for the Hunt-Bowman model is θ = (c₁, τ, β₀, β₁)^T with 4 independent parameters. Throughout, we assume that the design interval is user-specified and has the form [0, d̄]. Furthermore, we consider values of θ such that 0 < μ(d, θ) < 1, 0 ≤ τ < d̄ and β₁ > 0. As in §2, we denote the mean response by μ(d, θ) to emphasize its dependence on θ and sometimes simply μ(d) for short.

Hunt and Bowman⁽¹⁷⁾ used model (1) to fit data from a study that measured developmental effects of the chemical diethylhexyl phthalate on mice. In the experiment, the pregnant animals were exposed to one of five dose levels including the control dose at d = 0. Here, a dose level corresponds to administering the drug as a percentage of the animal’s diet. The number of affected fetuses was recorded for each animal and analysis results from⁽¹⁷⁾ showed a U-shape dose response at low dose levels. Figure 1 in §4 shows the mean function of the Hunt-Bowman model for various sets of values for θ and also the fitted mean response function from the exp+log model to be discussed. Throughout, our design setup is based on the Hunt and Bowman paper; for instance, our design interval is always [0, 0.15]. All designs are locally optimal in the sense that they depend on the nominal values of the model parameters. In other fields, these designs are sometimes called baseline designs.

Fig. 1.

Dose-response curves for the piecewise quadratic-logistic for different nominal values of θ: θ = (834, 0.035, 1.45, 38)^T (dotted), θ = (164, 0.04, 1.5, 44)^T (light gray), θ = (294, 0.037, 1.48, 44)^T (dark gray). The black solid line corresponds to the fitted response from the exp+logistic model when θ = (0.15, 85, 3.4, 45)^T.

3.1 Locally Optimal Designs for the Hunt-Bowman model

We now investigate the locally τ-optimal design, the locally D-optimal design and the locally h-optimal design for the Hunt-Bowman model. For θ = (c₁, τ, β₀, β₁)^T a direct calculation shows the vector f(d, θ) for the model is

$$f(d,\theta )={\left(d^2-\tau d,-c_{1}d,-\frac{e^{{\beta }_0}}{{(1+e^{{\beta }_0})}^2},0\right)}^T$$

if 0 ≤ d ≤ τ and

$$f(d,\theta )={\left(0,-{\beta }_1\frac{e^{{\beta }_0-{\beta }_1(d-\tau )}}{{(1+e^{{\beta }_0-{\beta }_1(d-\tau )})}^2},-\frac{e^{{\beta }_0-{\beta }_1(d-\tau )}}{{(1+e^{{\beta }_0-{\beta }_1(d-\tau )})}^2},\frac{(d-\tau )e^{{\beta }_0-{\beta }_1(d-\tau )}}{{(1+e^{{\beta }_0-{\beta }_1(d-\tau )})}^2}\right)}^T$$

if d > τ.

Table I shows locally D-optimal designs for different nominal values, including nominal values used in⁽¹⁷⁾ displayed in the first row. All have four doses, including the 0 dose, with equal proportions of observations at all doses.

Table I.

Locally D-optimal designs {d₀ = 0, d₁, d₂, d₃; 1/4, 1/4, 1/4, 1/4}for the Hunt-Bowman model for different nominal values. The D- and τ-efficiencies of the design ξ_u are given, along with the D-efficiencies of ${\xi }_0={\xi }_D^{\ast }({\theta }^{(0)})$ where θ⁽⁰⁾ =(170, 0.04, 1.46, 40)^T.

c1	τ	β0	β1	d1	d2	d3	effD(ξu)	effD(ξ0)	effτ (ξu)
170	0.04	1.46	40	0.020	0.04	0.0991	0.80	1	0.346
170	0.03	1.46	40	0.015	0.0404	0.0926	0.61	0.93	0.583
170	0.05	1.46	40	0.025	0.05	0.1090	0.86	0.70	0.405
170	0.04	1.26	40	0.020	0.0454	0.0976	0.81	0.98	0.420
170	0.04	1.66	40	0.020	0.04	0.1026	0.77	0.99	0.279
170	0.04	1.46	30	0.020	0.04	0.1188	0.75	0.94	0.205
170	0.04	1.46	50	0.020	0.0483	0.0901	0.76	0.88	0.541

The table also shows the D-efficiency of the design

$${\xi }_u=\{0,0.025,0.05,0.1,0.15;1/5,\dots ,1/5\}$$

that closely approximates the one implemented in the developmental toxicity study of diethylhexl phthalate (DEHP) reported in⁽¹⁷⁾. In what is to follow, we refer ξ_u as the implemented design for convenience. There was no rationale provided for the choice of ξ_u in their paper but we note that the design resembles a somewhat uniform design with equal weights over a set of log-uniformly spaced doses in Ω. Such designs may be intuitively appealing but it can be very inefficient, depending on the aims of the study. For example, row 1 in Table 1 lists the locally D-optimal design when the nominal value is θ⁽⁰⁾ = (170, 0.04, 1.46, 40) and shows that the τ-efficiency of ξ_u for estimating the threshold dose is only 34.6%. Here, the nominal value θ⁽⁰⁾ for θ is obtained from the estimates in the teratology data set described in⁽¹⁷⁾. The D-efficiency of the ξ_u for estimating the model parameters is 80% so this design is 20% less efficient than the locally D-optimal design ξ₀. As Table 1 shows the D-efficiencies of ξ_u can drop to 61% for other neighboring values of θ⁽⁰⁾. Even when there is good rationale for a uniform design, the choice for the number of points can be problematic⁽³³⁾.

Because a locally optimal design depends on nominal values of the parameters, it is instructive to study the effect of misspecification in the nominal values of the parameters. To this end, we calculate the D-efficiency of the design ${\xi }_0={\xi }_D^{\ast }({\theta }^{(0)})$ which is D-optimal for θ⁽⁰⁾ = (170, 0.04, 1.46, 40) and compute its efficiency for other values in the neighborhood of θ⁽⁰⁾. The column entitled eff_D(ξ₀) of Table I shows the locally D-optimal design ξ₀ is relatively robust to small misspecification of the nominal values displayed in the table because all efficiencies in the column are near unity. The biggest drop in D-efficiency occurs when the nominal values are in the third row of Table I. Even then the D-efficiency is still 70% for the range of nominal values shown in the table. Such sensitivity analysis is useful because in practice we do not know the true values of the model parameters and misspecification in the nominal values can result in unacceptable loss in efficiency.

The locally optimal designs for estimating τ are found from Lemma 2 in the appendix. Table I shows the D- and τ-efficiencies of the design ξ_u. The τ-efficiencies are uniformly low, implying that the implemented design in the DEHP study does not estimate the threshold value well at all. The second and third last columns also show the D-optimal designs are generally more robust to misspecification of the nominal values than the implemented design.

Numerical computation shows that locally h-optimal designs have 3 dose levels and their form is {0, τ/2, τ; w₀, 0.5, 0.5 − w₀} for the nominal values displayed in Table II. More design points are possible; for example, when θ_◇ = (170, 0.04, 1.86, 40)^T, the locally h-optimal design is ${\xi }_h^{\ast }({\theta }_{◇})=\{0,0.020,0.0479,0.125;0.3174,0.5029,0.1644,0.0153\}$. For the same set of nominal values as in Table I, we observe in Table II that the implemented design ξ_u always has lower than 50% efficiencies for detecting the presence of hormesis and for θ = (170, 0.03, 1.46, 40)^T this efficiency is only 16.4%.

Table II.

Locally h-optimal designs {d₀ = 0, d₁, d₂; w₀, w₁, w₂} for the Hunt-Bowman model and h-efficiencies of the design ξ_u for various nominal values.

c1	τ	β0	β1	d1	d2	w0	w1	w2	effh(ξu)
170	0.04	1.46	40	0.020	0.040	0.359	0.5	0.141	0.474
170	0.03	1.46	40	0.015	0.030	0.367	0.5	0.133	0.164
170	0.05	1.46	40	0.025	0.050	0.327	0.5	0.173	0.499
170	0.04	1.26	40	0.020	0.040	0.378	0.5	0.123	0.463
170	0.04	1.66	40	0.020	0.040	0.342	0.5	0.158	0.486
170	0.04	1.46	30	0.020	0.040	0.315	0.5	0.185	0.489
170	0.04	1.46	50	0.020	0.040	0.389	0.5	0.111	0.459

3.2 Criterion-robust Designs for the Hunt-Bowman Model

It is well known that optimal designs constructed under one criterion can perform poorly under another^(32,21). Consequently, it is always desirable to have a design that is robust under different criteria. This is especially so when there are explicit multi-objectives at the onset of the study. In this subsection, we first construct a criterion-robust design that provides relatively high efficiency for the two criteria: D- and τ-optimality. Formally, a criterion-robust design maximizes the minimum of D- and τ-efficiencies, that is

$$min\{{\text{eff}}_D(\xi ),{\text{eff}}_{\tau }(\xi )\}\to \underset{\xi }{max}.$$ (2)

Generalization of this robust criterion to 3 or more objectives is possible; in hormesis studies we may want to maximize the minimum of the efficiencies across all three criteria, i.e.

$$min\{{\text{eff}}_D(\xi ),{\text{eff}}_{\tau }(\xi ),{\text{eff}}_h(\xi )\}\to \underset{\xi }{max}.$$ (3)

We call (2) and (3) criterion (2) and criterion (3) respectively. Here and throughout, the following iterative algorithm is used to compute robust designs which are actually maximin designs. First, we maximize the optimality criterion within the class of all s-point designs where the initial value of s we choose is the number of parameters in the model. The resulting design is a s-point maximin optimal design. For optimization, we employ the Nelder-Mead algorithm in the matlab package. After the optimal s-point maximin design is found, we consider the class of all (s + 1)-point designs and find an optimal design within this class and repeat the procedure. At each iteration, we increase the number of points by one, until there is no change in the criterion value.

Numerical computation, in which we used θ = (170, 0.04, 1.46, 40)^T, shows that the criterion-robust design for criterion (2) is {0, 0.020, 0.040, 0.098, 0.104; 0.381, 0.099, 0.419, 0.097, 0.004} and its D-and τ-efficiencies are both equal to 0.799. The criterion (3) involves D-, h- and τ-optimality and the criterion-robust design is {0, 0.021, 0.040, 0.098, 0.112; 0.389, 0.249, 0.329, 0.031, 0.001}. Its D-, hand τ-efficiencies are all equal to 0.714, and both optimal designs are supported at five points.

4. ALTERNATIVE MODELS

There are two peculiarities of the Hunt-Bowman model. First, it has a derivative that is not continuous at the hormesis threshold dose level and second, its U-shape curve, by definition, has symmetry at low dose levels. The second restriction can be a serious limitation because non-symmetry may be an important feature in some applications; see, for example, Figure 4 in⁽³⁾. When hormesis is not present in the single-agent/response scenario, the Hunt-Bowman model simplifies to the simpler Schwartz’s threshold model⁽²⁶⁾:

$$\begin{array}{l}\mu (d)=\mu (d,{\beta }_0,{\beta }_1,\tau )\\ =\{\begin{array}{ll}\frac{1}{1+e^{{\beta }_0}}\hfill & 0\le d\le \tau ,\hfill \\ \frac{1}{1+e^{{\beta }_0-{\beta }_1(d-\tau )}}\hfill & d\ge \tau .\hfill \end{array}\end{array}$$ (4)

In practice, model assumptions are often questionable. Thus, one considers a variety of models and constructs optimal designs for obtaining the best estimates for several characteristics of the drug under various model assumptions. In⁽⁸⁾ it was shown that more flexible models can improve parameter estimation. As alternatives to the Hunt-Bowman model, we propose smooth analytic models that do not possess a threshold dose level parameter explicitly. Specifically, we use a mean function that is a sum of an exponential decay curve and a sigmoidal curve. In particular, we propose two models: one has the form

$$\begin{array}{l}\mu (d)=\mu (d,c_0,c_1,{\beta }_0,{\beta }_1)\\ =c_0{e}^{-c_{1}d}+\frac{1}{1+e^{{\beta }_0-{\beta }_{1}d}},\end{array}$$ (5)

which is a sum of an exponential decay model and a logistic model, and the other is

$$\begin{array}{l}\mu (d)=\mu (d,c_0,c_1,{\beta }_0,{\beta }_1,{\beta }_2)\\ =c_0{e}^{-c_{1}d}+\left(1-{\beta }_0{e}^{{\beta }_1{d}^{{\beta }_2}}\right),\end{array}$$ (6)

which is a sum of an exponential decay model and a Weibull model, which was suggested by⁽⁶⁾ for describing dose-response relationship in toxicity studies. For models (5) and (6) we consider parameter values for which 0 ≤ μ(d) ≤ 1 and μ(d) has a required shape for d ∈ [0, d̄]. Other possible sigmoidal growth models are the Gomperts, Richards and Morgan-Mercer-Flodin models given respectively by

$${\beta }_0{e}^{-{\beta }_1{e}^{-{\beta }_{2}d}},\frac{{\beta }_0}{{(1+{\beta }_1{e}^{-{\beta }_{2}d})}^{{\beta }_3}},\frac{{\beta }_1+{\beta }_0{d}^{{\beta }_3}}{{\beta }_2+d^{{\beta }_3}}$$

with β₀ = 1. Still another model is the Chen-Kodell’s model that assumes an increasing response

$$\mu (d)=\mu (d,{\beta }_0,{\beta }_1,{\beta }_2)=1-e^{{\beta }_0-{\beta }_1{d}^{{\beta }_2}}.$$ (7)

Several such dose-response relationships are displayed in Figure 1. We observe that models (5) and (6) are quite flexible for practical applications, and they have smooth response function unlike the Hunt-Bowman model. The hormesis threshold for these models is an implicit parameter and the vector b(θ) required in the τ-optimality criterion can be directly obtained from the implicit function theorem. Specifically, differentiating the identity μ(τ(θ), θ) = μ(0, θ) on θ and expressing ∂τ/∂θ via other terms we obtain that

$$b(\theta )={-\frac{\partial (\mu (d,\theta )-\mu (0,\theta ))}{\partial \theta }/\frac{\partial \mu (d,\theta )}{\partial d}|}_{d=\tau (\theta )}.$$

The resulting τ-optimal design minimizes b^T(θ)M⁻¹(ξ, θ)b(θ), which is proportional to the asymptotic variance of the estimated implicit threshold parameter in the model.

We fit some of the above models to the teratology data set digitized from Figure 1 in⁽¹⁷⁾. For convenience, we refer to models given in (5) and (6) as the exp+log and the exp+weib models respectively. Figure 2 displays the dose-response curves for these models and the observed proportions for each dose group.

Fig. 2.

The Hunt-Bowman, Schwartz, Chen-Kodell and the exp+log models have similar fits for their mean responses based on the teratology data.

Table III shows the estimates for the expected response probability at each dose level for the exp+log, exp+weib, Hunt-Bowman, Chen-Kodell and Schwartz’s models. We note that the sum of squares of errors (SSE) from the exp+log model is roughly the same as that from the Hunt-Bowman model, suggesting that smooth models can fit such data as well as threshold models. Other goodness of fit measures show similar results.

Table III.

Observed and fitted response probability μ(d_i) at the dose level d_i, i = 0, 1, 2, 3, 4 using various models.

dose	number of litters	exp+weib	exp+log	Hunt-Bowman	Chen-Kodell	Schwartz	observed proportions
		(6)	(5)	(1)	(7)	(4)
d0 = 0.000	30	0.1889	0.1889	0.1889	0.1554	0.1552	0.1889
d1 = 0.025	26	0.1162	0.1181	0.1162	0.1647	0.1552	0.1162
d2 = 0.050	26	0.2514	0.2423	0.2435	0.2407	0.2435	0.2514
d3 = 0.100	24	0.6961	0.7114	0.7115	0.6974	0.7115	0.6961
d4 = 0.150	25	0.9816	0.9504	0.9497	0.9823	0.9497	0.9816
SSE		4.6636	4.6958	4.6963	4.7614	4.7700

4.1 Locally Optimal Designs for the exp+log Model

Straightforward calculation yields that the vector f(d, θ) for the exp+log model is

$$f(d,\theta )={\left(e^{-c_{1}d},-c_0{de}^{-c_{1}d},-\frac{e^{{\beta }_0-{\beta }_{1}d}}{{(1+e^{{\beta }_0-{\beta }_{1}d})}^2},\frac{{de}^{{\beta }_0-{\beta }_{1}d}}{{(1+e^{{\beta }_0-{\beta }_{1}d})}^2}\right)}^T.$$

Table IV shows the locally D-optimal designs for the exp+log model and the relative efficiencies of the implemented design ξ_u in the DEHP study for various nominal values. We notice that the locally D-optimal design does not depend on the parameter c₀ because this parameter enters linearly in the mean response function. Numerical calculation shows that the locally D-optimal design has 4 points and always contains the zero dose. The table also shows the D-efficiencies of the locally D-optimal design ${\xi }_0={\xi }_D^{\ast }(\theta ){\mid }_{\theta =(0.15,89,3.2,41)}$ when other nominal values are used. These D-efficiencies indicate how sensitive the design ξ₀ is to misspecification of the nominal values. For the nominal values we looked at, all are at least 82% suggesting that ξ₀ is robust to misspecification of the nominal values. The corresponding D-efficiencies for the design ξ_u range from 57% to 72%, suggesting that this design is more costly to use when nominal values are misspecified. The last two columns shows the estimated threshold τ and the maximal efficiency of the implemented design ξ_u for estimating τ is 50.9% for the nominal values considered.

Table IV.

Locally D-optimal design {d₀ = 0, d₁, d₂, d₃; 1/4, 1/4, 1/4, 1/4} for the exp+log model for different nominal values. The D-efficiencies of designs ξ_u and ${\xi }_0={\xi }_D^{\ast }(\theta ){\mid }_{\theta =(0.15,89,3.2,41)}$ and the corresponding threshold τ are given at the penultimate last 3 columns. The last column shows the τ-efficiencies of the design ξ_u.

c0	c1	β0	β1	d1	d2	d3	effD(ξu)	effD(ξ0)	τ	effτ(ξu)
0.15	89	3.2	41	0.0109	0.0558	0.1051	0.65	1	0.042	0.479
0.15	70	3.2	41	0.0134	0.0579	0.1063	0.72	0.99	0.041	0.490
0.15	110	3.2	41	0.0090	0.0543	0.1043	0.57	0.99	0.042	0.468
0.15	89	2.4	41	0.0103	0.0433	0.0893	0.60	0.92	0.028	0.459
0.15	89	4.0	41	0.0112	0.0727	0.1233	0.65	0.90	0.058	0.428
0.15	89	3.2	30	0.0112	0.0734	0.1422	0.72	0.82	0.058	0.467
0.15	89	3.2	50	0.0106	0.0472	0.0870	0.59	0.90	0.034	0.509

Hormesis is ascertained via the hypothesis testing framework after identifying the vector h(θ) in §2. This vector is complicated for the exp+log model and the exp+weib model and we do not display it. Table V shows selected locally h-optimal designs for the exp+log model and the h-efficiencies of ξ₀ and ξ_u. Again, for the nominal values we investigated, the table shows these efficiencies for the implemented design ξ_u are unacceptably low, ranging from 10.7% to 29.6%; in contrast, the locally D-optimal design has at least 83.9% for detecting the presence of hormesis in the study. Tables 4 and 5 show that the implemented design ξ_u estimates both the threshold dose and the presence of hormesis poorly. The τ-efficiencies range from 42.8% to 50.9% and the h-efficiencies range from 10.5% to 29.6%. The locally τ-optimal design for the exp+log model is singular and takes all observations at 0 and τ with equal proportions, that is ${\xi }_{\tau }^{\ast }(\theta )=\{0,\tau (\theta );1/2,1/2\}$.

Table V.

Locally h-optimal design {d₀ = 0, d₁, d₂, d₃; w₀, w₁, w₂, w₃} for the exp+log model for different nominal values. The last two columns show that h-efficiencies of designs ξ_u and ${\xi }_0={\xi }_h^{\ast }(\theta ){\mid }_{\theta =(0.15,89,3.2,41)}$.

c0	c1	β0	β1	d1	d2	d3	w0	w1	w2	w3	effh(ξu)	effh(ξ0)
0.15	89	3.2	41	0.0108	0.0526	0.1187	0.371	0.501	0.087	0.041	0.193	1
0.15	70	3.2	41	0.0129	0.0564	0.1206	0.370	0.491	0.090	0.049	0.296	0.956
0.15	110	3.2	41	0.0091	0.0492	0.1174	0.372	0.508	0.085	0.035	0.107	0.954
0.15	89	2.4	41	0.0096	0.0430	0.1034	0.364	0.475	0.104	0.056	0.166	0.777
0.15	89	4.0	41	0.0118	0.0651	0.1362	0.377	0.528	0.066	0.029	0.216	0.943
0.15	89	3.2	30	0.0113	0.0648	0.1500	0.372	0.511	0.087	0.030	0.221	0.939
0.15	89	3.2	50	0.0103	0.0457	0.0986	0.370	0.493	0.089	0.048	0.186	0.839

4.2 Locally Optimal Designs for the exp+weib Model

Direct calculus yields that the vector f(d, θ) for the exp+weib model is

$$f(d,\theta )={\left(e^{-c_{1}d},-c_0{de}^{-c_{1}d},-e^{{\beta }_0-{\beta }_1{d}^{{\beta }_2}},d^{{\beta }_2}{e}^{{\beta }_0-{\beta }_1{d}^{{\beta }_2}},{\beta }_1{d}^{{\beta }_2}ln(d)e^{{\beta }_0-{\beta }_1{d}^{{\beta }_2}}\right)}^T.$$

The locally D-optimal design for this model does not depend on parameters c₀ and β₀ because they appear linearly in the mean function. Consequently, we do not vary their nominal values in Table VI that shows selected locally D-optimal designs and the D-efficiencies of the implemented design ξ_u. We observe that for the nominal values in the table, the locally D-optimal designs have 5 doses and always include the two extreme doses. The table shows locally D-optimal designs have at least 82% efficiencies for estimating the model parameters compared with at least 72% D-efficiencies for the implemented design ξ_u. Further, efficiency of ξ_u for estimating τ can be as low as 11%, suggesting that the implemented design ξ_u is a poor design to use for the study.

Table VI.

Locally D-optimal designs {d₀ = 0, d₁, d₂, d₃, d₄ = 0.15; 1/5,…, 1/5} for the exp+weib model for different nominal values. The D-efficiencies of designs ξ_u and ${\xi }_0={\xi }_D^{\ast }(\theta ){\mid }_{\theta =(0.9,10.5,0.55,65,1.8)}$ and the corresponding threshold τ are given in the last 3 penultimate columns. The last column shows the τ-efficiencies of the design ξ_u.

c0	c1	β0	β1	β2	d1	d2	d3	effD(ξu)	effD(ξ0)	τ	effτ(ξu)
0.9	10.5	0.55	65	1.8	0.0161	0.0535	0.1047	0.92	1	0.040	0.461
0.9	7.5	0.55	65	1.8	0.0160	0.0537	0.1050	0.92	0.99	0.028	0.431
0.9	13.5	0.55	65	1.8	0.0159	0.0532	0.1042	0.92	0.99	0.050	0.401
0.9	10.5	0.55	45	1.8	0.0167	0.0565	0.1092	0.91	0.99	0.059	0.372
0.9	10.5	0.55	85	1.8	0.0151	0.0509	0.0998	0.92	0.99	0.029	0.446
0.9	10.5	0.55	65	1.5	0.0091	0.0361	0.0827	0.72	0.82	0.007	0.110
0.9	10.5	0.55	65	2.1	0.0222	0.0665	0.1167	0.87	0.94	0.085	0.422

Table VII shows the locally h-optimal designs for the exp+weib model. These design have larger weights at the low dose levels and appear to be sensitive to the parameter β₂ and not sensitive to other parameters. Again it is clear from the table that locally D-optimal designs outperform the implemented design ξ_u by a wide margin in terms of assessing the presence of hormesis since h-efficiencies and τ-efficiencies of the design ξ_u for estimating the threshold are generally low and in average about 40%. These two tables show that ξ_u estimate τ and test for the presence of hormesis quite poorly. As in the exp+log model, our results show that the locally τ-optimal design for the exp+weib model requires that we take all observations at 0 and τ with equal proportions, that is ${\xi }_{\tau }^{\ast }(\theta )=\{0,\tau (\theta );1/2,1/2\}$.

Table VII.

Locally h-optimal design {d₀ = 0, d₁, d₂, d₃, d₄ = 0.15; w₀, w₁, w₂, w₃, w₄} for the exp+weib model for different nominal values of the parameters. The h-efficiencies of designs ξ_u and ${\xi }_0={\xi }_h^{\ast }(\theta ){\mid }_{\theta =(0.9,10.5,0.55,65,1.8)}$ are shown in the last 2 columns.

c0	c1	β0	β1	β2	d1	d2	d3	w0	w1	w2	w3	w4	effh(ξu)	effh(ξ0)
0.9	10.5	0.55	65	1.8	0.0129	0.0534	0.1105	0.299	0.389	0.146	0.109	0.056	0.424	1.000
0.9	7.5	0.55	65	1.8	0.0129	0.0535	0.1108	0.303	0.393	0.145	0.106	0.052	0.419	0.999
0.9	13.5	0.55	65	1.8	0.0127	0.0531	0.1101	0.296	0.382	0.145	0.114	0.063	0.431	0.997
0.9	10.5	0.55	45	1.8	0.0133	0.0565	0.1150	0.296	0.386	0.147	0.113	0.058	0.403	0.984
0.9	10.5	0.55	85	1.8	0.0121	0.0499	0.1054	0.302	0.391	0.143	0.107	0.057	0.431	0.968
0.9	10.5	0.55	65	1.5	0.0071	0.0354	0.0859	0.290	0.386	0.157	0.108	0.058	0.148	0.218
0.9	10.5	0.55	65	2.1	0.0185	0.0667	0.1214	0.319	0.404	0.137	0.096	0.044	0.398	0.726

5. ROBUST DESIGNS

In this section we construct criterion-robust and model-robust designs that offer some protection when we are uncertain on the design criterion and model assumptions. We first present designs that are robust to two and three optimality criteria for the exp+log model and the exp+weib model before we construct designs that are robust to model assumptions. All of these designs have to be determined numerically.

We recall that criterion (2) concerns estimating all parameters in the mean function and also the threshold parameter. Criterion (3) additionally seeks to detect the existence of hormesis. For the exp+log model with θ = (0.15, 89, 3.2, 41)^T, the criterion robust design for criterion (2) is {0, 0.013, 0.05, 0.107; 0.41, 0.12, 0.36, 0.11}, whose both the D- and τ-efficiencies are 82%. For criterion (3), the corresponding criterion robust design is {0, 0.012, 0.048, 0.114; 0.387, 0.264, 0.306, 0.043}, whose D-, τ- and h-efficiencies are 73%.

For the exp+weib model with θ = (0.9, 10.5, 0.55, 65, 1.8)^T, the criterion robust design for criterion (2) is {0, 0.027, 0.049, 0.106, 0.150; 0.38, 0.185, 0.26, 0.092, 0.083} and both its D- and τ-efficiencies are 75%. The corresponding criterion robust design for criterion (3) is {0, 0.015, 0.044, 0.108, 0.150; 0.375, 0.207, 0.31, 0.064, 0.044} and its D-, τ- and h-efficiencies are 71%. For the cases considered here, the criterion-robust designs for the exp+log model has four points and the criterion-robust designs for the exp+weib model has five points, regardless of the number of criteria involved. As expected, efficiencies always drop when an additional criterion is introduced because of more stringent demands on the design.

In developmental studies, there typically are several plausible dose-response models for describing the binary or non-binary outcomes. Consequently, it is desirable to design the study so that we have efficient estimates no matter which one of a few plausible models holds. Accordingly, we construct robust designs that maximizes the minimal D- and τ-efficiency for models for a few competing models and the maximization is either over a set of designs with a pre-determined of points or over the set of all continuous designs. The resulting design will ensure that we have the best possible efficiency for estimating model parameters and testing for the presence of hormesis as long as the true model is correctly identified as one of the plausible models. Specifically, we want to find a design that has the following property:

$$R(\xi \mid I):=\underset{i\in I}{min}min\{{\text{eff}}_D^{(i)}(\xi ),{\text{eff}}_{\tau }^{(i)}(\xi )\}\to \underset{\xi }{max}$$

where I is a set of models. We focus on two choices of I: I₂ is the set consisting two plausible models: the Hunt-Bowman and exp+log models and I₃ is the set consisting three plausible models: the Hunt-Bowman, exp+log and exp+weib models.

We briefly report robust designs found numerically using the following nominal values obtained from the teratology data set: We have θ = (170, 0.04, 1.46, 40)^T for the Hunt-Bowman model, θ = (0.15, 89, 3.2, 41)^T for the exp+log model, θ = (0.9, 10.5, 0.55, 65, 1.8)^T for the exp+weib model. The maximin robust designs depend on the set where the maximization is taken.

We found that the robust design that maximizes R(ξ|I₃) among all 5-point designs is the design that takes observations at dose levels 0, 0.015, 0.047, 0.107 and 0.150 with weights given by 0.394, 0.073, 0.401, 0.08 and 0.052. The minimal efficiency of this design is 67.3%. If we maximize the criterion among 6-point designs, the optimal design takes observations at dose levels 0, 0.013, 0.040, 0.061, 0.102 and 0.150 with weights given by 0.365, 0.094, 0.32, 0.07, 0.096 and 0.055. The minimal efficiency of this design is 73.6%. If we further enlarge this set to all designs with 7 or more points, the resulting robust design does not seem to provide a larger minimal efficiency.

For the case when we wish to maximize R(ξ|I₂), the maximin robust design requires doses at 0, 0.013, 0.04, 0.057 and 0.102 with weights given by 0.396, 0.11, 0.337, 0.048 and 0.109. The minimal efficiency of this design is 77.2%. Not surprisingly, this efficiency is higher than the two previous efficiencies because there are fewer competing models under consideration. Additional numerical results not given here show that robust designs just depend slightly on the nominal values of parameters used for their construction.

6. CONCLUSIONS

In this work, we discussed design issues for assessing hormetic effects and provided optimal designs for estimating threshold value, model parameters and whether hormesis exists. We proposed smooth models that are competitive with models that have an explicit threshold and found designs that are robust under a variation of design criteria and model assumptions. When we compared the constructed optimal designs with a design similar to the one implemented in a study reported by Hunt and Bowman⁽¹⁷⁾, the optimal designs have uniformly higher efficiencies for attaining the experimental goals. This means the proposed designs provide more accurate statistical inference than the implemented design for the sample size.

Some of the proposed optimal designs also enjoy invariant properties that enable to deduce how the locally optimal design changes when the dose interval is changed in a meaningful way. For example, consider the exp+log model with parameter θ = (c₀, c₁, β₀, β₁)^T on the dose interval [0, d̄] and we wish to determine how the optimal designs change when we expand the dose interval from [0, d̄] to [0, γd̄] and γ is a user-selected positive number. To this end, let $t_i^{\ast }(c_0,c_1,{\beta }_0,{\beta }_1,\overline{d})$ be the ith design point of the D-, τ- or h-optimal design on [0, d̄] with corresponding weight $w_i^{\ast }(c_0,c_1,{\beta }_0,{\beta }_1,\overline{d})$. It can be shown that the optimal design on the interval [0, γd̄] has the following design points and weights:

$$\begin{array}{l}\gamma t_i^{\ast }(c_0,c_1,{\beta }_0,{\beta }_1,\overline{d})=t_i^{\ast }(c_0,c_1/\gamma ,{\beta }_0,{\beta }_1/\gamma ,\gamma \overline{d}),\\ w_i^{\ast }(c_0,c_1,{\beta }_0,{\beta }_1,\overline{d})=w_i^{\ast }(c_0,c_1/\gamma ,{\beta }_0,{\beta }_1/\gamma ,\gamma \overline{d}).\end{array}$$

The corresponding results for the exp+weib model when the dose interval is changed from [0, d̄] to [0, γd̄] are

$$\begin{array}{l}\gamma t_i^{\ast }(c_0,c_1,{\beta }_0,{\beta }_1,{\beta }_2,\overline{d})=t_i^{\ast }(c_0,c_1/\gamma ,{\beta }_0,{\beta }_1/{\gamma }^{{\beta }_2},{\beta }_2,\gamma \overline{d}),\\ w_i^{\ast }(c_0,c_1,{\beta }_0,{\beta }_1,{\beta }_2,\overline{d})=w_i^{\ast }(c_0,c_1/\gamma ,{\beta }_0,{\beta }_1/{\gamma }^{{\beta }_2},{\beta }_2,\gamma \overline{d}).\end{array}$$

We close with a note that a common critique of optimal designs is that they have too few points to be useful in practice. For example, some of our optimal designs do not have enough points to detect lack of fit in the model. We remind readers that one of the main uses of optimal designs is to calibrate the worth of any design. If the researcher likes to have more design points and change the weights at some points, the researcher can use the optimal design as a guide how to adjust the design. Absent this guidance, practitioners tend to frequently use designs without good rationale resulting in waste of resources, as we demonstrated here with the use of the implemented design ξ_u reported in⁽¹⁷⁾. In general, the design should be selected carefully and not stray too far away from the optimum where its efficiencies become unacceptable.

APPENDIX

The Hunt-Bowman model defined on the interval Ω₀ = [0, d̄] does not satisfy regulatory conditions of the asymptotic theory because the response function is not necessarily differentiable at the threshold. However, for any δ > 0 the model is defined on Ω_δ = [0, τ − δ] ∪ [τ + δ, d̄] and satisfies the differentiability assumptions. Moreover, if ξ_δ is an optimal design on Ω_δ, it is easy to see that as δ → 0, the limit of ξ_δ converges to an optimal design computed for the region Ω₀. Thus for computational purposes, we can simply use the vector f(d, θ) defined on Ω₀ even though it is not continuous at d = τ.

The following technical result is a reformulation of the equivalence theorem for c-optimality given in⁽²⁴⁾.

Lemma 1

If f₁(d), …, f_m(d) are linearly independent continuous functions on the interval [0, d̄], the design ξ is c-optimal if and only if there exists a vector q ∈ ℝ^m, such that the generalized polynomial q^Tf(d) satisfies the following conditions for some ν > 0:

1. q^Tf(d_i) = (−1)ⁱ, i = 1, …, m,

2. |q^Tf(d)| ≤ 1 for all d ∈ [0, d̄],

3. F_w = νc,

where $F={({(-1)}^j{f}_i(d_j))}_{i,j=1}^{m,k}$ and w = (w₁, …, w_k). Moreover, c^TM⁻(ξ)c = 1/ν².

Lemma 1 is useful for finding a c-optimal design for our setup. Parts (i) and (ii), along with the vector q, provide a characterization of the dose levels required and part (iii) is particularly helpful for computing the optimal proportion of observations at each of the dose levels. The next lemma describes locally optimal designs for the Hunt-Bowman model.

Lemma 2

For the Hunt-Bowman model defined on the dose interval [0, d̄],

1. the locally τ-optimal design is singular and has design points points at 0 and τ, that is ${\xi }_{\tau }^{\ast }(\theta )=\{0,\tau (\theta );1/2,1/2\}$;

2. the locally D-optimal design does not depend on the parameter c₁ and contains 0 as a support point. Moreover, it has at most three design points on [0, τ] and at most two points on [τ, d̄]. If there are three points on [0, τ], then these points are 0, τ/2 and τ.

3. the locally h-optimal design has at least 3 design points.

Proof

Since τ is a component of the vector θ, we have b(θ) = (0, 1, 0, 0)^T. Part (iii) of Lemma 2 holds because νb(θ) = f(0, θ)/2 − f(τ, θ)/2 and ν = c₁τ. If we let $q=(0,-2/(\tau c_1),\frac{{(1+e^{{\beta }_0})}^2}{e^{{\beta }_0}},q_4)$, we have q^Tf(d) = 2d/τ − 1 on the interval [0, τ]. Note that q₄ can be chosen to ensure the inequality max |q^Tf(d)| < 1 holds on [τ, d̄]. Consequently, parts (i) and (ii) of Lemma 1 hold. This justifies case (i) of the proposition.

To prove case (ii) of Lemma 2, we note that the locally D-optimal design does not depend on parameter c₁ because the D-optimality criterion is a product of a power of c₁ and a function that does not depend on c₁. Further, we note that the function f^T(d)M⁻¹(ξ, θ)f(d) is a linear combination of monomials 1, d, d², d³, d⁴ on the interval [0, τ]. Consequently, this function can have at most 3 local maxima. By the equivalence theorem⁽²⁰⁾ it follows that the locally D-optimal design has at most 3 design points on [0, τ]. Similarly, the locally D-optimal design can have at most 2 design points on [τ, d̄] because the function f^T(d)M⁻¹(ξ, θ)f(d) has at most 2 local maxima on the interval [τ, d̄]. It follows from standard arguments in optimal design theory that 0, τ/2, τ are design points if the locally D-optimal design has three points on [0, τ] and 0 is a design point.

To prove part (iii) of Lemma 2, we note that h(θ) = (−τ, −c₁, 0, 0)^T. By inspection, part (iii) of Lemma 1 cannot hold for any 1- or 2-point designs and so the locally h-optimal design has at least three design points.