Optimal designs based on the maximum quasi-likelihood estimator

Abstract

We use optimal design theory and construct locally optimal designs based on the maximum quasi-likelihood estimator (MqLE), which is derived under less stringent conditions than those required for the MLE method. We show that the proposed locally optimal designs are asymptotically as efficient as those based on the MLE when the error distribution is from an exponential family, and they perform just as well or better than optimal designs based on any other asymptotically linear unbiased estimators such as the least square estimator (LSE). In addition, we show current algorithms for finding optimal designs can be directly used to find optimal designs based on the MqLE. As an illustrative application, we construct a variety of locally optimal designs based on the MqLE for the 4-parameter logistic (4PL) model and study their robustness properties to misspecifications in the model using asymptotic relative efficiency. The results suggest that optimal designs based on the MqLE can be easily generated and they are quite robust to mis-specification in the probability distribution of the responses.

1. Introduction

Model assumptions are frequently made for drawing statistical inference but they may not be tenable in practice. The optimal design constructed under wrong model assumptions can be very inefficient. When the model is nonlinear, a further complication is that nominal values for the model parameters are required before the optimal designs can be implemented. It is thus important to find a design that is robust to various forms of mis-specification in the statistical model.

Different approaches have been proposed to find designs robust to various departures from the model assumptions. They typically concern mis-specifications in the error distribution, the response mean function, and the form of heteroscedasticity in the model. For instance, Burridge and Sebastiani (1994) and Dette and Wong (1999) proposed optimal designs when the variance is a function of the mean response, and Chen et al. (2008) proposed optimal minimax designs for a heteroscedastic polynomial model. However the bulk of the work focused on a single violation from the model assumptions. Some examples are Läuter (1974); Stigler (1971); Lee (1987, 1988); Studden (1982); Song and Wong (1998a,b); Dette and Wong (1996, 1999); the first six considered model mis-specification in mean function and the last three concerned mis-specification in the heteroscedasity structure.

Much of the research in optimal design assumes that the design criterion is formulated in terms of the Fisher Information matrix. This matrix can only be computed under the full specification of the probability distribution for the response. In contrast, the quasi-likelihood method introduced by Weddernburn (1974) assumes only a functional relationship between the mean and the variance of the response. This is a tenable assumption in practical situations, see for example, Box and Hill (1974), Bickel (1978) and, Jobson and Fuller (1980). This suggests that designs based on the quasi-likelihood method are likely to be more robust to model assumptions than methods that require a full parametric probability model.

There is very little work on optimal designs based on the MqLE for nonlinear regression models despite numerous successful applications of the quasi-likelihood method for generalized linear models (McCullagh and Nelder, 1989). A clear exception is work by Niaparast (2009), and Niaparast and Schwabe (2013), who derived optimal designs based on the MqLE for Poisson regression models. The first paper focused on the theoretical construction of D-optimal designs for the Poisson models with random intercepts, and the second paper extended the work to a general mixed effects Poission model with random coefficients and so the paper deals with random slopes rather than random intercepts. However, their work concerns only finding locally D-optimal designs for the case when the response follows a Poisson distribution.

We consider a more general situation and develop theory to find various types of optimal designs based on the MqLE where the response may be a discrete or a continuous random variable. We also show such optimal designs can be generated from algorithms currently used to find optimal designs based on the Fisher Information matrix. We list advantages of such designs compared to other designs based on other efficient estimators such as the MLE and the LSE when the sample size is large and study the robustness properties of D-optimal designs to mis-specified nominal parameter values. We focus on locally optimal designs (Chernoff, 1953) where nominal values for the model parameters are required before they can be implemented. Bayesian optimal designs can also be directly constructed from our methodology but will not be discussed here. One of the key results is that we show here that asymptotically, locally optimal designs based on the MqLE also have the same property as those constructed based on the MLEs but without the normality distributional assumption. When responses are from a member of the class of exponential family distributions, we prove that the optimal designs based on the MqLE is asymptotically just as efficient as the optimal designs based on the MLE. Such designs are also asymptotically more efficient than the designs based on asymptotic best linear unbiased estimators, such as LSE. When responses are not from a member of the exponential family of distributions, we show optimal designs based on the MqLE also perform well based on an asymptotic relative efficiency measure.

Section 2 reviews preliminaries and discusses the asymptotic optimality of MqLEs. In Section 3, we review approximate designs and equivalence theorems for checking optimality of an approximate design. We then develop the theory and an algorithm to find a variety of optimal designs based on the MqLE. As an illustrative application, we construct locally optimal designs for dose response studies using the widely used 4-parameter logistic model (4PL-model). We construct optimal designs for estimating models and meaningful function of the model parameters in the 4PL-model and compare its performance with those based on MLEs and a uniform design with 10 points when there is mis-specification in (i) the nominal values of the model parameters, (ii) the heteroscedasticity structure, and (iii) the error distributions. Section 5 concludes with a summary of our work and recommendations.

2. Preliminaries

Suppose we have a dose response study and we have resources to take n observations at d distinct doses x₁, …, x_d with n_i replicates at x_i, i = 1, …, d and ${\sum }_{i=1}^d{n}_i=n$. Let y be the response at x with E(y) = μ and Var(y) = v(μ) where μ = μ(θ; x) and μ is a known function of an unknown r-dimensional parameter vector θ and dose x. For simplicity, we write μ(θ; x) as μ(θ) or simply μ if there is no confusion. Our stipulation Var(y) = v(μ) covers the situation of Niaparast (2009) but not the situation of Niaparast and Schwabe (2013) since the latter considered Var(y) = μ+μ²c(x) and c(x) is not a function of μ. Another difference is that the two papers assumed the n_i replications at x_i are not independent and we do here as in most dose response studies. Assuming that inf_μ v(μ) > 0, the log quasi-likelihood of y (with dispersion ϕ) is defined as

$$q(\mu ;y)\triangleq {\int }_{-\infty }^{\mu }\frac{y-t}{\varphi v(t)}dt.$$

Let $\stackrel{.}{q}\triangleq \frac{\partial }{\partial \mathit{\theta }}q(\mu ;y)$ (quasi-score) and $\ddot{q}\triangleq \frac{{\partial }^2}{\partial \mathit{\theta }\partial {\mathit{\theta }}^{\prime }}q(\mu ;y)$. Weddernburn (1974) showed that q(μ; y) behaves like a likelihood function in the sense that

$$E(\stackrel{.}{q})=0,E(\stackrel{.}{q}{\stackrel{.}{q}}^{\prime })=-E(\ddot{q})$$

and $-E(\ddot{q})=\frac{1}{\varphi v(\mu )}\stackrel{.}{\mu }(\mathit{\theta }){\stackrel{.}{\mu }}^{\prime }(\mathit{\theta })$ (quasi-information), where $\stackrel{.}{\mu }(\mathit{\theta })\triangleq \frac{\partial \mu }{\partial \mathit{\theta }}$. Given y₁, y₂, …, y_n, the maximum quasi-likelihood estimate (MqLE) of θ maximizes the joint quasi-likelihood ${\sum }_{i=1}^{n}q({\mu }_i;y_i)$.

More generally, let y = (y₁, …, y_n)′ be the vector of observations at the n doses and let μ = (μ₁, …, μ_n)′ be its mean vector. Without loss of generality, let the first d elements of y correspond to observations from the d distinct doses x₁, …, x_d and let μ_★ = (μ(θ; x₁), …, μ(θ; x_d))′ be the mean vector of the first d observations from the d distinct doses. Clearly μ_★ = Cμ, where C is the first d rows of the n-dimensional identity matrix I_n.

Let θ̂_n be the maximum quasi-likelihood estimator (MqLE) of θ, let $D_n^{\prime }(\mathit{\theta })=({\stackrel{.}{\mu }}_1,\dots ,{\stackrel{.}{\mu }}_n)$ be the full rank r×n matrix and let $V_n^{-1}(\mathit{\mu })$ be the inverse of the n×n diagonal matrix V_n(μ) = diag[v(μ₁), …, v(μ_n)]. By definition, θ̂_n satisfies

$$\sum _{i=1}^n\stackrel{.}{q}({\mu }_i;y_i)=\sum _{i=1}^n\frac{y_i-{\mu }_i}{\varphi v({\mu }_i)}\frac{\partial {\mu }_i}{\partial \mathit{\theta }}=\frac{1}{\varphi }{D}_n^{\prime }(\mathit{\theta })V_n^{-1}(\mathit{\mu })(\mathbf{y}-\mathit{\mu })=0,$$

assuming that ${\sum }_{i=1}^n\ddot{q}({\mu }_i;y_i)$ is positive definite. The MqLE of θ is then obtained iteratively via the quasi Newton-Raphson method, i.e., at the (k + 1)^th iteration with ${\widehat{\mu }}_i^{(k)}=\mu ({\widehat{\mathit{\theta }}}_n^{(k)};x_i)$,

$${\widehat{\mathit{\theta }}}_n^{(k+1)}={\widehat{\mathit{\theta }}}_n^{(k)}+{\{D_n^{\prime }({\widehat{\mathit{\theta }}}_n^{(k)})V_n^{-1}({\widehat{\mathit{\mu }}}^{(k)})D_n({\widehat{\mathit{\theta }}}_n^{(k)})\}}^{-1}{D}_n^{\prime }({\widehat{\mathit{\theta }}}_n^{(k)})V_n^{-1}({\widehat{\mathit{\mu }}}^{(k)})(\mathit{y}-{\widehat{\mathit{\mu }}}^{(k)}).$$

Observe that the MqLE is a Z-estimator with properties analogous to a MLE. Under similar regularity assumptions required for the MLE of θ, cf., Lehmann (1999), §7.1 & §7.3, (C1) – (C7), the MqLE θ̂_n as an M-estimator of θ is also asymptotically linear and unbiased satisfying

$${\widehat{\mathit{\theta }}}_n=\mathit{\theta }+{(D_n^{\prime }{V}_n^{-1}{D}_n)}^{-1}{D}_n^{\prime }{V}_n^{-1}(\mathit{y}-\mathit{\mu })+o_p(n^{-1/2}).$$ (1)

Here and elsewhere, we have suppressed the arguments in $D_n^{\prime }(\mathit{\theta })$ and $V_n^{-1}(\mathit{\mu })$ and write them as $D_n^{\prime }$ and $V_n^{-1}$, for simplicity, when there is no confusion. The symbol o_p(n^−1/2) denotes a random vector whose L^∞ norm is o(n^−1/2) in probability. We continue to use this notation for vectors and matrices for their L^∞ norm with the specified order when there is no confusion.

Eq(1) implies that $\sqrt{n}({\widehat{\mathit{\theta }}}_n-\mathit{\theta })\stackrel{d}{\to }N(0,\varphi {\mathrm{\Delta }}^{-1})$ if $\mathrm{\Delta }={lim}_{n\to \infty }{n}^{-1}(D_n^{\prime }{V}_n^{-1}{D}_n)$ exists, viz. Weddernburn (1974) or McCullagh (1983). Consequently, the MqLE of μ_★ is μ̂_★ = μ_★(θ̂_n). If we expand μ(θ) using a Taylor expansion about a neighborhood of θ₀, assuming sup_θ ||μ̇(θ)|| < ∞ and μ(θ) is twice differentiable, we have

$$\mu (\mathit{\theta })=\mu ({\mathit{\theta }}_0)+{\stackrel{.}{\mu }}^{\prime }({\mathit{\theta }}_0)(\mathit{\theta }-{\mathit{\theta }}_0)+O({\Vert \mathit{\theta }-{\mathit{\theta }}_0\Vert }^2)$$ (2)

and $\frac{\partial {\mathit{\mu }}_{★}}{\partial {\mathit{\theta }}^{\prime }}=D_{★}$, where $D_{★}^{\prime }=({\stackrel{.}{\mu }}_1,\dots ,{\stackrel{.}{\mu }}_d)$ is a r×d matrix. By the delta method, it follows that

$$\sqrt{n}({\widehat{\mathit{\mu }}}_{★}-{\mathit{\mu }}_{★})\stackrel{d}{\to }N(0,\varphi D_{★}{\mathrm{\Delta }}^{-1}{D}_{★}^{\prime }).$$

Here we state an asymptotic optimality of the MqLE and in A.1 of the appendix, we provide its justifications with technical details beyond that sketched out in McCullagh (1983).

Theorem 1

Let θ̃_n be an asymptotically linear unbiased estimator of θ₀ satisfying

$${\stackrel{\sim }{\mathit{\theta }}}_n={\mathit{\theta }}_0+{(L_n^{\prime }{L}_n)}^{-1}{L}_n^{\prime }{V}_n^{-1/2}[\mathit{y}-\mathit{\mu }({\mathit{\theta }}_0)]+o_p(n^{-1/2}),$$ (3)

where $L_n^{\prime }=(l_1,\dots ,l_n)$ is an arbitrary r×n full rank matrix. If $n^{-1}{\sum }_{i=1}^n{l}_i{l}_i^{\prime }$ tends to a positive definite matrix Λ (r×r and max_1≤i≤n ||l_i|| = o(n^1/2)), then μ̂_★ is asymptotically optimal in the sense that n{Var(μ̃_★)−Var(μ̂_★)} is asymptotically a non-negative definite matrix.

The implication of Theorem 1 is that under mild conditions, the MqLE μ̂_★ is the asymptotically best linear unbiased estimator of μ_★. This result may not be surprising because McCullagh (1983) had noted that if y have a distribution from the exponential family with pdf f(y; η) = exp{(yη − b(η))/aϕ − c(y; ϕ)} satisfying η = η(μ), E(y) = μ = μ(θ; x) and Var(y) = v = a(ϕ)v(μ), the MLE of θ, θ̌_n, has the approximation

$${\check{\mathit{\theta }}}_n=\mathit{\theta }+{(D_n^{\prime }{V}_n^{-1}{D}_n)}^{-1}{D}_n^{\prime }{V}_n^{-1}(\mathit{y}-\mathit{\mu })+o_p(n^{-1/2}).$$ (4)

It follows that the MLE θ̌_n has the same linear approximation as the MqLE θ̂_n when the observations come from an exponential family with canonical parameter η(μ) and variance function v(μ), see Eq(1). By the delta method, this implies that the MqLE μ̂_★ of μ_★ is asymptotically as efficient as the MLE μ̌_★ = μ(θ̌) of μ_★. Consequently, asymptotically linear unbiased estimators of a function of θ are not more accurate than MqLE for estimating the same function of θ.

Corollary 1

MqLE θ̂ is asymptotically superior to LSE θ̌^LSE for any designs under the regularity conditions.

Proof

Under the regularity conditions, LSE θ̌^LSE of θ satisfies

$${\check{\mathit{\theta }}}_n^{\mathit{LSE}}=\mathit{\theta }+{(D_n^{\prime }{D}_n)}^{-1}{D}_n^{\prime }(\mathit{y}-\mathit{\mu })+o_p(n^{-1/2}),$$

so θ̌^LSE is a member of the class of asymptotic best linear unbiased estimators with ${(L_n^{\prime }{L}_n)}^{-1}{L}_n^{\prime }={(D_n^{\prime }{D}_n)}^{-1}{D}_n^{\prime }{V}_n^{1/2}$ in Eq(4). By Theorem 1, n{Var(θ̌^LSE) − Var(θ̂)} is non-negative definite.

This implies that $n\{\mathit{Var}({\check{\mathit{\mu }}}_{★}^{\mathit{LSE}})-\mathit{Var}({\widehat{\mathit{\mu }}}_{★})\}$ is non-negative definite by virtue of the delta method, and so the MqLE μ̂_★ of μ_★ is superior to the LSE ${\check{\mathit{\mu }}}_{★}^{\mathit{LSE}}$ of μ_★.

Therefore the optimal designs based on the MqLE and the MLE have the same efficiency when the distribution of y is a member of exponential family distributions, and the optimal design based on the MqLE asymptotically performs just as well or better than optimal designs based on any other asymptotically linear unbiased estimators such as the least square estimator (LSE).

3. Locally optimal designs based on the MqLE

Optimal designs for nonlinear models are often constructed using strong parametric model assumptions and it is well known that they can be sensitive to the assumptions. Optimal designs based on the MqLE require less stringent assumptions and they can be found once the variance function Var(y) is formulated as a function of E(y). In this section, we first discuss optimality design criteria and approximate designs. We then construct locally optimal designs based on the MqLE and compare their performances with locally optimal design based on the MLE and a uniform design; the latter is popular because they are intuitive appealing and easy to construct by taking equal number of observations at equally spaced doses. As an illustrative application, we construct and compare optimal designs based on the MLE and MqLE for the 4-parameter logistic model, which is widely used in biometry.

The setup assumes that we have resources to take n observations from a model with

$$E(y_{ij})={\mu }_i=\mu (\mathit{\theta },x_i),\mathit{Var}(y_{ij})=v_i=\varphi v({\mu }_i);j=1,2,\dots ,n_i,i=1,2,\dots ,d.$$

Here θ is a r-dimensional vector of parameters and we have n_i replicates at each doses (or log doses) x_i with ${\sum }_{i=1}^d{n}_i=n$. The observation from the j^th replicate at the i^th dose is y_ij and its distribution is unknown except for the relationship between its mean and variance. All the observations are assumed to be independent and observations at the same dose are identically distributed. Given the design criterion and model assumptions, the design questions are how many doses are required, where these doses are and how many replications are required at each dose. In general, such questions are very difficult to answer, where the optimal designs can depend sensitively on the value of n, the regression model and the optimality criterion. To facilitate practitioners to use such designs, we would require an endless table of optimal designs! For nonlinear models, a longer list is required because for each set of nominal values of the model parameters, the locally optimal designs also depend on the set.

3.1. Approximate designs and design criteria

An increasingly common and effective approach to solve design questions is to use approximate designs, which are essentially probability measures defined on the given dose interval. For our problem, we denote such a design with d doses by $\xi ={\{(x_i,w_i)\}}_1^d$, where x_i is the i^th dose selected from a given dose interval and w_i = n_i/n is the proportion of the total observations allocated to x_i, i = 1, …, d. Given an optimality criterion, the design problem requires us to determine the optimal value of d and optimal choices for (x_i, w_i), i = 1, …, d. The implemented design takes roughly nw_i observations at x_i, subject to the constraint that each nw_i is rounded to a positive integer and they sum to n.

The advantages of working with approximate designs are well documented in the literature, see for example, the voluminous work of Kiefer (1985). The main ones are that the approach provides a unified framework for finding and verifying optimal designs under any concave (or convex) optimality criterion, and when a design is not optimal, the approach provides an easy evaluation of the proximity of the design to the optimum without knowing the optimum (Pazman, 1986). In addition, algorithms for finding a broad class of optimal designs are available. A typical application of this approach is to first calculate the Fisher information matrix for a given design and formulate the design criterion as a concave (or convex) function of this matrix.

For our setup, we recall that the quasi-information matrix is obtained from the negative of the expectation of the second derivatives of the log quasi-likelihood function with respect to the model parameters. Specifically, the normalized quasi-information matrix for θ is $\mathrm{\Delta }(\mathit{\theta };\xi )=n^{-1}{\varphi }^{-1}(D_n^{\prime }{V}_n^{-1}{D}_n)$, which can be written as $\mathrm{\Delta }(\mathit{\theta };\xi )=\frac{1}{\varphi }{\sum }_{i=1}^d{w}_{i}g(x_i)g^{\prime }(x_i)$ with ϕ as the dispersion parameter and $g(x_i)=v{({\mu }_i)}^{-1/2}{\left(\frac{\partial {\mu }_i}{\partial {\theta }_1},\frac{\partial {\mu }_i}{\partial {\theta }_2},\cdots ,\frac{\partial {\mu }_i}{\partial {\theta }_r}\right)}^{\prime }$ evaluated at the nominal values for θ. This quasi-information matrix is derived based only on the variance-mean relationship of the response distribution but has the same form as the Fisher information matrix. Both matrices become equal when responses come from a normal distribution with constant variance. As we show shortly, if the design criterion is a concave (convex) function of the quasi-information matrix, an equivalence theorem can be similarly developed for checking whether a design based on the MqLE is optimal among all designs on the given design interval. Equivalence theorems for verifying whether a design based on MLEs is optimal or not are available in design monographs such as Fedorov (1972), Pazman (1986), Pukelsheim (2006) or Fedorov and Leonov (2013).

A common objective is to find an efficient design to estimate the vector of parameters θ or some functions of model parameters. D-optimality seeks to minimize the volume of the confidence ellipsoid for θ by choice of a good design and so a D-optimal design provides the most accurate estimates for the model parameters. When there are several functions of the model parameters to estimate, C-optimality seeks to minimize the asymptotic average variance of the estimated functions. Consequently, the C-optimal design provides the most accurate inference for estimating the user-selected set of functions of the model parameters. If there is only a single function of the model parameters to estimate, C-optimality becomes c-optimality and a c-optimal design minimizes the asymptotic variance of the estimated function of interest.

We also consider designs for estimating s(s ≤ r) linear combinations of the parameters A′θ, where A is a r × s matrix of rank s. Here we are only interested in approximate designs ξ for which A′θ is estimable. The quasi-information matrix of a design ξ for estimating A′θ is M = (A′[Δ(θ; ξ)]⁻A)⁻¹, where Δ(θ; ξ)⁻ is a generalized inverse of Δ(θ, ξ).

A useful class of concave design criteria, indexed by a scalar t, for models with r parameters and formulated in terms of information matrix M is:

$${\mathit{\varphi }}_t(M)=\{\begin{array}{ll}{\lambda }_{\mathit{min}}(M)\hfill & t=-\infty ;\hfill \\ {(det[M])}^{1/r}\hfill & t=0;\hfill \\ {\left(\frac{1}{r}Tr[M^t]\right)}^{1/t}\hfill & t<0,\hfill \end{array}$$

where λ_min(M) denotes the minimum eigenvalue of M. Because the criteria are concave, standard arguments using the directional derivatives provide us with a tool to verify if a design is optimal among all approximate designs on the given design space. For example, it can be shown that for any t > −∞, if A is user-selected matrix, the design ξ_t is optimal among all designs on the given dose interval for estimating A′θ if and only if there is G = [Δ(θ; ξ_t)]⁻ such that

$$g^{\prime }(x)GA{(A^{\prime }GA)}^{-(t+1)}{A}^{\prime }{G}^{\prime }g(x)\le Tr[{(A^{\prime }GA)}^{-t}],$$ (5)

with equality at the dose levels of the design ξ_t (Pukelsheim, 2006). The locally optimal design becomes a locally D-optimal design when t = 0 and becomes a locally C-optimal design for estimating functions of the model parameters when t = −1. Locally c-optimal designs for estimating a function of the model parameters is a special case of the C-optimal design when s = 1. The function on the left hand side of (5) is sometimes called the sensitivity function and by examining whether its graph satisfies the conditions of the equivalence theorem, one can confirm the optimality of the design. All our optimal designs reported in this paper have been verified by an appropriate equivalence theorem.

D-optimal design maximizes ϕ₀(M). If we set A = I_r and t = 0 in (5), the equivalence theorem for D-optimality is obtained and the design ξ_D is D-optimal design if and only if for any dose x in the dose interval,

$$d(x,{\xi }_D)=g^{\prime }(x){[\mathrm{\Delta }(\mathit{\theta };{\xi }_D)]}^{-1}g(x)\le r,$$

with equality at the dose levels of the design ξ_D. Alternatively, we may want to have an efficient design for estimating several functions of the model parameters at the same time. In this case, we use the C-optimality criterion to find a design to minimize Tr[M⁻¹], which is equivalent to maximize ϕ₋₁(M). If the known functions of interest are C₁(θ), C₂(θ),…, C_s(θ), we set A = (Ċ₁(θ), Ċ₂(θ), …, Ċ_s(θ)), ${\stackrel{.}{C}}_i(\mathit{\theta })=\frac{\partial C_i(\mathit{\theta })}{\partial \mathit{\theta }}$ and t = −1. From (5), we have the design ξ_C is C-optimal design among all designs on the dose interval if and only if for any dose x in the dose interval,

$$d(x,{\xi }_C)={\Vert g^{\prime }(x){[\mathrm{\Delta }(\mathit{\theta };{\xi }_C)]}^{-}A\Vert }^2\le Tr[A^{\prime }{[\mathrm{\Delta }(\mathit{\theta };{\xi }_C)]}^{-}A],$$

with equality at the dose levels of the design ξ_C.

Let the vector of weights w = (w₁, …, w_d−1)′ because $w_d=1-{\sum }_{i=1}^{d-1}{w}_i$. The quasi-information matrix has the same form as the Fisher information matrix and we show in A.2 and A.3 of the appendix that the vector of weights w of the D and C-optimal designs can be obtained as follows:

Theorem 2

The vector of weights w of the D-optimal design are the non-negative roots of the equation $\frac{\partial \mathit{det}[\mathrm{\Delta }(\mathit{\theta };\xi )]}{\partial \mathit{w}}=0$. If the number of distinct dose levels in the locally D-optimal design is the same as the dimension of θ in μ(θ, x) (i.e., d = r), the D-optimal is equally weighted.

Theorem 3

The vector of weights w of the C-optimal design are the non-negative roots of the equation $\frac{\partial Tr(A^{\prime }{[\mathrm{\Delta }(\mathit{\theta };\xi )]}^{-1}A)}{\partial \mathit{w}}=0$.

Knowing a good upper bound on the number of support points of the optimal design can frequently simplify the search for the optimal design considerably. Ideas in Hyun et al. (2013) and Yang (2010) may be applicable to identify an upper bound on the number of points for the optimal design based on the MqLE. In our study, we provide an algorithm for finding optimal designs based on the MqLE without requiring knowledge of a useful upper bound.

3.2. Design efficiency and algorithms

Following convention, the efficiencies of a design ξ for estimating the r− dimensional vector θ and a set of interesting functions C(θ) based on the MqLE are, respectively, measured by

$$e_D(\xi )={\left(\frac{\mathit{det}[\mathrm{\Delta }(\mathit{\theta };\xi )]}{\mathit{det}[\mathrm{\Delta }(\mathit{\theta };{\xi }_D)]}\right)}^{\frac{1}{r}}$$

and

$$e_C(\xi )=\frac{Tr(A^{\prime }{[\mathrm{\Delta }(\mathit{\theta };{\xi }_C)]}^{-1}A)}{Tr(A^{\prime }{[\mathrm{\Delta }(\mathit{\theta };\xi )]}^{-1}A)}.$$

If this ratio is one-half, the interpretation is that the design ξ needs to be replicated twice for it to do as well as the optimal design. In practice, we want designs with high efficiencies.

Standard design algorithms such as those proposed by Fedorov (1972) and their many modifications thereof, can be used to search for a D-optimal or a C-optimal design based on the MLE. An improved version is the Yang-Biedermann-Tang algorithm (Yang et al., 2013), where they showed their algorithm can more effectively generate several types of optimal designs for nonlinear models with a few independent variables. Hyun et al. (2015) extended their algorithm to find multiple-objective optimal designs and this is the algorithm we used to find optimal designs in this work, after specializing our algorithm to find single-objective optimal designs. A distinguishing feature of our algorithm is that it first selects the starting design by optimizing dose levels via the Fedorov’s algorithm and obtains the optimal weights for the selected optimal doses using the Newton Raphson’s method based on Theorem 2 and 3. Hyun’s algorithm to search multiple-objective optimal designs is available in a R-package called VNM (Hyun et al., 2015) and the package can be freely downloaded from the R-archive. Interested readers may also write to the second author for the codes.

We now give examples using a popular model where heteroscedasticity is simply modeled via v(μ) = e^hμ, and h is some known constant; other functional forms for modeling the variance in terms of the mean, such as v(μ) = hμ², can also be directly incorporated in our algorithm. We note that when h = 0, the quasi-information matrix is identical to the Fisher information matrix when errors are independent and normally distributed with a constant variance. More generally, since the Fisher information matrix for θ based on the MLE under normal distribution is $F(\mathit{\theta };\xi )={\sum }_{i=1}^d{w}_{i}f(x_i)f^{\prime }(x_i)$ with $f(x_i)={\left[v_i^{-1}+\frac{1}{2}{\{\frac{{\stackrel{.}{v}}_i}{v_i}\}}^2\right]}^{1/2}{\left(\frac{\partial {\mu }_i}{\partial {\theta }_1},\frac{\partial {\mu }_i}{\partial {\theta }_2},\cdots ,\frac{\partial {\mu }_i}{\partial {\theta }_r}\right)}^{\prime }$, where ${\stackrel{.}{v}}_i=\frac{\partial v_i}{\partial {\mu }_i}$.

Let ${\xi }_D^0$ and ${\xi }_C^0$, respectively, be the locally D- and C-optimal designs based on the MLE when we have normally distributed responses with homoscedastic errors. Let ${\xi }_D^h$ and ${\xi }_C^h$, respectively, be the locally D- and C-optimal designs based on the MqLE for a heteroscedastic model. When errors are normally distributed with constant variance, we note that the latter two optimal designs based on the MqLE are identical to the optimal designs based on the LSE since MLE and LSE are the same.

3.3. Four-parameter logistic(4PL) model

The 4PL model is variously called the Hill model or the Emax model and is widely used across several disciplines. They include education testing as in item response theory (Hambleton and Swaminathan, 1985; Harris, 1989), enzyme kinetic studies (Dette et al., 2005), pharmaceutical studies (Bretz et al., 2010; Reeve and Turner, 2013) and in toxicology (Dinse, 2011; Gadakar and Call, 2015), to name a few. It is also used for curve-fitting analysis in bioassays or immunoassays such as ELISAs or dose-response curves. When some parameters of the model are set equal to certain values, the model reduces to the usual 2 parameter logistic model or the 3-parameter logistic model, which are also frequently used in the biomedical sciences. Because of the flexibility of the model, we select it as an exmeplary nonlinear model in biometrics to study the performance of optimal designs based on the MqLE.

The 4-parameter logistic model assumes symmetry around the inflection point and the mean of its continuous response is modeled by

$$\mu (\mathit{\theta },x)=\frac{{\theta }_1}{1+e^{{\theta }_{2}x+{\theta }_3}}+{\theta }_4,{\theta }_1>0,{\theta }_2\ne 0.$$ (6)

There are different parametrization of the models depending on the field and application at hand. Sometimes, the variable x may influence the choice of the parametrization. For example, the 4PL model (6) seems to be more widely used when the dose is measured in logarithmic units; otherwise, the Hill model seems to be the preferred choice. We choose the formulation in (6) because (i) its form provides more stable parameter estimates than other forms (Reeve and Turner, 2013) and (ii) it facilitates the numerical search of the optimal designs over a shorter dose interval.

An interesting dose to estimate is the ED₅₀, which is the dose expected to result in having a response halfway between the upper limit and the lower limit of the mean response. In (6), x is the log dose, θ₁ is the difference between the upper limit and the lower limit of the expected response, θ₂ is the negative value of the Hill’s slope that controls the steepness of the curve, θ₃ is −θ₂log(ED₅₀), and θ₄ is the lower limit of the response. Another interesting dose to estimate is MED(δ), the dose that is expected to result in a change of δ units in the mean response from the minimum dose and δ is user-specified. For example, when working with an inhibitory drug, it may be desirable to estimate the dose that would shrink the tumor size by some user-specified δ units. It is straightforward to show that for the 4PL model, ED₅₀ = −θ₃/θ₂ and $\mathit{MED}(\delta )=\{\mathit{log}(\frac{-\delta }{{\theta }_1+\delta })-{\theta }_3\}/{\theta }_2$ if θ₂ > 0 or $\mathit{MED}(\delta )=\{\mathit{log}(\frac{{\theta }_1-\delta }{\delta })-{\theta }_3\}/{\theta }_2$ if θ₂ < 0, where 0 < |δ| < θ₁ and δ must have an opposite sign from θ₂ (Hyun and Wong, 2015).

In what is to follow, we find locally D-optimal designs for estimating θ and locally C-optimal designs for estimating both ED₅₀ and MED(δ) simultaneously for the 4PL model. For illustrative purpose, we set δ = −1 for MED for the rest of the paper. A direct calculation shows that the quasi-information matrix for the 4PL model is $\mathrm{\Delta }(\mathit{\theta };\xi )={\sum }_{i=1}^d{w}_{i}g(x_i)g^{\prime }(x_i)/\varphi$, where

$$g(x)={[e^{h\mu (x,\mathit{\theta })}]}^{-1/2}\left(\frac{1}{1+e^{{\theta }_{2}x+{\theta }_3}},\frac{-{\theta }_{1}x{e}^{{\theta }_{2}x+{\theta }_3}}{{(1+e^{{\theta }_{2}x+{\theta }_3})}^2},\frac{-{\theta }_1{e}^{{\theta }_{2}x+{\theta }_3}}{{(1+e^{{\theta }_{2}x+{\theta }_3})}^2},1\right).$$

The 4PL model has a continuous response and we assume heteroscedasticity is modeled by v(μ) = e^hμ after noting that this assumption is not appropriate for the classical logistic model with binary responses. The locally optimal design does not depend on the of ϕ but on the value of h for the given nominal values of θ. The parameter ϕ enters the information matrix as a multiplicative constant and so our optimal designs do not depend on its nominal value. The nominal values for the remaining parameters were taken from Khinkis et al. (2003) who found locally D-optimal designs for studying characteristics of seven anti-cancer drugs using the above model. The dose interval used in their study was [log(.001), log(1000)]. Equivalence theorems for the D-and C-optimality criteria can be obtained by setting A′ = I₄ and A′ = (EḊ₅₀, MĖD)′, respectively in (5). Here $E{\stackrel{.}{D}}_{50}=\left(\begin{array}{llll}0,\hfill & \frac{{\theta }_3}{{\theta }_2^2},\hfill & -\frac{1}{{\theta }_2}\hfill & ,0\hfill \end{array}\right)$ and

$$M\stackrel{.}{E}D=\{\begin{array}{ll}\left(\begin{array}{cccc}\frac{-1}{({\theta }_1+\delta ){\theta }_2},& \frac{{\theta }_3-\mathit{log}(\frac{-\delta }{{\theta }_1+\delta })}{{\theta }_2^2},& -\frac{1}{{\theta }_2},& 0\end{array}\right),\hfill & \text{if}{\theta }_2>0\hfill \\ \left(\begin{array}{cccc}\frac{1}{({\theta }_1-\delta ){\theta }_2},& \frac{{\theta }_3-\mathit{log}(\frac{{\theta }_1-\delta }{\delta })}{{\theta }_2^2},& -\frac{1}{{\theta }_2},& 0\end{array}\right),\hfill & \text{if}{\theta }_2<0.\hfill \end{array}$$

Table 1 shows the locally D-and C-optimal designs based on the MqLE for various values of h in the variance function when the nominal value for the model parameters is θ = θ₁ = (1.563, 1.790, 8.442, 0.137)′. All locally D-optimal designs, as expected, are equally weighted with 4 doses but locally C-optimal designs may require 3 or 4 doses, depending on the values of heteroscedasticity parameter h. When h = 0, the locally optimal designs based on the MqLE ${\xi }_D^h$ and ${\xi }_C^h$ coincide with the corresponding optimal designs for the homoscedastic model. All locally optimal designs from our algorithm can be verified by examining the plot of the sensitivity function. For example, Figure 1 is the sensitivity plot $d(x,{\xi }_C^h)$ of the locally C-optimal design for estimating both the ED₅₀ and MED(δ = −1) in the 4PL model when θ = θ₁ and h = 1. The graph satisfies the conditions of the equivalence theorem and so confirms the optimality of the design based on the MqLE for estimating the ED₅₀ or MED in the 4PL model.

Table 1.

Locally D-optimal designs ( ${\xi }_D^h$) for estimating θ and locally C-optimal designs ( ${\xi }_C^h$) for estimating both ED₅₀ and MED(δ = −1) based on the MqLE when θ = θ₁ and ED₅₀ = −4.716 for the 4PL model with selected values of h in v(μ) = e^hμ. The optimal designs are obtained using the dose interval [log(.001), log(1000)] = [−6.91, 6.91]. The last two columns display the D-efficiencies (top rows) and C-efficiencies (bottom rows) of the locally homoscedastic D- and C-optimal designs ( ${\xi }_D^0,{\xi }_C^0$) and the uniform design at 10 points for different values of h.

h	${\xi }_D^h$	$e_D({\xi }_D^0)$	eD(ξU10)
h = −2	$\left(\begin{array}{cccc}-6.91& -5.55& -4.50& 6.91\\ 0.250& 0.250& 0.250& 0.250\end{array}\right)$	0.900	0.547
h = −1	$\left(\begin{array}{cccc}-6.91& -5.39& -4.28& 6.91\\ 0.250& 0.250& 0.250& 0.250\end{array}\right)$	0.974	0.596
h = 0	$\left(\begin{array}{cccc}-6.91& -5.21& -4.08& 6.91\\ 0.250& 0.250& 0.250& 0.250\end{array}\right)$	1.000	0.617
h = 1	$\left(\begin{array}{cccc}-6.91& -5.01& -3.88& 6.91\\ 0.250& 0.250& 0.250& 0.250\end{array}\right)$	0.973	0.607
h = 2	$\left(\begin{array}{cccc}-6.91& -4.80& -3.71& 6.91\\ 0.250& 0.250& 0.250& 0.250\end{array}\right)$	0.899	0.896
h = 5	$\left(\begin{array}{cccc}-5.43& -4.22& -3.23& 6.91\\ 0.250& 0.250& 0.250& 0.250\end{array}\right)$	0.532	0.530
h	${\xi }_C^h$	$e_c({\xi }_C^0)$	ec(ξU10)
h = −2	$\left(\begin{array}{cccc}-6.91& -5.62& -4.71& 6.91\\ 0.179& 0.058& 0.442& 0.321\end{array}\right)$	0.763	0.226
h = −1	$\left(\begin{array}{cccc}-6.91& -4.90& -4.49& -2.02\\ 0.275& 0.241& 0.255& 0.229\end{array}\right)$	0.962	0.241
h = 0	$\left(\begin{array}{ccc}-6.91& -4.67& -2.70\\ 0.378& 0.445& 0.177\end{array}\right)$	1.000	0.202
h = 1	$\left(\begin{array}{ccc}-6.91& -4.63& -2.91\\ 0.496& 0.391& 0.113\end{array}\right)$	0.937	0.158
h = 2	$\left(\begin{array}{ccc}-6.91& -4.65& -2.85\\ 0.611& 0.322& 0.067\end{array}\right)$	0.792	0.126
h = 5	$\left(\begin{array}{cccc}-6.91& -4.29& -3.20& 2.11\\ 0.821& 0.119& 0.045& 0.016\end{array}\right)$	0.457	0.222

Figure 1.

Plot of $d(x,{\xi }_C^1)$ for the 4PL model when θ = (1.563, 1.790, 8.442, 0.137)′ and v(μ) = e^hμ with h = 1 confirms optimality of the design ${\xi }_C^1$.

Table 1 also displays the D-and C-efficiencies of optimal designs based on MqLEs and the uniform design ξ_U10 with 10 points evenly spread out across the dose range [log(.001), log(1000)]. A uniform design is a popular choice among practitioners because it is simple to implement and intuitively appealing. An issue in the choice of a uniform design is the number of points it has. Larger number of points have been shown to result in lower efficiencies (Wong and Lachenbruch, 1996) and so an informed decision on the number of points it has is required. We choose 10 doses on a log scale here because this design allows us to conduct a lack of fit test to assess model adequacy. Another reason is that there are real studies that uses 10 doses to fit toxicology data using the 4PL model. The take home message from Table 1 is that the homoscedastic optimal designs ${\xi }_D^0$ and ${\xi }_C^0$ have high efficiencies when there is only small mis-specification in the heteroscedastic assumption, via the values of h, and they become inefficient when the amount of heteroscedasticity is increased. In addition, the ED₅₀ is always located between the middle two points when there are 4 optimal design points or close to the middle points when there are 3 optimal design points. It seems that the middle two design points or the middle points move toward to the right of the ED₅₀ dose as the value of h increases. The 10-point uniform design ξ_U10 performs much worse under the same situation. For the range of values of h considered for the 10-point uniform design, the C-efficiencies for estimating both ED₅₀ and MED(δ = −1) average about 19%, with some as low as 12%. For D-efficiencies, they average 59% with some as low as 31.5%. This reaffirms that the choice of an appropriate uniform design is important; in this case, a uniform design with fewer points such as 7 or 8 points is likely to perform better than one with 10 points (Wong and Lachenbruch, 1996).

We recall that MLE provides the best performance when the error distribution is correctly specified. The MqLE is asymptotically as efficient as the MLE and optimal designs based on MLE or MqLE are the same if responses are from a member of the exponential family distribution. We now investigate the robustness properties of optimal designs based on the MqLE when responses do not come from a member of the exponential class of family distributions. We do this by using an asymptotic efficiency measure and consider two distributions with heteroscedasticity. Let μ_i = μ(θ, x_i) and v_i = ϕe^hμ_i. The two distributions are

• D1= a normal distribution with E(y_ij) = μ_i and V ar(y_ij) = v_i;

• D2= a gamma distribution with the shape parameter $\alpha ={\mu }_i^2/v_i$ and scale parameter β = μ_i/v_i.

The asymptotic efficiency compares the covariance matrix of the MqLE θ̂ with the one based on MLE θ̌, and so it is measured by

$$\mathit{ARE}(\widehat{\mathit{\theta }},\check{\mathit{\theta }})={(\mathit{det}[\mathrm{\Delta }(\mathit{\theta };\xi )]/\mathit{det}[M^{\ast }(\mathit{\theta };\xi )])}^{1/4},$$

where M^*(θ; ξ) is the Fisher information matrix based on the MLE θ̌ under the given distribution. For D1, M^*(θ; ξ) = F(θ; ξ) and for D2, M^*(θ; ξ) is given in the appendix A.4. The given Fisher information matrices for D1 and D2 are generic versions for any functional forms of v_i. We first compare the asymptotic efficiencies of the MqLE and MLE using the same design, i.e. the uniform design ξ_U10. Then the locally D-optimal design ${\xi }_D^h$ is used to construct the MqLE and its asymptotic efficiency is evaluated again to see how much gain was realized using the optimal design compared to efficiency gains from the MLE using ξ_U10.

Table 2 shows the asymptotic efficiencies of the MqLE for various ϕ and h values under D1 and D2 with θ = θ₁. The asymptotic efficiency of the MqLE depends on the heteroscedasticity of the responses since it is changed by both ϕ and h. When the same design ξ_U10 is used for both estimators, the MLE performs better than the MqLE as we expected, but for D1, the MqLE performs close to the MLE when ϕ ≤ 0.2 and h ≤ 1, and as discussed earlier, the MqLE becomes identical to the MLE when h = 0. The asymptotic efficiency of the MqLE is increased almost double fold when its optimal design ${\xi }_D^h$ is used. It is also observed that the MqLE obtained from ${\xi }_D^h$ is more efficient than the MLE found from ξ_U10 for selected values of h and ϕ under D1. For example, when ϕ = 0.1, h = −2, and ${\xi }_D^h$ is used to obtain the MqLE, ARE(θ̂, θ̌) = 1.718 and this tells that the MLE needs about 72% more subjects to do as well as the MqLE. For D2, we did not observe the same outperformance of MqLE obtained from ${\xi }_D^h$, but the optimal designs produce the MqLE that performs about as well as the MLE when ϕ < 0.2 and h < 1. The overall results suggest that when the distribution of the response is not a member of exponential family distributions, the MqLE still performs reasonably well compared with the MLE for some cases. In addition, the performance of the MqLE can be dramatically increased by using the optimal designs based on the MqLE.

Table 2.

Asymptotic efficiencies of the MqLE when ξ_U10 or ${\xi }_D^h$ is used under D1 and D2 with h = −2,−1, 0, 1, 2, ϕ = 0.1, 0.2, 0.5, 1.0, and θ = θ₁ = (1.563, 1.790, 8.442, 0.137)′.

Distr.	ϕ	ξU10					${\xi }_D^h$
		h= −2	h = −1	h = 0	h = 1	h = 2	h = −2	h = −1	h = 0	h = 1	h = 2
D1	0.1	0.939	0.975	1	0.875	0.388	1.718	1.637	1.621	1.441	0.676
	0.2	0.889	0.953	1	0.781	0.259	1.625	1.599	1.621	1.286	0.451
	0.5	0.772	0.891	1	0.596	0.134	1.410	1.495	1.621	0.981	0.233
	1.0	0.634	0.806	1	0.432	0.075	1.177	1.352	1.621	0.711	0.131
D2	0.1	0.529	0.493	0.446	0.373	0.249	0.967	0.867	0.734	0.614	0.434
	0.2	0.409	0.371	0.321	0.244	0.138	0.747	0.622	0.520	0.402	0.240
	0.5	0.272	0.237	0.189	0.122	0.057	0.497	0.397	0.306	0.201	0.098
	1.0	0.192	0.162	0.118	0.066	0.028	0.351	0.271	0.191	0.109	0.049

Table 3 compares robustness properties of locally D-optimal designs based on the MqLE and MLE when there is mis-specifications in the nominal values for θ. They displays D-efficiencies of ${\xi }_D^h$ with h = −2, 0, 2,5 in the variance function assuming that the true values of the nominal values for the 4PL model are θ = θ₁. We recall that ${\xi }_D^h$ with h = 0 is the optimal design based on the MLE. Both types of optimal designs seem to be sensitive to nominal values and seem more so when there is large heteroscedasticity in the model, as in the case when h = 5. As expected, they perform poorly when there is a large deviation from the true values, especially for θ₂ and θ₃. The optimal designs seem to be more sensitive to θ₂ than θ₃. Since θ₂ is the slope controlling parameter with a very narrow range of possible values, locally optimal designs become much less efficient for a small departure from the true value of θ₂ compared to the case for θ₃. For this model, it is known that ${\xi }_D^0$ does not depend on θ₁ and θ₄ (Li and Majumdar, 2008). However, we observe that when h ≠ 0, ${\xi }_D^h$ depends on θ₁ and θ₄ and ${\xi }_D^h$ has quite high D-efficiencies for a broad of values for these parameters. The performance of the design ${\xi }_D^h$ becomes less efficient as the value of h increases. In addition, Table 4 shows the robustness properties of the same designs when there is a mis-specification in the ED₅₀ under the assumption that θ = θ₁. The same patterns are observed here like Table 3. The optimal designs work very poorly when there is a large departure from the true value of the ED₅₀.

Table 3.

D-efficiencies of locally D-optimal designs ( ${\xi }_D^h$) based on the MqLE for h = −2, 0, 2, 5 when their nominal values for the 4-PL Model are mis-specified to be θ₂, …, θ₉ and the vector of true value is θ₁ = (1.563, 1.790, 8.442, 0.137)′.

θ = (θ1, θ2, θ3, θ4)	$e_D({\xi }_D^0)$	$e_D({\xi }_D^{-2})$	$e_D({\xi }_D^2)$	$e_D({\xi }_D^5)$
θ1 = (1.563, 1.790, 8.442, 0.137)	1.000	1.000	1.000	1.000
θ2 = (1.563, 2, 10, 0.137)	0.933	0.945	0.909	0.867
θ3 = (1.563, 3, 8, 0.137)	0.017	0.010	0.022	0.020
θ4 = (1.563, 1.5, 6, 0.137)	0.769	0.786	0.718	0.595
θ5 = (1.563, 1.5, 4, 0.137)	0.192	0.195	0.154	0.072
θ6 = (1.563, 1, 8, 0.137)	0.593	0.686	0.483	0.108
θ7 = (1.563, 0.5, 1, 0.137)	0.306	0.381	0.212	0.069
θ8 = (2, 1.790, 8.442, 1)	1.000	0.993	0.993	0.918
θ9 = (3, 1.790, 8.442, 0.137)	1.000	0.930	0.960	0.637

Table 4.

D-efficiencies of locally D-optimal designs ( ${\xi }_D^h$) based on the MqLE for h = −2, 0, 2, 5 when there is a mis-specification in ED₅₀ and the true value is ED₅₀ = −4.716 under θ₁ = (1.563, 1.790, 8.442, 0.137)′.

ED50	$e_D({\xi }_D^0)$	$e_D({\xi }_D^{-2})$	$e_D({\xi }_D^2)$	$e_D({\xi }_D^5)$
−5.5	0.732	0.731	0.695	0.600
−5	0.955	0.955	0.949	0.932
−4	0.744	0.743	0.716	0.661
−3	0.221	0.209	0.196	0.085
1	< 0.001	< 0.001	< 0.001	< 0.001

4. Conclusions

In this work, we showed how to construct optimal designs based on the MqLE, which are derived under less stringent model assumptions than those for the MLE. Using only the functional relationship between the mean and variance of the response variable, we showed that the MqLE is asymptotically best linear unbiased estimator and when the response has a distribution belonging to the exponential family, MqLE is as efficient as MLE. Consequently, the proposed locally optimal designs based on the MqLE have asymptotic optimality properties comparable to those based on the MLE. Often, optimal designs are obtained based on the LSE which is the same as the MLE under normal distribution with homoscedasticity. We proved that the optimal design based on the MqLE is asymptotically more efficient than the optimal designs based on the LSE.

Although the MqLE has good asymptotic optimality, it is rarely used to construct optimal designs. We provided some theoretical results and show these optimal designs based the MqLE are quite generally robust to distributional assumptions in a dose-response study settings. In particular, we found two types of locally optimal designs based on the MqLE for the 4PL model: D-optimal design for estimating the model parameters and C-optimal design for estimating both the ED₅₀ and MED at the same time. They were compared with the corresponding optimal designs based on the MLE for a homoscedastic model and with a uniform design with 10 doses. Our results show that optimal designs based on the MqLE can produce estimators that perform well under various model mis-specifications. We focused on the 4PL model but the methodology can be easily applied to find other types of optimal designs for different nonlinear models, as long as the optimality criterion is a concave or convex function of the information matrix. We also showed such designs can be constructed using current algorithms.

In conclusion, the robustness properties of a design to model assumptions should always be assessed before implementation. Ideally the implemented design has acceptable efficiencies under a variety of violations in the model assumptions and a change in the design criterion. While broad conclusions are usually difficult to establish, especially when there are multiple objectives in the study, it is important to carry out such an investigation for specific violations in the model assumptions to better understand sensitivities of the design to model perturbations before implementation. Our proposed designs here provide an alternative to optimal designs based on MLE when model assumptions are questionable.

Highlights.

• Propose optimal designs based on the MqLE that require less stringent model assumptions.

• Discuss asymptotic optimality of the MqLE.

• Develop the theory and an algorithm to find various optimal designs based on the MqLE.

• The proposed designs here provide an alternative to optimal designs based on the MLE when model assumptions are questionable.

5. Appendix

A.1. Proof of Theorem 1

Proof

Clearly θ̂_n in Eq(1) belongs to the class of asymptotically linear unbiased estimators. Each component of the n-dimensional random vector $V_n^{-1/2}[{\mathit{y}}_n-\mathit{\mu }({\mathit{\theta }}_0)]$ has a univariate distribution with mean 0 and variance ϕ, so by the Lindeberg-Feller CLT,

$$\sqrt{n}({\stackrel{\sim }{\mathit{\theta }}}_n-{\mathit{\theta }}_0)=\sqrt{n}{(L_n^{\prime }{L}_n)}^{-1}{L}_n^{\prime }{V}_n^{-1/2}[{\mathit{y}}_n-\mathit{\mu }({\mathit{\theta }}_0)]+o_p(1)\stackrel{d}{\to }N(0,\varphi {\mathrm{\Lambda }}^{-1}).$$

By the delta method, $\sqrt{n}[{\mathit{\mu }}_{★}({\stackrel{\sim }{\mathit{\theta }}}_n)-{\mathit{\mu }}_{★}({\mathit{\theta }}_0)]\stackrel{d}{\to }N(0,D_{★}\varphi {\mathrm{\Lambda }}^{-1}{D}_{★}^{\prime })$ and along with Eq(2), we have μ(θ̃_n) = μ(θ₀)+D(θ₀)(θ̃_n−θ₀)+ε with ||ε||_∞ = O_p(||θ̃_n − θ₀||²), which implies that

$${(L_n^{\prime }{L}_n)}^{-1}{L}_n^{\prime }{V}_n^{-1/2}[\mathit{\mu }({\stackrel{\sim }{\mathit{\theta }}}_n)-\mathit{\mu }({\mathit{\theta }}_0)]={(L_n^{\prime }{L}_n)}^{-1}{L}_n^{\prime }{V}_n^{-1/2}D({\stackrel{\sim }{\mathit{\theta }}}_n-{\mathit{\theta }}_0)+{(L_n^{\prime }{L}_n)}^{-1}{L}_n^{\prime }{V}_n^{-1/2}\mathit{\epsilon }.$$

Since $n^{-1}{\sum }_{i=1}^n{l}_i{l}_i^{\prime }\to \mathrm{\Lambda }$, max_1≤i≤n ||l_i|| = o(n^1/2) and inf_μ v(μ) > 0, we have

$$\Vert {(L_n^{\prime }{L}_n)}^{-1}{L}_n^{\prime }{V}_n^{-1/2}\mathit{\epsilon }\Vert \lesssim \{\frac{1}{\sqrt{n}}\underset{1\le i\le n}{max}\Vert {\mathrm{\Lambda }}^{-1}{l}_i\Vert \}\{\Vert \sqrt{n}({\stackrel{\sim }{\mathit{\theta }}}_n-{\mathit{\theta }}_0)\Vert \}\{\Vert {\stackrel{\sim }{\mathit{\theta }}}_n-{\mathit{\theta }}_0\Vert \}=o_p(\Vert {\stackrel{\sim }{\mathit{\theta }}}_n-{\mathit{\theta }}_0\Vert ).$$

We observe that Eq(2) and Eq(3) imply ${(L_n^{\prime }{L}_n)}^{-1}{L}^{\prime }{V}_n^{-1/2}[\mathit{\mu }({\stackrel{\sim }{\mathit{\theta }}}_n)-\mathit{\mu }({\mathit{\theta }}_0)]={\stackrel{\sim }{\mathit{\theta }}}_n-{\mathit{\theta }}_0+o(1)$, so ${(L_n^{\prime }{L}_n)}^{-1}{L}_n^{\prime }{V}_n^{-1/2}{D}_n=I_p+o(1)$, where I_p is the p×p identity matrix. To complete the proof, let $M_n=(V_n^{-1/2}{D}_n){[{(V_n^{-1/2}{D}_n)}^{\prime }(V_n^{-1/2}{D}_n)]}^{-1}{(V_n^{-1/2}{D}_n)}^{\prime }$, which is an idempotent and symmetric matrix. It follows that {I_n−M_n} is also symmetric and idempotent, and hence non-negative definite. Further,

$$\underset{n\to \infty }{lim}n(\mathit{Var}({\stackrel{\sim }{\mathit{\mu }}}_{★})-\mathit{Var}({\widehat{\mathit{\mu }}}_{★}))=\varphi \underset{n\to \infty }{lim}n\{D_{★}{(L^{\prime }L)}^{-1}{L}^{\prime }\}\{I_n-M_n\}{\{D_{★}{(L^{\prime }L)}^{-1}{L}^{\prime }\}}^{\prime },$$

and the desired conclusion follows.

The justifications for the next two results rely on a result from Hyun et al. (2011) who showed that if M(θ; ξ) is the Fisher information matrix, then for any matrix B, the matrix with (i, j)^th element equal to

$$\frac{{\partial }^2\mathit{det}[B{[M(\mathit{\theta };\xi )]}^{-1}{B}^{\prime }]}{\partial w_i\partial w_j}$$

is a non-negative definite matrix.

A.2. Proof of Theorem 2

Proof

Maximizing the D-optimality criterion det[Δ(θ; ξ)] is equivalent to minimizing det[[Δ(θ; ξ)]⁻¹]. The optimal design points are non-negative roots of the equation

$$\frac{\partial \mathit{det}[{[\mathrm{\Delta }(\mathit{\theta };\xi )]}^{-1}]}{\partial w_i}=0$$

provided that they minimize the inverse of the determinant of the quasi-information matrix, which is true if the matrix with (i, j)^th element equal to

$$\frac{{\partial }^2\mathit{det}[{[\mathrm{\Delta }(\mathit{\theta };\xi )]}^{-1}]}{\partial w_i\partial w_j}$$

is non-negative definite.

The key step is to recognize that the quasi-information matrix Δ(θ; ξ) has the same form as the Fisher information matrix, and it is also a full rank non-negative definite matrix. By setting B = I_r, the above result from Hyun et al. (2011) implies the determinant of the quasi-information matrix is maximized.

A.3. Proof of Theorem 3

Proof

The C-optimality criterion is Tr(A′[Δ(θ; ξ)]⁻¹A) and we want to minimize it by choice of a design ξ. The proof is similar to Theorem 2 after we first note that the weights of the C-optimal design are non-negative roots of the equation $\frac{\partial Tr(A^{\prime }{[\mathrm{\Delta }(\mathit{\theta };\xi )]}^{-1}A)}{\partial w_i}=0$. Then we use the invariance property of the Tr function under cyclic permutation and the same result from Hyun et al. (2011) to show the matrix with (i, j)^th element equal

$$\frac{{\partial }^{2}Tr(A^{\prime }{[\mathrm{\Delta }(\mathit{\theta };\xi )]}^{-1}A)}{\partial w_i\partial w_j}$$

is non-negative definite.

A.4. The Fisher information matrix for the MLE θ̌ under Gamma distribution

Let the shape parameter and scale parameter of the Gamma distribution be $\alpha ={\mu }_i^2/v_i$ and β = μ_i/v_i, respectively. If y has such a distribution, then E(y) = α/β and Var(y) = α/β². For the gamma distribution, the log likelihood function is the following:

$$\sum _{i=1}^{n}l({\mu }_i;y_i)=\sum _{i=1}^n\left\{\frac{{\mu }_i^2}{v_i}\mathit{log}\left(\frac{{\mu }_i}{v_i}\right)-\mathit{log}\left(\mathrm{\Gamma }(\frac{{\mu }_i^2}{v_i})\right)+\left(\frac{{\mu }_i^2}{v_i}-1\right)\mathit{log}(y_i)-\frac{{\mu }_i^2{y}_i}{v_i}\right\}.$$

Assuming we have a random sample of size n at d distinct log doses, a direct calculation shows the Fisher information matrix for θ̌ is

$$M^{\ast }(\mathit{\theta };\xi )=\sum _{i=1}^d{w}_{i}f(x_i)f^{\prime }(x_i),$$

where $f(x_i)={[P_1+P_2-P_3]}^{1/2}{\left(\frac{\partial {\mu }_i}{\partial {\theta }_1},\frac{\partial {\mu }_i}{\partial {\theta }_2},\cdots ,\frac{\partial {\mu }_i}{\partial {\theta }_r}\right)}^{\prime }$ with

$$\begin{array}{l}{P}_1=\frac{2v_i^2-{\mu }_i{v}_i{\ddot{v}}_i-4{\mu }_i{v}_i{\stackrel{.}{v}}_i+2{({\mu }_i{\stackrel{.}{v}}_i)}^2}{v_i^3}\left(\mathit{log}(v_i)+{\psi }_0-2\mathit{log}({\mu }_i)\right);\\ P_2=\frac{2{\mu }_i{v}_i-{\mu }_i^2{\stackrel{.}{v}}_i}{v_i^2}\left(\frac{{\stackrel{.}{v}}_i}{v_i}+{\psi }_1\frac{2{\mu }_i{v}_i-{\mu }_i^2{\stackrel{.}{v}}_i}{v_i^2}-\frac{1}{{\mu }_i}\right);\end{array}$$

and

$$P_3=\frac{1}{v_i}\left(1-\frac{{\mu }_i{\stackrel{.}{v}}_i}{v_i}\right).$$

Here ${\stackrel{.}{v}}_i=\frac{\partial v_i}{\partial {\mu }_i};{\ddot{v}}_i=\frac{{\partial }^2{v}_i}{\partial {\mu }_i^2};{\psi }_0=\frac{\partial }{\partial {\mu }_i}\mathit{log}\left(\mathrm{\Gamma }(\frac{{\mu }_i^2}{v_i})\right)$ and ${\psi }_1=\frac{\partial }{\partial {\mu }_i}{\psi }_0$.