Optimal designs for health risk assessments using fractional polynomial models

Abstract

Fractional polynomials (FP) have been shown to be more flexible than polynomial models for fitting data from an univariate regression model with a continuous outcome but design issues for FP models have lagged. We focus on FPs with a single variable and construct D-optimal designs for estimating model parameters and I-optimal designs for prediction over a user-specified region of the design space. Some analytic results are given, along with a discussion on model uncertainty. In addition, we provide an applet to facilitate users find tailor made optimal designs for their problems. As applications, we construct optimal designs for three studies that used FPs to model risk assessments of (a) testosterone levels from magnesium accumulation in certain areas of the brains in songbirds, (b) rats subject to exposure of different chemicals, and (c) hormetic effects due to small toxic exposure. In each case, we elaborate the benefits of having an optimal design in terms of cost and quality of the statistical inference.

Appendix

This appendix provides proofs for Theorem 1: D-optimal designs for FP1 and FP2 models and Theorem 2: I-optimal designs for the FP1 models.

Proof (Theorem 1: construction of D-optimal designs)

[Theorem 1: construction of D-optimal designs]

Suppose that a design $\xi$ for a FP1 model defined on $[\epsilon ,a]$ with 3 points at $\epsilon \le s_1< s_2 < s_3 \le a$. From the General Equivalence Theorem of Kiefer and Wolfowitz (1960), if $\xi$ were D-optimal, its sensitivity function c(x) must satisfy $c(x) =f^T(x) M^{-1}(\xi ) f(x) -2 \le 0$ for all $x \in [\epsilon ,a]$ with equality at the support points. If $p \ne 0$, the component functions in c(x) are $1,x^p,x^{2p}$. These 3 component functions form a Tchebycheff system on the interval $[\epsilon ,a]$ because the determinant of the matrix

$$\begin{aligned} \begin{array}{|ccc|} 1 & x_1^p &x_1^{2p} \\ 1 & x_2^p & x_2^{2p} \\ 1 & x_3^p & x_3^{2p} \\ \end{array}=-(x_1^p-x_2^p) (x_1^p-x_3^p) (x_2^p-x_3^p), \end{aligned}$$

has the same sign for any $\epsilon \le x_1< x_2 < x_3 \le a$. It follows that for each $0 \ne p \in {\mathcal {P}}$, the sensitivity function has at most 2 zeros and since the D-optimal design for a FP1 model has at least 2 points for it to have a nonsingular information matrix, the D-optimal design is equally supported at 2 points (Pukelsheim 1993; Fedorov 1972). A similar argument shows that the same conclusion applies when $p=0$ and the component functions are $\{1,\ln [x],\ln [x]^2\}$. Direct calculus shows that the sensitivity function of the design equally supported at the two end-points is $c(x) =4 \left( a^p-x^p\right) \left( \epsilon ^p-x^p\right) /\left( a^p-\epsilon ^p\right) ^2$ when $p\ne 0$. Clearly, this function satisfies the conditions required in the equivalence theorem for D-optimality in (3) and so the design is D-optimal for FP1 models. A similar argument applies when $p=0$.

Note that the previous reasoning about $p \ne 0$ is valid for any real number value of p. Thus the result proved here is more general than for $p \in {\mathcal {P}}$. The same applies for the following results.

For the FP2 models, we first establish that the D-optimal design $\xi _D$ has three points. For such models, the sensitivity function c(x) has at most 6 component functions, i.e. $1, x^p, x^{2p}, x^q, x^{2q}$ and $x^{p+q}$. In selected cases, such as when $p=-0.5$ and $q=-1$, the sensitivity function has only 5 component functions. A direct calculation shows that the associated Wronskians for this system of functions are positive for any values of p and q and so the component functions form a Tchebycheff system (Gasull et al. 2012). It follows that there are at most 5 zeroes (counting multiplicities). The interior support points have multiplicity two, because the maximum value of the sensitivity function of the D-optimal design has to be less than or equal to zero in the interval with the maximum value attained at the support points. This implies only three support points are possible, either two interior points and one extreme point of the design interval or one interior support point and the two extreme points of the interval $[\epsilon ,a]$. Because the number of support points is the same as the number of parameters, the D-optimal design is equally weighted (Pukelsheim 1993). We next argue that the optimal designs have to include the two extreme points.

Suppose the equally weighted design $\xi$ is supported at $s_1<s_2<a$, where $\epsilon <s_1$. The determinant of the information matrix of this design for model FP2(p, q) with $0 \ne p\ne q \ne 0$ is

$$\begin{aligned} |M(\xi ) |= \frac{1}{27}\left| \begin{array}{ccc} 1 &{} s_1^p &{} s_1^q\\ 1 &{} s_2^p &{} s_2^q\\ 1 &{} a^p &{} a^q\\ \end{array} \right| ^2 = \frac{1}{27} D^2>0 \end{aligned}$$

since D is always either positive or negative for any values of $s_1,s_2$ and a because we have a Chebyshev system. Further,

$$\begin{aligned} \frac{\partial D}{\partial s_1}=q s_1^{q-1}(a^p-s_2^p) -p s_1^{p-1}(a^q - s_2^q) \ne 0 \end{aligned}$$

implies that

$$\begin{aligned} \partial |M(\xi ) |/\partial s_1= (2/27) D \partial D/\partial s_1\ne 0 \end{aligned}$$

and so D is a decreasing function of $s_1$. Consequently, $\epsilon$ is a support point of the D-optimal design. We note that the last equation holds if and only if $s_1^{q-p}\frac{q (a^p-s_2^p) }{p (a^q - s_2^q) }\ne 1$, i.e. $\left( \frac{s_1}{c} \right) ^{q-p} \ne 1.$ For the first equivalence we note that $ps_1^{p-1}(a^q-s_2^q) \ne 0$. The last equivalence obtains because by the mean value theorem, there exists a $c \in [s_2,a]$ such that $(a^p-s_2^p) /(a^q - s_2^q) =c^{p-q} p/q$. Consequently, since $s_1<c$ and $q>p$ without loss of generality, $(s_1/c) ^{q-p}<1$ proving the result. It follows that $\partial |M(\xi ) |/\partial s_1$ is negative, otherwise the optimal design is singular. It is easy to verify that the determinant of the information matrix is a decreasing function of $s_1$ and its maximum is obtained at $s_1=\epsilon$, the left end-point of the design space. Similar reasoning leads to the optimal design as being supported at $\epsilon<s_2<s_3$ and its determinant is maximized when $s_3=a$. The upshot is that the D-optimal design is equally supported at the two end-points of the design space and at an interior point. The above arguments apply to other cases of p and q (p and q unequal and one of them zero, p and q equal and $p=q=0$). In either case, the interior support point s is the unique root of the derivative of the sensitivity function.$\square$

Proof (Theorem 2: construction for I-optimal designs)

[Theorem 2: construction for I-optimal designs]

The previous reasoning can be used to directly prove that I-optimal designs for FP1 models are unequally supported at the end-points of the design space. We note that the component functions in the sensitivity function of (4) are $1,x^p,x^{2p}$ when $p \ne 0$ and $\{1,\ln [x],\ln [x]^2\}$ when $p=0$, and they form a Tchebycheff system on $[\epsilon ,a]$. Similarly, the weights in Sect. 3.2 are found by finding the roots of the sensitivity function of the I-optimal design evaluated at $x=\epsilon$ and $x=a$.

In addition, a direct calculation, from 1/w given in 1., shows the result 3. for $p>0$ and $\epsilon =0$. $\square$