Experiment design for nonparametric models based on minimizing Bayes Risk: application to voriconazole$^{1}$

Abstract

An experimental design approach is presented for individualized therapy in the special case where the prior information is specified by a nonparametric (NP) population model. Here, a NP model refers to a discrete probability model characterized by a finite set of support points and their associated weights. An important question arises as to how to best design experiments for this type of model. Many experimental design methods are based on Fisher information or other approaches originally developed for parametric models. While such approaches have been used with some success across various applications, it is interesting to note that they largely fail to address the fundamentally discrete nature of the NP model. Specifically, the problem of identifying an individual from a NP prior is more naturally treated as a problem of classification, i.e., to find a support point that best matches the patient’s behavior. This paper studies the discrete nature of the NP experiment design problem from a classification point of view. Several new insights are provided including the use of Bayes Risk as an information measure, and new alternative methods for experiment design. One particular method, denoted as MMopt (multiple-model optimal), will be examined in detail and shown to require minimal computation while having distinct advantages compared to existing approaches. Several simulated examples, including a case study involving oral voriconazole in children, are given to demonstrate the usefulness of MMopt in pharmacokinetics applications.

Appendix: Two-support-point problem with close parameters

Definition 1

(Two-support-point problem) A general class of two-support-point problems is defined of the form,

$$\begin{aligned} y\left( t_{k}\right) =\mu \left( t_{k},\,a\right) +\sigma _{k} n_{k}, \end{aligned}$$

(58)

where $a\in \Omega \buildrel \Delta \over =\{a_{1},\,a_{2}\}$ is a random variable taking on values $a_{1}$ and $a_{2};\,z(t,\,a_{i}),\, i=1,\,2$ are the support point responses, each assumed to be continuous over the closed interval $t\in {\mathcal {T}}\buildrel \Delta \over =[t_{A},\,t_{B}];\,y(t_{k})$ is the noisy measurement of $z(t,\,a)$ taken at discrete time $t_{k};\,n_{k}\sim N(0,\,1)$ is the measurement noise at time instant $t_{k};\,\sigma _{k}$ scales $n_{k}$ to a desired level; and the prior support point probabilities are specified as $p_{1}$ for $a_{1}$ and $p_{2}$ for $a_{2}$ where $p_{1}>0, \,p_{2}>0,$ and $p_{1}+p_{2}=1.$

The experiment design problem is to find the set U of n optimal sampling times

$$\begin{aligned} U=\left\{ t_{1},\ldots ,t_{n}\right\} , \end{aligned}$$

(59)

on a specified time interval $t\in {\mathcal {T}}\buildrel \Delta \over =[t_{A},\,t_{B}]$ that act to best identify the parameter vector $a\in \{a_{1},\,a_{2}\}.$

Definition 2

(Restricted two-support-point problem) A restricted two-support-point problem is defined by Definition 1 under the additional conditions that the noise is independent of time, $\sigma _{k}=\sigma ;$ there is only one sample to be taken, $n=1,\,U=\{t_{1}\};$ and the prior support point probabilities are uniformly distributed, i.e., $p_{1}=p_{2}=0.5.$ Furthermore, it is assumed that the function $\mu (t,\,a)$ is sufficiently smooth to admit a first-order Taylor expansion in the vicinity of parameter values $a_{1}$ and $a_{2}$ for all $t\in {\mathcal {T}}.$

In what follows, the asymptotic order notation ${\mathcal {O}}(\epsilon ^{k})$ is used to indicate a term that decreases to zero as $\epsilon ^{k}.$ Specifically, $\lim _{\epsilon \rightarrow 0}{\mathcal {O}}(\epsilon ^{k})=0$ and there exists a constant M such that [9],

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\left| {\mathcal {O}}\left( \epsilon ^{k}\right) /\epsilon ^{k}\right| =M<\infty . \end{aligned}$$

(60)

Furthermore, the notational dependence of $\mu (t,\,a)$ on time t will be suppressed and denoted more simply as $\mu (a).$

Theorem 2

The costs associated with the ED, D [evaluated on the mean parameter value ${\overline{a}}={1\over 2}(a_{1}+a_{2})],$ EID and ELD optimal designs for the restricted two-support-point problem of Definition 2 can be written in terms of the response separation $r(t)=|\mu (a_{2})-\mu (a_{1})|$ for sufficiently small parameter uncertainty $\Delta a=a_{2}-a_{1}$ as,

$$\begin{aligned}&J_{ED}=0.5 \left| M\left( a_{1}\right) \right| +0.5 \left| M\left( a_{2}\right) \right| =\frac{r(t)^{2}}{\sigma ^{2}\Delta a^{2}}+{\mathcal {O}}(\Delta a),\end{aligned}$$

(61)

$$\begin{aligned}&J_{D}=|M({\overline{a}})|=\frac{r(t)^{2}}{\sigma ^{2}\Delta a^{2}}+{\mathcal {O}}(\Delta a),\end{aligned}$$

(62)

$$\begin{aligned}&J_{EID}=0.5 \left| M\left( a_{1}\right) \right| ^{-1}+0.5 \left| M\left( a_{2}\right) \right| ^{-1}= \frac{\sigma ^{2}\Delta a^{2}}{r(t)^{2}}(1+{\mathcal {O}}(\Delta a)),\end{aligned}$$

(63)

$$\begin{aligned}&J_{ELD}=0.5 \ln \left| M\left( a_{1}\right) \right| +0.5 \ln \left| M\left( a_{2}\right) \right| = 0.5\ln \left( \frac{r(t)^{4}}{\sigma ^{4}\Delta a^{4}}+{\mathcal {O}}(\Delta a)\right) . \end{aligned}$$

(64)

Proof of Eq. ( 61 ): Consider the response $\mu (a)$ as a function of the parameter a. Let $\mu (a_{2})$ be expressed in terms of the Taylor expansion about $a_{1}$

$$\begin{aligned} \mu \left( a_{2}\right) =\mu \left( a_{1}\right) +\frac{\partial \mu }{\partial a}\Big|_{a_{1}}\left( a_{2}-a_{1}\right) +{\mathcal {O}}\left( a_{2}-a_{1}\right) ^{2} . \end{aligned}$$

(65)

Letting $\Delta a\buildrel \Delta \over =a_{2}-a_{1}$ yields upon rearranging,

$$\begin{aligned} \mu \left( a_{2}\right) -\mu \left( a_{1}\right) =\frac{\partial \mu }{\partial a}\Big| _{a_{1}}\Delta a+{\mathcal {O}}(\Delta a)^{2} . \end{aligned}$$

(66)

Squaring both sides of (66) gives

$$\begin{aligned} \left( \mu \left( a_{2}\right) -\mu \left( a_{1}\right) \right) ^{2}=(\Delta a)^{2}\left( \frac{\partial \mu }{\partial a}\Big| _{a_{1}}\right) ^{2} +{\mathcal {O}}(\Delta a)^{3}\Big. . \end{aligned}$$

(67)

Similarly, let $\mu (a_{1})$ be expressed in terms of the Taylor expansion about $a_{2}$

$$\begin{aligned} \mu \left( a_{1}\right) =\mu \left( a_{2}\right) +\frac{\partial \mu }{\partial a}\Big| _{a_{2}}\left( a_{1}-a_{2}\right) +{\mathcal {O}}(\Delta a)^{2} , \end{aligned}$$

(68)

which can be rearranged as

$$\begin{aligned} \mu \left( a_{2}\right) -\mu \left( a_{1}\right) =\frac{\partial \mu }{\partial a}\Big| _{a_{2}}\Delta a +{\mathcal {O}}(\Delta a)^{2} . \end{aligned}$$

(69)

Squaring both sides of (69) yields

$$\begin{aligned} \left( \mu \left( a_{2}\right) -\mu \left( a_{1}\right) \right) ^{2}=(\Delta a)^{2}\left( \frac{\partial \mu }{\partial a}\Big| _{a_{2}}\right) ^{2} +{\mathcal {O}}(\Delta a)^{3}\Big. . \end{aligned}$$

(70)

Adding one half of (67) to one half of (70) gives

$$\begin{aligned} \left( \mu \left( a_{2}\right) -\mu \left( a_{1}\right) \right) ^{2}=(\Delta a)^{2}\left( \frac{1}{2}\left( \frac{\partial \mu }{\partial a}\Big| _{a_{1}}\right) ^{2} +\frac{1}{2}\left( \frac{\partial \mu }{\partial a}\Big| _{a_{2}}\right) ^{2}\right) +{\mathcal {O}}(\Delta a)^{3}\Big. \Big. . \end{aligned}$$

(71)

Dividing both sides by $\sigma ^{2}(\Delta a)^{2}$ gives,

$$\begin{aligned} \frac{1}{2\sigma ^{2}}\left( \frac{\partial \mu }{\partial a}\Big| _{a_{1}}\right) ^{2} +\frac{1}{2\sigma ^{2}}\left( \frac{\partial \mu }{\partial a}\Big| _{a_{2}}\right) ^{2} =\frac{(\mu (a_{2})-\mu (a_{1}))^{2}}{\sigma ^{2}\Delta a^{2}}+{\mathcal {O}}(\Delta a)\Big. \Big. . \end{aligned}$$

(72)

Proof of Eq. ( 62 ): Combining relations ${\overline{a}}={1\over 2}(a_{1}+a_{2})$ and $\Delta a=a_{2}-a_{1},$ gives

$$\begin{aligned} {\overline{a}}=a_{1}+{1\over 2}\Delta a, \end{aligned}$$

(73)

$$\begin{aligned} {\overline{a}}=a_{2}-{1\over 2}\Delta a. \end{aligned}$$

(74)

Let $\mu (a_{2})$ be expressed in terms of a Taylor expansion of $\mu (a)$ about ${\overline{a}}$

$$\begin{aligned} \mu \left( a_{2}\right) =\mu ({\overline{a}})+\frac{\partial \mu }{\partial a}\Big| _{{\overline{a}}}\left( a_{2}-{\overline{a}}\right) +{\mathcal {O}}\left( a_{2}-{\overline{a}}\right) ^{2} . \end{aligned}$$

(75)

Rearranging (74) gives $a_{2}-{\overline{a}}={1\over 2}\Delta a$ which is substituted into (75) to give

$$\begin{aligned} \mu \left( a_{2}\right) =\mu ({\overline{a}})+\frac{\partial \mu }{\partial a}\Big| _{{\overline{a}}}\left( {1\over 2}\Delta a\right) +{\mathcal {O}}(\Delta a)^{2} \end{aligned}$$

(76)

Similarly, let $\mu (a_{1})$ be expressed in terms of a Taylor expansion of $\mu (a)$ about ${\overline{a}}$

$$\begin{aligned} \mu \left( a_{1}\right) =\mu ({\overline{a}})+\frac{\partial \mu }{\partial a}\Big| _{{\overline{a}}}\left( a_{1}-{\overline{a}}\right) +{\mathcal {O}}\left( a_{1}-{\overline{a}}\right) ^{2} . \end{aligned}$$

(77)

Rearranging (73) gives $a_{1}-{\overline{a}}={-}{1\over 2}\Delta a$ which is substituted into (77) to give

$$\begin{aligned} \mu \left( a_{1}\right) =\mu ({\overline{a}})-\frac{\partial \mu }{\partial a}\Big| _{{\overline{a}}}\left( {1\over 2}\Delta a\right) +{\mathcal {O}}(\Delta a)^{2} . \end{aligned}$$

(78)

Subtracting (78) from (76) and rearranging gives

$$\begin{aligned} \mu \left( a_{2}\right) -\mu \left( a_{1}\right) =\frac{\partial \mu }{\partial a}\Big| _{{\overline{a}}}\Delta a +{\mathcal {O}}(\Delta a)^{2} . \end{aligned}$$

(79)

Squaring both sides of (79) and dividing by $\sigma ^{2}(\Delta a)^{2}$ gives upon rearranging,

$$\begin{aligned} \frac{1}{\sigma ^{2}}\left(\frac{\partial \mu }{\partial a}\Big| _{{\overline{a}}}\right)^2= \frac{(\mu (a_{2})-\mu (a_{1}))^{2}}{\sigma ^{2}\Delta a^{2}}+{\mathcal {O}}(\Delta a)\end{aligned}.$$

(80)

Proof of Eq. ( 63 ): Reciprocating both sides of Eq. (67), using the relation $\frac{1}{1+\epsilon }= 1-\epsilon +{\mathcal {O}}(\epsilon ^{2})$ for small $\epsilon ,$ and rearranging gives,

$$\frac{{\Updelta {a^{2}}}}{{{{(\mu ({a_{2}}) - \mu ({a_{1}}))}^{2}}}} = {\left( {\left. {\frac{{\partial \mu }}{{\partial a}}} \right|_{{a_{1}}}^{2}} \right)^{ - 1}}(1 - {\mathcal {O}}(\Updelta a)).$$

(81)

Likewise, reciprocating both sides of Eq. (70) and rearranging gives,

$$\frac{{\Updelta {a^{2}}}}{{{{(\mu ({a_{2}}) - \mu ({a_{1}}))}^{2}}}} = {\left( {\left. {\frac{{\partial \mu }}{{\partial a}}} \right|_{{a_{2}}}^{2}} \right)^{ - 1}}(1 - {\mathcal {O}}(\Updelta a)).$$

(82)

Adding (81) and (82) and multiplying both sides by $\sigma ^{2}/2$ gives upon rearranging

$$\begin{aligned} 0.5 \left( \frac{1}{\sigma ^{2}}\frac{\partial \mu }{\partial a}\Big| _{a_{1}}^{2}\right) ^{-1} +\,0.5\left( \frac{1}{\sigma ^{2}}\frac{\partial \mu }{\partial a}\Big| _{a_{2}}^{2}\right) ^{-1}= \frac{\sigma ^{2}\Delta a^{2}}{(\mu (a_{2})-\mu (a_{1}))^{2}}(1+{\mathcal {O}}(\Delta a)) . \end{aligned}$$

(83)

Proof of Eq. ( 64 ): Multiplying (67) and (70) yields,

$$\begin{aligned} \left( \mu \left( a_{2}\right) -\mu \left( a_{1}\right) \right) ^{4}= \Delta a^{4}\left( \frac{\partial \mu }{\partial a}\Big| _{a_{1}}\right) ^{2} \left( \frac{\partial \mu }{\partial a} \Big| _{a_{2}}\right) ^{2} +{\mathcal {O}}(\Delta a)^{5}\Big. \Big. . \end{aligned}$$

(84)

Dividing both sides by $\sigma ^{4}\Delta a^{4}$ gives upon rearranging

$$\begin{aligned} \left( \frac{1}{\sigma ^{2}}\frac{\partial \mu }{\partial a}\Big| _{a_{1}}^{2}\right) \left( \frac{1}{\sigma ^{2}}\frac{\partial \mu }{\partial a}\Big| _{a_{2}}^{2}\right) = \frac{(\mu (a_{2})-\mu (a_{1}))^{4}}{\sigma ^{4}\Delta a^{4}}+{\mathcal {O}}(\Delta a)\Big. \Big. . \end{aligned}$$

(85)

Taking ${1\over 2}\ln (\cdot )$ of both sides gives

$$\begin{aligned} 0.5\ln \left( \frac{1}{\sigma ^{2}}\frac{\partial \mu }{\partial a}\Big| _{a_{1}}^{2}\right) +0.5\ln \left( \frac{1}{\sigma ^{2}}\frac{\partial \mu }{\partial a}\Big| _{a_{2}}^{2}\right) = 0.5\ln \left( \frac{(\mu (a_{2})-\mu (a_{1}))^{4}}{\sigma ^{4}\Delta a^{4}} +{\mathcal {O}}(\Delta a)\right) \Big. \Big. . \end{aligned}$$

(86)

Corollary 1

As the parameter uncertainty $\Delta a=a_{2}-a_{1}$ becomes small in the restricted two-support-point problem of Definition 2, the objective functions $J_{ED},\,J_{D}$ and $J_{ELD}$ approach a monotonically increasing function of the response separation r(t), while $J_{EID}$ approaches a monotonically decreasing function of the response separation r(t).

Proof

The result follows by the properties of relations (61), (62), (63), (64) as $\Delta a$ becomes small, noting that functions 1 / x and $\ln (x)$ are monotonically decreasing and increasing in x,  respectively. $\square$

Corollary 1 indicates that as the parameter uncertainty $\Delta a=a_{2}-a_{1}$ becomes small in the restricted two-support-point problem of Definition 2, the objectives of maximizing $J_{ED},\,J_{D},\,J_{ELD}$ and minimizing $J_{EID}$ asymptotically approach the single common objective of maximizing the response separation $r(t)=|\mu (a_{2},\,t)-\mu (a_{1},\,t)|$ over time $t\in {\mathcal {T}}.$ These asymptotic properties are examined numerically in the “Two-support-point example” section.