V-optimality of designs in random effects Poisson regression models

Abstract

The knowledge of the Fisher information is a fundamental tool to judge the quality of an experiment. Unlike in linear and generalized linear models without random effects, there is no closed form for the Fisher information in the situation of generalized linear mixed models, in general. To circumvent this problem, we make use of the quasi-information in this paper as an approximation to the Fisher information. We derive optimal designs based on the V-criterion, which aims to minimize the average variance of prediction of the mean response. For this criterion, we obtain locally optimal designs in two specific cases of a Poisson straight line regression model with either random intercepts or random slopes.

Appendices

Appendix A Proof of the convexity of the V-criterion

Let $\xi _1$ and $\xi _2$ be two designs. For any $\alpha \in [0,1]$, Niaparast and Schwabe (2013) showed

$$\begin{aligned} {\mathfrak {M}}_{\varvec{\beta }}((1-\alpha )\xi _1+\alpha \xi _2)\ge (1-\alpha ){\mathfrak {M}}_{\varvec{\beta }}(\xi _1)+\alpha {\mathfrak {M}}_{\varvec{\beta }}(\xi _2) \end{aligned}$$

(B1)

in the sense of Loewner ordering of non-negative definiteness. This requires that the matrices $\varvec{C}_\xi$ of variance correction terms are non-negative definite which follows from the corresponding property of $\varvec{C}_i$ mentioned in Sect. 2.

By standard inversion formulas the convexity of the inverse follows,

$$\begin{aligned} {\mathfrak {M}}^{-1}_{\varvec{\beta }}((1-\alpha )\xi _1+\alpha \xi _2)\le (1-\alpha ){\mathfrak {M}}^{-1}_{\varvec{\beta }}(\xi _1)+\alpha {\mathfrak {M}}^{-1}_{\varvec{\beta }}(\xi _2) . \end{aligned}$$

Hence, because $\textrm{tr}(\varvec{A}\varvec{B})$ is linear in $\varvec{A}$, the V-criterion is a convex functional in $\xi$.

Appendix B Proof of Theorem 1

In order to obtain an equivalence theorem we consider the directional derivative of the trace $\phi (\xi )=\textrm{tr}({\mathfrak {M}}_{\varvec{\beta }}^{-1}(\xi )\varvec{B})$. The directional derivative of $\phi (\xi )$ in the direction of $\xi '$, $F_{\phi }(\xi ,\xi ')$ is

$$\begin{aligned} F_{\phi }(\xi ,\xi ')=\frac{\textrm{d}}{\textrm{d} \alpha }\textrm{tr}(({\mathfrak {M}}_{\varvec{\beta }}^{-1}((1-\alpha ){\xi }+\alpha {\xi '}))\varvec{B})|_{\alpha =0^+} \end{aligned}$$

(B1)

Since $(\varvec{A}^{-1}_{\xi }+\varvec{C}_{\xi })^{-1}=\varvec{A}_{\xi }-\varvec{A}_{\xi }(\varvec{I}+\varvec{C}_{\xi }\varvec{A}_{\xi })^{-1}\varvec{C}_{\xi }\varvec{A}_{\xi }$ where $\varvec{I}$ is the identity matrix, the quasi-information matrix in Eq. (2) can be represented as

$$\begin{aligned} {\mathfrak {M}}_{\varvec{\beta }}(\xi )=\varvec{F}^\textrm{T}_{\xi }\left( \varvec{A}_{\xi }-\varvec{A}_{\xi }(\varvec{I}+{\varvec{C}}_{\xi }\varvec{A}_{\xi })^{-1}\varvec{C}_{\xi }\varvec{A}_{\xi }\right) \varvec{F}_{\xi }. \end{aligned}$$

(B2)

As in Niaparast and Schwabe (2013) we will derive a representation of the quasi-information matrix of the convex combination of $\xi$ and $\xi '$. For this we define the weighted joint intensity matrix $\varvec{A}_{\xi , \xi '}(\alpha )=\left( \begin{array}{cc}(1-\alpha )\varvec{A}_{\xi }&{}0\\ 0&{}\alpha \varvec{A}_{\xi '}\end{array}\right)$ for two designs $\xi$ and $\xi '$, $0 \le \alpha \le 1$, where $\varvec{A}_{\xi }$ and $\varvec{A}_{\xi '}$ are diagonal matrices with the weighted response means $(m_{j}\mu _{j})$ as diagonal elements corresponding to $\xi$ and $\xi '$ respectively, $\varvec{F}_{\xi ,\xi '}=\left( \begin{array}{cc}\varvec{F}^\textrm{T}_{\xi }&\varvec{F}^\textrm{T}_{\xi '}\end{array}\right) ^\textrm{T}$ the joint reduced design matrix for the designs $\xi$ and $\xi '$ and by $\varvec{C}_{\xi ,\xi '}=\left( \begin{array}{cc}\varvec{C}_{\xi }&{}\varvec{\Gamma }_{\xi ,\xi '}\\ \varvec{\Gamma }^\textrm{T}_{\xi ,\xi '}&{}\varvec{C}_{\xi '} \end{array}\right)$ the combined correction matrix, which contains the mixed correction terms for $\xi$ and $\xi '$ in $\varvec{\Gamma }_{\xi ,\xi '}=(c(x,x'))$, where x and $x'$ are the support points of $\xi$ and $\xi '$, respectively.

Then regarding (B2), the quasi-information matrix of the convex combination of $\xi$ and $\xi '$ is as follow

$$\begin{aligned} \varvec{{\mathfrak {M}}}_{\varvec{\beta }}{((1-\alpha ){\xi }+\alpha {\xi '})}= & {} \varvec{F}^\textrm{T}_{\xi ,\xi '}[\varvec{A}_{\xi ,\xi '}(\alpha )\\{} & {} - \varvec{A}_{\xi ,\xi '}(\alpha )(\varvec{I}+\varvec{C}_{\xi ,\xi '}\varvec{A}_{\xi ,\xi '}(\alpha ))^{-1}\varvec{C}_{\xi ,\xi '}\varvec{A}_{\xi ,\xi '}(\alpha )]{\varvec{F}_{\xi ,\xi '}}. \end{aligned}$$

Then,

$$\begin{aligned} F_{\phi }(\xi ,\xi ')= & {} \textrm{tr}(\frac{\textrm{d}}{\textrm{d}\alpha }(\varvec{F}^\textrm{T}_{\xi ,\xi '}[\varvec{A}_{\xi ,\xi '}(\alpha )-\varvec{A}_{\xi ,\xi '}(\alpha )(\varvec{I}+\varvec{C}_{\xi ,\xi '}\varvec{A}_{\xi ,\xi '}(\alpha ))^{-1}\nonumber \\{} & {} \varvec{C}_{\xi ,\xi '}\varvec{A}_{\xi ,\xi '}(\alpha )]\varvec{F}_{\xi ,\xi '})\varvec{B})|_{\alpha =0^+}\nonumber \\= & {} \textrm{tr}[-\varvec{{\mathfrak {M}}}_{\varvec{\beta }}^{-1}((1-\alpha ){\xi }+\alpha {\xi '})(\varvec{F}^\textrm{T}_{\xi ,\xi '}([\varvec{A}'_{\xi ,\xi '}(\alpha )-\varvec{A}'_{\xi ,\xi '}(0)\nonumber \\{} & {} (\varvec{I}+\varvec{C}_{\xi ,\xi '}\varvec{A}_{\xi ,\xi '}(\alpha ))^{-1}\varvec{C}_{\xi ,\xi '}\varvec{A}_{\xi ,\xi '}(\alpha )+\varvec{A}_{\xi ,\xi '}(\alpha )\nonumber \\{} & {} (\varvec{I}+\varvec{C}_{\xi ,\xi '}\varvec{A}_{\xi ,\xi '}(\alpha ))^{-1} \varvec{C}_{\xi ,\xi '}\varvec{A}'_{\xi ,\xi '}(\alpha ) (\varvec{I}+\varvec{C}_{\xi ,\xi '}\varvec{A}_{\xi ,\xi '}(\alpha ))^{-1}\nonumber \\{} & {} \varvec{C}_{\xi ,\xi '}\varvec{A}_{\xi ,\xi '}(\alpha )-\varvec{A}_{\xi ,\xi '}(\alpha )(\varvec{I}+\varvec{C}_{\xi ,\xi '}\varvec{A}_{\xi ,\xi '}(\alpha ))^{-1}\varvec{C}_{\xi ,\xi '}\varvec{A}'_{\xi ,\xi '}(\alpha )]\nonumber \\{} & {} \varvec{F}_{\xi ,\xi '})\varvec{{\mathfrak {M}}}_{\varvec{\beta }}^{-1}((1-\alpha ){\xi }+\alpha {\xi '})]\varvec{B})|_{\alpha =0^+}, \end{aligned}$$

(B3)

where $\varvec{A}'_{\xi ,\xi '}(\alpha )$ is the derivative of $\varvec{A}_{\xi ,\xi '}(\alpha )$ w.r.t. $\alpha$. We obtain for $\alpha =0$ that

$(\varvec{I}+\varvec{C}_{\xi ,\xi '}\varvec{A}_{\xi ,\xi '}(0))= \left( \begin{array}{cc} \varvec{I}+\varvec{C}_{\xi }\varvec{A}_{\xi }&{}0\\ \varvec{\Gamma }^\textrm{T}_{\xi ,\xi '}\varvec{A}_{\xi }&{}\varvec{I} \end{array}\right)$ is lower block triangular as well its inverse $(\varvec{I}+\varvec{C}_{\xi ,\xi '}\varvec{A}_{\xi ,\xi '}(0))^{-1}=\left( \begin{array}{cc} (\varvec{I}+\varvec{C}_{\xi }\varvec{A}_{\xi })^{-1}&{}0\\ \varvec{-\Gamma }^\textrm{T}_{\xi ,\xi '}\varvec{A}_{\xi }(\varvec{I}+\varvec{C}_{\xi }\varvec{A}_{\xi })^{-1}&{}\varvec{I }\end{array}\right)$.

We obtain by multiplication of the block matrix

$$\begin{aligned} \varvec{F}^\textrm{T}_{\xi ,\xi '}\varvec{A}'_{\xi ,\xi '}(0)\varvec{F}_{\xi ,\xi '}= -\varvec{F}^\textrm{T}_{\xi }\varvec{A}_{\xi }\varvec{F}_{\xi }+\varvec{F}^\textrm{T}_{\xi '}\varvec{A}_{\xi '}\varvec{F}_{\xi '} \end{aligned}$$

and also we obtain

$\varvec{F}^\textrm{T}_{\xi ,\xi '}\varvec{A}'_{\xi ,\xi '}(0)(\varvec{I}+\varvec{C}_{\xi ,\xi '}\varvec{A}_{\xi ,\xi '}(0))^{-1}\varvec{C}_{\xi ,\xi '}\varvec{A}_{\xi ,\xi '}(0)\varvec{F}_{\xi ,\xi '}$

$$\begin{aligned}= & {} -\varvec{F}^\textrm{T}_{\xi }\varvec{A}_{\xi }(\varvec{C}_{\xi }\varvec{A}_{\xi }+\varvec{I})^{-1}\varvec{C}_{\xi }\varvec{A}_{\xi }\varvec{F}_{\xi }-\varvec{F}^\textrm{T}_{\xi '}\varvec{A}_{\xi '}\varvec{\Gamma }^\textrm{T}_{\xi ,\xi '}\varvec{A}_{\xi }(\varvec{C}_{\xi }\varvec{A}_{\xi }+\varvec{I})^{-1}\varvec{C}_{\xi }\varvec{A}_{\xi }\varvec{F}_{\xi }\\{} & {} \quad +\varvec{F}^\textrm{T}_{\xi '}\varvec{A}_{\xi '}\varvec{\Gamma }^\textrm{T}_{\xi ,\xi '}\varvec{A}_{\xi '}\varvec{F}_{\xi }. \end{aligned}$$

By the same way we have

$\varvec{F}^\textrm{T}_{\xi ,\xi '}\varvec{A}_{\xi ,\xi '}(0)(\varvec{I}+\varvec{C}_{\xi ,\xi '}\varvec{A}_{\xi ,\xi '}(0))^{-1}\varvec{C}_{\xi ,\xi '}\varvec{A}'_{\xi ,\xi '}(0)(\varvec{I}+\varvec{C}_{\xi ,\xi '}\varvec{A}_{\xi ,\xi '}(0))^{-1}\varvec{C}_{\xi ,\xi '}\varvec{A}_{\xi ,\xi '}(0)\varvec{F}_{\xi ,\xi '}$

$$\begin{aligned}= & {} -\varvec{F}^\textrm{T}_{\xi }\varvec{A}_{\xi }(\varvec{C}_{\xi }\varvec{A}_{\xi }+\varvec{I})^{-1}\varvec{C}_{\xi }\varvec{A}_{\xi }(\varvec{C}_{\xi }\varvec{A}_{\xi }+\varvec{I})^{-1}\varvec{C}_{\xi }\varvec{A}_{\xi }\varvec{F}_{\xi }\\{} & {} -\varvec{F}^\textrm{T}_{\xi }\varvec{A}_{\xi }(\varvec{C}_{\xi }\varvec{A}_{\xi }+\varvec{I})^{-1} \varvec{\Gamma }_{\xi ,\xi '}\varvec{A}_{\xi '}\varvec{\Gamma }^\textrm{T}_{\xi ,\xi '}\varvec{A}_{\xi }(\varvec{C}_{\xi }\varvec{A}_{\xi }+\varvec{I})^{-1}\varvec{C}_{\xi }\varvec{A}_{\xi }\varvec{F}_{\xi }\\{} & {} +\varvec{F}^\textrm{T}_{\xi }\varvec{A}_{\xi }(\varvec{C}_{\xi }\varvec{A}_{\xi }+\varvec{I})^{-1}\varvec{\Gamma }_{\xi ,\xi '}\varvec{A}_{\xi '}\varvec{\Gamma }^\textrm{T}_{\xi ,\xi '}\varvec{A}_{\xi }\varvec{F}_{\xi }, \end{aligned}$$

and also

$\varvec{F}^\textrm{T}_{\xi ,\xi '}\varvec{A}_{\xi ,\xi '}(0)(\varvec{I}+\varvec{C}_{\xi ,\xi '}\varvec{A}_{\xi ,\xi '}(0))^{-1}\varvec{C}_{\xi ,\xi '}\varvec{A}'_{\xi ,\xi '}(0)\varvec{F}_{\xi ,\xi '}$

$$\begin{aligned}= & {} -\varvec{F}^\textrm{T}_{\xi }\varvec{A}_{\xi }(\varvec{C}_{\xi }\varvec{A}_{\xi }+\varvec{I})^{-1}\varvec{C}_{\xi }\varvec{A}_{\xi }\varvec{F}_{\xi }+\varvec{F}^\textrm{T}_{\xi }\varvec{A}_{\xi }(\varvec{C}_{\xi }\varvec{A}_{\xi }+\varvec{I})^{-1} \varvec{\Gamma }_{\xi ,\xi '}\varvec{A}_{\xi '}\varvec{F}_{\xi '}. \end{aligned}$$

Inserting these results into (B3) we obtain

By $(\varvec{A}^{-1}_{\xi }+\varvec{C}_{\xi })^{-1}=\varvec{A}_{\xi }-\varvec{A}_{\xi }(\varvec{C}_{\xi }\varvec{A}_{\xi }+\varvec{I})^{-1}\varvec{C}_{\xi }\varvec{A}_{\xi }$, it follows that

The directional derivative ${F}_{\phi }(\xi ,\xi ')$ is linear in $\xi '$. Therefore, it suffices to consider one-point designs $\xi _x$ which assign all observations to a single setting x. For such one-point designs the directional derivative reduces to

According to the general equivalence theorem a design $\xi ^*$ is optimal, if and only if

$$\begin{aligned} F_{\phi }(\xi ^* ,\xi _x)\ge 0;~\forall x \in {\mathcal {X}} \end{aligned}$$

Hence, the result follows.