On Efficient Design of Pilot Experiment for Generalized Linear Models

Abstract

The experimental design for a generalized linear model (GLM) is important but challenging since the design criterion often depends on model specification including the link function, the linear predictor, and the unknown regression coefficients. Prior to constructing locally or globally optimal designs, a pilot experiment is usually conducted to provide some insights on the model specifications. In pilot experiments, little information on the model specification of GLM is available. Surprisingly, there is very limited research on the design of pilot experiments for GLMs. In this work, we obtain some theoretical understanding of the design efficiency in pilot experiments for GLMs. Guided by the theory, we propose to adopt a low-discrepancy design with respect to some target distribution for pilot experiments. The performance of the proposed design is assessed through several numerical examples.

Appendix

Derivation of the discrepancy in (10).

We first consider the case $d=1$. We integrate the kernel once:

$$\begin{aligned} \int _{-1}^1 K(t,x) \, \mathrm{d}F_{unif }(t)=&\frac{1}{2}\int _{-1}^1 \left[ 1+\frac{1}{2}(|t|+|x|-|t-x|)\right] \, \mathrm{d}t\\ =&\frac{1}{2}\left[ 2+|x|+\frac{1}{2}-\frac{1}{2}\left( \int _{-1}^x(x-t)\mathrm{d}t+\int _x^1 (t-x)\mathrm{d}t\right) \right] \\ =&\frac{1}{2}\left[ \frac{5}{2}+|x|-\frac{1}{2}\left( x^2+1\right) \right] \\ =&\frac{1}{2}\left( 2+|x|-\frac{1}{2}x^2\right) . \end{aligned}$$

Then we integrate once more:

$$\begin{aligned} {\int _{-1}^1 \int _{-1}^1 K(t,x) \, \mathrm{d}F_{unif }(t) \mathrm{d}F_{unif }(x)}&= \int _{-1}^{1} \frac{1}{4}\left( 2+|x|-\frac{1}{2}x^2\right) \, \mathrm{d}x\\&= \frac{7}{6}. \end{aligned}$$

Generalizing this to the d-dimensional case yields

$$\begin{aligned}&\int _{[-1,1]^d\times [-1,1]^d} K({\varvec{x}},{\varvec{t}}) \, \mathrm{d}F_{unif }({\varvec{x}})\mathrm{d}F_{unif }({\varvec{t}}) = \left( \frac{7}{6}\right) ^d, \\&\int _{[-1,1]^d}K({\varvec{x}},{\varvec{x}}_i) \, \mathrm{d}F_{unif }({\varvec{x}}) = \frac{1}{2^d}\prod \limits _{j=1}^d \left( 2+|x_{ij}|-\frac{1}{2}x_{ij}^2\right) . \end{aligned}$$

Thus, the discrepancy of a design $\xi$ for the uniform distribution on $[-1,1]^d$ is

$$\begin{aligned} D^2(\xi ; F_unif )&= \left( \frac{7}{6}\right) ^d - \frac{1}{2^{d-1}n}\sum _{i=1}^m n_i \prod _{j=1}^d \left[ 2+|x_{ij}|-\frac{x_{ij}^2}{2} \right] \nonumber \\&\qquad + \frac{1}{n^2}\sum _{i,k=1}^m n_in_k\prod _{j=1}^d\left[ 1+\frac{1}{2}\left( |x_{ij}|+|x_{kj}|-|x_{ij}-x_{kj}| \right) \right] , \end{aligned}$$

Derivation of the discrepancy in (11).

Following the same procedure as the derivation of $D^2(\xi ; F_unif )$,

$$\begin{aligned}&\int _{-1}^1 K(t,x) \, \mathrm{d}F_{asin }(t) = 1+\frac{1}{\pi }+\frac{1}{2}|x|-\frac{1}{\pi }\left( x\arcsin (x)+\sqrt{1-x^2}\right) , \\&{\int _{-1}^1 \int _{-1}^1 K(t,x) \, \mathrm{d}F_{asin }(t) \mathrm{d}F_{asin }(x)} = 1+\frac{2}{\pi }-\frac{4}{\pi ^2}, \end{aligned}$$

and thus, the discrepancy of a design $\xi$ for the arcsine distribution on $[-1,1]^d$ is

$$\begin{aligned} D^2(\xi ; F_asin )&= \left( 1+\frac{2}{\pi }-\frac{4}{\pi ^2}\right) ^d - \frac{2}{n}\sum _{i=1}^m n_i \prod _{j=1}^d \left[ 1+\frac{1}{\pi }\right. \\&\qquad \left. +\frac{1}{2}|x_{ij}|-\frac{1}{\pi }\left( x_{ij}\arcsin (x_{ij}) +\sqrt{1-x_{ij}^2} \right) \right] \nonumber \\&\quad + \frac{1}{n^2}\sum _{i,k=1}^m n_in_k\prod _{j=1}^d\left[ 1+\frac{1}{2}\left( |x_{ij}|+|x_{kj}|-|x_{ij}-x_{kj}| \right) \right] . \end{aligned}$$