Analytic solutions for locally optimal designs for gamma models having linear predictors without intercept

Abstract

The gamma model is a generalized linear model for gamma-distributed outcomes. The model is widely applied in psychology, ecology or medicine. Recently, Gaffke et al. (J Stat Plan Inference 203:199–214, 2019) established a complete class and an essentially complete class of designs for gamma models to obtain locally optimal designs in particular when the linear predictor includes an intercept term. In this paper we extend this approach to gamma models having linear predictors without intercept. For a specific scenario sets of locally D- and A-optimal designs are established. It turns out that the optimality problem can be transformed to one under gamma models with intercept leading to a reduction in the dimension of the experimental region. On that basis optimality results can be transferred from one model to the other and vice versa. Additionally by means of the general equivalence theorem optimality can be characterized for multiple regression by a system of polynomial inequalities which can be solved analytically or by computer algebra. Thus necessary and sufficient conditions can be obtained on the parameter values for the local D-optimality of specific designs. The robustness of the derived designs with respect to misspecification of the initial parameter values is examined by means of their local D-efficiencies.

Appendix

Lemma

Consider a two-factor gamma model with intercept such that $\varvec{f}(\varvec{x})=(1, x_1, x_2)^\mathsf{T}$ on the experimental region $\mathcal {X}=[a,b]^2$, $a,b \in \mathbb {R}$, $a<b$ with the intensity function $u(\varvec{x},\varvec{\beta })=1/(\beta _0+\beta _1x_1+\beta _2x_2)^2$. The vertices of $\mathcal {X}$ are given by $(a,a)^\mathsf{T}$, $(b,a)^\mathsf{T}$, $(a,b)^\mathsf{T}$ and $(b,b)^\mathsf{T}$ with the corresponding intensities $u_1=1/(\beta _0+\beta _1\,a+\beta _2\,a)^2$, $u_2=1/(\beta _0+\beta _1\,b+\beta _2\,a)^2$, $u_3=1/(\beta _0+\beta _1\,a+\beta _2\,b)^2$ and $u_4=1/(\beta _0+\beta _1\,b+\beta _2\,b)^2$. Let $\varvec{\beta }=(\beta _0,\beta _1,\beta _2)^\mathsf{T}$ be a parameter point such that $\varvec{f}^\mathsf{T}(\varvec{x})\varvec{\beta }>0$ for all $\varvec{x}\in \mathcal {X}$, or equivalently, $\beta _0+\beta _1\,a+\beta _2\,a>0$, $\beta _0+\beta _1\,b+\beta _2\,a>0$, $\beta _0+\beta _1\,a+\beta _2\,b>0$ and $\beta _0+\beta _1\,b+\beta _2\,b>0$. Then the unique locally D-optimal design $\xi ^*$ (at $\varvec{\beta }$) is as follows.

• (i) If $u_{(1)}^{-1}\,\ge u_{(2)}^{-1}+u_{(3)}^{-1}+u_{(4)}^{-1}$ then $\xi ^*$ is a three-point design supported by the three vertices whose intensity values are given by $u_{(2)}$, $u_{(3)}$, $u_{(4)}$, with equal weights 1 / 3.

• (ii) If $u_{(1)}^{-1}\,< u_{(2)}^{-1}+u_{(3)}^{-1}+u_{(4)}^{-1}$ then $\xi ^*$ is a four-point design supported by the four vertices $(a,a)^\mathsf{T}$, $(b,a)^\mathsf{T}$, $(a,b)^\mathsf{T}$, $(b,b)^\mathsf{T}$ with corresponding weights $\omega _1^*,\,\omega _2^*,\,\omega _3^*,\,\omega _4^*\,>0$ and $\sum _{k=1}^4\omega _k^*=1$.

Proof

The proof can be demonstrated by making use of the results of Gaffke et al. (2019) who derived the D-optimal designs for a gamma model with intercept on the standardized experimental region $\mathcal {Z}=[0,1]^2$. To this end, denote $\varvec{f}_{\varvec{\beta }}(\varvec{x})=(\varvec{f}^\mathsf{T}(\varvec{x})\varvec{\beta })^{-1}\varvec{f}(\varvec{x})$ so we have $\varvec{M}(\varvec{x},\varvec{\beta })=\varvec{f}_{\varvec{\beta }}(\varvec{x})\varvec{f}_{\varvec{\beta }}^\mathsf{T}(\varvec{x})$. Based on equivariance (see Radloff and Schwabe 2016) a D-optimal design $\xi ^*$ on $\mathcal {X}$ given in the lemma can be derived by transformation of a respective D-optimal design $\xi ^{**}$ on $\mathcal {Z}=[0,1]^2$. Here, $\textstyle {x_j\rightarrow z_j=\frac{x_j}{b-a}-\frac{a}{b-a},j=1,2}$. For the transformation matrix

we have $\textstyle \varvec{f}(\varvec{z})=\varvec{B}\varvec{f}(\varvec{x})=(1,z_1,z_2)^\mathsf{T}$ with $\textstyle \varvec{\tilde{\beta }}=\big (\varvec{B}^\mathsf{T}\big )^{-1}\varvec{ \beta }=(\tilde{\beta }_0,\,\tilde{\beta }_{1},\,\tilde{\beta }_{2})^\mathsf{T}$  where $\textstyle \tilde{\beta }_0=\beta _0+(-a/(b-a))\,(\beta _1+\beta _2)$ ,  $\textstyle \tilde{\beta }_{1}=\beta _1(1/(b-a))$  and   $\textstyle \tilde{\beta }_{2}=\beta _2(1/(b-a))$. It follows that $\textstyle \varvec{f}_{\varvec{\tilde{\beta }}}(\varvec{z})=\bigl (\varvec{f}^\mathsf{T}(\varvec{z})\varvec{\tilde{\beta }}\bigr )^{-1}\varvec{f}(\varvec{z})$ and the information matrix is given by $\textstyle \varvec{\tilde{M}}(\varvec{z},\varvec{\tilde{\beta }})=\varvec{f}_{\varvec{\tilde{\beta }}}(\varvec{z})\varvec{f}_{\varvec{\tilde{\beta }}}^\mathsf{T}(\varvec{z})$. It is easily seen that $\varvec{M}(\varvec{x},\varvec{\beta }) =\varvec{B}^{-1}\varvec{\tilde{M}}(\varvec{z},\varvec{\tilde{\beta }})\varvec{B}^{-1}$, thus the derived D-optimal designs on $\mathcal {X}$ and $\mathcal {Z}$, respectively are equivariant. Then the results follow from Theorem 4.2 in Gaffke et al. (2019). $\square$

Proof of Theorem 3.3

First we give an outline of the proof. The proof is obtained by making use of the condition of the equivalence theorem (Theorem 2.1, part (a)). By that we develop a system of feasible inequalities evaluated at the vertices $\varvec{v}_i$ for all $i=1,\ldots ,8$. For simplification of presentation in the case $\beta _1\ne 0$ we use the ratio $\gamma =\beta /\beta _1$ for which the range is given by $(-\infty ,-1)\cup (-1/4,\infty )$. It turns out that some of the inequalities are implied by some others and thus the resulting system is reduced to an equivalent system of only a few inequalities. The intersection of the set of solutions of the system with the range of $\gamma$ leads to the optimality condition (subregion) of the corresponding optimal design. For minimally supported designs given in cases (i), (ii), (iv) we display the $3\times 3$ design matrix $\varvec{F}$, its inverse $\varvec{F}^{-1}$ and the $3\times 3$ weight matrix $\varvec{V}$. Note that for $\beta _1\ne 0$ the intensities $u_i=u(\varvec{v}_i,\varvec{\beta }), i=1,\ldots , 8$ are equal to $u_{1}=\beta _1^{-2}\big (1\,+2\,\gamma \big )^{-2}$,   $u_{2}=\beta _1^{-2}\big (2\,+2\,\gamma \big )^{-2}$,   $u_{3}=u_{4}=\beta _1^{-2}\big (1+3\,\gamma \big )^{-2}$,   $u_{5}=\beta _1^{-2}\big (1+4\,\gamma \big )^{-2}$,   $u_{6}=u_{7}=\beta _1^{-2}\big (2+3\,\gamma \big )^{-2}$,   $u_{8}=\beta _1^{-2}\big (2+4\,\gamma \big )^{-2}$, respectively.

Proof of part (i): The $3\times 3$ design matrix $\varvec{F}=[\varvec{v}_2, \varvec{v}_3,\varvec{v}_4]^\mathsf{T}$ is given by

$$\begin{aligned} \varvec{F}= & {} \left( \begin{array}{ccc} 2 &{}\quad 1 &{}\quad 1 \\ 1 &{}\quad 2 &{}\quad 1 \\ 1 &{}\quad 1 &{}\quad 2 \\ \end{array} \right) \,\, \text{ with } \,\, \varvec{F}^{-1}=\frac{1}{4} \left( \begin{array}{rrr} 3 &{}\quad -1 &{}\quad -1 \\ -1 &{}\quad 3 &{}\quad -1 \\ -1 &{}\quad -1 &{}\quad 3 \\ \end{array} \right) \,\, \text{ and } \text{ weight } \text{ matrix } \\&\varvec{V}=\mathrm {diag}\big (u_{2},u_{3},u_{4}\big ). \end{aligned}$$

Hence, the condition of the equivalence theorem is given by

$$\begin{aligned}&\varvec{f}^\mathsf{T}(\varvec{x})\varvec{F}^{-1} \varvec{V}^{-1}\big (\varvec{F}^\mathsf{T}\big )^{-1}\varvec{f}(\varvec{x})\le \big (\beta _1x_1+\beta _2x_2+\beta _3x_3\big )^2\,\,\,\,\,\,\forall \varvec{x} \in \{1,2\}^3. \end{aligned}$$

(6.3)

For the case $\beta >0$, $\beta _1= 0$, condition (6.3) is equivalent to

$$\begin{aligned}&4\big (3x_1-(x_2+x_3)\big )^2+9\bigl (\big ( 3x_2-(x_1+x_3)\big )^2+\big (3x_3-(x_1+x_2)\big )^2\bigr )\\&\quad \le 16\big (x_2+x_3\big )^2\,\,\forall \varvec{x}\in \{1,2\}^3, \end{aligned}$$

which is independent of $\beta$ and is satisfied for all vertices $\varvec{v}_i$ for all $i=1,\ldots ,8$ and equality holds for the support. For the other cases, i.e., $\beta \ge -3\beta _1$, $\beta _1< 0$  or  $\beta > \beta _1/5$, $\beta _1>0$ condition (6.3) is equivalent to

$$\begin{aligned}&\big (3x_1-(x_2+x_3)\big )^2(2\,+2\,\gamma )^{2}+\bigl (\big ( 3x_2-(x_1+x_3)\big )^2+\big (3x_3-(x_1+x_2)\big )^2\bigr )(1+3\,\gamma \big )^{2}\nonumber \\&\le 16\big (x_1+\gamma (x_2+x_3)\big )^2\,\,\forall \varvec{x}\in \{1,2\}^3. \end{aligned}$$

(6.4)

By some lengthy but straightforward calculations, the above inequalities can be reduced to

$$\begin{aligned} 15 \gamma ^2+2\gamma -1 \ge 0 \text{ and } 3 \gamma ^2+10\gamma +3 \ge 0 \end{aligned}$$

where the first inequality comes from vertex $\varvec{v}_5$ and the second inequality comes from the vertices $\varvec{v}_6$ and $\varvec{v}_7$. The l.h.s. of each inequality is a polynomial in $\gamma$ of degree 2 and the sets of solutions are given by $(-\infty ,-1/3]\cup [1/5,\infty )$ and $(-\infty ,-3]\cup [-1/3,\infty )$, respectively. Note that the bounds are the roots of the respective polynomials. Hence, by considering the intersection of both sets with the range of $\gamma$, the design $\xi _1^*$ is locally D-optimal if $\gamma \in (-\infty ,-3]\cup [1/5,\infty )$ which is equivalent to the optimality subregion $\beta \ge -3\beta _1$, $\beta _1< 0$   or    $\beta > \beta _1/5$, $\beta _1>0$ given in part (i) of the theorem.

Proof of part (ii): The $3\times 3$ design matrix $\varvec{F}=[\varvec{v}_3, \varvec{v}_4,\varvec{v}_5]^\mathsf{T}$ is given by

$$\begin{aligned} \varvec{F}= \left( \begin{array}{ccc} 1&{}\quad 2&{}\quad 1\\ 1&{}\quad 1&{}\quad 2\\ 1&{}\quad 2&{}\quad 2 \end{array} \right) \,\, \text{ with } \,\, \varvec{F}^{-1}= \left( \begin{array}{rrr} 2&{}\quad 2&{}\quad -3\\ 0&{}\quad -1&{}\quad 1\\ -1&{}\quad 0&{}\quad 1 \end{array} \right) \,\, \text{ and } \text{ weight } \text{ matrix } \,\,\varvec{V}=\mathrm {diag}\big (u_{3},u_{4},u_{5}\big ). \end{aligned}$$

Hence, the condition of the equivalence theorem is equivalent to

$$\begin{aligned}&\big ( \big (2\,x_1-x_2\big )^2+\big (2\,x_1-x_3\big )^2 \big )\,\big (1+3\,\gamma \big )^{2}+ \big ( x_3+x_2-3\,x_1\big )^2\,\big (1+4\,\gamma \big )^{2}\\&\quad \le \big (x_1+\gamma (x_2+x_3)\big )^2\,\,\forall \varvec{x}\in \{1,2\}^3, \end{aligned}$$

and similar to part (i) the above inequalities reduce to

$$\begin{aligned}&69 \gamma ^2+38\gamma +5 \le 0 \end{aligned}$$

which arises from vertex $\varvec{v}_2$. The set of solutions of the polynomial determined by the l.h.s. is given by $[-1/3,-5/23]$. By considering the intersection with the range of $\gamma$, the design $\xi _2^*$ is locally D-optimal if $\gamma \in (-1/4,-5/23]$.

Proof of part (iii): Consider design $\xi _3^*$. Let the associated weights be denoted as $\omega _2^*=(5+23\,\gamma )/(16\,(1+4\,\gamma ))$,   $\omega _3^*=\omega _4^*=9\,(1+3\gamma )^2/(32\,(1+\gamma )(1+4\,\gamma ))$,   $\omega _5^*=(1-\gamma -20\,\gamma ^2)/(8\,(1+\gamma )(1+4\,\gamma ))$ where   $\gamma =\beta /\beta _1$. Note that $\omega _2^*>0$ for all $\gamma >-5/23$, $\omega _3^*=\omega _4^*>0$ for all $\gamma \in \mathbb {R}$ and $\omega _5^*>0$ for all $\gamma \in (-1/4,1/5)$, and it is obvious that $\omega _2^*,\omega _3^*,\omega _5^*$ are positive over the interval $(-5/23,1/5)$ and satisfy $\omega _2^*+\omega _3^*+\omega _4^*+\omega _5^*=1$. The $4\times 3$ design matrix is given by $\varvec{F}=[\varvec{v}_2, \varvec{v}_3,\varvec{v}_4,\varvec{v}_5]^\mathsf{T}$ with weight matrix $\varvec{V}=\mathrm {diag}\big (s_{2},s_{3},s_{4},s_{5}\big )$ where $s_{i}=\omega _{i}^*u_{i}, i=2,3,4,5$ and $s_{3}=s_{4}$. The information matrix is given by

$$\begin{aligned} \varvec{M}\bigl (\xi _3^*, \varvec{\beta }\bigr )=\left( \begin{array}{lll} 4\,s_{2}+2\,s_{3}+s_{5}&{}2\,s_{2}+3\,s_{3}+2\,s_{5}&{}2\,s_{2}+3\,s_{3}+2\,s_{5}\\ 2\,s_{2}+3\,s_{2}+2\,s_{5}&{}s_{2}+5\,s_{3}+4\,s_{5}&{}s_{2}+4\,s_{3}+4\,s_{5}\\ 2\,s_{2}+3\,s_{3}+2\,s_{5}&{}s_{2}+4\,s_{3}+4\,s_{4}&{}s_{2}+5\,s_{3}+4\,s_{5} \end{array} \right) \end{aligned}$$

and hence $\det \varvec{M}\bigl (\xi _3^*, \varvec{\beta }\bigr )=16\,s_{2}\,s_{3}^{2}+18\,s_{2}\,s_{3}\,s_{5}+s_{3}^{2}s_{5}$. Define the following quantities

$$\begin{aligned}&c_{1}=\frac{s_{3}(2\,s_{2}+9\,s_{3}+8\,s_{5})}{16\,s_{2}\,s_{3}^{2}+18\,s_{2}\,s_{3}\,s_{5}+s_{3}^{2}s_{5}},\,\, c_{2}=\frac{-s_{3}(2\,s_{2}+3\,s_{3}+2\,s_{5})}{16\,s_{2}\,s_{3}^{2}+18\,s_{2}\,s_{3}\,s_{5}+s_{3}^{2}s_{5}},\\&c_{3}=\frac{10\,s_{2}\,s_{3}+9\,s_{2}\,s_{5}+s_{3}^{2}+s_{3}\,s_{5}}{16\,s_{2}\,s_{3}^{2}+18\,s_{2}\,s_{3}\,s_{5}+s_{3}^{2}s_{5}},\,\,c_{4}=\frac{-6\,s_{2}\,s_{3}+9\,s_{2}\,s_{5}-s_{3}^{2}}{16\,s_{2}\,s_{3}^{2}+18\,s_{2}\,s_{3}\,s_{5}+s_{3}^{2}s_{5}}. \end{aligned}$$

Then the inverse of the information matrix is given by

$$\begin{aligned} \varvec{M}^{-1}\bigl (\xi _3^*, \varvec{\beta }\bigr )=\left( \begin{array}{l@{\quad }l@{\quad }l} c_1&{}c_2&{}c_2\\ c_2&{}c_3&{}c_4\\ c_2&{}c_4&{}c_3 \end{array} \right) . \end{aligned}$$

Hence, the condition of the equivalence theorem can be rewritten as

$$\begin{aligned}&c_1\,x_1^2+c_3\,(x_2^2+x_3^2)+2\,c_2\,(x_1\,x_2+x_1\,x_3)+2\,c_4\,x_2\,x_3 \le 3\,\big (x_1+\gamma \,(x_2+x_3)\big )^2\,\, \\&\quad \forall \,\,\varvec{x}\in \{1,2\}^3 \end{aligned}$$

which is equivalent to the following system of inequalities

$$\begin{aligned} c_1+4 c_2+2c_3+2 c_4\le 3 \left( 1+2 \gamma \right) ^2 \text{ and } 4 c_1+12 c_2+5 c_3+4 c_4\le 3 \left( 2+3 \gamma \right) ^2 \end{aligned}$$

where the first inequality arises form the vertices $\varvec{v}_1$ and $\varvec{v}_8$ and the second inequality comes from the vertices $\varvec{v}_6$ and $\varvec{v}_7$. Because of the complexity of the system above we employed computer algebra using Wolfram Mathematica 11.3 (see Wolfram Research, Inc. 2018) to obtain the set of solutions for $\gamma$ as given in part (iii).

Proof of part (iv): The $3\times 3$ design matrix $\varvec{F}=[\varvec{v}_2, \varvec{v}_6,\varvec{v}_7]^\mathsf{T}$ is given by

$$\begin{aligned} \varvec{F}= & {} \left( \begin{array}{l@{\quad }l@{\quad }l} 2&{}1&{}1\\ 2&{}1&{}2\\ 2&{}2&{}1 \end{array} \right) \,\, \text{ with } \,\, \varvec{F}^{-1}= \frac{1}{2}\left( \begin{array}{r@{\quad }r@{\quad }r} 3&{}-1&{}-1\\ -2&{}0&{}2\\ -2&{}2&{}0 \end{array} \right) \,\, \\&\quad \text{ and } \text{ weight } \text{ matrix } \,\,\varvec{V}=\mathrm {diag}\big (u_{2},u_{6},u_{7}\big ). \end{aligned}$$

The condition of the equivalence theorem is equivalent to

$$\begin{aligned}&\left( \left( x_2-\frac{x_1}{2}\right) ^2+ \left( x_3-\frac{x_1}{2}\right) ^2\right) \left( 2+ 3 \gamma \right) ^2 + \left( \frac{3 x_1}{2}-x_2-x_3\right) ^2 \\&\quad \left( 2+2 \gamma \right) ^2\le \big (x_1+\gamma (x_2+x_3)\big )^2\,\,\forall \varvec{x}\in \{1,2\}^3, \end{aligned}$$

and the above inequalities reduce to

$$\begin{aligned} 90\, \gamma ^2+168 \gamma +72\le 0 \text{ and } 6 \gamma ^2+16 \gamma +8\le 0 \end{aligned}$$

where the first inequality arises form the vertices $\varvec{v}_3$ and $\varvec{v}_4$ and the second inequality comes from vertex $\varvec{v}_8$. In analogy to parts (i) and (ii) the sets of solutions are given by $[-1.2,-2/3]$ and $[-2,-2/3]$, respectively where the bounds are the roots of the respective polynomials. Hence, by considering the intersection of both sets with the range of $\gamma$, the design $\xi _4^*$ is locally D-optimal if $\gamma \in [-1.2,-1)$. $\square$