Approximate theory-aided robust efficient factorial fractions under baseline parametrization

Abstract

With reference to a baseline parametrization, we explore highly efficient, fractional factorial designs for inference on the main effects and, perhaps, some interactions. Our tools include approximate theory together with certain, carefully devised discretization procedures. The robustness of these designs to possible model misspecification is investigated using a minimaxity approach. Examples are given to demonstrate that our technique works well even when the run size is quite small.

Appendix

Proof of Lemma 1

By (10), the lemma is obvious if either $M(p)$ or $M(\tilde{p})$ is singular. Suppose they are both nonsingular. Then by (5) and (11), on simplification,

$$\begin{aligned} M((1-\varepsilon )p+\varepsilon \tilde{p})=(1-\varepsilon )M(p)+\varepsilon M(\tilde{p})+\varepsilon (1-\varepsilon )Z^\mathrm{T}(\tilde{p}-p)(\tilde{p}-p)^\mathrm{T}Z. \end{aligned}$$

(26)

i.e., $M((1-\varepsilon )p+\varepsilon \tilde{p})-\{(1-\varepsilon )M(p)+\varepsilon M(\tilde{p})\}$ is nnd. Since $M(p)$ and $M(\tilde{p})$ are both nonsingular, the result now follows from (10). $\square$

Proof of Lemma 2

We proceed along the lines of Silvey (1980, pp. 18–20), with more elaborate arguments to cope with the nonlinearity of $M(p)$ in the elements of $p$. Let $\tilde{p}=(\tilde{p}_1 ,\ldots ,\tilde{p}_v )^\mathrm{T}$ and $p$ be any design measures such that $M(p)$ is nonsingular. The lemma will follow arguing as in Silvey (1980) if we can show that

$$\begin{aligned} \phi (\tilde{p})-\phi (p)\ge \mathop {\lim }\limits _{\varepsilon \rightarrow 0+} \{\phi ((1-\varepsilon )p+\varepsilon {}\tilde{p})-\phi (p)\}/\varepsilon , \end{aligned}$$

(27)

and

$$\begin{aligned}&\mathop {\lim }\limits _{\varepsilon \rightarrow 0+} \{\phi ((1-\varepsilon )p+\varepsilon \tilde{p})-\phi (p)\}/\varepsilon \nonumber \\&\quad =\Sigma _{k=1}^v \tilde{p}_k [\mathrm{tr}\{M(p)\}^{-1}-e_k^\mathrm{T}\{M(p)\}^{-1}\{M(p)\}^{-1}e_k ], \end{aligned}$$

(28)

where $e_k =z_k -Z^\mathrm{T}p, \,1\le k\le v$. The truth of (27) is not hard to see from Lemma 1. It remains to prove (28). To that effect, observe that by (26), $M((1-\varepsilon )p+\varepsilon \tilde{p})=(1-\varepsilon )M(p)+\varepsilon Q(\varepsilon )$, where

$$\begin{aligned} Q(\varepsilon )=M(\tilde{p})+(1-\varepsilon )Z^\mathrm{T}(\tilde{p}-p)(\tilde{p}-p)^\mathrm{T}Z \end{aligned}$$

is nnd, for $0<\varepsilon <1$. Write $g=Z^\mathrm{T}(\tilde{p}-p)$ and note that by (5) and (11),

$$\begin{aligned} M(\tilde{p})=\Sigma _{k=1}^v \tilde{p}_k (z_k -Z^\mathrm{T}\tilde{p})(z_k -Z^\mathrm{T}\tilde{p})^\mathrm{T}=\Sigma _{k=1}^v \tilde{p}_k (e_k -g)(e_k -g)^\mathrm{T}. \end{aligned}$$

Therefore, $Q(\varepsilon )=U(\varepsilon )U(\varepsilon )^\mathrm{T}$, where $U(\varepsilon )$ consists of the $v+1$ columns $\tilde{p}_k^{1/2} (e_k -g)$, $1\le k\le v$, and $(1-\varepsilon )^{1/2}g$. Thus for $0<\varepsilon <1$, the inverse of $M((1-\varepsilon )p+\varepsilon {}\tilde{p})$ is given by

$$\begin{aligned}&(1-\varepsilon )^{-1}\left[ \{M(p)\}^{-1}-\eta \{M(p)\}^{-1}U(\varepsilon )\{I_{v+1} \right. \\&\quad \left. +\eta U(\varepsilon )^\mathrm{T}(M(p))^{-1}U(\varepsilon )\}^{-1}U(\varepsilon )^\mathrm{T}\{M(p)\}^{-1}\right] , \end{aligned}$$

where $\eta =\varepsilon /(1-\varepsilon )$. Hence by (10), after some simplification, the left-hand side of (28) equals

$$\begin{aligned} \mathrm{tr}\{M(p)\}^{-1}-\mathrm{tr}\left[ \{M(p)\}^{-1}U_0 U_0^\mathrm{T} \{M(p)\}^{-1}\right] , \end{aligned}$$

where $U_0 =\lim _{\varepsilon \rightarrow 0+} U(\varepsilon )$. Since $\Sigma _{k=1}^v \tilde{p}_k e_k =g$ and, as a consequence, $U_0 U_0^\mathrm{T} =\Sigma _{k=1}^v \tilde{p}_k e_k e_k^\mathrm{T}$, the truth of (28) is now evident. $\square$

Proof of Lemma 3

• (a) Note the followings facts: (i) As in (9), $Z_d^\mathrm{T} L_N P_d =Z^\mathrm{T}\Delta (r_d )P$. (ii) If $X_d = \quad [1_N \;Z_d ]$, then $L_N X_d =[0_N \;\;L_N Z_d ]$, so that as in (9), $Z^\mathrm{T}\Delta (r_d )X=Z_d^\mathrm{T} L_N X_d = \quad [0_q \;\;H_d ]$, using (7); therefore, $H_d^{-1} Z^\mathrm{T}\Delta (r_d )X=[0_q \;\;I_q ]$. (iii) By the definition of $P$, the matrix $PP^\mathrm{T}$ is the orthogonal projector on $C^{\bot }(X)$, i.e., $PP^\mathrm{T}=I_v -X(X^\mathrm{T}X)^{-1}X^\mathrm{T}$. By (i)–(iii),

  $$\begin{aligned} H_d^{-1} Z_d^\mathrm{T} L_N P_d P_d^\mathrm{T} L_N Z_d H_d^{-1}&= H_d^{-1} Z^\mathrm{T}\Delta (r_d)PP^\mathrm{T}\Delta (r_d )ZH_d^{-1} \\&= H_d^{-1} Z^\mathrm{T}\Delta (r_d )\{I_v -X(X^\mathrm{T}X)^{-1}X^\mathrm{T}\}\Delta (r_d)ZH_d^{-1} \\&= V_d -[0_q \;\;I_q ](X^\mathrm{T}X)^{-1}[0_q \;\;I_q ]^\mathrm{T}=V_d -W^*, \end{aligned}$$

  where $W^*$is the square submatrix of $(X^\mathrm{T}X)^{-1}$ as given by its last $q$ rows and columns. Part (a) is now immediate, noting that $W^*=W$, by (5) and the definition of $X$.

• (b) By (5), after some algebra,

  $$\begin{aligned} \Delta (r_d )\Delta (r_d )-\Delta (r_d )=(I_v -N^{-1}r_d 1_v^\mathrm{T} )\{D(r_d )D(r_d )-D(r_d )\}(I_v -N^{-1}1_v r_d^\mathrm{T} ), \end{aligned}$$

  (29)

  which is nnd, because $D(r_d )D(r_d )-D(r_d )=\mathrm{diag}(r_{d1}^2 -r_{d1} ,\ldots ,r_{dv}^2 -r_{rv} )$ and each $r_{di}$ is a nonnegative integer. Therefore, recalling (9), $V_d -H_d^{-1} =H_d^{-1} Z^\mathrm{T}\{\Delta (r_d )\Delta (r_d )-\Delta (r_d )\}ZH_d^{-1}$ is nnd. If $d$ is binary, then (29) vanishes, and so $V_d =H_d^{-1}$. $\square$