Augmented Block Designs for Unreplicated Trials

Abstract

This paper is concerned with augmented block designs for unreplicated trials for which the underlying model comprises fixed block and fixed treatment effects. Explicit expressions for the average scaled variances and the maximum variances of estimates of the pairwise differences between controls, between unreplicated test lines and between controls and unreplicated test lines are developed and demonstrate the crucial role of the control design in constructing the attendant A- and MV-optimal designs. The results extend quite naturally to p-rep block designs and a novel algorithm for generating such designs is introduced. Examples which illustrate the implications of the findings are also presented. Supplementary materials accompanying this paper appear online.

Appendix

1.1 A.1 Derivation of the Generalized Inverse Matrix $C^{-}$

Consider the treatment information matrix C written in partitioned form as

$$\begin{aligned} C= \left[ \begin{array}{cc} C_{11} &{} C_{12} \\ C_{21} &{} C_{22} \\ \end{array} \right] = \left[ \begin{array}{cc} R - \displaystyle \frac{1}{k} N_c^T N_c &{} - \displaystyle \frac{1}{k} N_c^T \otimes 1_{k_s}^T \\ - \displaystyle \frac{1}{k} N_c \otimes 1_{k_s} &{} ~~~ I_b \otimes (I_{k_s} - \displaystyle \frac{1}{k} J_{k_s}) \\ \end{array} \right] . \end{aligned}$$

Following Pringle and Rayner (1971, p.46), the formula for a g-inverse of C is given by

$$\begin{aligned} C^{-} = \left[ \begin{array}{cc} (C_{11}^*)^{-} &{} -(C_{11}^*)^{-} \, C_{12} \, C_{22}^{-} \\ -C_{22}^{-} \, C_{21} \, (C_{11}^*)^{-} &{} ~~~C_{22}^{-} + C_{22}^{-} \, C_{21} \, (C_{11}^*)^{-} \, C_{12} \, C_{22}^{-} \end{array} \right] , \end{aligned}$$

where $C_{11}^*=C_{11} - C_{12} C_{22}^{-} C_{21}$. Note that this inverse is of the same form as that of a nonsingular $2 \times 2$ partitioned matrix. The formula for the latter is well-known and is stated, for example, in the paper by Lu and Shiou (2002, Theorem 2.1).

It now follows that, since estimates of pairwise differences of the standard varieties are not influenced by test lines, the matrix $(C_{11}^*)^-$ can be taken to be G, any generalized inverse of the information matrix for treatments in the control design. Note that this result can also be obtained formally. Specifically

$$\begin{aligned} C_{11}^*= & {} R - \frac{1}{k} N_c^{T} N_c - \frac{1}{k} (N_c^{T} \otimes 1_{k_s}^{T}) [I_b \otimes (I_{k_s} + \frac{1}{k_c} J_{k_s}) \frac{1}{k} (N_c \otimes 1_{k_s})\\= & {} R - \frac{1}{k} N_c^{T} N_c - \frac{1}{k^2} (k_s+\frac{1}{k_c} k_s^2) N_c^{T} N_c\\= & {} R - \frac{1}{k_c} N_c^{T} N_c. \end{aligned}$$

Furthermore the inverse of the matrix $C_{22}$ is readily derived as

$$\begin{aligned} C_{22}^{-1} = I_b \otimes (I_{k_s} - \displaystyle \frac{1}{k} \, J_{k_s})^{-1} = I_b \otimes (I_{k_s} + \displaystyle \frac{1}{(k-k_s)} \, J_{k_s}) = I_b \otimes (I_{k_s} + \displaystyle \frac{1}{k_c} \, J_{k_s}) \end{aligned}$$

and thus $C_{22}^{-}$ is simply $C_{22}^{-1}$. In addition, it follows from straightforward algebra involving Kronecker products that

$$\begin{aligned} -(C_{11}^*)^{-} \, C_{12} \, C_{22}^{-}= & {} \left( G \otimes 1 \right) \, ( \displaystyle \frac{1}{k} \, N_c^T \otimes 1_{k_s}^T ) \, [ I_b \otimes (I_{k_s} + \displaystyle \frac{1}{k_c} \, J_{k_s}) ] \\= & {} \displaystyle \frac{1}{k} \, (G \, N_c^T) \otimes [1_{k_s}^T \, (I_{k_s} + \displaystyle \frac{1}{k_c} \, J_{k_s}) ]\\= & {} \displaystyle \frac{1}{k_c} \, (G \, N_c^T) \otimes 1_{k_s}^T \end{aligned}$$

and that

$$\begin{aligned} C_{22}^{-} + C_{22}^{-} C_{21} (C_{11}^*)^- C_{12} C_{22}^{-}= & {} I_b \otimes (I_{k_s} + \displaystyle \frac{1}{k_c} J_{k_s}) + \displaystyle \frac{1}{k_c} (N_c \otimes 1_{k_s}) [\displaystyle \frac{1}{k_c} (G N_c^T) \otimes 1_{k_s}^T]\\= & {} I_{t_s} + I_b \otimes \displaystyle \frac{1}{k_c} J_{k_s} + \displaystyle \frac{1}{k_c} [( N_c G N_c^T) \otimes \displaystyle \frac{1}{k_c} J_{k_s}]\\= & {} I_{t_s} + [I_b + \displaystyle \frac{1}{k_c} ( N_c G N_c^T)] \otimes \displaystyle \frac{1}{k_c} J_{k_s}. \end{aligned}$$

Thus the partitioned g-inverse of the information matrix C can be assembled as

$$\begin{aligned} C^{-} = \left[ \begin{array}{cc} G &{} \displaystyle \frac{1}{k_c} (G N_c^T) \otimes 1_{k_s}^T \\ \displaystyle \frac{1}{k_c} (G N_c) \otimes 1_{k_s} &{} ~~~I_{t_s} + [I_b + \displaystyle \frac{1}{k_c} ( N_c G N_c^T)] \otimes \displaystyle \frac{1}{k_c} J_{k_s} \\ \end{array} \right] . \end{aligned}$$

1.2 A.2 Derivation of the A-optimality Criteria $A_{tt}$, $A_{cc}$ and $A_{ct}$

1.2.1 A.2.1 The Criterion $A_{cc}$

Consider the $\left( {\begin{array}{c}t_c\\ 2\end{array}}\right) \times t_c$ matrix $T_c$ with rows comprising an element 1, an element -1 and all other elements 0 and constructed in such a way that the matrix represents all pairwise treatment differences between the controls. Then $T_c^T T_c =t_c I - J$ (Dey 2010, p.39) and

$$\begin{aligned} A_{cc} = \frac{1}{\left( {\begin{array}{c}t_c\\ 2\end{array}}\right) } tr(T_c G T_c^T) = \frac{1}{\left( {\begin{array}{c}t_c\\ 2\end{array}}\right) } tr(G \; T_c^T T_c) = \frac{2}{t_c-1} tr[G ( I - \frac{1}{t_c} J)]. \end{aligned}$$

1.2.2 A.2.2 The Criterion $A_{tt}$

Consider the $\left( {\begin{array}{c}t_s\\ 2\end{array}}\right) \times t_s$ matrix H which represents all pairwise treatment differences between the test lines in the same way as $T_c$ for pairwise differences between controls. Then $H^T H = t_s I_{t_s}-J_{t_s}$ and $A_{tt}= \displaystyle \frac{1}{\left( {\begin{array}{c}t_s\\ 2\end{array}}\right) } tr(H C_{22}^{-} H^T ) = \displaystyle \frac{1}{\left( {\begin{array}{c}t_s\\ 2\end{array}}\right) } tr(C_{22}^{-} H^T H)$. Thus

$$\begin{aligned} A_{tt}= \displaystyle \frac{1}{\left( {\begin{array}{c}t_s\\ 2\end{array}}\right) } tr \left\{ \left[ I_{t_s} + (I_b + \displaystyle \frac{1}{k_c} N_c \, G \, N_c^T) \otimes \displaystyle \frac{1}{k_c} J_{k_s} \right] \; (t_s I_{t_s}-J_{t_s}) \right\} . \end{aligned}$$

Consider now terms in $A_{tt}$ identified as

$$\begin{aligned} t_1=tr[I_{t_s} (t_s I_{t_s}-J_{t_s})] = t_s (t_s-1) \end{aligned}$$

and, by taking $t_s I_{t_s}-J_{t_s} = t_s I_b \otimes I_{k_s} - J_b \otimes J_{k_s}$, as

$$\begin{aligned} t_2 = tr[I_b \otimes \frac{1}{k_c} J_{k_s} ) (t_s I_b \otimes I_{k_s} - J_b \otimes J_{k_s}) ] = \frac{t_s^2 (b-1)}{b k_c} \end{aligned}$$

and as

$$\begin{aligned}&t_3 = tr [ (\frac{1}{k_c} (N_c G N_c^T) \otimes \frac{1}{k_c} J_{k_s} ) (t_s I_b \otimes I_{k_s} - J_b \otimes J_{k_s} )]\\&\quad = \frac{t_s^2}{b k_c^2} [ tr(N_c G N_c^T) - \frac{1}{b} 1_b^T (N_c G N_c^T) 1_b]. \end{aligned}$$

Then

$$\begin{aligned}&A_{tt} = \frac{2}{t_s (t_s -1)} (t_1+t_2+t_3)\\&\quad = 2 \left\{ 1 + \frac{t_s (b-1)}{ (t_s-1) b k_c} + \frac{t_s}{ (t_s-1) b k_c^2 } \left[ tr( N_c \, G \, N_c^T) - \frac{1}{b} 1_b^T ( N_c \, G \, N_c^T) 1_b \right] \right\} . \end{aligned}$$

1.2.3 A.2.3 The Criterion $A_{ct}$

The matrix defining pairwise differences between controls and unreplicated test lines is given by $A= \left[ I_{t_c} \otimes 1_{t_s} ~~~ - 1_{t_c} \otimes I_{t_s} \right] .$ Thus

$$\begin{aligned} A^T A = \left[ \begin{array}{cc} t_s I_{t_c} &{} -J_{t_c \times t_s} \\ -J_{t_s \times t_c} &{} ~~~ t_c I_{t_s} \end{array} \right] \end{aligned}$$

and an expression for $tr(A \, C^- \, A^T )=tr(C^- A^T \, A)$ is required. Consider therefore the matrix product

$$\begin{aligned} M = \left[ \begin{array}{cc} G &{} \displaystyle \frac{1}{k_c} (G \, N_c^T) \otimes 1_{k_s}^T \\ \displaystyle \frac{1}{k_c} (N_c \, G) \otimes 1_{k_s} &{} ~~~~ I_{t_s} + [I_b + \displaystyle \frac{1}{k_c} ( N_c \, G \, N_c^T)] \otimes \displaystyle \frac{1}{k_c} J_{k_s} \\ \end{array} \right] \left[ \begin{array}{cc} t_s I_{t_c} &{} -J_{t_c \times t_s} \\ -J_{t_s \times t_c} &{} ~~~ t_c I_{t_s} \end{array} \right] . \end{aligned}$$

Now

$$\begin{aligned} M_{11}= & {} t_s G - [\displaystyle \frac{1}{k_c} (G \, N_c^T) \otimes 1_{k_s}^T] J_{t_s \times t_c} \\= & {} t_s G - [\displaystyle \frac{1}{k_c} (G \, N_c^T) \otimes 1_{k_s}^T] (J_{b \times t_c} \otimes 1_{k_s}) \\= & {} t_s G - \displaystyle \frac{k_s}{k_c} \, G \, N_c^T J_{b \times t_c} \end{aligned}$$

and thus $tr (M_{11})=t_s \, tr(G) - \displaystyle \frac{k_s}{k_c} \, 1_{t_c}^T \, G \, N_c^T \, 1_{b}$. Further

$$\begin{aligned} M_{22}= & {} - [\displaystyle \frac{1}{k_c} ( N_c \, G) \otimes 1_{k_s}] J_{t_c \times t_s} + t_c \left[ I_{t_s} + [I_b + \displaystyle \frac{1}{k_c} ( N_c \, G \, N_c^T)] \otimes \displaystyle \frac{1}{k_c} J_{k_s} \right] \\= & {} - [ \displaystyle \frac{1}{k_c} ( N_c \, G) \otimes 1_{k_s}] (J_{t_c \times b} \otimes 1_{k_s}^T) + t_c \left[ I_{t_s} + [I_b + \displaystyle \frac{1}{k_c} ( N_c \, G \, N_c^T)] \otimes \displaystyle \frac{1}{k_c} J_{k_s} \right] \\= & {} t_c \left[ I_{t_s} + [I_b + \displaystyle \frac{1}{k_c} ( N_c \, G \, N_c^T)] \otimes \displaystyle \frac{1}{k_c} J_{k_s} \right] - \displaystyle \frac{1}{k_c} ( N_c \, G J_{t_c \times b}) \otimes J_{k_s} \end{aligned}$$

and thus $tr (M_{22})=t_c \, t_s + \displaystyle \frac{t_c \, t_s}{k_c} + \displaystyle \frac{t_c k_s}{k_c^2} tr(N_c \, G \, N_c^T) - \displaystyle \frac{k_s}{k_c} 1_{b}^T \, N_c G \,1_{t_c}$. Overall therefore

$$\begin{aligned} A_{ct}= & {} \frac{1}{t_c \, t_s} \left[ t_c \, t_s + \displaystyle \frac{t_c \, t_s}{k_c} + t_s \, tr(G) + \displaystyle \frac{t_c \, k_s}{k_c^2} tr(N_c G N_c^T) - \displaystyle 2 \frac{k_s}{k_c} tr(1_{t_c}^T \, G \, N_c^T \, 1_b) \right] \\= & {} 1 + \displaystyle \frac{1}{k_c} + \displaystyle \frac{1}{t_c} \, tr(G) + \displaystyle \frac{1}{b k_c^2} tr(N_c G N_c^T) - \displaystyle \frac{2}{t_c^2 } 1_{t_c}^T \, G \, N_c^T \, 1_b. \end{aligned}$$

1.3 A.3 Variances of the Estimates of Pairwise Treatment Differences

It follows straightforwardly from the g-inverse of the information matrix for the controls, that is the matrix G, that the variance of estimates of pairwise comparisons between the controls is given by

$$\begin{aligned} g_{ii}+g_{jj} - 2 g_{ij}, \end{aligned}$$

where the terms $g_{ij}, i,j=1, \ldots , t_c,$ denote elements of the g-inverse G. In addition, the variance of the estimate of the pairwise difference between two test lines in the same block is precisely 2.

In order to obtain variances of the estimates for pairwise differences between the unreplicated test lines in different blocks and between the controls and the unreplicated test lines, consider a design comprising the controls and a single test line in each block. Then the g-inverse of the treatment information matrix C is given by

$$\begin{aligned} C_{1}^{-} = \left[ \begin{array}{cc} G &{} \displaystyle \frac{1}{k_c} G N_c^T \\ \displaystyle \frac{1}{k_c} N_c G &{} ~~~ (1+ \displaystyle \frac{1}{k_c}) I_b + \displaystyle \frac{1}{k_c^2} ( N_c G N_c^T) \\ \end{array} \right] \end{aligned}$$

and hence by

$$\begin{aligned} C_{1}^{-} = \left[ \begin{array}{cc} 0 &{} 0 \\ 0 &{} ~~~ (1+ \displaystyle \frac{1}{k_c}) I_b \\ \end{array} \right] + \left[ \begin{array}{c} I \\ \displaystyle \frac{1}{k_c} N_c \end{array} \right] G \left[ \begin{array}{cc} I&\displaystyle \frac{1}{k_c} N_c^T \end{array} \right] . \end{aligned}$$

It thus follows that the variance of the estimate of a pairwise difference between test lines in different blocks $d_1$ and $d_2$ is given by the $\left( {\begin{array}{c}b\\ 2\end{array}}\right)$ expressions

$$\begin{aligned} 2 + \frac{2}{k_c} + \frac{1}{k_c^2} (n_{d_1} - n_{d_2})^T G (n_{d_1} - n_{d_2}) \end{aligned}$$

and that between the ith control and a test line in block d by the $b \times t_c$ expressions

$$\begin{aligned} 1 +\frac{1}{k_c} + (e_{t_c,i} - \frac{1}{k_c} n_d)^T G (e_{t_c,i} - \frac{1}{k_c} n_d), \end{aligned}$$

where $n_d^T$ represents the dth row of the incidence matrix of the control treatments $N_c$, $d=1, \ldots , b$ and $i=1, \ldots , t_c$. Note that, in the latter expression, $e_{t_c,i} - \frac{1}{k_c} n_d$ is the coefficient of a contrast of control treatment effects and, more specifically,

$$\begin{aligned} e_{t_c,i} - \frac{1}{k_c} n_d = \frac{1}{k_c} \sum _{j \in B_d\setminus i} ( e_{t_c,i} - e_{t_c,j} ), \end{aligned}$$

where $B_d$ is the set of all treatments in block d. Thus if the control treatment and the test line are in the same block, the set $B_d\setminus i$ comprises $k_c-1$ treatments and if the control treatment and the test line are in different blocks it comprises $k_c$ treatments.

1.4 A.4 Globally A-optimal Designs

1.4.1 A.4.1 The Setting $v=b (k-1)+1$

The globally A-optimal block designs of Bailey and Cameron (2013, Sect. 10.1) with b blocks of size k and the number of treatments v equal to the maximum permitted for connectivity, that is $v=b (k-1)+1$, are ‘queen-bee’ designs with one treatment occurring in all b blocks and the remaining treatments occurring precisely once. The incidence matrices of these designs can thus be written compactly as $N=\left[ \begin{array}{cc} 1_b &{} 1_{k-1}^T \otimes I_b\\ \end{array} \right]$ and can be readily partitioned to yield $A_{tot}$-optimal augmented block designs for the setting with $t_c = b (k_c-1)+1$. For example, if the number of controls in each block is two, that is $k_c=2$, the $b \times (b+1)$ incidence matrix for the $b+1$ controls can be taken to be $N_c= \left[ \begin{array}{cc} 1_b &{} I_b\\ \end{array} \right]$ and that for the test lines to be $N_s = \left[ 1_{k-2}^T \otimes I_b \right]$. More generally, the incidence matrix for the setting with $k_c$ controls in each block is given by $N_c= \left[ \begin{array}{cc} 1_b &{} 1_{k_c-1}^T \otimes I_b\\ \end{array} \right]$ and that for the unreplicated test lines by $N_s= \left[ \begin{array}{c} 1_{k-k_c}^T \otimes I_b\\ \end{array} \right] .$

1.4.2 A.4.2 The Setting $v=b (k-1)$

The designs of Bailey and Cameron (2013, Sect. 10.2) for $v=b (k-1)$ are somewhat more complicated than those for $v=b (k-1)+1$ in that their structures depend on both b and k. For example, consider a setting with $b \ge 5$ and $k \ge 6$. Then the globally A-optimal block designs have incidence matrices of the form

$$\begin{aligned} N= \left[ \begin{array}{cc} 1_2 &{}1_2\\ 1_{b-2} &{} 0_{b-2}\\ \end{array} \begin{array}{c} 0_{2,(k-1)(b-2)}\\ 1_{k-1}^T \otimes I_{b-2} \\ \end{array} \begin{array}{c} 1_{k-2}^T \otimes I_2\\ 0_{(b-2),2 (k-2)} \\ \end{array} \right] . \end{aligned}$$

Thus, for an $A_{tot}$-optimal augmented block design with $k_c=2$, the $b \times b$ incidence matrix for the controls can be extracted from N and is given by $N_c = \left[ \begin{array}{cc} 1_2 &{}1_2\\ 1_{b-2} &{} 0_{b-2}\\ \end{array} \begin{array}{c} 0_{2,(b-2)}\\ I_{b-2} \\ \end{array} \right]$ and that for the unreplicated test lines by $N_s = \left[ \begin{array}{c} 0_{2,(k-2)(b-2)}\\ 1_{k-2}^T \otimes I_{b-2} \\ \end{array} \begin{array}{c} 1_{k-2}^T \otimes I_2\\ 0_{(b-2),2 (k-2)} \\ \end{array}\right] .$ The structure of $N_c$ is interesting in that one treatment occurs in all b blocks and one block is repeated twice. For general $k_c \ge 3$, the incidence matrix for the controls, $N_c$, mirrors that of the original block design but with $k_c$ replacing k.