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4 Randomized Block Designs
Our statistical set-up for a randomized block design consists of n experimental units divided into b blocks of size kj each, 1 ≤ j ≤ b, kj > v. Let nij be the number of times treatment i is observed in block j, then Pv
i=1
Pni
j=1nij = Pb
j=1kj = n, the total number of observations. The one-way elimination of heterogeneity model specifies the transformed expectation of response to be g(E(Yijh)) = µ + τi + βj, where g is the canonical link function, µ is the overall mean, τi is the ith treatment effect, Pv
i=1τi = 0, and βj is the effect of block j, Pb
j=1βj = 0, i = 1, · · · , v;
j = 1, · · · , b; h = 1, · · · , nij, if treatment i is applied to unit h in block j. Both τi’s and βj’s are fixed effects.
Analogous to the completely randomized design, in block designs we also have scalar functions applied elementwise to the vectors so that ηij is rewritten as ηij = xTi β, where β = (µ, τi, · · · , τv, β1, · · · , βb), and xTi denotes the ith row of design matrix X = diag(1n11, 1n12, · · · , 1nvb)(1v×1⊗ 1b×1 : 1v×1⊗ Ib×b: Iv×v⊗ 1b×1), the two-way layout additive model without interaction, and ⊗ denotes the Kronecker product.
The response variables Yijh’s are independently distributed from a member of the natural exponential family. That is, f (yij; θij) = exp(yijθij − b(θij) + c(yij)), where θij is the natural parameter, and b(·) and c(·) are known functions.
The Fisher information matrix I for estimating (µ, τ1, · · · , τv, β1, · · · , βb) can be obtained through standard methods, and is a straightforward extension of what we derived in Section 2, which yields
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focus is on estimating treatment contrasts, the information matrix M for the es-timation of (τ1, · · · , τv)T can be derived as M = I22− I32TI33−1I32 = ((mil)), whereThe above result follows using the similar arguments as we used in Section 2 for a completely randomization design, or specifically, by substituting I21∗ = [I21 I23], I12∗ = [I12 I32], and I11∗ for I21, I12 and I11, respectively, where
Because the entries of M are too complicated to derive a general expression for the product, not to mention the sum of inverses, of the v − 1 non-zero eigenvalues, we deal mainly with the problem of finding the D- and A-optimal block designs for v = 2 and v = 3. For v = 2, we will show the direct analogy between a randomized
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block design and a completely randomized design in terms of D-optimality, and this makes it clear that the steps in the construction of D-optimal block designs for v = 2 parallel those in the construction of D-optimal designs.
4.1 D-optimal Designs for v = 2
To find a D-optimal design for v = 2 and b blocks with each block size kj given, we begin by deriving the only non-zero eigenvalue of M, which is 2Pb
j=1(n1jn2jw1jw2j)/
(n1jw1j+ n2jw2j) by an easy computation.
It is clear that nij and wij appear in the jth blocks only, indicating the problem of finding a D-optimal block design for v = 2 with each block size kj fixed, 1 ≤ j ≤ b, reduces to that of finding D-optimal designs individually within each block.
Now, we let wij = cijw1j, c1j = 1 and c2j = cj > 1, 1 ≤ j ≤ b, without loss of generality. The above eigenvalue becomes 2Pb
j=1cjw1j(n1jn2j)/(n1j+ cjn2j) = 2Pb
j=1cjw1j D(n1j, n2j), say. Our goal is to find values of n1j and n2j, 1 ≤ j ≤ b, that maximize D(n1j, n2j) subject to the given block size kj, the constraint that P2
i=1
Pb
j=1nij = n is fixed, and the cj’s given.
It is easily seen that the maximization of D(n1j, n2j) is achieved by maximizing each of the b terms separately, so the D-optimal block design can be determined by making use of the results given in Section 3.3 upon replacing n by kj and c by cj. Hence, for v = 2 with c1j = 1 and c2j = cj > 1, and b blocks with each block size kj given, design having n1j =√
cjn2j, 1 ≤ j ≤ b, is a D-optimal approximate design.
The preceding result motivates a conjecture that, when cj = c and kj = k, ∀ j, arrangingPb
j=1n1j, the number of times treatment 1 is observed in the experiment
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j=1n2j as possible, and then allocating the resulting Pb
j=1n1j to b blocks as equal as possible, is a D-optimal exact design.
This conjecture ends up with being proven false, however, by a rigorous proof omitted. An illustrative example is provided below instead.
Example 8. Consider v = 2, b = 6, k = 5, and c = 9. It follows from the above that a design having n1j = 3.75 and n2j = 1.25, or equivalently, Pb
j=1n1j = 22.5 and Pb
j=1n2j = 7.5, is a D-optimal approximate design. Now, consider the three designs d1, d2, and d3given below. It is easy to calculate values ofPb
j=1D(n1j, n2j) for each design, which are 1.8462, 1.8242, and 1.8022, respectively.
d1 : 4 4 4 4 4 4
1 1 1 1 1 1 , d2 : 4 4 4 4 4 3
1 1 1 1 1 2 , d3 : 4 4 4 4 3 3 1 1 1 1 2 2 .
We now proceed with the consideration to exact design counterpart. For v = 2 and b blocks with each block size kj given, kj ≥ 3, j = 1, · · · , b, and c1j = 1, c2j =
by straightforward computation. Then it follows from the above that the difference between optimal n1j’s, 1 ≤ j ≤ b, derived from the approximate block design and from the exact block design is less than 1 per block.
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4.2 D-optimal Designs for v = 3
The standard way of calculating determinant gives the product of the two non-zero eigenvalues of M, which is λ1λ2 = 3(m12m13+ m12m23+ m13m23), where
We begin by proving the D-optimality of the block design for v = 3 when the treatment variances are the same within a block.
Theorem 24. For v = 3 and b blocks with each block size kj given, and w1j = w2j = w3j, 1 ≤ j ≤ b, design having nij = [kj/3] or [kj/3] + 1 is a D-optimal block design.
Proof. Without loss of generality, suppose wij and nij are fixed, 2 ≤ j ≤ b and w11 = w21= w and n11− n21 ≥ 2. Let n∗11 = n11− 1, n∗21= n21+ 1, that is,
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Since m∗12 > m12, m∗23 > m23, and m∗13 + m∗23 = m13+ m23, it remains only to show that m∗13m∗23 > m13m23in proving (m∗12m∗13+ m∗12m23+ m∗13m23) − (m12m13+ m12m23+ m13m23) > 0. Because
|m∗13− m∗23| − |m13− m23| = n31ww31((n∗11− n∗21) − (n11− n21)) (n11+ n21)w + n31w31 < 0,
m∗12m∗13+m∗12m∗23+m∗13m∗23= m∗12(m13+m23)+m∗13m∗23> m12(m13+m23)+m13m23, and thus the proof is complete.
As we continue to study the relationship between nij and wij, it is easily seen from (2) that the determination of (n11, n21, n31) depends not only on (w11, w21, w31), but also on (n12, n22, n32) and (w12, w22, w32), if we take the simplest case for b = 2 as an example. This intertwining structure makes it unlikely to prove any conjec-ture by fixing (w12, w22, w32), say, only without imposing any other assumption on (w12, w22, w32), even when w11 < w21 < w31 is assumed. Accordingly we perform some exhaustive searches instead and numerical evidences conjecture that a design satisfying the condition of (n1j, n2j, n3j) inversely proportional to (w1j, w2j, w3j) is better in terms of D-optimality, as long as (w11, w21, w31) and (w12, w22, w32) follow the same order.
D-optimal block designs with various orderings of (w12, w22, w32) and k2, while keeping (w11, w21, w31) and k1 fixed, are given in Table 1.
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Table 1: Optimal nij for k2 = 13 and k2 = 31
k1 = 11 k2 = 13 k2 = 31 Cases (w12, w22, w32) (n11, n21, n31) (n12, n22, n32) (n12, n22, n32) 1 (12, 6, 20) (2, 5, 4) (5, 4, 4) (11, 12, 8) 2 (12, 6, 30) (2, 5, 4) (5, 5, 3) (11, 12, 8) 3 (6, 12, 20) (4, 4, 3) (5, 4, 4) (13, 10, 8) 4 (6, 12, 30) (5, 3, 3) (5, 5, 3) (13, 11, 7)
Because ηij is assumed to follow the usual additive model without interaction, it is reasonable to assume (w11, w21, w31) and (w12, w22, w32) have the same ordering, and we are in the progress of proof for the case b = 2 under this assumption.
We end this section by briefly addressing the derivation of λ−11 + λ−12 for the determination of optimal designs when v = 3 with b blocks. As for v = 2, the A-and D-optimal block designs are identical for there is only one non-zero eigenvalue of M.
4.3 A-optimal Designs
For v = 3, λ1 + λ2 can be obtained by summing the 3 determinants of the first order principal minors of M, which leads to λ1+ λ2 = 2(m12+ m13+ m23). Hence, λ−11 +λ−12 = 2(m12+m13+m23)/3(m12m13+m12m23+m13m23), where m12, m13and m23have the same definition as in (2). The problem of finding the A-optimal block design now reduced to minimizing (m12+m13+m23)/(m12m13+m12m23+m13m23) satisfying nij > 0 and P3
i=1
Pb
j=1nij = n, with kj and wij’s given and n fixed, i = 1, 2, 3, 1 ≤ j ≤ b.
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