In the above experiment, one may also want to study the effect of different age groups of cows and breeds. Suppose that there are four age groups of cows and four breeds. This can be done by arranging the 16 treatment combinations in a 4×4 array such that each row represents a different breed and each column represents a different age group of cows as shown in Table 3.1 (this table is taken from [7]). The rows and columns of the design can be thought of as blocking factors and such a design is called a full row-column factorial design.
Age 1 Age 2 Age 3 Age 4 Breed 1 1111 0100 0010 1001 Breed 2 0001 1010 1100 0111 Breed 3 1000 0011 0101 1110 Breed 4 0110 1101 1011 0000
Table 3.1: A regular row-column factorial design of type I4(4,4; 2).
In Table 3.1 we have given an example of a row-column factorial design in which each treatment combination appears exactly once. However, in general, the treatment combinations can appear more than once. Formally,
Definition 3.3. An m×n row-column factorial designqk is any arrangement of the elements of α ×[q]k in an m ×n array. Necessarily, mn must be a multiple of qk.
A blocking factor is a partition (usually equipartition) of the treatment combinations of the factorial design. While blocking factors help study some additional information, balance in the treatment combinations within each block can help avoid estimation bias. Therefore the design needs a more care- ful arrangement of treatment combinations. There are several structured ap- proaches in constructing factorial designs with blocking factors ([1, 2, 8, 9, 13]).
However, as mentioned in [11], designs with two forms of blocking have not been investigated thoroughly.
Consider the design in Table 3.1 (with two forms of blocking). If we look closely we can see that in each row (or column) at any particular position i ∈ [4] the entries 0 and 1 each appear twice. Thus half of the entries for a particular drug in each block are at a high level and half of them are at a low. These type of regularity properties in the design allows the unbiased estimation of the main effects; the four drugs, age and breed.
On the other hand, if we fix the first and the fourth coordinates, then pairs 01 and 10 do not appear in the first row. Similarly in the columns, if we choose
the second and the third coordinate, the design lacks this property. Therefore the interaction factorD1D4 confounds with the age groups and the interaction factor D2D3 is not estimable across the breeds.
We say that a row-column factorial design is of strength t if each of its columns is an orthogonal array of type OA(m, k, q, t) and each of its rows is a transpose of anOA(n, k, q, t). We denote such a design byIk(m, n, q, t). In the context of experimental design, in anIk(m, n, q, t) all subsets of interactions of size at most t can be estimated without confounding by the row and column blocking factors. Thus, the design in Table 3.1 is an I4(4,4,2,1) but not an I4(4,4,2,2). Here we want to remind the reader that, in Chapter 4 a regular row-column design Ik(m, n, q) is equivalent to Ik(m, n, q,1).
In experimental design, there is a wide range of designs that can be con- sidered row-column designs. Almost all of them have the property of being arranged in a rectangular array and typically (but not always) the rows and columns act as blocking factors. One of the earliest examples of a row-column factorial design can be found in [15] in which the following design was featured:
11111 10011 00110 01010 00101 10000 01001 11100 10010 11110 01011 00111 10001 00100 11101 01000 11011 10100 00001 01101 00010 10111 01110 11011 10101 11001 01100 00000 10110 00011 11010 01111 00011 01000 11010 10001 11111 01101 10100 00110 01001 00010 10000 11011 01100 11110 00111 10101 00100 01111 11011 10110 11011 01010 10011 00001 01110 00101 10111 11100 01011 11001 00000 10010 Table 3.2: A Row-column design for a 25-Factorial with 2 replicates
In practical use, sometimes a row-column design of strength 0 (non-regular) is also useful. The design in Table 3.3 containing six replicates of 4×8 fac- torial was used by CSIRO Division of Forestry [19]. The two factors were the
four salt-irrigation levels and eight different lots of seeds. It was known that the effect of growth varies with respect to the distance from the walls of the glasshouse and also from north to south. Therefore, the row-column design was constructed to analyse this two-dimensional physical effect.
3 8 2 2 1 7 4 3 1 5 2 1 3 6 4 4
1 4 3 1 2 3 1 8 2 6 4 2 4 5 2 7
4 1 4 7 3 5 2 4 3 3 1 6 2 8 1 2
2 7 1 3 4 6 3 2 4 8 3 4 1 1 2 5
1 5 1 7 2 3 3 2 2 8 3 4 4 6 4 1
3 7 4 8 3 6 2 5 4 2 1 3 1 1 2 4
2 6 3 1 1 8 4 4 3 3 4 5 2 7 1 2
4 3 2 2 4 7 1 6 1 4 2 1 3 8 3 5
4 5 1 8 2 3 1 7 3 2 3 4 2 1 4 6
2 2 2 7 4 8 3 6 1 4 4 1 1 5 3 3
1 6 3 1 3 5 2 8 4 7 1 3 4 2 2 4
3 8 4 3 1 1 4 4 2 6 2 5 3 7 1 2
4 3 2 2 3 6 3 5 1 4 1 7 4 8 2 1
3 2 4 6 1 1 2 7 2 8 4 5 3 4 1 3
2 6 1 5 2 3 4 4 4 1 3 8 1 2 3 7
1 8 3 1 4 7 1 6 3 3 2 4 2 5 4 2
2 4 3 7 4 1 1 3 1 6 2 5 3 2 4 8
4 6 1 5 3 8 4 7 2 3 3 4 2 1 1 2
3 3 2 8 1 4 3 6 4 2 1 1 4 5 2 7
1 7 4 4 2 6 2 2 3 1 4 3 1 8 3 5
4 7 1 2 2 6 3 1 2 4 4 5 3 8 1 3
3 2 3 7 4 1 2 5 4 3 1 6 1 4 2 8
2 3 4 8 3 5 1 7 1 1 2 2 4 6 3 4
1 5 2 1 1 8 4 4 3 6 3 3 2 7 4 2
Table 3.3: Row-column design for a 4×8 Factorial with 6 replicates
Unbiased estimation of factorial effects sometimes required the treatment combinations to be arranged in a way that coincides with the structure of a quasi-Latin rectangle.
Definition 3.4. A quasi-Latin rectangle LR(m, n;q) is anm×n array based on a symbol set of size q, where q > m, nand q divides the product mn, such that each symbol appears λ times in the array and no symbol appears more than once in any row or column.
A quasi-Latin rectangle in whichm=n is called a quasi-Latin square.
Example 3.2. Table 3.4 and Table 3.5 are, respectively, a quasi-Latin square and a quasi-Latin rectangle based on the symbol set {0,1, . . . ,7}:
6 5 4 0
1 0 6 7
2 7 1 3
5 2 3 4
Table 3.4: LR(4,4; 8)
1 2 3 4 5 6
7 0 1 2 3 4
5 6 7 0 1 2
3 4 5 6 7 0
Table 3.5: LR(4,6; 8)
Quasi-Latin rectangles have been used to construct row-column factorial designs ([3]). Here if we consider each vector as a symbol then a row-column factorial design can be thought of a quasi-Latin rectangle. For example, the following 23 row-column design (I3(4,4,2,1)) featured in [3] corresponds to the quasi-Latin square given in Table 3.4.
0,1,1 1,0,1 1,1,0 0,0,0 1,0,0 0,0,0 0,1,1 1,1,1 0,1,0 1,1,1 1,0,0 0,0,1 1,0,1 0,1,0 0,0,1 1,1,0 Table 3.6: An I3(4,4,2,1) design.
John and Lewis ([12]) describe a technique to generate row-column designs by using a generalized cyclic method. Some other examples and techniques to construct row-column designs are also given in [5], [6] and [4]. Wang [18] defines a technique to construct a row-column design involvingkfactors each with two levels, such that all the main effects are estimable. The procedure involves careful selection of treatment combinations to generate the first row and the first column of the design and then complete the table using component-wise addition modulo 2, to get an Ik(2M,2N,2,1) wherek =M+N. For example,
11111,10100,01100 and 11110,11101 were used to span the first column and the first row, respectively, of the following design:
00000 11110 11101 00011
11111 00001 00010 11100
10100 01010 01001 10111
01100 10010 10001 01111
01011 10101 10110 01000
10011 01101 01110 10000
11000 00110 00101 11011
00111 11001 11010 00100
Table 3.7: An I5(23,22,2,1) design.
A generalization of row-column design, called generalized confounded row- column design (GCRC), was introduced by Datta et al. [7], in which the intersection of rows and columns can contain multiple treatment combinations.
Their construction also utilizes the structure of a Latin square of suitable order.
For example, the following design featured in [7] uses a Latin square of order four. Here each block corresponds to a cell of a Latin square.
0000 1011 1000 0100 0001 1010 0010 1001
0111 1100 0011 1111 0110 1101 0101 1110
1000 0100 0001 1010 0010 1001 0000 1011
0011 1111 0110 1101 0101 1110 0111 1100
0001 1010 0010 1001 0000 1011 1000 0100
0110 1101 0101 1110 0111 1100 0011 1111
0010 1001 0000 1011 1000 0100 0001 1010
0101 1110 0111 1100 0011 1111 0110 1101
Table 3.8: GCRC based on 24−factorial with block size 22
In [11], Godolphin described a method to construct row-column designs with two levels such that all main effects are estimable and the design also maximizes the possible number of two-factor interactions of interest. The construction uses a similar approach as in [7] by selecting specific treatment combinations to generate the first row and the first column of the design. The generating treatment combinations are grouped in the form of a matrix called array generator matrix (AGM). In [11], the authors have also described the properties of theAGM swhich dictate the estimability of the main effects and two-factor interactions in the subsequent design. Consider the followingAGM to construct a 25−factorial in a 22×23 array:
G= Gc Gr
!
=
1 1 1 1 0 1 1 0 0 1 1 0 1 1 0 0 1 1 0 1 0 1 0 0 1
Since the matrix G has rank 5 (full) therefore the resulting design will consist of a full 25 factorial, and the number of treatment combinations in Gc and Gr corresponds to the dimension of the resulting array. Now by using the treatment combinations in Gc and Gr to span, respectively, the first column and the first row of the design and then completing the table using addition modulo 2 we get the following design:
00000 10110 01101 11011 01001 11111 00100 10010 11110 01000 10011 00101 10111 00001 11010 01100 11001 01111 10100 00010 10000 00110 11101 01011 00111 10001 01010 11100 01110 11000 00011 10101
Table 3.9: 22×23 full factorial row-column design
Since each column in the sub-matricesGc andGr is non-zero, the resulting array contains exactly half the number of zeroes and ones in each column and row (where the position is fixed). Thus the design in Table 3.9 is an
I5(22,23,2,1). This property in the design enables the estimability of all main effects. The first two columns in Gc are the same, this leads to the appearance of only two types of pairs, either {10,01} or {11,00}, in each column at the first and second coordinate. Thus the interaction factor F1F2 is confounded in the columns. Similarly, F3F4 is also confounded in the columns.
In matrix Gr, the first column is the same as the fourth and the second is the same as the fifth. This causes the interaction factors F1F4 and F2F5 to be confounded in the columns. In other words, the design in Table 3.9 lacks the strength 2 property in these positions and therefore these interaction factors are confounded.
In [11], the author has also provided an upper bound ω (defined below) on the maximal number of interaction factors that can be estimated together with all the main effects in a 2p×2q row-column design.
Let n = α(2p −1) +β, where α and β are integers and 0 ≤ β ≤ 2p −2.
Then
ω = n
2
!
−αβ−(2p−1) α 2
!
. (3.1)
If we plug in the values of the parameters of the design in Table 3.9 we get ω = 8. However, the design only estimates the six interaction factors out of ten. If we take the AGM matrix such that each column ofGc is distinct then the resulting design would allow the estimation of the maximum number of interaction factors together with all main effects.
Example 3.3. [11] The following AGM matrix G1 has the same Gc subma- trix as in G but the Gr contains no identical columns. Therefore, the same interaction factors are confounded in the columns and there is no interaction factor confounded in the rows.
G= Gc Gr
!
=
1 1 1 1 0 1 1 0 0 1 1 0 1 0 1 0 1 1 1 0 0 1 1 0 1
00000 10101 01110 11011 01101 11000 00011 10110 11110 01011 10000 00101 10011 00110 11101 01000 11001 01100 10111 00010 10100 00001 11010 01111 00111 10010 01001 11100 01010 11111 00100 10001
Table 3.10: An I5(22,23,2,1) design.
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