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Notation for Two-Factor Experiments
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Assuming that we have a treatments and b blocks, we construct Table 9-4, where it is supposed that there is one experimental value (for example, yield per acre) corresponding to each treatment and block. For treatment j and block k we denote this value by xjk. The mean of the entries in the jth row is denoted by xj., where j 1, . . . , a, #
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CHAPTER 9 Analysis of Variance
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while the mean of the entries in the kth column is denoted by x.k, where k # is denoted by x. In symbols, # xj. # 1 x , b ka jk 1
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1, . . . , b. The overall, or grand, mean
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x.k #
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1 a a xjk,
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1 x ab a jk j,k
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(27)
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Table 9-4 Blocks 1 Treatments 1 2 ( a x11 x21 ( xa1 x.1 # 2 x12 x22 ( xa2 x.2 # c c c ( ( ( c c b x1b x2b xab x.b # x1. # x2. # xa. #
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Variations for Two-Factor Experiments
As in the case of one-factor experiments, we can define variations for two-factor experiments. We first define the total variation, as in (3), to be v By writing the identity xjk x # (xjk xj. # v where ve variation due to error or chance vr vc x.k # ve a (xjk
a (xjk
x)2 #
(28)
x) # vr xj. #
(xj. # vc x.k #
x) #
(x.k #
x) #
(29)
and then squaring and summing over j and k, we can show that (30) x)2 #
variation between rows (treatments)
b a (xj. #
j 1 b
x)2 #
variation between columns (blocks)
a a (x.k #
x)2 #
The variation due to error or chance is also known as the residual variation. The following are short formulas for computation, analogous to (10), (11), and (12). v vr vc ve a xjk
t2 ab t2 ab t2 ab vc
(31)
1 t2 b ja j. 1 1 2 a a t.k
k 1 b
(32)
(33) (34)
where tj. is the total of entries of the jth row, tk is the total of entries in the kth column, and t is the total of all . entries.
CHAPTER 9 Analysis of Variance
Analysis of Variance for Two-Factor Experiments
For the mathematical model of two-factor experiments, let us assume that the random variables Xjk whose values are the xjk can be written as Xjk m aj bk
(35)
Here m is the population grand mean, aj is that part of Xjk due to the different treatments (sometimes called the treatment effects), bk is that part of Xjk due to the different blocks (sometimes called the block effects), and jk is that part of Xjk due to chance or error. As before, we can take the jk as independent normally distributed random variables with mean zero and variance s2, so that the Xjk are also independent normally distributed variables with variance s2. Under suitable assumptions on the means of the Xjk, we have a aj
a bk
(36)
which makes m 1 E(Xjk) ab a j,k
Corresponding to the results (16) through (18), we can prove that E(Vr) E(Vc) E(Ve) E(V) (a (b (a (ab 1)s2 1)s2 1)(b 1)s2 b a a2 j
(37) (38) (39) a a b2 k
a a b2 k
1)s2 b a a2 j
(40)
There are two null hypotheses that we would want to test: H(1): All treatment (row) means are equal, i.e., aj 0 H(2): 0 All block (column) means are equal, i.e., bk 0, j 0, k 1, . . . , a 1, . . . , b
We see from (39) that, without regard to H(1) or H(2), a best (unbiased) estimate of s2 is provided by 0 0 S2 e
Ve 1)(b
i.e.,
E(S 2) e
(41)
Also, if the hypotheses H(1) and H(2) are true, then 0 0 S2 r
Vr a 1
, S2 c
Vc b 1
V ab 1
(42)
will be unbiased estimates of s2. If H(1) and H(2) are not true, however, we have from (37) and (38), respectively, 0 0 E(S 2) r
b a a b
1a j 1a k
a2 j
(43)
E(S 2) c
b2 k
(44)
The following theorems are similar to Theorems 9-1 and 9-2. Theorem 9-4 Ve >s2 is chi-square distributed with (a or H(2). 0 1)(b 1) degrees of freedom, without regard to H(1) 0
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