barcode font reporting services Analysis of Variance in Software

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CHAPTER 9 Analysis of Variance
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Theorem 9-5 Under the hypothesis H(1), Vr >s2 is chi-square distributed with a 1 degrees of freedom. Under 0 the hypothesis H(2), Vc >s2 is chi-square distributed with b 1 degrees of freedom. Under both 0 hypotheses H(1) and H(2),V>s2 is chi-square distributed with ab 1 degrees of freedom. 0 0
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2 To test the hypothesis H(1) it is natural to consider the statistic Sr >S 2 since we can see from (43) that S 2 is exe r 0 2 if the row (treatment) means are significantly different. Similarly, to test pected to differ significantly from s ^ ^ ^ ^ ^ ^ (2) the hypothesis H0 , we consider the statistic S 2 >S 2. The distributions of S 2 >S 2 and S 2 >S 2 are given in the followc e r e c e ing analog to Theorem 9-3.
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Theorem 9-6 Under the hypothesis H(1) the statistic S 2 >S 2 has the F distribution with a 1 and (a 1)(b r e 0 ^ ^ degrees of freedom. Under the hypothesis H(2) the statistic S 2 >S 2 has the F distribution with b c e 0 and (a 1)(b 1) degrees of freedom.
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The theorem enables us to accept or reject H(1) and H(2) at specified significance levels. For convenience, as in 0 0 the one-factor case, an analysis of variance table can be constructed as shown in Table 9-5. Table 9-5 Variation Between Treatments, vr b a (xj. #
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Degrees of Freedom
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Mean Square vr a 1
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^ s r >s 2 e with a 1 (a 1)(b 1) degrees of freedom ^2
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x)2 #
Between Blocks, vc a a (x.k #
x)2 #
vr b 1
^ s c >s 2 e with b 1 (a 1)(b 1) degrees of freedom ^2
Residual or Random, ve v vr vc Total, v vr
1)(b
ve 1)(b
ve x)2 #
a (xjk
Two-Factor Experiments with Replication
In Table 9-4 there is only one entry corresponding to a given treatment and a given block. More information regarding the factors can often be obtained by repeating the experiment, a process called replication. In that case there will be more than one entry corresponding to a given treatment and a given block. We shall suppose that there are c entries for every position; appropriate changes can be made when the replication numbers are not all equal. Because of replication an appropriate model must be used to replace that given by (35), page 320. To obtain this, we let Xjkl denote the random variable corresponding to the jth row or treatment, the kth column or block, and the lth repetition or replication. The model is then given by Xjkl m aj bk gjk
where m, aj, bk are defined as before, jkl are independent normally distributed random variables with mean zero and variance s2, while gjk denote row-column or treatment-block interaction effects (often simply called interactions). Corresponding to (36) we have a aj
a bk
a gjk
a gjk
(45)
CHAPTER 9 Analysis of Variance
As before, the total variation v of all the data can be broken up into variations due to rows vr, columns vc, and random or residual error ve: v where v vr vc vi ve vr
j,k,l a
vi x)2 # x)2 #
(46) (47)
a (xjkl bc a (xj.. #
j 1 b
(48)
ac a (x.k. #
x)2 # xj.. # xjk.)2 # x.k. # x)2 #
(49) (50) (51)
c a (xjk. #
a (xjkl
j,k,l
In these results the dots in subscripts have meanings analogous to those given before (page 319). For example, xj.. # 1 x bc a jkl k,l 1 x b a # jk. k (52)
Using the appropriate number of degrees of freedom (df) for each source of variation, we can set up the analysis of variation table, Table 9-6.
Table 9-6 Variation Between Treatments, vr Degrees of Freedom Mean Square vr a 1 F
^ s r >s 2 e with a 1, ab(c 1) degrees of freedom ^2
Between Blocks, vc
vc b 1 vi 1)(b
^ s c >s 2 e with b 1, ab(c 1) degrees of freedom ^2 ^ s i >s 2 e with (a 1)(b 1), ab(c 1) degrees of freedom ^2
Interaction, vi
1)(b
Residual or Random, ve Total, v
ab(c
ve ab(c 1)
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