barcode font reporting services We shall denote by xj. the mean of the measurements in the jth row. We have # xj. # 1 x b ka jk 1 in Software

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We shall denote by xj. the mean of the measurements in the jth row. We have # xj. # 1 x b ka jk 1
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1, 2, c, a
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The dot in xj. is used to show that the index k has been summed out. The values xj. are called group means, treat# # ment means, or row means. The grand mean, or overall mean, is the mean of all the measurements in all the groups and is denoted by x, i.e., # x # 1 x ab a jk j,k 1 x ab ja ka jk 1 1
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CHAPTER 9 Analysis of Variance
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Total Variation. Variation Within Treatments. Variation Between Treatments
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We define the total variation, denoted by v, as the sum of the squares of the deviations of each measurement from the grand mean x, i.e., # Total variation By writing the identity, xjk x # (xjk xj.) # (xj. # x) # (4) v a (xjk
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and then squaring and summing over j and k, we can show (see Problem 9.1) that a (xjk
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x)2 #
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a (xjk
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xj.)2 #
# a (xj.
x)2 #
or a (xjk
x)2 #
a (xjk
xj.)2 #
b a (xj. #
x)2 #
We call the first summation on the right of (5) or (6) the variation within treatments (since it involves the squares of the deviations of xjk from the treatment means xj.) and denote it by vw. Therefore, # vw a (xjk
xj.)2 #
The second summation on the right of (5) or (6) is called the variation between treatments (since it involves the squares of the deviations of the various treatment means xj. from the grand mean x) and is denoted by vb. Therefore, # # vb Equations (5) or (6) can thus be written v vw vb (9) a (xj.
b a (xj.
Shortcut Methods for Obtaining Variations
To minimize the labor in computing the above variations, the following forms are convenient: v vb vw a x2 jk
t2 ab t2 ab
(10) (11) (12)
1 t2 b a j. j v vb
where t is the total of all values xjk and tj. is the total of all values in the jth treatment, i.e., t a xjk
a xjk
(13)
In practice it is convenient to subtract some fixed value from all the data in the table; this has no effect on the final results.
Linear Mathematical Model for Analysis of Variance
We can consider each row of Table 9-1 as representing a random sample of size b from the population for that particular treatment. Therefore, for treatment j we have the independent, identically distributed random variables Xj1, Xj2, . . . , Xjb, which, respectively, take on the values xj1, xj2, . . . , xjb. Each of the Xjk (k 1, 2, . . . , b)
CHAPTER 9 Analysis of Variance
can be expressed as the sum of its expected value and a chance or error term: Xjk mj
(14)
The jk can be taken as independent (relative to j as well as to k), normally distributed random variables with mean zero and variance s2. This is equivalent to assuming that the Xjk ( j 1, 2, . . . , a; k l, 2, . . . , b) are mutually independent, normal variables with means mj and common variance s2. Let us define the constant m by m 1 a a mj
We can think of m as the mean for a sort of grand population comprising all the treatment populations. Then (14) can be rewritten as (see Problem 9.18) Xjk m aj
where
a aj
(15)
The constant aj can be viewed as the special effect of the jth treatment. The null hypothesis that all treatment means are equal is given by (H0: aj 0; j 1, 2, . . . , a) or equivalently by (H0: mj m; j 1, 2, . . . , a). If H0 is true, the treatment populations, which by assumption are normal, have a common mean as well as a common variance. Then there is just one treatment population, and all treatments are statistically identical.
Expected Values of the Variations
The between-treatments variation Vb, the within-treatments variation Vw, and the total variation V are random variables that, respectively, assume the values vb, vw, and v as defined in (8), (7), and (3). We can show (Problem 9.19) that E(Vb) E(Vw) E(V) From (17) it follows that EB (a a(b (ab 1)s2 1)s2 1)s2 b a a2 j b a a2 j
(16) (17) (18)
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