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SSY = SS0 + SSA + SSE

N = 1 + (a - 1) + N - a

12 = 1 + 2 + 9

The ANOVA is shown in Table 20.9. Notice that the variation due to processors is insignificant as compared to that due to modeling errors. Confidence intervals for processor effects can be obtained after computing their standard deviations using expressions given in Table 20.6. To understand how these expressions are derived, consider the effect of processor Z. Since,

The error in α3 is the weighted sum of errors in the observations on the right-hand side of the preceding equation:

Since eij’s are normally distributed with zero mean and an estimated variance of s2e, the variance of eα3 (and hence that of α3) is

It is easy to verify that the value computed here is the same as that obtained from the expression in Table 20.6.

The key results related to the analysis of one-factor experiments are summarized in Box 20.1.

Box 20.1 Analysis of One-Factor Experiments

1.  Model: yij = µ + αj + eij; the effects are computed so that = 0.
2.  Effects:
3.  Allocation of variation: SSE can be calculated after computing other terms below:

4.  Degrees of freedom: SSY = SS0 + SSA + SSE
ar = 1 + (a - 1) + a(r - 1)
5.  Mean squares: MSA = SSA/(a - 1); MSE = SSE/[a(r - 1)]
6.  Analysis of variance: MSA/MSE should be greater than F[1 - α;a-1,a(r - 1)].
7.  Standard deviation of errors:
8.  Standard deviation of parameters:
9.  Contrast of effects , where

10.  All confidence intervals are computed using t[1 - α/2;a(r - 1)]
11.  Model assumptions:
(a)  Errors are IID normal variates, with zero mean.
(b)  Errors have the same variance for all factor levels.
(c)  The effect of the factor and errors are additive.
12.  Visual tests.
(a)  The scatter plot of errors versus predicted responses should not have any trend.
(b)  The normal quantile-quantile plot of errors should be linear.
(c)  Spread of y values for all levels of the factor should be comparable.
  If any test fails or if the ratio ymax/ymin is large, multiplicative models or transformations should be investigated.

EXERCISE

20.1 For a single-factor design, suppose we want to write an expression for αj in terms of yij’s:

αj = a11jy11 + a12jy12 + ... + arajyra

What are the values of the a..j’s? From the preceding expression, the error in αj is seen to be

eαj = a11je11 + a12je12 + ... + arajera

Assuming errors eij are normally distributed with zero mean and variance σ2e, write an expression for the variance of eαj . Verify that your answer matches that in Table 20.5.


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