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Chapter 5 Experiments to Study Variances 1 Random Effects Models for Variances Section 5.1 pp 148-151 Traditional definitions Fixed - treatment level is reproducible, our interest is the means and our inference is to just the levels tested. Examples ..... Random - treatment level is a random draw from a larger population, it is not reproducible, our interest is the variance(s) and our inference is to the entire population. Examples ..... Newer (and much better) definition If the effect level can reasonable be assumed to represent a probability distribution then the effect is random. If it does not represent a probability distribution then the effect is fixed. Period. (p.92 SLM) Examples .....

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Chapter 5 Experiments to Study Variances 2 Casting Example Bar Casting 1 Casting 2 Casting 3 1 88.0 85.9 94.2 2 88.0 88.6 91.5 ! ! ! ! 10 93.0 87.5 92.5
Chapter 5 Experiments to Study Variances 3 A Statistical Model for Variance Components Section 5.2 pp 151-152 For a fixed model For a random model The objective is to decompose the total variance into identifiable components. ANOVA for a one-way random effects model Source df SS MS EMS Among Groups t-1 SS Among (SSA) MS Among (MSA) Within Groups N-t SS Within (SSW) MS Within (MSW) Total N-1 SS Total (SST)

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Chapter 5 Experiments to Study Variances 4 Point Estimates of Variance Components Section 5.3 pp 152-153 We assume that the random effect of the model are distributed normally and thus significance can be tested based on null and alternative hypotheses such as I show below for the among group component. We get the MSA and MSW from SAS.
Chapter 5 Experiments to Study Variances 5 Data a; do casting= 1 to 3; do bars = 1 to 10; input strength @@; output; end; end; datalines; 88.0 88.0 94.8 90.0 93.0 89.0 86.0 92.9 89.0 93.0 85.9 88.6 90.0 87.1 85.6 86.0 91.0 89.6 93.0 87.5 94.2 91.5 92.0 96.5 95.6 93.8 92.5 93.2 96.2 92.5 ;;;; proc glm; class casting; model strength = casting; random casting; run;

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Chapter 5 Experiments to Study Variances 6 The GLM Procedure Sum of Source DF Squares Mean Square F Value Pr > F Model 2 147.8846667 73.9423333 12.71 0.0001 Error 27 157.1020000 5.8185926 Corr Total 29 304.9866667 R-Square Coeff Var Root MSE strength Mean 0.484889 2.654632 2.412176 90.86667 Source DF Type III SS Mean Square F Value Pr > F casting 2 147.8846667 73.9423333 12.71 0.0001 Source Type III Expected Mean Square casting Var(Error) + 10 Var(casting)
Chapter 5 Experiments to Study Variances 7 proc varcomp data=a method=type1; class casting; model strength = casting; run; Variance Components Estimation Procedure Class Levels Values casting 3 1 2 3 Number of observations 30 Dependent Variable: strength Type 1 Analysis of Variance Source DF SS MS Expected Mean Square casting 2 147.884667 73.942333 Var(Error) + 10 Var(casting) Error 27 157.102000 5.818593 Var(Error) Corr Total 29 304.986667 Type 1 Estimates Variance Component Estimate Var(casting) 6.81237 Var(Error) 5.81859 E = 12.63096

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Chapter 5 Experiments to Study Variances 8 Interval Estimates for Variance Components Section 5.4 pp 153-155 We will discuss two methods for the construction of confidence intervals about variance components. The first method is the one you are already familiar with where one uses large sample normal approximation to construct the confidence intervals based on the standard errors for each of the variance components. The normal approximation is what the book shows you in
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