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raneffs10 - Statistics 514 Experiments with Random Effects...

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Statistics 514: Experiments with Random Effects Lecture 10: Experiments with Random Effects Montgomery, Chapter 12 or 13 Page 1
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Statistics 514: Experiments with Random Effects Example 1 A textile company weaves a fabric on a large number of looms. It would like the looms to be homogeneous so that it obtains a fabric of uniform strength. A process engineer suspects that, in addition to the usual variation in strength within samples of fabric from the same loom, there may also be significant variations in strength between looms. To investigate this, she selects four looms at random and makes four strength determinations on the fabric manufactured on each loom. The layout and data are given in the following. Observations looms 1 2 3 4 1 98 97 99 96 2 91 90 93 92 3 96 95 97 95 4 95 96 99 98 Page 2
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Statistics 514: Experiments with Random Effects Random Effects vs Fixed Effects Consider factor with numerous possible levels Want to draw inference on population of levels Not concerned with any specific levels Example of difference (1=fixed, 2=random) 1. Compare reading ability of 10 2nd grade classes in NY Select a = 10 specific classes of interest. Randomly choose n students from each classroom. Want to compare τ i (class-specific effects). 2. Study the variability among all 2nd grade classes in NY Randomly choose a = 10 classes from large number of classes. Randomly choose n students from each classroom. Want to assess σ 2 τ (class to class variability). Inference broader in random effects case Levels chosen randomly inference on population Page 3
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Statistics 514: Experiments with Random Effects Random Effects Model (CRD) Similar model (as in the fixed case) with different assumptions y ij = μ + τ i + ij i = 1 , 2 . . . a j = 1 , 2 , . . . n i μ - grand mean τ i - i th treatment effect (random) ij N(0 , σ 2 ) Instead of τ i = 0 , assume τ i N(0 , σ 2 τ ) { τ i } and { ij } independent Var( y ij ) = σ 2 τ + σ 2 σ 2 τ and σ 2 are called variance components Page 4
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Statistics 514: Experiments with Random Effects Statistical Analysis The basic hypotheses are: H 0 : σ 2 τ = 0 vs. H 1 : σ 2 τ > 0 Same ANOVA table (as before) Source SS DF MS F 0 Between SS tr a - 1 MS tr F 0 = MS tr MS E Within SS E N - a MS E Total SS T = SS tr + SS E N - 1 E(MS E )= σ 2 E(MS tr )= σ 2 + 2 τ Under H 0 , F 0 F a - 1 ,N - a Same test as before Conclusions, however, pertain to entire population Page 5
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Statistics 514: Experiments with Random Effects Estimation Usually interested in estimating variances Use mean squares (known as ANOVA method) ˆ σ 2 = MS E ˆ σ 2 τ = (MS tr - MS E ) /n If unbalanced, replace n with n 0 = 1 a - 1 a i =1 n i - a i =1 n 2 i a i =1 n i Estimate of σ 2 τ can be negative - Supports H 0 ? Use zero as estimate? - Validity of model? Nonlinear?
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