lect10 - IE 410 Design of Experiments Notes for Lecture 10...

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IE 410 Design of Experiments Notes for Lecture 10 OC Curves, Power of the ANOVA test, and Selection of Sample Size
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Recall our previous discussions of β error: Given a Hypothesis test specified by : - H 0 and H 1 - A test statistic with known distribution - α and the corresponding cutoff value Calculate β and related quantities
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β = Pr{Fail to reject H 0 | H 0 false} = Pr{Test Stat < Cutoff | μ ‘s not all equal}
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To calculate β we need to: 1. Specify H 1 most precisely. That is, quantify "how false H 0 is" 2. Given the quantification of 1., we need to know the distribution of the test statistic when H 0 is false as specified.
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Fixed Effects ANOVA: The way to specify how false H 0 is: Recall H 0 : τ 1 = τ 2 = . .... = τ a = 0 Define Φ 2 = n* i τ i 2 / a* σ 2 Φ 2 measures how false H 0 is in some sense If H 0 true, Φ 2 = 0
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1. The more different the treatment means, the larger in magnitude are the τ i 's, and the bigger Φ 2 will be. 2. The distribution of the test statistic F 0 assuming a value for Φ 2 is specified is known. It is called the Non-central F distribution.
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To calculate β , you must specify how false H 0 is. Thus you must specify Φ 2 . This is not easy to do. There are a few heuristic ways of specifying Φ 2
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Example 1. Hypothesize values for μ 1 ,..., μ a that you think are a significant difference. e.g. "If μ 1 =1.2 μ 2 =1.5 μ 3 =2.0 Then I consider those means to be different, and would like to be able detect such differences with reasonable probability.
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Thus: μ = (1.9+1.5+2.6)/3 = 2.0 τ 1 = -.1 τ 2 = -.5 τ 3 = .6 i τ i 2 = .01+.25+.36 = .62 Φ 2 = n*0.62 / 3* σ 2
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Example 2. Specify the difference between any 2
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This note was uploaded on 08/06/2008 for the course IE 410 taught by Professor Storer during the Fall '04 term at Lehigh University .

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lect10 - IE 410 Design of Experiments Notes for Lecture 10...

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