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lect3 - IE 410 Notes for Lecture 3 THE ANALYSIS OF...

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IE 410 Notes for Lecture 3 THE ANALYSIS OF VARIANCE (or ANOVA) One Sample T-test: H 0 : μ = μ 0 Two-Sample T-test: H 0 : μ 1 = μ 2 (Two populations have the same mean)
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Example: Which tennis ball brand bounces higher, Penn or Wilson? Get (say) 3 balls of each brand: Drop each from 10 feet onto concrete OBSERVE: height of bounce (the response variable Y)
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Single Factor Analysis of Variance (a.k.a. One-way ANOVA) H 0 : μ 1 = μ 2 = μ 3 = ..... = μ k Test if the mean of several populations are all the same H 1 : at least one mean is different
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Example : Does fertilizer type effect yield from an acre of corn? Factor = "fertilizer type" Factor levels ( treatments) = types "1", "2", "3", & "4" H 0 : μ 1 = μ 2 = μ 3 = μ 4 H 1 : at least one different Y(i,j) = yield (bushels) from the jth field treated with fertilizer i
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Basic Model of a data set: Assume the factor (call it Factor A) is at " a " levels, and that n observations are taken within each level Let y(i,j) be the response for the jth observation within the ith level of Factor A.
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FACTOR A 1 2 3 ....... a ----------------------------------------- y(1,1) y(2,1) y(3,1) ...... y(a,1) y(1,2) y(2,2) y(3,2) y(a,2) . . Obs. . . y(1,n) y(2,n) y(3,n) ..... y(a,n)
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Each y(i,j) is a random variable. Once the expt has been conducted, we will have numbers that are outcomes of the random variables. If we repeated the expt, we would get different outcomes Sampling is assumed to be RANDOM
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Statistical Models: Two Basic Types depending on the situation FIXED vs. RANDOM Effects models Example : y(i,j) =SAT score of randomly students from University i Factor A = University at levels "Lehigh", "Laughyette", "Michigan" Randomly select students from each school, and record their SAT score in the dataset.
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FIXED Effects: Interested only in the 3 schools, Lehigh, Laff, and UofM H 0 : μ lehigh = μ laff = μ UofM
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RANDOM Effects Model Suppose you wanted to test whether or not the mean SAT scores were the same at all American universities. How would you do it ? From the population of American Universities, you might randomly select "a" universities to study. From each of these universities, you might randomly select students, and record their scores in the dataset. H 0 : Mean SAT scores are equal at all American Univ’s
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Fixed Effects: Interested in making inference only about those treatments (factor levels) included in the study. Random Effects: Interested in making inference about a large population of treatments from which those studied are a representitive random sample.
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Basic Relevance: For the One-way ANOVA i) The statistical tests are the same for Fixed and Random ii) The power of the tests is different For Multi-factor ANOVA the tests are different as well.
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