Lecture15 - AMS 572 Class Notes November 19, 2010 Chapter...

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AMS 572 Class Notes November 19, 2010 Chapter 12 Analysis of Variance (ANOVA) One-way ANOVA (fixed factors) * Goal: compare the means from a (a≥2) different populations. * It is an extension of the pooled variance t-test. * Assumptions: (i) Equal (unknown) population variances (ii) Normal populations (iii) Independent samples : these ’s are not all equal. Assumptions: a population, , i=1,2,…,a. is unknown. Samples: a independent samples. Data:
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Balanced design: Unbalanced design: otherwise Derivation of the test (1) PQ, can be derived (2) * Union-intersection method. Best method for this type of test as in other regression analysis related tests. Please see AMS 570/571 text book, and also the book by G.A.F. Seber: Linear Regression Model, published by John Wiley for details. (3) LRT (Likelihood Ratio test) Test Statistic: Total sample size . Sample mean: grand mean Balanced design: ,
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Theorem Let (1) (2) (3) and are independent. Definition , where and they are independent. , . When is true: , () is true: .
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Intuitively, we reject in favor of if , where C is determined by the significance level as usual: . When a=2, Note: If . (One can prove this easily using the definitions of the t- and F-distributions) If we reject the ANOVA hypothesis, then we should do the pairwise comparisons.
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The multiple comparison problem FWE: (Family wise error rate) =P(reject at least 1 true null hypothesis) Tukey’s Studentized Range test (* It is the preferred method to ensure the FWE) At FWE , reject if Finally, the name ANOVA came from the partitioning of the variations: For more details, please refer to the text book.
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Attachment: Handout. Ronald Fisher (1890-1962) Sir Ronald Aylmer Fisher was a British statistician, evolutionary biologist, and geneticist. He has been described as: “a genius who almost single-handedly created the foundations for modern statistical science” and “the greatest of Darwin's successors”. Fisher was born in East Finchley in London, to George and Katie Fisher. Although Fisher had very poor eyesight, he was a precocious student, winning the Neeld Medal (a competitive essay in Mathematics) at Harrow School at the age of 16. Because of his poor eyesight, he was tutored in mathematics without the aid of paper and pen, which developed his ability to visualize problems in geometrical terms, as opposed to using algebraic manipulations. He was legendary in being able to produce mathematical results without setting down the intermediate steps. In 1909 he won a scholarship to Gonville and Caius College, Cambridge, and graduated with a degree in mathematics in 1913. During his work as a statistician at the Rothamsted Agricultural Experiment Station, UK, Fisher pioneered the principles of the design of experiments and elaborated his studies of " analysis of variance ". In addition to "analysis of variance", Fisher invented the technique of maximum likelihood and originated the concepts of sufficiency, ancillarity, Fisher's linear discriminator and Fisher information. The contributions Fisher made also included
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Lecture15 - AMS 572 Class Notes November 19, 2010 Chapter...

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