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Lecture_2010_09_22

Lecture_2010_09_22 - Stat 430/830 Analysis of Variance...

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1 Stat Stat 430/830: Analysis of Variance Lecture September 22, 2010 Nonparametric Methods Nonparametric Methods Kruskal –Wallis Test Used to test the hypothesis that a treatments are identical vs. the alternative that some treatments generate observations that are larger than others. - Rank observations y ij in ascending order and assign the rank, R ij . For tied observations assign the average rank of the tied observations. - Let R i . the sum of the ranks in the i treatment Kruskal Wallis Test The test statistics is with n i number of observations on the i treatment and N the total number of observations. If there are no ties then Under moderate ties this result provides good approximation + - - = + - = ∑∑ = = = 4 ) 1 ( 1 1 4 ) 1 ( 1 2 1 1 2 2 1 2 2 . 2 N N R N S N N n R S H a i n j ij a i i i i = + - + = + = a i i i N n R N N H and N N S 1 2 . ) 1 ( 3 ) 1 ( 12 12 / ) 1 ( Kruskal Wallis Test One can also do an ANOVA on the ranks. It can be shown that the F test for the ANOVA on the ranks satisfies Therefore, ANOVA on ranks is equivalent to the Kruskal Wallis Test H ely Approximat a 2 1 ~ - χ ) /( ) 1 ( ) 1 /( a N H N a H F o - - - - =
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2 Variance Stabilizing Transformations Variance Stabilizing Transformations If the variance is proportional to a power of the mean, e.g., σ ∝ μ α , then it can be shown that w = y 1- α has constant variance Box Box - Cox Transformation Cox Transformation { } . selecting in help can likelihood profile the of plot A )). L( maximized that one the (e.g., )} L( arg{max selecting suggest Cox Box model. the of squres of sum error the is SS and y of geometric the is y , y y z where z SS N cons L by given is for likelihood profile The x y where for y for y y tion transforma the suggest Cox - Box valid. not is variance constant of assumption the but x, for proposed is model linear a Suppose E E λ λ λ λ λ λ α λ λ λ λ λ λ λ λ λ - = - = + = = - = - & & 1 ) ( ) ( ) ( ) ( / ) ( log 2 ) ( . 0 ) log( 0 / ) 1 ( Example Example Peak Discharge Data The plot provides the profile likelihood L( λ ) for the discharge data. L( λ ) is maximized when λ = 0.535 One can select λ = 0.5 or the square root as the transformation for the data Determining Sample Size Determining Sample Size In designing an experiment one needs to determine the sample size (e.g., # of replicates per factor level). One needs to specify: The desired level of power, 1- β The departure from the null hypothesis - usually expressed as An estimate of the residual variance, σ 2 = = a i i D 1 2 ) ( τ τ
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3 Distribution of Quadratic Forms Distribution of Quadratic Forms ) , , ( ~ / / ) 0 , ( ~ ) , ( ~ ) , ( ) , ( ) , ( . ) , ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 φ χ φ χ σ μ μ φ φ χ σ σ μ μ Σ μ φ φ χ Σ μ μ φ Σ φ χ Σ Σ Σ Σ Σ μ s r F s Q r Q F t, independen are s Q and r Q If ' with n ~ y y' Q I N ~ y if b) ' with n ~ y y' Q a) particular In A ' and ) (A matrix of rank k where k ~ Q then A A A (e.g., idempotent is ) (A matrix the and ) , ( N ~ y If form quadratic a is Ay y' Q Then constants known of matrix symmetric A and variables random of vector a y If 1 2 1 n 1 - 1 - n nxn n,1 = = = = = = = = = Determining Sample Size Determining Sample Size
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