Lecture_2010_09_22

Lecture_2010_09_22 - 1 Stat Stat 430/830 430/830 Analysis...

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Unformatted text preview: 1 Stat Stat 430/830: 430/830: Analysis Analysis of of Variance 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 Kruskal Wallis Test 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 Kruskal Wallis Test 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---- = 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 ) ( . ) log( / ) 1 ( Example Example – Peak Discharge Data 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 3 Distribution of Quadratic Forms Distribution of Quadratic Forms ) , , ( ~ / / ) , ( ~ ) , ( ~ ) , ( ) , ( ) , ( ....
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Lecture_2010_09_22 - 1 Stat Stat 430/830 430/830 Analysis...

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