This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 BoxCox 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 ) , , ( ~ / / ) , ( ~ ) , ( ~ ) , ( ) , ( ) , ( ....
View
Full
Document
 Fall '03
 StefanSteiner
 Variance, Latin Square design, Yij, Incomplete Block Designs, Block Effect, Graeco Latin Square Design

Click to edit the document details