Lecture 3 2016 09 15 - Stat 430/830 Lecture Outline...

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Stat 430/830Lecture September 15, 2016
OutlineNon-parametric analysis of one-factor designsVariance stabilization transformationsRandom effects modelSample size
Nonparametric Methods For Comparing K TreatmentsKruskal –Wallis TestUsed to test the hypothesis that atreatments are identical vs. the alternative that some treatments generate observations that are larger than others.-Rank all observations yijin ascending order and assign the rank, Rij. For tied observations assign the average rank of the tied observations.- Let Ri.the sum of the ranks in the i treatment
Kruskal Wallis TestThe test statistics iswith ninumber of observations on the itreatment and Nthe total number of observations. If there are no ties then Under moderate ties this result provides good approximation4)1(114)1(121122122.2NNRNSNNnRSHainjijaiiiiaiiiNnRNNHNNS12.2)1(3)1(12and12/)1(
Kruskal Wallis TestOne can also do an ANOVA on the ranks. It can be shown that the F test for the ANOVA on the ranks satisfiesTherefore, ANOVA on ranks is equivalent to the Kruskal Wallis TestHely Approximata21~)/()1()1/(aNHNaHFo
Example –Discharge Measure Methods
Example –Discharge Measure Methods
Example –Discharge Measure MethodsSourceDFSum of SquaresMean SquareF ValuePr > FModel31057.333333352.44444476.48<.0001Error2092.1666674.608333Corrected Total231149.500000R-SquareCoeff VarRoot MSERanks Mean0.91982017.173622.14670312.50000
Variance Stabilizing TransformationsIf the variance is proportional to a power of the mean, e.g.,   , then it can be shown thatw = y1-has constant variance
Box-Cox Transformation.selectinginhelpcanlikelihoodprofiletheofplotA)).L(maximizedthatonethe(e.g.,)}L(arg{maxselectingsuggestCoxBoxmodel.theofsquresofsumerrortheisSSandy ofgeometrictheisy,yyzwherezSSNconsLbygivenisforlikelihoodprofileThexywhereforyforyytiontransformathesuggestCox-Boxvalid.notisvarianceconstantofassumptionthebutx,forproposedismodellinearaSupposeEE1)()()()(/)(log2)(.0

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