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# APP NONPARAMETRIC STAT MATH 324

JMU

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This 18 page Class Notes was uploaded by Eunice Schoen on Saturday September 26, 2015. The Class Notes belongs to MATH 324 at James Madison University taught by Steven Garren in Fall. Since its upload, it has received 56 views. For similar materials see /class/214030/math-324-james-madison-university in Mathematics (M) at James Madison University.

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Date Created: 09/26/15

Chapter 3 K Sample Methods January 87 2009 3 KSample Methods General hypothesis test H0 F1x F2x Ha S or 2 with strict inequality holding for at least one x and at least one pair Lj Special ease The populations in Ha might differ only by location ie7 Ha Fz 7 m In this special case of Ha7 we may write Xij Mi 8m where Xij is the jth observation for the 2th treatrnent7 and the EMS are independent and have the same distribution 1 Section 311 The F Statistic January 87 2009 2 311 The F Statistic Suppose k 2 2 and the 617 are independent N07U We test for equality of means using an F statistic7 Via one way analysts of variance ANOVA7 from Math 321 Let X be the overall mean Let X 517 and m be the sample mean7 sample standard deviation7 and sample size7 respectively7 for the 2th population7 for t 17 quot71 De ne the F test statistic by El MOE X2 F 2210 Us need not memorize Under H07 the F statistic has an F distribution with k 7 1 degrees of freedom for the numerator and N7 k degrees of freedom for the denominator7 where N is the total sample size Regarding power7 do we prefer degrees of freedom to be large or small Section 311 The F Statistic January 87 2009 3 Graph the pdf for the following F distributions FZOA7 F4207 F1140 lntuitively7 variation among the treatment means variation among individuals within the same treatment Do we prefer a large value of F or a small value of F At which tail would rejection of H0 occur Problem 311 corn and tobacco quite hypothetical A farmer has three brands of fertilizer7 and is testing if the different brands of fertilizer result in different mean yields of corn Each fertilizer will be tested on four plots of land7 so tweer similar plots of land will be involved in the experiment In a separate independent experiment7 the three brands of fertilizer also are tested on a tobacco crop The data are given in the table below Section 311 The F Statistic January 87 2009 4 a Graph the data7 and explain which crop7 com or tobacco7 seems to be more in uenced by the brand of fertilizer lntuitively argue which crop should have the larger value of F Fertilizer Corn yield Tobacco yield 1 26 25 1 23 15 1 24 33 1 28 52 2 14 35 2 13 17 2 15 23 2 12 49 3 39 28 3 37 19 3 38 50 3 35 37 gt Z scan2 problern311 7 T7 T gt fertilizer z 7 1 l gt corn z 7 2 l gt tobacco z 7 3 gt plot fertilizer7 corn gt plot fertilizer7 tobacco Section 311 The F Statistic January 87 2009 5 b How many degrees of freedom are associated with each test c Assume independent and identically distributed normal errors 81 in the model XijMi5ij7 j17771i 17273 Test if the mean corn yield is the same for all three brands of fertilizer versus the alternative that at least two of the mean corn yields are different gt permftest corn7 fertilizer7 0 Section 311 The F Statistic January 87 2009 6 1 Under the assumption of normal errors7 test if the mean tobacco yield is the same for all three brands of fertilizer versus the alternative that at least two of the mean tobacco yields are different gt perrnftest tobacco7 fertilizer7 0 Suppose the error terms with nite standard deviation are NOT norrnally distributed7 but the sample size of each treatment is large ls the approximation to the F distribution still valid Section 312 Steps in Carrying Out the Permutation F Statistic January 87 2009 7 312 Steps in Carrying Out the Permutation F Statistic For a permutation F test7 we use the same F statistic7 but we do not approximate the distribution of the F statistic by an F distribution lnstead7 pvalues are based on the permutation distribution of the F statistic To obtain this pvalue 0 Compute the value of F based on the original data Permute the treatments or responses to obtain a permuted F statistic The observed F statistic is compared to the values of the F statistic under either all permutations or a large number of simulated permutations The pvalue is the proportion of permuted F statistics which are at least as large as the observed F statistic How many groupings of the treatments are possible for the sake of obtaining the exact pvalue from the permutation F test Section 312 Steps in Carrying Out the Permutation F Statistic January 87 2009 8 Revisit Problem 311 corn and tobacco However7 this time do NOT assume that the three populations for crop yield are approximately normal How many groupings of the treatments are possible for the sake of obtaining the exact pvalue from the permutation F test a Test if the mean com yield is the same for all three brands of fertilizer versus the alternative that at least two of the mean com yields are different Section 312 Steps in Carrying Out the Permutation F Statistic January 87 2009 9 b Test if the mean tobacco yield is the same for all three brands of fertilizer versus the alternative that at least two of the mean tobacco yields are different Section 312 Steps in Carrying Out the Permutation F Statistic January 87 2009 F Chisquare and normal distributions Plot the xi x3 and x3 probability density functions Plot the xi xg and x3 probability density functions Plot the xi x30 and x30 probability density functions Section 312 Steps in Carrying Out the Permutation F Statistic January 87 2009 11 For large degrees of freedom7 what distribution does a X2 distribution approximate 0 Recall that the ANOVA F test has k 7 1 degrees of freedom for the numerator and N7 k degrees of freedom for the denominator7 where k is the number of treatments and N is the sample size Consider k 2 treatments and sample sizes of N 3 or N 4 or N 5 Plot the FLL7 F12 and F13 probability density functions For xed k 2 but large N7 what distribution does an F distribution approximate For any xed k but large N7 what distribution does k 71F approximate7 where F has an F distribution with k 7 1 and N 7 k degrees of freedom Section 32 The Kruskal Wallis Test January 87 2009 12 32 The Kruskal Wallis Test Are the permutation F test and ANOVA F test heavily in uenced by outliers The Kruskal Wallis test statistic is based on ranks Here7 the populations are assumed to be identical7 except for possibly location7 and we test for equality of medians or means7 if nite Idea Convert the original N observations to their appropriate ranks 17 27 37 7 N Then7 perform the permutation F test using these ranks to obtain the right tailed p value This p value is the same one obtained when permuting the Kruskal Wallis test statistic The Kruskal Wallis test statistic is 12 k N 1 2 T gt lt KW m lt15 7 need not memorize7 where Rl is the average rank for the ith sample The modi ed formula for KW based on ties is given in section 322 Section 32 The Kruskal Wallis Test January 87 2009 13 To obtain the Kruskal Wallis 10 yalue 0 Compute the value of KW based on the original ranks Permute the treatments or ranks to obtain a permuted KW statistic The observed KW statistic is compared to the values of the KW statistic under either all permutations or a large number of simulated permutations The p Value is the proportion of permuted KW statistics at least as large as the observed KW statistic The textbook shows that a strictly monotone increasing relationship exists between the permutation F statistic based on ranks and the Kruskal Wallis test statistic For large sample sizes with xed k under H07 the KW test statistic is approximately X2 distributed7 with how many degrees of freedom ls the Kruskal Wallis test heavily in uenced by outliers Section 32 The Kmskal Wallis Test January 87 2009 14 Revisit Problem 311 corn and tobacco a Scan in the data7 and perform tests based on the TOBACCO data gt Z scan2 problem311 7 T7 T gt fertilizer Z 7 1 gt tobacco Z 7 3 b Determine the value of the Kruskal Wallis test statistic7 without using the macro kruskaltesti gt ranks ranktobacco gtnic47474 gtNsumni gt meanranki c meanranks147 meanranks587 meanranks912 Section 32 The Kruskal Wallis Test January 87 2009 15 gtKW12NN1sumnigtkmeanranki7N12A2 c Determine the p Value of the Kruskal Wallis test7 using the macro permftest7 d How many degrees of freedom are associated with the Kruskal Wallis test e Determine the asymptotic pvalue of the Kruskal Wallis test statistic7 without using the macro kruskaltest7 Section 32 The Kmskal Wallis Test January 87 2009 16 f Obtain the Kruskal Wallis statistic and pvalue using kruskaltesti gt 7kruskaltest Problem 321 Birth conditions and IQ Steel 19597 Biometrics pre sented the data below for testing whether certain conditions are associated with a lowering of IQ The lQ score was obtained for 24 girls of which six each are healthy7 mom39s7 premature7 and Rh negative Section 32 The Kruskal Wallis Test January 87 2009 17 Healthy 103 111 136 106 122 114 Anoxic 119 100 97 89 112 86 Rh negative 89 132 86 114 114 125 Premature 92 114 86 119 131 94 a Graph the data lntuitively7 do the four populations of birth conditions seem to differ regarding HQ gt 1Q scan2 problem321 7 F7 T Read in data as a vector gt birthcondition C rep1767 rep2767 rep3767 rep476 gt plot birthcondition7 1Q b Assume independent and identically distributed errors 817 in the model XijMi5ij7 j177ni 91774 Formulate the null and alternative hypotheses for testing if the mean IQ scores are the same for all four types of birth conditions versus the alternative that at least two of the mean IQ scores are different Section 32 The Kruskal Wallis Test January 87 2009 18 c How many degrees of freedom are associated with the Kruskal Wallis test 1 Perform the Kruskal Wallis test e How many degrees of freedom are associated with the ANOVA F test f Perform the ANOVA F test

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