Nonparametric Statistics Tölfræði sem ekki byggir á mati stika Chapter 13 Nonparametric Statistics Tölfræði sem ekki byggir á mati stika

Sign Test for Paired Samples Formerkjapróf fyrir paraúrtök Suppose that paired or matched random samples are taken from a population, and the differences equal to 0 are discarded, leaving n observations. Calculate the difference for each pair of observations and record the sign of the difference. The sign test is used to test: Hugsum okkur að pöruð eða samstæð slembiúrtök séu tekin úr þýði, og þeim sem hafa mismun jafnan núlli sé hent, svo eftir sitji n athuganir. Reiknum mismun fyrir hvert par athugana og skráum formerki mismunar. Formerkjaprófið er notað til að prófa eftirfarandi: Where  is the proportion of nonzero observations in the population that are positive. The test-statistic S for the sign test is simply, Þar sem phi er það hlutfall athugana frábrugðnar núlli í þýðinu sem eru pósitífar. Útreiknað prófgildi S fyrir formerkjaprófið er einfaldlega, And S has a binomial distribution with  = 0.5 and n = the number of nonzero differences. Og S hefur tvíkostadreifingu (tvíliðudreifingu) með phi=0.5 og n=fjölda athugana með mismun frábrugðinn núlli.

Determining p-value for a Sign Test The p-value for a Sign Test is found using the binomial distribution with n = number of nonzero differences, S = number of positive differences, and  = 0.5 For an upper-tail test, H1:  > 0.5, p-value = P(x  S) For a lower-tail test, H1:  < 0.5, p-value = P(x  S) For a two-tail test, H1:   0.5, 2(p-value)

Product Preference Example for Sign Test (Example 13.1) Taster Rating Difference Sign of Difference Original Product New Product (Original -New) A B C D E F G H 6 4 5 8 3 7 9 -2 -5 1 -6 -3 -4 - +

The Sign Test: Normal Approximation (Large Samples) If the number n of nonzero sample observations is large, then the sign test is based on the normal approximation to the binomial with mean and standard deviation The test statistic is Where S* is the test-statistic corrected for continuity defined as: (S stjarna er útreiknað gildi fyrir prófið sem leiðrétt hefur verið fyrir samfelldni, og skilgreint sem eftirfarandi) For a two-tail test, S* = S + 0.5, if S <  or S* = S - 0.5, if S >  For upper-tail test, S* = S – 0.5 For lower-tail test, S* = S + 0.5

The Wilcoxon Signed Rank Test for Paired Samples The Wilcoxon Signed Rank Test can be employed when a random sample of matched pairs of observations is available Hægt að nota þegar fáanlegt er slemiúrtak athugana fyrir samstæð pör. Assume that the population distribution of the differences in these paired samples is symmetric Gerum ráð fyrir að þýðisdreifing fyrir mismun úrtakspara sé samhverf, and we want to test the null hypothesis that this distribution is centered at 0 og við viljum prófa núlltilgátu um að miðja dreifingar sé í núlli. Discarding pairs for which the difference is 0, we rank the remaining n absolute differences in ascending order (hækkandi röð ) with ties assigned the average of the ranks they occupy (sætistala bundin við meðaltal þess sætis sem þau tilheyra). The sums of the ranks corresponding to positive and negative differences are calculated, and the smaller of these sums is the Wilcoxon Signed Rank Statistic T, that is Summur jákæðra og neikvæðra sætistalna eru reiknaðar, og sú lægri notuð sem reiknað gildi fyrir Wilcoxon Formerkis Sætisgildi, það er T = min(T+, T- ) jafna (13.8) bls 539 Where T+ = the sum of the positive ranks T- = the sum of the negative ranks n = the number of nonzero differences The null hypothesis is rejected if T is less than or equal to the value in the Appendix table. Sjá glæru 8

The Wilcoxon Signed Rank Test: Normal Approximation (Large Samples) Under the null hypothesis that the population differences are centered on 0, the Wilcoxon Signed Rank Test has mean and variance given by Fyrir núlltilgátu and Then, for large n, the distribution of the random variable, Z, is approximately standard normal where If the number n of nonzero differences is large and T is the observed value of the Wilcoxon statistic, then the following test have significance level ,

The Wilcoxon Signed Rank Test: Normal Approximation (Large Samples) (continued) If the alternative hypothesis is one-sided, reject the null hypothesis if Fyrir einhala valtilgátu If the alternative hypothesis is two-sided, reject the null hypothesis if Fyrir tvíhala valtilgátu

Mann-Whitney U Statistic Assume that apart from any possible differences in central location, that two population distributions are identical. Suppose that n1 observations are available from the first population and n2 observations from the second. The two samples are pooled and the observations are ranked in ascending order with ties assigned the average of the next available ranks Úrtökin tvö eru sameinuð og athugunum gefnar sætistölur í hækkandi röð (raðað í hækkandi röð), þar sem athuganir með sömu sætistölu fá meðalgildi næstu sæta (sjá töflu 13.4 bls 546). Let R1 denote the sum of the ranks of the observations from the first population Látum R1 vera summu sætistala fyrir athuganir úr fyrsta þýði. The Mann-Whitney U statistic is then defined as Þá er Mann Whitney U reiknigildi skilgreint sem

Mann-Whitney U Test: Normal Approximation Assuming that the null hypothesis that the central locations of the two population distributions are the same, the Mann-Whitney U, has mean and variance Then for large sample sizes (both at least 10), the distribution of the random variable is approximated by the normal distribution.

Decision Rules for the Mann-Whitney Test It is assumed that the two population distributions are identical, apart from any possible differences in central location. Gert er ráð fyrir að þýðisdreifingarnar séu eins, að undanskildum einhverjum mögulegum mismun í miðlægri staðsetningu. In testing the null hypothesis that the two populations have the same central location, the decision rule for a given significance level is Við prófun núll tilgátu um að bæði þýðin hafi sömu miðlægu staðsetningu, er ákvörðunarreglan við sérhvert gefið alfa eftirfarandi For a one-sided upper-tailed alternative hypothesis, the decision rule is: Fora one-sided lower-tailed hypothesis, the decision rule is: For a two-sided alternative hypothesis, the decision rule is:

Wilcoxon Rank Sum Statistic T Suppose that n1 observations are available from the first population and n2 observations from the second. Gerum ráð fyrir að n1 athuganir séu fyrirliggjandi úr fyrsta þýði og n2 athugandir úr seinna þýði. The two samples are pooled and the observations are ranked in ascending order, with ties assigned the average of the next available ranks. Úrtökin tvö eru sameinuð og athugunum gefnar sætistölur í hækkandi röð (raðað í hækkandi röð), þar sem sameiginlegar sætistölur fá gildi sem er meðaltal næstu fáanlegra sæta (sjá töflu 13.4 bls 546). Let T denote the sum of the ranks of the observations from the first population (T in the Wilcoxon Rank Sum Test is the same as R1 in the Mann-Whitney U Test) Látum T tákna summu sætistalna athugana úr fyrsta þýði. Assuming that the null hypothesis is to be true, The Wilcoxon Rank Sum Statistic T has and Then, for large n, (n1  10 and n2  10) the distribution of the random variable, is approximated by the normal distribution.

Spearman’s Rank Correlation Raðfylgni Spearmans Suppose that a random sample (x1 , y1), . . .,(xn, yn) of n pairs of observations is taken. Gerum ráð fyrir að tekið sé slembiúrtak (x1 , y1), . . .,(xn, yn) fyrir n pör athugana. If the xi and yi are each ranked in ascending order and the sample correlation of these ranks is calculated, the resulting coefficient is called Spearman’s Rank Correlation Coefficient. Ef xi og yi er raðað hverju fyrir sig í hækkandi röð sætistalna og úrtaksfylgni sætistalna þessara raða reiknuð, þá er stuðulinn sem fæst kallaður raðfylgnistuðull Spearmans. If there are no tied ranks, an equivalent formula for computing this coefficient is Ef ekki eru neinar fastar sætistölur (um hvora röð gildir að engin sæti hafa (eru bundin) sömu sætistölu, þá má reikna stuðulinn með eftirfarandi jöfnu Where the di are the differences of the ranked pairs. Þar sem di er mismunur sætistalna para í röðum.

Spearman’s Rank Correlation (continued) The following tests of the null hypothesis H0 of no association in the population have significance level  Eftirfarandi próf fyrir núlltilgátuna H0 um engin tengsl í þýði hafa marktæknikröfu  To test against the alternative of positive association jákvæð tengsl, the decision rule is To test against the alternative of negative association neikvæð tengsl, the decision rule is To test against the two-sided alternative of some association, the decision rule is

Key Words Mann-Whitney U Test Sign Test Spearmann’s Rank Correlation Normal Approximation Statistic Sign Test Paired Samples P-value Population Median Spearmann’s Rank Correlation Coefficient Test Wilcoxon Rank Sum Test Statistic Wilcoxon Signed Rank Test Normal Approximation