應(yīng)用統(tǒng)計(jì)學(xué)-非參數(shù)

1、BEO225 Applied Statistics for Business,Week Five –Non-parametric tests for independent & paired samples,SPSS非參數(shù)檢驗(yàn),前面進(jìn)行的假設(shè)檢驗(yàn)和方差分析，大都是在數(shù)據(jù)服從正態(tài)分布或近似地服從正態(tài)分布的條件下進(jìn)行的。但是如果總體的分布未知，如何進(jìn)行總體參數(shù)的檢驗(yàn)，或者如何檢驗(yàn)總體服從一個(gè)指定的分布，都可以歸結(jié)為非參數(shù)檢驗(yàn)方法

2、。,SPSS非參數(shù)檢驗(yàn),在總體分布未知的情況下，利用樣本數(shù)據(jù)對(duì)總體的分布或各總體的分布是否有顯著差異進(jìn)行推斷。單樣本非參數(shù)檢驗(yàn)兩獨(dú)立樣本的非參數(shù)檢驗(yàn)多獨(dú)立樣本的非參數(shù)檢驗(yàn)兩配對(duì)樣本的非參數(shù)檢驗(yàn)多配對(duì)樣本的非參數(shù)檢驗(yàn),Comparing two or more populations,本章主要內(nèi)容,單樣本的非參數(shù)檢驗(yàn)兩獨(dú)立樣本非參數(shù)檢驗(yàn)兩配對(duì)樣本非參數(shù)檢驗(yàn),Non-parametric tests,Non-parametr

3、ic tests do not require that the data is normal (or close to normal).If the data is skewed or heavy or light tailed, a non-parametric test may be appropriate. In parametric tests, we test characteristics of a populatio

4、n looking at one of its parameters i.e. mean.For ex, whether µex,male > µex,femaleBut in non-parametric tests, we test whether two population locations differ (populations medians).,Non-parametric tests fo

5、r independent samples,If the data is not normal A t test may give misleading results. The nonparametric statistical techniques presentedare designed to compare two populations; deal with ranked data;analyse the rela

6、tive location of the populations studied (medians).In testing the locations we will not refer to any parameter, thus the procedure’s name.,Nonparametric methods can replace the parametric methods used for numerical data

7、 when the population is not normal.In nonparametric tests we hypothesise on the population locations (not necessarily their means).,,,,,,Two populations – same location,Two populations – different locations,,,,,Wilcoxon

8、 rank sum test for independent samples,The problem characteristics this test is dealing with are:the problem objective is to compare two populationsthe data are either ranked or numerical, but not normalthe samples ar

9、e independent.Ranking tests deal with “l(fā)ocations” rather than mean values.If two populations share the same location they are, in effect, a single population.Another way of saying it is that we are comparing two distr

10、ibutions of data.Null hypothesis:The null hypothesis is that two distributions are identical.,SPSS兩獨(dú)立樣本非參數(shù)檢驗(yàn),(一)目的由獨(dú)立樣本數(shù)據(jù)推斷兩總體的分布是否存在顯著差異(或兩樣本是否來(lái)自同一總體)。 (二)基本假設(shè)H0:兩總體分布無(wú)顯著差異(兩樣本來(lái)自同一總體)(三)數(shù)據(jù)要求樣本數(shù)據(jù)和分組標(biāo)志,SPSS兩獨(dú)立樣本非參數(shù)

11、檢驗(yàn),(四)基本方法1.曼-惠特尼U檢驗(yàn)(Mann-Whitney U):平均秩檢驗(yàn)將兩樣本數(shù)據(jù)混合并按升序排序求出其秩對(duì)兩樣本的秩分別求平均如果兩樣本的平均秩大致相同,則認(rèn)為兩總體分布無(wú)顯著差異,Example 1, Wilcoxon rank sum test,Consider the sales of a sample of 10 male and 10 female sales executives.Null hyp

12、othesis:The distributions of sales in the relevant target population are the same for males and females. How does the test work? Under the null hypothesis, whether a sales executive is male or female is irrelevant. i.

13、e. There is only one population.Procedure:We rank the sales data 1 = lowest up to 20 = highest.We add the ranks for each group to find TM and TF, sum of ranks,Example 1, Wilcoxon rank sum test,Rationale:If sales for

14、males and females belong to the same distribution, the sample of males should have about as many high ranks as the sample of females, and as many low ranks.If the samples are the same size (nM = nF), under the null hypo

15、thesis, TM ? TF.TM >> TF or TM << TF suggests the null hypothesis is not true.If nM ? nF, under the null mean(RankM) ? mean(RankF).,Data & Ranks assigned,male445576568758904559 60rank 1 4

16、11 514 616 2 7 8female63887892939984709450rank 91512171820131019 3TM = 74, TF = 136.,Example 1, Wilcoxon rank sum test,Results:Obviously TM << TF but is the differen

17、ce sufficient for us to reject the null hypothesis?If we have “l(fā)arge” samples, we can use the normal distribution.But we don’t.The test statistic is the rank total, T, for the first group.First group?Any group.,Exam

18、ple 1, Wilcoxon rank sum test,Sample size:Large:nA, nB > 10.Expected (or average) value of T:Standard deviation:,,The Z test statistic:Then the same as a t test.,Example 1, Wilcoxon rank sum test,Sample si

19、ze:“Small” samples: Critical values:For nA = nB = 10, on a 2-sided test at ? = 0.05, the critical values are TL = 78 and TU = 132.Result:TM = 74 < TL ? reject HO.Interpretation:At 5% significance level, there

20、is sufficient evidence to conclude that the distributions of the sales of male and female sales executives are not identical in the target population.,Ho: The distributions of sales in the relevant target population are

21、the same for males and females. H1: The distributions of sales in the relevant target population are not the same for males and females.,Table E8, Appendices P 755, Berenson et al (2013),Critical Vaules of the Wilcoxon

22、 Rank Sum Test for Independent Samples.Test statistics is T = TA , where TA is the rank sum of the sample with the smaller sample size.? = .025 one tail, ? = .05 two-tail,Decision rule,,0,132,78,TM,TM = 74,Reject Ho i

23、fT < TL or TU < TIf T < TL the location of the population A (male executives) is to the left of BIf TU < T, the location of the population A (male executives) is to the right of B,Reject,Reject,Example 1

24、(cont.), using SPSS,In SPSSAnalyse/Nonparametric Tests/Independent Samples.Select Statistics / Mann-Whitney U.Procedure:First, check the data.Does it seem normal?To check:Descriptive Statistics / Descriptives. Se

25、lect Statistics / Skewness and Kurtosis.Select Charts / Histograms and With normal curve.,,,Descriptive Statistics,,10,-0.067,,.072,,-1.506,,.0342,,10,,,,Sales,,Valid N (listwise),,Statistic,,Statistic,,Std. Error,,Stat

26、istic,,Std. Error,,N,,Skewness,,Kurtosis,,,,,,,,,,,,,,,,,,,,Data has strangepeak(s)(Kurtosis << 0),,Data is less concentrated around the mean than a normal distribution.,Data is fairly symmetric (Skewness ? 0

27、),Data is obviously not normally distributed.,Example 1 (cont.),Which test?Obviously the data is not normally distributed.In view of this and the small number of observations (20), the non-parametric test seems appropr

28、iate.,,Rank total of males.,,Rank total of females.,,Average rank total of males,Average rank total of females.,Mean ranks are different , but different enough to reject the null hypothesis?,Check sig value to test Ho,Ca

29、n reject the null that males and females have the same distributionof sales results because Sig < 0.05.,Ho: males and females have the same distribution of sales results H1: males and females do not have the sam

30、e distribution of sales results,Conclusion: At 5% significance level, there is sufficient evidence to conclude that the distributions of the sales of male and female sales executives are not identical in the target popul

31、ation.,Non-parametric tests for paired samples,Wilcoxon signed rank sum testThis is used whenThe objective is to compare 2 populationsThe data are numerical but not normally distributedThe samples are matched pairsW

32、e begin as in the t test for paired samples.We calculate Di = X1i - X2i.Null hypothesisThe null hypothesis is that the differences are symmetric around zero.Under the null hypothesis, the distribution of negative dif

33、ferences is the same as the distribution of positive differences.,SPSS兩配對(duì)樣本非參數(shù)檢驗(yàn),正負(fù)符號(hào)檢驗(yàn)(sign)將樣本2的各樣本值減去樣本1的各樣本值.如果差值為正,則記為正號(hào);如果差值為負(fù),則記為負(fù)號(hào)如果正號(hào)的個(gè)數(shù)與負(fù)號(hào)的個(gè)數(shù)相當(dāng),則認(rèn)為無(wú)顯著變化.否則,認(rèn)為有顯著變化例如:采用新訓(xùn)練方法前后的最好成績(jī)比較,兩配對(duì)樣本W(wǎng)ilcoxon符號(hào)秩檢驗(yàn),首先，按

34、照符號(hào)檢驗(yàn)的方法，用正負(fù)號(hào)分別表示兩組對(duì)應(yīng)樣本數(shù)據(jù)差值情況。然后，將差值變量進(jìn)行升序排序，并求出差值變量的秩。分別計(jì)算正號(hào)秩及統(tǒng)計(jì)量W+ 和負(fù)號(hào)秩及統(tǒng)計(jì)量W-,基本思想,Wilcoxon signed rank test cont…,ProcedureConsider the differences Di = X1i - X2i.Look at the differences and ignore the negative si

35、gns. Delete any zero differences from the sample.Rank the differences (ignoring the signs) 1 = lowest to n = highest.Attach signs to the ranks.Add the positive ranks to get T+ and the negative ranks to get T-.,Ration

36、ale:If HO is true, T+ ? T-The test statistic:T = min{T+, T-}. For n > 50, T has a normal distribution so proceed as for a t test.,Example 2,The following table shows exports figures of a product to Country X be

37、fore and after a bi-lateral trade agreement is signed.,Estimate the difference b/w sales figures & assign ranks ignoring the + or - sign,Now estimate the total of positive & negative ranks (T),Decision criteriaO

38、btain critical values from Table E9, Berenson et al (2013) Appendices (p 756)On a 2-sided test with ? = 0.05 and n = 10, critical values are 8 & 47Rule is that reject HO if T < 8ResultSince T = 17, we cannot

39、reject the null hypothesisConclusionAt 5% significance level, we are unable to conclude that there is any difference in exports of the product to Country X in the population distribution of before and after the trade

40、agreement.,,Table E9 Berenson et al (2013) Appendices (p756),Wilcoxon signed rank sum test Using SPSS,In SPSS Analyse/Nonparametric Tests/Related Samples.Select:Nominate before and after variables.Normality?With onl

41、y a few observations, non-normality could be a serious problem.Check the distribution of differences.,,,Descriptive Statistics,,10,-1.260,,1.002,,2.183,,1.375,,10,,,,Sales,,Valid N (listwise),,Statistic,,Statistic,,Std.

42、 Error,,Statistic,,Std. Error,,N,,Skewness,,Kurtosis,,,,,,,,,,,,,,,,,,,,,Not close to 0.,Not close to 0,Data does not appear to be normally distributed.In view of this and the small number of observations (10), the non

43、-parametric test seems appropriate.,,,Ranks,,3,a,,5.67,,17.00,,7,b,,5.43,,38.00,,0,c,,,,10,,,,,Negative Ranks,,Positive Ranks,,Ties,,Total,,,Before - After,,N,,Mean Rank,,Sum of Ranks,,,,,AFTER < BEFORE,a.,,AFTER >

44、 BEFORE,b.,AFTER = BEFORE,c.,,,,,,,,,,Wilcoxon Signed Ranks Test,Number of negative differences,Number of positive differences,Test statistic,Smaller of the rank totals,Test Statistics,b,,-1.071,a,,0.284,,Z,,Asymp. Si

45、g. (2-tailed),,,Based on negative ranks.,a.,,,Wilcoxon Signed Ranks Test,b.,,,,,Before - After,,,,,Cannot reject the null that there is no Difference in the distributions ofbefore and after results because Sig >

46、0.05.,Hypotheses,HO: the distribution of differences (after - before) is symmetric around 0 (no difference).HA: the distribution of differences (after - before) is not symmetric around 0 (difference).,Advise re tests us

47、ing SPSS,Mostly, SPSS produces sig values for two-sided tests.Note that for chi-square and F, there is no such thing as a two sided test.Therefore, use the sig value given in SPSS and compare against ? ? 0.05.Decision