Nonparametric tests for location One-sample case Paired samples Two independent samples Statistics in Geophysics: Inferential Statistics III Steffen Unkel Department of Statistics Ludwig-Maximilians-University Munich, Germany Winter Term 2013/14 1/10 Nonparametric tests for location One-sample case Paired samples Two independent samples Background Tests not requiring assumptions involving specific parametric distributions for the data or for the sampling distribution of the test statistics are called nonparametric Nonparametric methods are appropriate if we know or suspect that the parametric assumption(s) required for a particular test are not met; a test statistic that is suggested or dictated by the problem at hand is a complicated function of the data, and its sampling distribution is unknown and/or cannot be derived analytically Only a few nonparametric tests for location will be presented here Winter Term 2013/14 2/10 Nonparametric tests for location One-sample case Paired samples Two independent samples One-sample Wilcoxon signed-rank test Let X1 , , Xn be a random sample with continuous cdf FX (·) Suppose that it is desired to test that the 0.5 quantile, xmed , of the population sampled from is a specific value, say δ0 Consider the test problems: (a) H0 : xmed = δ0 vs H1 : xmed = δ0 (b) H0 : xmed ≥ δ0 vs H1 : xmed < δ0 (c) H0 : xmed ≤ δ0 vs H1 : xmed > δ0 For i = , n, let Di = Xi − δ0 and define Zi = if Di > if Di < Winter Term 2013/14 3/10 Nonparametric tests for location One-sample case Paired samples Two independent samples One-sample Wilcoxon signed-rank test Test statistic n W+ = Ri Zi , i=1 where Ri is the rank of |Di | Rejection region: + + (a) W + > w1−α/2 or W + < wα/2 (b) W + < wα+ + (c) W + > w1−α , where wα+ denotes the α-quantile of the distribution of W + Winter Term 2013/14 4/10 Nonparametric tests for location One-sample case Paired samples Two independent samples One-sample Wilcoxon signed-rank test For sufficiently large samples: Approximation by N n(n+1) , n(n+1)(2n+1) 24 Test statistic: Z= W+ − n(n+1) a ∼ N (0, 1) n(n+1)(2n+1) 24 Rejection region: (a) Z > z1−α/2 or Z < zα/2 (b) Z < zα (c) Z > z1−α , where zα is the α-quantile of the standard normal distribution Winter Term 2013/14 5/10 Nonparametric tests for location One-sample case Paired samples Two independent samples Wilcoxon signed-rank test for paired data We assume that the sampling situation is such that we observe paired data (X1 , Y1 ), , (Xn , Yn ) For i = 1, , n, the differences Di = Xi − Yi arise from a continuous distribution and each pair (Xi , Yi ) is chosen randomly and independent The null hypothesis is that the median difference, δ, between pairs of observations is zero Consider the test problems: (a) H0 : δ = vs H1 : δ = (b) H0 : δ ≥ vs H1 : δ < (c) H0 : δ ≤ vs H1 : δ > Winter Term 2013/14 6/10 Nonparametric tests for location One-sample case Paired samples Two independent samples Wilcoxon signed-rank test for paired data Define if Di > 0 if Di < Zi = Test statistic: n W + = Ri Zi , i=1 where Ri is the rank of |Di | Rejection region: + + (a) W + > w1−α/2 or W + < wα/2 + + (b) W < wα + (c) W + > w1−α , where wα+ denotes the α-quantile of the distribution of W + Winter Term 2013/14 7/10 Nonparametric tests for location One-sample case Paired samples Two independent samples Wilcoxon rank-sum test Given two samples of independent data, the aim is to test for a possible difference in location The null hypothesis is that the two data samples have been drawn from the same distribution Under H0 there are n + m observations making up a single distribution, where n (m) denote the number of observations in sample (sample 2) The test statistic is a function of the ranks of the data values within the n + m observations that are pooled under H0 Winter Term 2013/14 8/10 Nonparametric tests for location One-sample case Paired samples Two independent samples Wilcoxon rank-sum test Let X1 , , Xn and Y1 , , Ym be two random samples from populations with continuous cdfs FX (·) and FY (·), respectively Consider the test problems: (a) H0 : xmed = ymed vs H1 : xmed = ymed (b) H0 : xmed ≥ ymed vs H1 : xmed < ymed (c) H0 : xmed ≤ ymed vs H1 : xmed > ymed Arrange the n + m observations of the pooled sample X1 , , Xn , Y1 , , Ym in ascending order Define Vi = if the i-th order statistic belongs to the X sample if the i-th order statistic belongs to the Y sample Winter Term 2013/14 9/10 Nonparametric tests for location One-sample case Paired samples Two independent samples Wilcoxon rank-sum test Test statistic: n+m Wn,m = n iVi = i=1 R(Xi ) , i=1 where R(Xi ) is the rank of Xi in the pooled sample Rejection region: (a) Wn,m > w1−α/2 (n, m) or Wn,m < wα/2 (n, m) (b) Wn,m < wα (n, m) (c) Wn,m > w1−α (n, m), where wα denotes the α-quantile of the distribution of Wn,m For sufficiently large samples: Approximation by N (n(n + m + 1)/2, nm(n + m + 1)/12) Winter Term 2013/14 10/10 ... Paired samples Two independent samples Background Tests not requiring assumptions involving specific parametric distributions for the data or for the sampling distribution of the test statistics are... X1 , , Xn , Y1 , , Ym in ascending order Define Vi = if the i-th order statistic belongs to the X sample if the i-th order statistic belongs to the Y sample Winter Term 2013/14 9/10 Nonparametric... distribution Winter Term 2013/14 5/10 Nonparametric tests for location One-sample case Paired samples Two independent samples Wilcoxon signed-rank test for paired data We assume that the sampling situation