mwtie_fr: Analogue of mwtie_xy for settings with grouped data

In EQUIVNONINF: Testing for Equivalence and Noninferiority

Description

Implementation of the asymptotically distribution-free test for equivalence of discrete distributions from which grouped data are obtained. Hypothesis formulation is in terms of the Mann-Whitney-Wilcoxon functional generalized to the case that ties between observations from different distributions may occur with positive probability. For details see Wellek S (2010) Testing statistical hypotheses of equivalence and noninferiority. Second edition, p.155.

Usage

mwtie_fr(k,alpha,m,n,eps1_,eps2_,x,y) 

Arguments

k	total number of grouped values which can be distinguished in the pooled sample
alpha	significance level
m	size of Sample 1
n	size of Sample 2
eps1_	absolute value of the left-hand limit of the hypothetical equivalence range for π_+/(1-π_0) - 1/2
eps2_	right-hand limit of the hypothetical equivalence range for π_+/(1-π_0) - 1/2
x	row vector with the m observations making up Sample1 as components
y	row vector with the n observations making up Sample2 as components

Details

Notation: π_+ and π_0 stands for the functional defined by π_+ = P[X>Y] and π_0 = P[X=Y], respectively, with X\sim F \equiv cdf of Population 1 being independent of Y\sim G \equiv cdf of Population 2.

Value

alpha	significance level
m	size of Sample 1
n	size of Sample 2
eps1_	absolute value of the left-hand limit of the hypothetical equivalence range for π_+/(1-π_0) - 1/2
eps2_	right-hand limit of the hypothetical equivalence range for π_+/(1-π_0) - 1/2
WXY_TIE	observed value of the U-statistics – based estimator of π_+/(1-π_0)
SIGMAH	square root of the estimated asymtotic variance of W_+/(1-W_0)
CRIT	upper critical bound to |W_+/(1-W_0) - 1/2 - (\varepsilon^\prime_2-\varepsilon^\prime_1)/2|/\hat{σ}
REJ	indicator of a positive [=1] vs negative [=0] rejection decision to be taken with the data under analysis

Author(s)

Stefan Wellek <[e-mail]>
Peter Ziegler <[e-mail]>

References

Wellek S, Hampel B: A distribution-free two-sample equivalence test allowing for tied observations. Biometrical Journal 41 (1999), 171-186.

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, Par. 6.4.

Examples

x <- c(1,1,3,2,2,3,1,1,1,2,1,2,2,2,1,2,1,3,2,1,2,1,1,1,1,1,1,1,1,1,1,1,2,1,3,1,3,2,1,1,
       2,1,2,1,1,2,2,1,2,1,1,1,1,1,2,2,1,2,2,1,3,1,2,1,1,2,2,1,2,2,1,1,1,3,2,1,1,1,2,1,
       3,3,3,1,2,1,2,2,1,1,1,2,2,1,1,2,1,1,2,3,1,3,2,1,1,1,1,2,2,2,1,1,2,2,3,2,1,2,1,1,
       2,2,1,2,2,2,1,1,2,3,2,1,3,2,1,1,1,2,2,2,2,1,2,2,1,1,1,1,2,1,1,1,2,1,2,2,1,2,2,2,
       2,1,1,2,1,2,2,1,1,1,1,3,1,1,2,2,1,1,1,2,2,2,1,2,3,2,2,1,2,1,2,1,1,2,1,2,2,1,1,1,
       2,2,2,2)
y <- c(2,1,2,2,1,1,2,2,2,1,1,2,1,3,3,1,1,1,1,1,1,2,2,3,1,1,1,3,1,1,1,1,1,1,1,2,2,3,2,1,
       2,2,2,1,2,1,1,2,2,1,2,1,1,1,1,2,1,2,1,1,3,1,1,1,2,2,2,1,1,1,1,2,1,2,1,1,2,2,2,2,
       2,1,1,1,3,2,2,2,1,2,3,1,2,1,1,1,2,1,3,3,1,2,2,2,2,2,2,1,2,1,1,1,1,2,2,1,1,1,1,2,
       1,3,1,1,2,1,2,1,2,2,2,1,2,2,2,1,1,1,2,1,2,1,2,1,1,1,2,1,2,2,1,1,1,1,2,2,3,1,3,1,
       1,2,2,2,1,1,1,1,2,1,1,3,2,2,3,1,2,2,1,1,2,1,1,2,1,2,2,1,2,1,2,2,2,1,1,1,1,1,1,1,
       1,1,1,2,1,3,2,2,1,1,1,2,2,1,1,2,1,2,1,2,2,2,1,2,3,1,1,2,1,2,2,1,1,1,1,2,2,2,1,1,
       3,2,1,2,2,2,1,1,1,2,1,2,2,1,2,1,1,2)
mwtie_fr(3,0.05,204,258,0.10,0.10,x,y)

Example output

Loading required package: BiasedUrn
 k = 3   alpha = 0.05   m = 204   n = 258   eps1_ = 0.1   eps2_ = 0.1   WXY_TIE = 0.5209283   SIGMAH = 0.04275121   CRIT = 0.7054544   REJ = 1

EQUIVNONINF documentation built on July 12, 2021, 5:08 p.m.