mwtie_fr {EQUIVNONINF}

Analogue of mwtie_xy for settings with grouped data

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 \pi_+/(1-\pi_0) - 1/2

eps2_

right-hand limit of the hypothetical equivalence range for \pi_+/(1-\pi_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: \pi_+ and \pi_0 stands for the functional defined by \pi_+ = P[X>Y] and \pi_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 \pi_+/(1-\pi_0) - 1/2

eps2_

right-hand limit of the hypothetical equivalence range for \pi_+/(1-\pi_0) - 1/2

WXY_TIE

observed value of the U-statistics – based estimator of \pi_+/(1-\pi_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{\sigma}

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, \S 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)