Testing for equivalence and noninferiority

Description

The package makes available in R the complete set of programs accompanying S. Wellek's (2010) monograph "Testing Statistical Hypotheses of Equivalence and Noninferiority. Second Edition" (Chapman&Hall/CRC).

Note

In order to keep execution time of all examples below the limit set by the CRAN administration, in a number of cases the function calls shown in the documentation contain specifications which are insufficient for real applications. This holds in particular true for the width sw of search grids, which should be chosen to be .001 or smaller. Similarly, the maximum number of interval halving steps to be carried out in finding maximally admissible significance levels should be set to values >= 10.

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

Maintainer: Stefan Wellek <stefan.wellek@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2015.

Examples

bi2ste1(397,397,0.0,0.025,0.511,0.384)
bi2ste2(0.0,0.025,0.95,0.8,0.80,1.0)

Internal Equiv_Noninf Functions

Description

Internal Equiv_Noninf functions

Details

These functions are not to be called by the user

Critical constants and power of the UMP test for equivalence of a single binomial proportion to some given reference value

Description

The function computes the critical constants defining the uniformly most powerful (randomized) test for the problem p \le p_1 or p \ge p_2 versus p_1 < p < p_2, with p denoting the parameter of a binomial distribution from which a single sample of size n is available. In the output, one also finds the power against the alternative that the true value of p falls on the midpoint of the hypothetical equivalence interval (p_1 , p_2).

Usage

bi1st(alpha,n,P1,P2)

Arguments

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 4.3.

Examples

bi1st(.05,273,.65,.75)

Power of the exact Fisher type test for equivalence

Description

The function computes exact values of the power of the randomized UMPU test for equivalence in the strict (i.e. two-sided) sense of two binomial distributions and the conservative nonrandomized version of that test. It is assumed that the samples being available from both distributions are independent.

Usage

bi2aeq1(m,n,rho1,rho2,alpha,p1,p2)

Arguments

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 6.6.4.

Examples

bi2aeq1(302,302,0.6667,1.5,0.05,0.5,0.5)
  

Sample sizes for the exact Fisher type test for equivalence

Description

The function computes minimum sample sizes required in the randomized UMPU test for equivalence of two binomial distributions with respect to the odds ratio. Computation is done under the side condition that the ratio m/n has some predefined value \lambda.

Usage

bi2aeq2(rho1,rho2,alpha,p1,p2,beta,qlambd)

Arguments

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 6.6.4.

Examples

bi2aeq2(0.5,2.0,0.05,0.5,0.5,0.60,1.0)

Determination of a maximally raised nominal significance level for the nonrandomized version of the exact Fisher type test for equivalence

Description

The objective is to raise the nominal significance level as far as possible without exceeding the target significance level in the nonrandomized version of the test. The approach goes back to R.D. Boschloo (1970) who used the same technique for reducing the conservatism of the traditional nonrandomized Fisher test for superiority.

Usage

bi2aeq3(m,n,rho1,rho2,alpha,sw,tolrd,tol,maxh)

Arguments

Details

It should be noted that, as the function of the nominal level, the size of the nonrandomized test is piecewise constant. Accordingly, there is a nondegenerate interval of "candidate" nominal levels serving the purpose. The limits of such an interval can be read from the output. In terms of execution time, bi2aeq3 is the most demanding program of the whole package.

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Boschloo RD: Raised conditional level of significance for the 2 x 2- table when testing the equality of two probabilities. Statistica Neerlandica 24 (1970), 1-35.

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 6.6.5.

Examples

bi2aeq3(50,50,0.6667,1.5000,0.05,0.01,0.000001,0.0001,5)

Objective Bayesian test for noninferiority in the two-sample setting with binary data and the odds ratio as the parameter of interest

Description

Implementation of the construction described on pp. 179–181 of Wellek S (2010) Testing statistical hypotheses of equivalence and noninferiority. Second edition.

Usage

bi2by_ni_OR(N1,N2,EPS,SW,NSUB,ALPHA,MAXH)

Arguments

Details

The program uses 96-point Gauss-Legendre quadrature on each of the NSUB intervals into which the range of integration is partitioned.

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Statistical methods for the analysis of two-arm non-inferiority trials with binary outcomes. Biometrical Journal 47 (2005), 48–61.

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 6.6.2.

Examples

bi2by_ni_OR(10,10,1/3,.0005,10,.05,12)

Objective Bayesian test for noninferiority in the two-sample setting with binary data and the difference of the two proportions as the parameter of interest

Description

Implementation of the construction described on pp. 185-6 of Wellek S (2010) Testing statistical hypotheses of equivalence and noninferiority. Second edition.

Usage

bi2by_ni_del(N1,N2,EPS,SW,NSUB,ALPHA,MAXH)

Arguments

Details

The program uses 96-point Gauss-Legendre quadrature on each of the NSUB intervals into which the range of integration is partitioned.

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Statistical methods for the analysis of two-armed non-inferiority trials with binary outcomes. Biometrical Journal 47 (2005), 48–61.

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 6.6.3.

Examples

bi2by_ni_del(20,20,.10,.01,10,.05,5)

Determination of a corrected nominal significance level for the asymptotic test for equivalence of two unrelated binomial proportions with respect to the difference \delta of their population counterparts

Description

The program computes the largest nominal significance level which can be substituted for the target level \alpha without making the exact size of the asymptotic testing procedure larger than \alpha.

Usage

bi2diffac(alpha,m,n,del1,del2,sw,tolrd,tol,maxh)

Arguments

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 6.6.6.

Examples

bi2diffac(0.05,20,20,0.40,0.40,0.1,1e-6,1e-4,3)

Exact rejection probability of the asymptotic test for equivalence of two unrelated binomial proportions with respect to the difference of their expectations at any nominal level under an arbitrary parameter configuration

Description

The program computes exact values of the rejection probability of the asymptotic test for equivalence in the sense of -\delta_1 < p_1-p_2 < \delta_2, at any nominal level \alpha_0. [The largest \alpha_0 for which the test is valid in terms of the significance level, can be computed by means of the program bi2diffac.]

Usage

bi2dipow(alpha0,m,n,del1,del2,p1,p2)

Arguments

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 6.6.6.

Examples

bi2dipow(0.0228,50,50,0.20,0.20,0.50,0.50)

Power of the exact Fisher type test for relevant differences

Description

The function computes exact values of the power of the randomized UMPU test for relevant differences between two binomial distributions and the conservative nonrandomized version of that test. It is assumed that the samples being available from both distributions are independent.

Usage

bi2rlv1(m,n,rho1,rho2,alpha,p1,p2)

Arguments

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 11.3.3.

Examples

bi2rlv1(200,300,.6667,1.5,.05,.25,.10)
  

Sample sizes for the exact Fisher type test for relevant differences

Description

The function computes minimum sample sizes required in the randomized UMPU test for relevant differences between two binomial distributions with respect to the odds ratio. Computation is done under the side condition that the ratio m/n has some predefined value \lambda.

Usage

bi2rlv2(rho1,rho2,alpha,p1,p2,beta,qlambd)

Arguments

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 11.3.3.

Examples

                                     
bi2rlv2(.6667,1.5,.05,.70,.50,.50,2.0)  

Critical constants for the exact Fisher type UMPU test for equivalence of two binomial distributions with respect to the odds ratio

Description

The function computes the critical constants defining the uniformly most powerful unbiased test for equivalence of two binomial distributions with parameters p_1 and p_2 in terms of the odds ratio. Like the ordinary Fisher type test of the null hypothesis p_1 = p_2, the test is conditional on the total number S of successes in the pooled sample.

Usage

bi2st(alpha,m,n,s,rho1,rho2)

Arguments

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 6.6.4.

Examples

bi2st(.05,225,119,171, 2/3, 3/2)

Power of the exact Fisher type test for noninferiority

Description

The function computes exact values of the power of the randomized UMPU test for one-sided equivalence of two binomial distributions and its conservative nonrandomized version. It is assumed that the samples being available from both distributions are independent.

Usage

bi2ste1(m, n, eps, alpha, p1, p2)

Arguments

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, 6.6.1.

Examples

bi2ste1(106,107,0.5,0.05,0.9245,0.9065)

Sample sizes for the exact Fisher type test for noninferiority

Description

Sample sizes for the exact Fisher type test for noninferiority

Usage

bi2ste2(eps, alpha, p1, p2, bet, qlambd)

Arguments

Details

The program computes the smallest sample sizes m,n satisfying m/n = \lambda required for ensuring that the power of the randomized UMPU test does not fall below \beta.

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, 6.6.1.

Examples

bi2ste2(0.5,0.05,0.9245,0.9065,0.80,1.00)

Determination of a maximally raised nominal significance level for the nonrandomized version of the exact Fisher type test for noninferiority

Description

The objective is to raise the nominal significance level as far as possible without exceeding the target significance level in the nonrandomized version of the test. The approach goes back to R.D. Boschloo (1970) who used the same technique for reducing the conservatism of the traditional nonrandomized Fisher test for superiority.

Usage

bi2ste3(m, n, eps, alpha, sw, tolrd, tol, maxh)

Arguments

Details

It should be noted that, as the function of the nominal level, the size of the nonrandomized test is piecewise constant. Accordingly, there is a nondegenerate interval of "candidate" nominal levels serving the purpose. The limits of such an interval can be read from the output.

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Boschloo RD: Raised conditional level of significance for the 2 x 2- table when testing the equality of two probabilities. Statistica Neerlandica 24 (1970), 1-35.

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S6.6.2.

Examples

 bi2ste3(50, 50, 1/3, 0.05, 0.05, 1e-10, 1e-8, 10) 

Function to compute corrected nominal levels for the Wald type (asymptotic) test for one-sided equivalence of two binomial distributions with respect to the difference of success rates

Description

Implementation of the construction described on pp. 183-5 of Wellek S (2010) Testing statistical hypotheses of equivalence and noninferiority. Second edition.

Usage

bi2wld_ni_del(N1,N2,EPS,SW,ALPHA,MAXH)

Arguments

Details

The program computes the largest nominal significance level to be used for determining the critical lower bound to the Wald-type statistic for the problem of testing H:p_1 \le p_2 - \varepsilon versus K: p_1 < p_2 - \varepsilon.

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 6.6.3.

Examples

bi2wld_ni_del(25,25,.10,.01,.05,10) 

Exact confidence bounds to the relative excess heterozygosity (REH) exhibited by a SNP genotype distribution

Description

Implementation of the interval estimation procedure described on pp. 305-6 of Wellek S (2010) Testing statistical hypotheses of equivalence and noninferiority. Second edition.

Usage

cf_reh_exact(X1,X2,X3,alpha,SW,TOL,ITMAX)

Arguments

Details

The program exploits the structure of the family of all genotype distributions, which is 2-parameter exponential with \log(REH) as one of these parameters.

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S, Goddard KAB, Ziegler A: A confidence-limit-based approach to the assessment of Hardy-Weinberg equilibrium. Biometrical Journal 52 (2010), 253-270.

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 9.4.3.

Examples

               
cf_reh_exact(34,118,96,.05,.1,1E-4,25)

Mid-p-value - based confidence bounds to the relative excess heterozygosity (REH) exhibited by a SNP genotype distribution

Description

Implementation of the interval estimation procedure described on pp. 306-7 of Wellek S (2010) Testing statistical hypotheses of equivalence and noninferiority. Second edition.

Usage

cf_reh_midp(X1,X2,X3,alpha,SW,TOL,ITMAX)

Arguments

Details

The mid-p algorithm serves as a device for reducing the conservatism inherent in exact confidence estimation procedures for parameters of discrete distributions.

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Agresti A: Categorical data Analysis (2nd edn). Hoboken, NJ: Wiley, Inc., 2002, Section 1.4.5.

Wellek S, Goddard KAB, Ziegler A: A confidence-limit-based approach to the assessment of Hardy-Weinberg equilibrium. Biometrical Journal 52 (2010), 253-270.

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 9.4.3.

Examples

                      
cf_reh_midp(137,34,8,.05,.1,1E-4,25)

Critical constants and power against the null alternative of the UMP test for equivalence of the hazard rate of a single exponential distribution to some given reference value

Description

The function computes the critical constants defining the uniformly most powerful test for the problem \sigma \le 1/(1 + \varepsilon) or \sigma\ge (1 + \varepsilon) versus 1/(1 + \varepsilon) < \sigma < (1 + \varepsilon), with \sigma denoting the scale parameter [\equiv reciprocal hazard rate] of an exponential distribution.

Usage

exp1st(alpha,tol,itmax,n,eps) 

Arguments

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 4.2.

Examples

exp1st(0.05,1.0e-10,100,80,0.3)

Critical constants and power of the UMPI (uniformly most powerful invariant) test for dispersion equivalence of two Gaussian distributions

Description

The function computes the critical constants defining the optimal test for the problem \sigma^2/\tau^2 \le \varrho_1 or \sigma^2/\tau^2 \ge \varrho_2 versus \varrho_1 < \sigma^2/\tau^2 < \varrho_2, with (\varrho_1,\varrho_2) as a fixed nonempty interval around unity.

Usage

fstretch(alpha,tol,itmax,ny1,ny2,rho1,rho2) 

Arguments

Value

Note

If the two independent samples under analysis are from exponential rather than Gaussian distributions, the critical constants computed by means of fstretch with \nu_1 = 2m, \nu_2 = 2n, can be used for testing for equivalence with respect to the ratio of hazard rates. The only difference is that the ratio of sample means rather than variances has to be used as the test statistic then.

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 6.5.

Examples

fstretch(0.05, 1.0e-10, 50,40,45,0.5625,1.7689)

Critical constants of the exact UMPU test for approximate compatibility of a SNP genotype distribution with the Hardy-Weinberg model

Description

The function computes the critical constants defining the uniformly most powerful unbiased test for equivalence of the population distribution of the three genotypes distinguishable in terms of a single nucleotide polymorphism (SNP), to a distribution being in Hardy-Weinberg equilibrium (HWE).
The test is conditional on the total count S of alleles of the kind of interest, and the parameter \theta, in terms of which equivalence shall be established, is defined by \theta = \frac{\pi_2^2}{\pi_1(1-\pi_1-\pi_2)}, with \pi_1 and \pi_2 denoting the population frequence of homozygotes of the 1st kind and heterozygotes, respectively.

Usage

gofhwex(alpha,n,s,del1,del2) 

Arguments

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Tests for establishing compatibility of an observed genotype distribution with Hardy-Weinberg equilibrium in the case of a biallelic locus. Biometrics 60 (2004), 694-703.

Goddard KAB, Ziegler A, Wellek S: Adapting the logical basis of tests for Hardy-Weinberg equilibrium to the real needs of association studies in human and medical genetics. Genetic Epidemiology 33 (2009), 569-580.

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 9.4.2.

Examples

gofhwex(0.05,475,429,1-1/1.96,0.96)

Critical constants of the exact UMPU test for absence of a substantial deficit of heterozygotes as compared with a HWE-compliant SNP genotype distribution [noninferiority version of the test implemented by means of gofhwex]

Description

The function computes the critical constants defining the UMPU test for one-sided equivalence of the population distribution of a SNP, to a distribution being in Hardy-Weinberg equilibrium (HWE).
A substantial deficit of heterozygotes is defined to occur when the true value of the parametric function \omega = \frac{\pi_2/2}{\sqrt{\pi_1\pi_3}} [called relative excess heterozygosity (REH)] falls below unity by more than some given margin \delta_0.
Like its two-sided counterpart [see the description of the R function gofhwex], the test is conditional on the total count S of alleles of the kind of interest.

Usage

gofhwex_1s(alpha,n,s,del0) 

Arguments

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, pp. 300-302.

Examples

gofhwex_1s(0.05,133,65,1-1/1.96)

Establishing approximate independence in a two-way contingency table: Test statistic and critical bound

Description

The function computes all quantities required for carrying out the asymptotic test for approximate independence of two categorial variables derived in \S 9.2 of Wellek S (2010) Testing statistical hypotheses of equivalence and noninferiority. Second edition.

Usage

gofind_t(alpha,r,s,eps,xv) 

Arguments

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 9.2.

Examples

xv <- c(8, 13, 15, 6,  19, 21, 31, 7) 
gofind_t(0.05,2,4,0.15,xv)

Establishing goodness of fit of an observed to a fully specified multinomial distribution: test statistic and critical bound

Description

The function computes all quantities required for carrying out the asymptotic test for goodness rather than lack of fit of an observed to a fully specified multinomial distribution derived in \S 9.1 of Wellek S (2010) Testing statistical hypotheses of equivalence and noninferiority. Second edition.

Usage

gofsimpt(alpha,n,k,eps,x,pio) 

Arguments

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 9.1.

Examples

x<- c(17,16,25,9,16,17)
pio <- rep(1,6)/6
gofsimpt(0.05,100,6,0.15,x,pio)

Mann-Whitney test for equivalence of two continuous distributions of arbitrary shape: test statistic and critical upper bound

Description

Implementation of the asymptotically distribution-free test for equivalence of two continuous distributions in terms of the Mann-Whitney-Wilcoxon functional. For details see Wellek S (2010) Testing statistical hypotheses of equivalence and noninferiority. Second edition, \S 6.2.

Usage

mawi(alpha,m,n,eps1_,eps2_,x,y) 

Arguments

Details

Notation: \pi_+ stands for the Mann-Whitney functional defined by \pi_+ = P[X>Y], with X\sim F \equiv cdf of Population 1 being independent of Y\sim G \equiv cdf of Population 2.

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: A new approach to equivalence assessment in standard comparative bioavailability trials by means of the Mann-Whitney statistic. Biometrical Journal 38 (1996), 695-710.

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 6.2.

Examples

x <- c(10.3,11.3,2.0,-6.1,6.2,6.8,3.7,-3.3,-3.6,-3.5,13.7,12.6)
y <- c(3.3,17.7,6.7,11.1,-5.8,6.9,5.8,3.0,6.0,3.5,18.7,9.6)
mawi(0.05,12,12,0.1382,0.2602,x,y)

Determination of a corrected nominal significance level for the asymptotic test for noninferiority in the McNemar setting

Description

The program computes the largest nominal significance level which can be substituted for the target level \alpha without making the exact size of the asymptotic testing procedure larger than \alpha.

Usage

mcnasc_ni(alpha,n,del0,sw,tol,maxh)

Arguments

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 5.2.3.

Examples

mcnasc_ni(0.05,50,0.05,0.05,0.0001,5)

Bayesian test for noninferiority in the McNemar setting with the difference of proportions as the parameter of interest

Description

The program determines through iteration the largest nominal level \alpha_0 such that comparing the posterior probability of the alternative hypothesis K_1: \delta > -\delta_0 to the lower bound 1-\alpha_0 generates a critical region whose size does not exceed the target significance level \alpha. In addition, exact values of the power against specific parameter configurations with \delta = 0 are output.

Usage

mcnby_ni(N,DEL0,K1,K2,K3,NSUB,SW,ALPHA,MAXH)

Arguments

Details

The program uses 96-point Gauss-Legendre quadrature on each of the NSUB intervals into which the range of integration is partitioned.

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 5.2.3.

Examples

mcnby_ni(25,.10,.5,.5,.5,10,.05,.05,5)

Computation of the posterior probability of the alternative hypothesis of noninferiority in the McNemar setting, given a specific point in the sample space

Description

Evaluation of the integral on the right-hand side of Equation (5.24) on p. 88 of Wellek S (2010) Testing statistical hypotheses of equivalence and noninferiority. Second edition.

Usage

mcnby_ni_pp(N,DEL0,N10,N01)

Arguments

Details

The program uses 96-point Gauss-Legendre quadrature on each of 10 subintervals into which the range of integration is partitioned.

Value

Note

The program uses Equation (5.24) of Wellek S (2010) corrected for a typo in the middle line which must read

\int_{\delta_0}^{(1+\delta_0)/2}\Big[ B\big(n_{01}+1/2,n-n_{01}+1\big)\,\, p_{01}^{n_{01}-1/2}(1-p_{01})^{n-n_{01}}

.

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 5.2.3.

Examples

mcnby_ni_pp(72,0.05,4,5)

Determination of a corrected nominal significance level for the asymptotic test for equivalence of two paired binomial proportions with respect to the difference of their expectations (McNemar setting)

Description

The program computes the largest nominal significance level which can be substituted for the target level \alpha without making the exact size of the asymptotic testing procedure larger than \alpha.

Usage

 mcnemasc(alpha,n,del0,sw,tol,maxh)

Arguments

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 5.2.2.

Examples

mcnemasc(0.05,50,0.20,0.05,0.0005,5)

Exact rejection probability of the asymptotic test for equivalence of two paired binomial proportions with respect to the difference of their expectations (McNemar setting)

Description

The program computes exact values of the rejection probability of the asymptotic test for equivalence in the sense of -\delta_0 < p_{10}-p_{01} < \delta_0, at any nominal level \alpha. [The largest \alpha for which the test is valid in terms of the significance level, can be computed by means of the program mcnemasc.]

Usage

 mcnempow(alpha,n,del0,p10,p01)

Arguments

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, p.84.

Examples

mcnempow(0.024902,50,0.20,0.30,0.30)

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

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

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

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)

Distribution-free two-sample equivalence test for tied data: test statistic and critical upper bound

Description

Implementation of the asymptotically distribution-free test for equivalence of discrete distributions 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, \S 6.4.

Usage

mwtie_xy(alpha,m,n,eps1_,eps2_,x,y) 

Arguments

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

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

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)
y <- c(2,1,2,2,1,1,2,2,2,1,1,2)
mwtie_xy(0.05,10,12,0.10,0.10,x,y)

Bayesian posterior probability of the alternative hypothesis of probability-based individual bioequivalence (PBIBE)

Description

Implementation of the algorithm presented in \S 10.3.3 of Wellek S (2010) Testing statistical hypotheses of equivalence and noninferiority. Second edition.

Usage

po_pbibe(n,eps,pio,zq,s,tol,sw,ihmax)

Arguments

Details

The program uses 96-point Gauss-Legendre quadrature.

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Bayesian construction of an improved parametric test for probability-based individual bioequivalence. Biometrical Journal 42 (2000), 1039-52.

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 10.3.3.

Examples

po_pbibe(20,0.25,0.75,0.17451,0.04169, 10e-10,0.01,100)

Bayesian posterior probability of the alternative hypothesis in the setting of the one-sample t-test for equivalence

Description

Evaluation of the integral appearing on the right-hand side of equation (3.6) on p. 38 of Wellek S (2010) Testing statistical hypotheses of equivalence and noninferiority. Second edition

Usage

postmys(n,dq,sd,eps1,eps2,tol)

Arguments

Details

The program uses 96-point Gauss-Legendre quadrature.

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 3.2.

Examples

postmys(23,0.16,3.99,0.5,0.5,1e-6)

Confidence innterval inclusion test for average bioequivalence: exact power against an arbitrary specific alternative

Description

Evaluation of the integral on the right-hand side of equation (10.11) of p. 317 of Wellek S (2010) Testing statistical hypotheses of equivalence and noninferiority. Second edition

Usage

pow_abe(m,n,alpha,del_0,del,sig)

Arguments

Details

The program uses 96-point Gauss-Legendre quadrature.

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 10.2.1.

Examples

pow_abe(12,13,0.05,log(1.25),log(1.25)/2,0.175624)

Nonconditional power of the UMPU sign test for equivalence and its nonrandomized counterpart

Description

The program computes for each possible value of the number n_0 of zero observations the power conditional on N_0 = n_0 and averages these conditional power values with respect to the distribution of N_0. Equivalence is defined in terms of the logarithm of the ratio p_+/p_-, where p_+ and p_- denotes the probability of obtaining a positive and negative sign, respectively.

Usage

 powsign(alpha,n,eps1,eps2,poa)

Arguments

Value

Note

A special case of the test whose power is computed by this program, is the exact conditional equivalence test for the McNemar setting (cf. Wellek 2010, pp. 76-77).

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 5.1.

Examples

    
powsign(0.06580,50,0.847298,0.847298,0.26)

Signed rank test for equivalence of an arbitrary continuous distribution of the intraindividual differences in terms of the probability of a positive sign of a Walsh average: test statistic and critical upper bound

Description

Implementation of the paired-data analogue of the Mann-Whitney-Wilcoxon test for equivalence of continuous distributions. The continuity assumption relates to the intraindividual differences D_i. For details see Wellek S (2010) Testing statistical hypotheses of equivalence and noninferiority. Second edition,\S 5.4.

Usage

sgnrk(alpha,n,qpl1,qpl2,d) 

Arguments

Details

q_+ is the probability of getting a positive sign of the so-called Walsh-average of a pair of within-subject differences and can be viewed as a natural paired-observations analogue of the Mann-Whitney functional \pi_+ = P[X>Y].

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 5.4.

Examples

d <- c(-0.5,0.333,0.667,1.333,1.5,-2.0,-1.0,-0.167,1.667,0.833,-2.167,-1.833,
       4.5,-7.5,2.667,3.333,-4.167,5.667,2.333,-2.5)
sgnrk(0.05,20,0.2398,0.7602,d)

Generalized signed rank test for equivalence for tied data: test statistic and critical upper bound

Description

Implementation of a generalized version of the signed-rank test for equivalence allowing for arbitrary patterns of ties between the within-subject differences. For details see Wellek S (2010) Testing statistical hypotheses of equivalence and noninferiority. Second edition, \S 5.5.

Usage

srktie_d(n,alpha,eps1,eps2,d) 

Arguments

Details

Notation: q_+ and q_0 stands for the functional defined by q_+ = P[D_i+D_j>0] and q_0 = P[D_i+D_j=0], respectively, with D_i and D_j as the intraindividual differences observed in two individuals independently selected from the underlying bivariate population.

Value

Note

The function srktie_d can be viewed as the paired-data analogue of mwtie_xy

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 5.5.

Examples

d <- c(0.8,0.2,0.0,-0.1,-0.3,0.3,-0.1,0.4,0.6,0.2,0.0,-0.2,-0.3,0.0,0.1,0.3,-0.3,
       0.1,-0.2,-0.5,0.2,-0.1,0.2,-0.1)
srktie_d(24,0.05,0.2602,0.2602,d)

Analogue of srktie_d for settings where the distribution of intraindividual differences is concentrated on a finite lattice

Description

Analogue of the function srktie_d tailored for settings where the distribution of the within-subject differences is concentrated on a finite lattice. For details see Wellek S (2010) Testing statistical hypotheses of equivalence and noninferiority. Second edition, pp.112-3.

Usage

srktie_m(n,alpha,eps1,eps2,w,d) 

Arguments

Details

Notation: q_+ and q_0 stands for the functional defined by q_+ = P[D_i+D_j>0] and q_0 = P[D_i+D_j=0], respectively, with D_i and D_j as the intraindividual differences observed in two individuals independently selected from the underlying bivariate population.

Value

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, pp. 112-114.

Examples

d <- c(0.8,0.2,0.0,-0.1,-0.3,0.3,-0.1,0.4,0.6,0.2,0.0,-0.2,-0.3,0.0,0.1,0.3,-0.3,
       0.1,-0.2,-0.5,0.2,-0.1,0.2,-0.1)
srktie_m(24,0.05,0.2602,0.2602,0.1,d)

Critical constants and power against the null alternative of the one-sample t-test for equivalence with an arbitrary, maybe nonsymmetric choice of the limits of the equivalence range

Description

The function computes the critical constants defining the uniformly most powerful invariant test for the problem \delta/\sigma_D \le \theta_1 or \delta/\sigma_D \ge \theta_2 versus \theta_1 < \delta/\sigma_D < \theta_2, with (\theta_1,\theta_2) as a fixed nondegenerate interval on the real line. In addition, tt1st outputs the power against the null alternative \delta = 0.

Usage

tt1st(n,alpha,theta1,theta2,tol,itmax) 

Arguments

Value

Note

If the output value of ERR2 is NA, the deviation of the rejection probability at the right-hand boundary of the hypothetical equivalence interval from \alpha is smaller than the smallest real number representable in R.

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 5.3.

Examples

tt1st(36,0.05, -0.4716,0.3853,1e-10,50)

Critical constants and power against the null alternative of the two-sample t-test for equivalence with an arbitrary, maybe nonsymmetric choice of the limits of the equivalence range

Description

The function computes the critical constants defining the uniformly most powerful invariant test for the problem (\xi-\eta)/\sigma \le -\varepsilon_1 or (\xi-\eta)/\sigma \ge \varepsilon_2 versus -\varepsilon_1 < (\xi-\eta)/\sigma < \varepsilon_2, with \xi and \eta denoting the expected values of two normal distributions with common variance \sigma^2 from which independent samples are taken. In addition, tt2st outputs the power against the null alternative \xi = \eta.

Usage

tt2st(m,n,alpha,eps1,eps2,tol,itmax) 

Arguments

Value

Note

If the output value of ERR2 is NA, the deviation of the rejection probability at the right-hand boundary of the hypothetical equivalence interval from \alpha is smaller than the smallest real number representable in R.

Author(s)

Stefan Wellek <stefan.wellek@zi-mannheim.de>
Peter Ziegler <peter.ziegler@zi-mannheim.de>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 6.1.

Examples

tt2st(12,12,0.05,0.50,1.00,1e-10,50)