Package 'EQUIVNONINF'

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 <[email protected]>
Peter Ziegler <[email protected]>

Maintainer: Stefan Wellek <[email protected]>

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)

---

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

alpha	significance level
n	sample size
P1	lower limit of the hypothetical equivalence range for the binomial parameter $p$
P2	upper limit of the hypothetical equivalence range for $p$

Value

alpha	significance level
n	sample size
P1	lower limit of the hypothetical equivalence range for the binomial parameter $p$
P2	upper limit of the hypothetical equivalence range for $p$
C1	left-hand limit of the critical interval for the observed number $X$ of successes
C2	right-hand limit of the critical interval for $X$
GAM1	probability of rejecting the null hypothesis when it turns out that $X=C_1$
GAM2	probability of rejecting the null hypothesis for $X=C_2$
POWNONRD	Power of the nonrandomized version of the test against the alternative $p = (p_1+p_2)/2$
POW	Power of the randomized UMP test against the alternative $p = (p_1+p_2)/2$

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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

m	size of Sample 1
n	size of Sample 2
rho1	lower limit of the hypothetical equivalence range for the odds ratio
rho2	upper limit of the hypothetical equivalence range for the odds ratio
alpha	significance level
p1	true success rate in Population 1
p2	true success rate in Population 2

Value

m	size of Sample 1
n	size of Sample 2
rho1	lower limit of the hypothetical equivalence range for the odds ratio
rho2	upper limit of the hypothetical equivalence range for the odds ratio
alpha	significance level
p1	true success rate in Population 1
p2	true success rate in Population 2
POWNR	Power of the nonrandomized version of the test
POW	Power of the randomized UMPU test

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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

rho1	lower limit of the hypothetical equivalence range for the odds ratio
rho2	upper limit of the hypothetical equivalence range for the odds ratio
alpha	significance level
p1	true success rate in Population 1
p2	true success rate in Population 2
beta	target value of power
qlambd	sample size ratio $m/n$

Value

rho1	lower limit of the hypothetical equivalence range for the odds ratio
rho2	upper limit of the hypothetical equivalence range for the odds ratio
alpha	significance level
p1	true success rate in Population 1
p2	true success rate in Population 2
beta	target value of power
qlambd	sample size ratio $m/n$
M	minimum size of Sample 1
N	minimum size of Sample 2
POW	Power of the randomized UMPU test attained with the computed values of m,n

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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

m	size of Sample 1
n	size of Sample 2
rho1	lower limit of the hypothetical equivalence range for the odds ratio
rho2	upper limit of the hypothetical equivalence range for the odds ratio
alpha	significance level
sw	width of the search grid for determining the maximum of the rejection probability on the common boundary of the hypotheses
tolrd	horizontal distance from 0 and 1, respectively of the left- and right-most boundary point to be included in the search grid
tol	upper bound to the absolute difference between size and target level below which the search for a corrected nominal level terminates
maxh	maximum number of interval halving steps to be carried out in finding the maximally raised nominal level

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

m	size of Sample 1
n	size of Sample 2
rho1	lower limit of the hypothetical equivalence range for the odds ratio
rho2	upper limit of the hypothetical equivalence range for the odds ratio
alpha	significance level
sw	width of the search grid for determining the maximum of the rejection probability on the common boundary of the hypotheses
tolrd	horizontal distance from 0 and 1, respectively of the left- and right-most boundary point to be included in the search grid
tol	upper bound to the absolute difference between size and target level below which the search for a corrected nominal level terminates
maxh	maximum number of interval halving steps to be carried out in finding the maximally raised nominal level
ALPH_0	current trial value of the raised nominal level searched for
NHST	number of interval-halving steps performed up to now
SIZE	size of the critical region corresponding to $\alpha_0$

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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 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

N1	size of Sample 1
N2	size of sample 2
EPS	noninferiority margin to the difference of success probabilities
SW	width of the search grid for determining the maximum of the rejection probability on the common boundary of the hypotheses
NSUB	number of subintervals for partitioning the range of integration
ALPHA	target significance level
MAXH	maximum number of interval halving steps to be carried out in finding the maximally admissible nominal level

Details

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

Value

N1	size of Sample 1
N2	size of sample 2
EPS	noninferiority margin to the difference of success probabilities
NSUB	number of subintervals for partitioning the range of integration
SW	width of the search grid for determining the maximum of the rejection probability on the common boundary of the hypotheses
ALPHA0	result of the search for the largest admissible nominal level
SIZE0	size of the critical region corresponding to $\alpha_0$
SIZE_UNC	size of the critical region of the test at uncorrected nominal level

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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)

---

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

N1	size of sample 1
N2	size of sample 2
EPS	noninferiority margin to the deviation of the odds ratio from unity
SW	width of the search grid for determining the maximum of the rejection probability on the common boundary of the hypotheses
NSUB	number of subintervals for partitioning the range of integration
ALPHA	target significance level
MAXH	maximum number of interval halving steps to be carried out in finding the maximally admissible nominal level

Details

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

Value

N1	size of sample 1
N2	size of sample 2
EPS	noninferiority margin to the deviation of the odds ratio from unity
NSUB	number of subintervals for partitioning the range of integration
SW	width of the search grid for determining the maximum of the rejection probability on the common boundary of the hypotheses
ALPHA0	result of the search for the largest admissible nominal level
SIZE0	size of the critical region corresponding to $\alpha_0$
SIZE_UNC	size of the critical region of the test at uncorrected nominal level

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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)

---

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

alpha	significance level
m	size of Sample 1
n	size of Sample 2
del1	absolute value of the lower limit of the hypothetical equivalence range for $p_1-p_2$
del2	upper limit of the hypothetical equivalence range for $p_1-p_2$
sw	width of the search grid for determining the maximum of the rejection probability on the common boundary of the hypotheses
tolrd	horizontal distance of the left- and right-most boundary point to be included in the search grid
tol	upper bound to the absolute difference between size and target level below which the search for a corrected nominal level terminates
maxh	maximum number of interval halving steps to be carried out in finding the maximally raised nominal level

Value

alpha	significance level
m	size of Sample 1
n	size of Sample 2
del1	absolute value of the lower limit of the hypothetical equivalence range for $p_1-p_2$
del2	upper limit of the hypothetical equivalence range for $p_1-p_2$
sw	width of the search grid for determining the maximum of the rejection probability on the common boundary of the hypotheses
tolrd	horizontal distance of the left- and right-most boundary point to be included in the search grid
tol	upper bound to the absolute difference between size and target level below which the search for a corrected nominal level terminates
maxh	maximum number of interval halving steps to be carried out in finding the maximally raised nominal level
NH	number of interval-halving steps actually performed
ALPH_0	value of the raised nominal level obtained after NH steps
SIZE0	size of the critical region corresponding to $\alpha_0$
ERROR	error indicator answering the question of whether or not the sufficient condition for the correctness of the result output by the program, was satisfied

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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

alpha0	nominal significance level
m	size of Sample 1
n	size of Sample 2
del1	absolute value of the lower limit of the hypothetical equivalence range for $p_1-p_2$
del2	upper limit of the hypothetical equivalence range for $p_1-p_2$
p1	true value of the success probability in Population 1
p2	true value of the success probability in Population 2

Value

alpha0	nominal significance level
m	size of Sample 1
n	size of Sample 2
del1	absolute value of the lower limit of the hypothetical equivalence range for $p_1-p_2$
del2	upper limit of the hypothetical equivalence range for $p_1-p_2$
p1	true value of the success probability in Population 1
p2	true value of the success probability in Population 2
POWEX0	exact rejection probability under $(p_1,p_2)$ of the test at nominal level $\alpha_0$ for equivalence of two binomial distributions with respect to the difference of the success probabilities
ERROR	error indicator answering the question of whether or not the sufficient condition for the correctness of the result output by the program, was satisfied

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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

m	size of Sample 1
n	size of Sample 2
rho1	lower limit of the hypothetical equivalence range for the odds ratio
rho2	upper limit of the hypothetical equivalence range for the odds ratio
alpha	significance level
p1	true success rate in Population 1
p2	true success rate in Population 2

Value

m	size of Sample 1
n	size of Sample 2
rho1	lower limit of the hypothetical equivalence range for the odds ratio
rho2	upper limit of the hypothetical equivalence range for the odds ratio
alpha	significance level
p1	true success rate in Population 1
p2	true success rate in Population 2
POWNR	power of the nonrandomized version of the test
POW	power of the randomized UMPU test

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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

rho1	lower limit of the hypothetical equivalence range for the odds ratio
rho2	upper limit of the hypothetical equivalence range for the odds ratio
alpha	significance level
p1	true success rate in Population 1
p2	true success rate in Population 2
beta	target value of power
qlambd	sample size ratio $m/n$

Value

rho1	lower limit of the hypothetical equivalence range for the odds ratio
rho2	upper limit of the hypothetical equivalence range for the odds ratio
alpha	significance level
p1	true success rate in Population 1
p2	true success rate in Population 2
beta	target value of power
qlambd	sample size ratio $m/n$
M	minimum size of Sample 1
N	minimum size of Sample 2
POW	power of the randomized UMPU test attained with the computed values of m, n

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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

alpha	significance level
m	size of Sample 1
n	size of Sample 2
s	observed total count of successes
rho1	lower limit of the hypothetical equivalence range for the odds ratio $\varrho = \frac{p_1(1-p_2)}{p_2(1-p_1)}$
rho2	upper limit of the hypothetical equivalence range for $\varrho$

Value

alpha	significance level
m	size of Sample 1
n	size of Sample 2
s	observed total count of successes
rho1	lower limit of the hypothetical equivalence range for the odds ratio $\varrho = \frac{p_1(1-p_2)}{p_2(1-p_1)}$
rho2	upper limit of the hypothetical equivalence range for $\varrho$
C1	left-hand limit of the critical interval for the number $X$ of successes observed in Sample 1
C2	right-hand limit of the critical interval for $X$
GAM1	probability of rejecting the null hypothesis when it turns out that $X=C_1$
GAM2	probability of rejecting the null hypothesis for $X=C_2$

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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

m	size of Sample 1
n	size of Sample 2
eps	noninferiority margin to the odds ratio $\varrho$, defined to be the maximum acceptable deviation of the true value of $\varrho$ from unity
alpha	significance level
p1	success rate in Population 1
p2	success rate in Population 2

Value

m	size of Sample 1
n	size of Sample 2
eps	noninferiority margin to the odds ratio $\varrho$, defined to be the maximum acceptable deviation of the true value of $\varrho$ from unity
alpha	significance level
p1	success rate in Population 1
p2	success rate in Population 2
POWNR	power of the nonrandomized version of the test
POW	power of the randomized UMPU test

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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

eps	noninferiority margin to the odds ratio
alpha	significance level
p1	success rate in Population 1
p2	success rate in Population 2
bet	target power value
qlambd	sample size ratio $m/n$

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

eps	noninferiority margin to the odds ratio
alpha	significance level
p1	success rate in Population 1
p2	success rate in Population 2
bet	target power value
qlambd	sample size ratio $m/n$
M	minimum size of Sample 1
N	minimum size of Sample 2
POW	power of the randomized UMPU test attained with the computed values of m, n

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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

m	size of Sample 1
n	size of Sample 2
eps	noninferiority margin to the odds ratio $\varrho$, defined to be the maximum acceptable deviation of the true value of $\varrho$ from unity
alpha	target significance level
sw	width of the search grid for determining the maximum of the rejection probability on the common boundary of the hypotheses
tolrd	horizontal distance from 0 and 1, respectively, of the left- and right-most boundary point to be included in the search grid
tol	upper bound to the absolute difference between size and target level below which the search for a corrected nominal level terminates
maxh	maximum number of interval-halving steps to be carried out in finding the maximally raised nominal level

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

m	size of Sample 1
n	size of Sample 2
eps	noninferiority margin to the odds ratio $\varrho$, defined to be the maximum acceptable deviation of the true value of $\varrho$ from unity
alpha	target significance level
sw	width of the search grid for determining the maximum of the rejection probability on the common boundary of the hypotheses
tolrd	horizontal distance from 0 and 1, respectively, of the left- and right-most boundary point to be included in the search grid
tol	upper bound to the absolute difference between size and target level below which the search for a corrected nominal level terminates
maxh	maximum number of interval-halving steps to be carried out in finding the maximally raised nominal level
ALPH_0	current trial value of the raised nominal level searched for
NHST	number of interval-halving steps performed up to now
SIZE	size of the critical region corresponding to $\alpha_0$

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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$6.6.2.

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

N1	size of Sample 1
N2	size of Sample 2
EPS	noninferiority margin to the difference of success probabilities
SW	width of the search grid for determining the maximum of the rejection probability on the common boundary of the hypotheses
ALPHA	target significance level
MAXH	maximum number of interval-halving steps

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

N1	size of Sample 1
N2	size of Sample 2
EPS	noninferiority margin to the difference of success probabilities
SW	width of the search grid for determining the maximum of the rejection probability on the common boundary of the hypotheses
ALPHA	target significance level
MAXH	maximum number of interval-halving steps
ALPHA0	corrected nominal level
SIZE0	size of the critical region of the test at nominal level ALPHA0
SIZE_UNC	size of the test at uncorrected nominal level ALPHA
ERR_IND	indicator taking value 1 when it occurs that the sufficient condition allowing one to restrict the search for the maximum of the rejection probability under the null hypothesis to its boundary, fails to be satisfied; otherwise the indicator retains its default value 0.

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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

X1	count of homozygotes of the first kind [$\leftrightarrow$ genotype AA]
X2	count of heterozygotes [$\leftrightarrow$ genotype AB]
X3	count of homozygotes of the second kind [$\leftrightarrow$ genotype BB]
alpha	1 - confidence level
SW	width of the search grid for determining an interval covering the parameter point at which the conditional distribution function takes value $\alpha$ and $1-\alpha$, respectively
TOL	numerical tolerance to the deviation between the computed confidence limits and their exact values
ITMAX	maximum number of interval-halving steps

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

X1	count of homozygotes of the first kind [$\leftrightarrow$ genotype AA]
X2	count of heterozygotes [$\leftrightarrow$ genotype AB]
X3	count of homozygotes of the second kind [$\leftrightarrow$ genotype BB]
alpha	1 - confidence level
SW	width of the search grid for determining an interval covering the parameter point at which the conditional distribution function takes value $\alpha$ and $1-\alpha$, respectively
TOL	numerical tolerance to the deviation between the computed confidence limits and their exact values
ITMAX	maximum number of interval-halving steps
C_l_exact	exact conditional lower $(1-\alpha)$-confidence bound to REH
C_r_exact	exact conditional upper $(1-\alpha)$-confidence bound to REH

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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

X1	count of homozygotes of the first kind [$\leftrightarrow$ genotype AA]
X2	count of heterozygotes [$\leftrightarrow$ genotype AB]
X3	count of homozygotes of the second kind [$\leftrightarrow$ genotype BB]
alpha	1 - confidence level
SW	width of the search grid for determining an interval covering the parameter point at which the conditional distribution function takes value $\alpha$ and $1-\alpha$, respectively
TOL	numerical tolerance to the deviation between the computed confidence limits and their exact values
ITMAX	maximum number of interval-halving steps

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

X1	count of homozygotes of the first kind [$\leftrightarrow$ genotype AA]
X2	count of heterozygotes [$\leftrightarrow$ genotype AB]
X3	count of homozygotes of the second kind [$\leftrightarrow$ genotype BB]
alpha	1 - confidence level
SW	width of the search grid for determining an interval covering the parameter point at which the conditional distribution function takes value $\alpha$ and $1-\alpha$, respectively
TOL	numerical tolerance to the deviation between the computed confidence limits and their exact values
ITMAX	maximum number of interval-halving steps
C_l_midp	lower $(1-\alpha)$-confidence bound to REH based on conditional mid-p-values
C_r_midp	upper $(1-\alpha)$-confidence bound to REH based on conditional mid-p-values

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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

alpha	significance level
tol	tolerable deviation from $\alpha$ of the rejection probability at either boundary of the hypothetical equivalence interval
itmax	maximum number of iteration steps
n	sample size
eps	margin determining the hypothetical equivalence range symmetrically on the log-scale

Value

alpha	significance level
tol	tolerable deviation from $\alpha$ of the rejection probability at either boundary of the hypothetical equivalence interval
itmax	maximum number of iteration steps
n	sample size
eps	margin determining the hypothetical equivalence range symmetrically on the log-scale
IT	number of iteration steps performed until reaching the stopping criterion corresponding to TOL
C1	left-hand limit of the critical interval for $T =\sum_{i=1}^n X_i$
C2	right-hand limit of the critical interval for $T =\sum_{i=1}^n X_i$
ERR1	deviation of the rejection probability from $\alpha$ under $\sigma = 1/(1 + \varepsilon)$
POW0	power of the randomized UMP test against the alternative $\sigma = 1$

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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

alpha	significance level
tol	tolerable deviation from $\alpha$ of the rejection probability at either boundary of the hypothetical equivalence interval
itmax	maximum number of iteration steps
ny1	number of degrees of freedom of the estimator of $\sigma^2$
ny2	number of degrees of freedom of the estimator of $\tau^2$
rho1	lower equivalence limit to $\sigma^2/\tau^2$
rho2	upper equivalence limit to $\sigma^2/\tau^2$

Value

alpha	significance level
tol	tolerable deviation from $\alpha$ of the rejection probability at either boundary of the hypothetical equivalence interval
itmax	maximum number of iteration steps
ny1	number of degrees of freedom of the estimator of $\sigma^2$
ny2	number of degrees of freedom of the estimator of $\tau^2$
rho1	lower equivalence limit to $\sigma^2/\tau^2$
rho2	upper equivalence limit to $\sigma^2/\tau^2$
IT	number of iteration steps performed until reaching the stopping criterion corresponding to TOL
C1	left-hand limit of the critical interval for $T = \frac{n-1}{m-1} \sum_{i=1}^m (X_i-\overline{X})^2 / \sum_{j=1}^{n-1} (Y_j-\overline{Y})^2$
C2	right-hand limit of the critical interval for $T = \frac{n-1}{m-1} \sum_{i=1}^m (X_i-\overline{X})^2 / \sum_{j=1}^{n-1} (Y_j-\overline{Y})^2$
ERR	deviation of the rejection probability from $\alpha$ under $\sigma^2/\tau^2 = \varrho_1$
POW0	power of the UMPI test against the alternative $\sigma^2/\tau^2 = 1$

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 <[email protected]>
Peter Ziegler <[email protected]>

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

alpha	significance level
n	number of genotyped individuals
s	observed count of alleles of the kind of interest
del1	absolute value of the lower equivalence limit to $\theta/4 - 1$
del2	upper equivalence limit to $\theta/4 - 1$

Value

alpha	significance level
n	number of genotyped individuals
s	observed count of alleles of the kind of interest
del1	absolute value of the lower equivalence limit to $\theta/4 - 1$
del2	upper equivalence limit to $\theta/4 - 1$
C1	left-hand limit of the critical interval for the observed number $X_2$ of heterozygotes
C2	right-hand limit of the critical interval for the observed number $X_2$
GAM1	probability of rejecting the null hypothesis when it turns out that $X_2=C_1$
GAM2	probability of rejecting the null hypothesis for $X_2=C_2$

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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

alpha	significance level
n	number of genotyped individuals
s	observed count of alleles of the kind of interest
del0	noninferiority margin for $\omega$, which has to satisfy $\omega > 1-\delta_0$ under the alternative hypothesis to be established

Value

alpha	significance level
n	number of genotyped individuals
s	observed count of alleles of the kind of interest
del0	noninferiority margin for $\omega$, which has to satisfy $\omega > 1-\delta_0$ under the alternative hypothesis to be established
C	left-hand limit of the critical interval for the observed number $X_2$ of heterozygotes
GAM	probability of rejecting the null hypothesis when it turns out that $X_2=C$

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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

alpha	significance level
r	number of rows of the contingency table under analysis
s	number of columns of the contingency table under analysis
eps	margin to the Euclidean distance between the vector $\mathbf{\pi}$ of true cell probabilities and the associated vector of products of marginal totals
xv	row vector of length $r\cdot s$ whose $(i-1)s + j$-th component is the entry in cell $(i,j)$ of the $r\times s$ contingency table under analysis $i=1,\ldots,r$, $j=1,\ldots,s$.

Value

n	size of the sample to which the input table relates
alpha	significance level
r	number of rows of the contingency table under analysis
s	number of columns of the contingency table under analysis
eps	margin to the Euclidean distance between the vector $\mathbf{\pi}$ of true cell probabilities and the associated vector of products of marginal totals
X(r, s)	observed cell counts
DSQ_OBS	observed value of the squared Euclidean distance
VN	square root of the estimated asymtotic variance of $\sqrt{n}DSQ\_OBS$
CRIT	upper critical bound to $\sqrt{n}DSQ\_OBS$
REJ	indicator of a positive [=1] vs negative [=0] rejection decision to be taken with the data under analysis

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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

alpha	significance level
n	sample size
k	number of categories
eps	margin to the Euclidean distance between the vectors $\mathbf{\pi}$ and $\mathbf{\pi}_0$ of true and hypothesized cell probabilities
x	vector of length $k$ with the observed cell counts as components
pio	prespecified vector of cell probabilities

Value

alpha	significance level
n	sample size
k	number of categories
eps	margin to the Euclidean distance between the vectors $\mathbf{\pi}$ and $\mathbf{\pi}_0$ of true and hypothesized cell probabilities
X(1, K)	observed cell counts
PI0(1, K)	hypothecized cell probabilities
DSQPIH_0	observed value of the squared Euclidean distance
VN_N	square root of the estimated asymtotic variance of $\sqrt{n}DSQPIH\_0$
CRIT	upper critical bound to $\sqrt{n}DSQPIH\_0$
REJ	indicator of a positive [=1] vs negative [=0] rejection decision to be taken with the data under analysis

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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

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 $\pi_+ - 1/2$
eps2_	right-hand limit of the hypothetical equivalence range for $\pi_+ - 1/2$
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: $\pi_+$ 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

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 $\pi_+ - 1/2$
eps2_	right-hand limit of the hypothetical equivalence range for $\pi_+ - 1/2$
W+	observed value of the $U$-statistics estimator of $\pi_+$
SIGMAH	square root of the estimated asymtotic variance of $W_+$
CRIT	upper critical bound to $|W_+ - 1/2 - (\varepsilon^\prime_2-\varepsilon^\prime_1)/2|/\hat{\sigma}$
REJ	indicator of a positive [=1] vs negative [=0] rejection decision to be taken with the data under analysis

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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

alpha	significance level
n	sample size
del0	absolute value of the noninferiority margin for $\delta := p_{10}-p_{01}$, with $p_{10}$ and $p_{01}$ denoting the probabilities of discordant pairs of both kinds
sw	width of the search grid for determining the maximum of the rejection probability on the common boundary of the hypotheses
tol	upper bound to the absolute difference between size and target level below which the search for a corrected nominal level terminates
maxh	maximum number of interval halving steps to be carried out in finding the maximally raised nominal level

Value

alpha	significance level
n	sample size
del0	absolute value of the noninferiority margin for $\delta := p_{10}-p_{01}$, with $p_{10}$ and $p_{01}$ denoting the probabilities of discordant pairs of both kinds
sw	width of the search grid for determining the maximum of the rejection probability on the common boundary of the hypotheses
ALPH_0	value of the corrected nominal level obtained after nh steps
SIZE_UNC	exact size of the rejection region of the test at uncorrected nominal level $\alpha$
SIZE0	exact size of the rejection region of the test at nominal level $\alpha_0$
NH	number of interval-halving steps actually performed

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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

N	sample size
DEL0	noninferiority margin to the difference of the parameters of the marginal binomial distributions under comparison
K1	Parameter 1 of the Dirichlet prior for the family of trinomial distributions
K2	Parameter 2 of the Dirichlet prior for the family of trinomial distributions
K3	Parameter 3 of the Dirichlet prior for the family of trinomial distributions
NSUB	number of subintervals for partitioning the range of integration
SW	width of the search grid for determining the maximum of the rejection probability on the common boundary of the hypotheses
ALPHA	target significance level
MAXH	maximum number of interval halving steps to be carried out in finding the maximally raised nominal level

Details

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

Value

N	sample size
DEL0	noninferiority margin to the difference of the parameters of the marginal binomial distributions under comparison
K1	Parameter 1 of the Dirichlet prior for the family of trinomial distributions
K2	Parameter 2 of the Dirichlet prior for the family of trinomial distributions
K3	Parameter 3 of the Dirichlet prior for the family of trinomial distributions
NSUB	number of subintervals for partitioning the range of integration
SW	width of the search grid for determining the maximum of the rejection probability on the common boundary of the hypotheses
ALPHA	target significance level
MAXH	maximum number of interval halving steps to be carried out in finding the maximally raised nominal level
ALPHA0	result of the search for the largest admissible nominal level
SIZE0	size of the critical region corresponding to $\alpha_0$
SIZE_UNC	size of the critical region of test at uncorrected nominal level $\alpha$
POW	power against 7 different parameter configurations with $\delta =0$

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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

N	sample size
DEL0	noninferiority margin to the difference of the parameters of the marginal binomial distributions under comparison
N10	count of pairs with $(X,Y) = (1,0)$
N01	count of pairs with $(X,Y) = (0,1)$

Details

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

Value

N	sample size
DEL0	noninferiority margin to the difference of the parameters of the marginal binomial distributions under comparison
N10	count of pairs with $(X,Y) = (1,0)$
N01	count of pairs with $(X,Y) = (0,1)$
PPOST	posterior probability of the alternative hypothesis $K_1: \delta > -\delta_0$ with respect to the noninformative prior determined according to Jeffrey's rule

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 <[email protected]>
Peter Ziegler <[email protected]>

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

alpha	significance level
n	sample size
del0	upper limit set to $|p_{10}-p_{01}|$ under the alternative hypothesis of equivalence, with $p_{10}$ and $p_{01}$ denoting the probabilities of discordant pairs of both kinds
sw	width of the search grid for determining the maximum of the rejection probability on the common boundary of the hypotheses
tol	upper bound to the absolute difference between size and target level below which the search for a corrected nominal level terminates
maxh	maximum number of interval halving steps to be carried out in finding the maximally raised nominal level

Value

alpha	significance level
n	sample size
del0	upper limit set to $|p_{10}-p_{01}|$ under the alternative hypothesis of equivalence, with $p_{10}$ and $p_{01}$ denoting the probabilities of discordant pairs of both kinds
sw	width of the search grid for determining the maximum of the rejection probability on the common boundary of the hypotheses
ALPH_0	value of the corrected nominal level obtained after nh steps
NH	number of interval-halving steps actually performed
ERROR	error indicator messaging "!!!!!" if the sufficient condition for the correctness of the result output by the program was found violated

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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

alpha	nominal significance level
n	sample size
del0	upper limit set to $|\delta|$ under the alternative hypothesis of equivalence
p10	true value of $P[X=1,Y=0]$
p01	true value of $P[X=0,Y=1]$

Value

alpha	nominal significance level
n	sample size
del0	upper limit set to $|\delta|$ under the alternative hypothesis of equivalence
p10	true value of $P[X=1,Y=0]$
p01	true value of $P[X=0,Y=1]$
POW	exact rejection probability of the asymptotic McNemar test for equivalence at nominal level $\alpha$
ERROR	error indicator messaging "!!!!!" if the sufficient condition for the correctness of the result output by the program was found violated

Author(s)

Stefan Wellek <[email protected]>
Peter Ziegler <[email protected]>

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

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 <[email protected]>
Peter Ziegler <[email protected]>

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)

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