equivalenceTest: eqivalenceTest: A package for evaluating equivalence of the...

In equivalenceTest: Equivalence Test for the Means of Two Normal Distributions

Description

We implemented two equivalence tests which evaluate equivalence in the means of two normal distributions. The first is discussed by \insertCitetsong2017development;textualequivalenceTest and the second by \insertCiteweng2018improved;textualequivalenceTest.

Details

Let X_{I,i}\sim_{IID} N(μ_I,σ_I) for I=T,R and i=1,...,n_I, where T stands for test distribution and R for reference distribution. The equivalence test here considers the following hypotheses,

H_0: |μ_T - μ_R| ≥ δ \;\mathrm{versus}\;H_1:|μ_T - μ_R| < δ,

where δ is the equivalence margin.

Let \hat{μ}_I and \hat{σ}_I^2 be the sample mean and unbiased sample variance estimates respectively for I=T,R. \insertCitetsong2017development;textualequivalenceTest define the follows test statistics,

τ_1=\frac{\hat{μ}_T-\hat{μ}_R+δ}{√{\hat{σ}_T^2/n_T^*+\hat{σ}_R^2/n_R^*}},

and

τ_2=\frac{\hat{μ}_T-\hat{μ}_R-δ}{√{\hat{σ}_T^2/n_T^*+\hat{σ}_R^2/n_R^*}},

where n_T^*=min\{n_T,1.5n_R\} and n_R^*=min\{n_R,1.5n_R\} are possibly adjusted sample sizes proposed by \insertCitedong2017adjustment;textualequivalenceTest.

The null hypothesis H_0 is rejected at nominal size α if both τ_1 > t_{1-α,df^*} and τ_2 < -t_{1-α,df^*} where t_{1-α,df^*} is the (1-α)-th quantile of the t-distribution with degree of freedom df^*, which is approximated by the Satterthwaite method with sample size adjusted and given as follows,

df^*=\frac{≤ft(\frac{\hat{σ}_T^2}{n_T^*}+\frac{\hat{σ}_R^2}{n_T^*}\right)^2}{\frac{1}{n_B-1} ≤ft(\frac{\hat{σ}_T^2}{n_T^*}\right)^2+\frac{1}{n_R-1} ≤ft(\frac{\hat{σ}_R^2}{n_R^*}\right)^2}.

The above assumes that δ is a predetermined constant. However, in many studies, such constant is not available, and δ must be determined by the study data. A popular choice is δ=k\hat{σ_R}. In this case, the above test may not control type I error well.

Replacing δ by kσ_R, the hypotheses becomes

H_0^\prime: |μ_T - μ_R| ≥ kσ_R \;\mathrm{versus}\;H_a^\prime |μ_T - μ_R| < kσ_R.

\insertCite

weng2018improved;textualequivalenceTest proposed an improved Wald test with the following test statistics,

τ_1^\prime=\frac{\hat{μ}_T-\hat{μ}_R+k\hat{σ}_R}{√{\frac{\tilde{σ}_{T,1}^2}{n_T^*}+≤ft(\frac{1}{n_R^*}+\frac{k^2V_{n_R}}{n_R-1}\right)\tilde{σ}_{R,1}^2}},

τ_2^\prime=\frac{\hat{μ}_T-\hat{μ}_R-k\hat{σ}_R}{√{\frac{\tilde{σ}_{T,2}^2}{n_T^*}+≤ft(\frac{1}{n_R^*}+\frac{k^2V_{n_R}}{n_R-1}\right)\tilde{σ}_{R,2}^2}},

where V_{n_R} = n_R-1-2\frac{Γ^22(n_R/2)}{Γ^2((n_R-1)/2)} and \tilde{σ}_{T,i} and \tilde{σ}_{R,i} are the restricted maximum likelihood estimator of σ_T and σ_R respectively with the constraint μ_T - μ_R = (-1)^i σ_R.

The null hypothesis H_0^\prime is rejected at nominal size α if both τ_1^\prime > z_{1-α} and τ_2^\prime < -z_{1-α} where z_{1-α} is the (1-α)-th quantile of the standard normal distribution.

For more details, see the cited reference.

References

\insertAllCited

equivalenceTest documentation built on May 1, 2019, 8:18 p.m.