eqivalenceTest: A package for evaluating equivalence of 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 Tsong et al. (2017) and the second by Weng et al. (2018).
Details
Let XI,i∼IIDN(μI,σI) for I=T,R and i=1,...,nI, where T stands for test distribution and R for reference distribution.
The equivalence test here considers the following hypotheses,
H0:∣μT−μR∣≥δversusH1:∣μT−μR∣<δ,
where δ is the equivalence margin.
Let μ^I and σ^I2 be the sample mean and unbiased sample variance estimates respectively for I=T,R.
Tsong et al. (2017) define the follows test statistics,
τ1=σ^T2/nT∗+σ^R2/nR∗μ^T−μ^R+δ,
and
τ2=σ^T2/nT∗+σ^R2/nR∗μ^T−μ^R−δ,
where nT∗=min{nT,1.5nR} and nR∗=min{nR,1.5nR} are possibly adjusted sample sizes proposed by Dong et al. (2017).
The null hypothesis H0 is rejected at nominal size α if both τ1>t1−α,df∗ and τ2<−t1−α,df∗ where t1−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,
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=kσR^. In this case, the above test may not control type I error well.
Replacing δ by kσR, the hypotheses becomes
H0′:∣μT−μR∣≥kσRversusHa′∣μT−μR∣<kσR.
Weng et al. (2018) proposed an improved Wald test with the following test statistics,
where VnR=nR−1−2Γ2((nR−1)/2)Γ22(nR/2) and σ~T,i and σ~R,i are the restricted maximum likelihood estimator of σT and σR respectively with the constraint μT−μR=(−1)iσR.
The null hypothesis H0′ is rejected at nominal size α if both τ1′>z1−α and τ2′<−z1−α where z1−α is the (1−α)-th quantile of the standard normal distribution.
For more details, see the cited reference.
References
Dong X, Weng Y, Tsong Y (2017).
“Adjustment for unbalanced sample size for analytical biosimilar equivalence assessment.”
Journal of biopharmaceutical statistics, 27(2), 220–232.
Tsong Y, Dong X, Shen M (2017).
“Development of statistical methods for analytical similarity assessment.”
Journal of biopharmaceutical statistics, 27(2), 197–205.
Weng Y, Tsong Y, Shen M, Wang C (2018).
“Improved Wald Test for Equivalence Assessment of Analytical Biosimilarity.”
International Journal of Clinical Biostatistics and Biometrics, 4(1), 1–10.