equivalenceTest {equivalenceTest}

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 X_{I,i}\sim_{IID} N(\mu_I,\sigma_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: |\mu_T - \mu_R| \ge \delta \;\mathrm{versus}\;H_1:|\mu_T - \mu_R| < \delta,

where \delta is the equivalence margin.

Let \hat{\mu}_I and \hat{\sigma}_I^2 be the sample mean and unbiased sample variance estimates respectively for I=T,R. Tsong et al. (2017) define the follows test statistics,

\tau_1=\frac{\hat{\mu}_T-\hat{\mu}_R+\delta}{\sqrt{\hat{\sigma}_T^2/n_T^*+\hat{\sigma}_R^2/n_R^*}},

and

\tau_2=\frac{\hat{\mu}_T-\hat{\mu}_R-\delta}{\sqrt{\hat{\sigma}_T^2/n_T^*+\hat{\sigma}_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 Dong et al. (2017).

The null hypothesis H_0 is rejected at nominal size \alpha if both \tau_1 > t_{1-\alpha,df^*} and \tau_2 < -t_{1-\alpha,df^*} where t_{1-\alpha,df^*} is the (1-\alpha)-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{\left(\frac{\hat{\sigma}_T^2}{n_T^*}+\frac{\hat{\sigma}_R^2}{n_T^*}\right)^2}{\frac{1}{n_B-1} \left(\frac{\hat{\sigma}_T^2}{n_T^*}\right)^2+\frac{1}{n_R-1} \left(\frac{\hat{\sigma}_R^2}{n_R^*}\right)^2}.

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

Replacing \delta by k\sigma_R, the hypotheses becomes

H_0^\prime: |\mu_T - \mu_R| \ge k\sigma_R \;\mathrm{versus}\;H_a^\prime |\mu_T - \mu_R| < k\sigma_R.

Weng et al. (2018) proposed an improved Wald test with the following test statistics,

\tau_1^\prime=\frac{\hat{\mu}_T-\hat{\mu}_R+k\hat{\sigma}_R}{\sqrt{\frac{\tilde{\sigma}_{T,1}^2}{n_T^*}+\left(\frac{1}{n_R^*}+\frac{k^2V_{n_R}}{n_R-1}\right)\tilde{\sigma}_{R,1}^2}},

\tau_2^\prime=\frac{\hat{\mu}_T-\hat{\mu}_R-k\hat{\sigma}_R}{\sqrt{\frac{\tilde{\sigma}_{T,2}^2}{n_T^*}+\left(\frac{1}{n_R^*}+\frac{k^2V_{n_R}}{n_R-1}\right)\tilde{\sigma}_{R,2}^2}},

where V_{n_R} = n_R-1-2\frac{\Gamma^22(n_R/2)}{\Gamma^2((n_R-1)/2)} and \tilde{\sigma}_{T,i} and \tilde{\sigma}_{R,i} are the restricted maximum likelihood estimator of \sigma_T and \sigma_R respectively with the constraint \mu_T - \mu_R = (-1)^i \sigma_R.

The null hypothesis H_0^\prime is rejected at nominal size \alpha if both \tau_1^\prime > z_{1-\alpha} and \tau_2^\prime < -z_{1-\alpha} where z_{1-\alpha} is the (1-\alpha)-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.