Correction

This article refers to:

A Simple Mantel–Haenszel Type Test for Noninferiority

Article title: A Simple Mantel-Haenszel Type Test for Noninferiority

Authors: Koti, K.M.

Journal: Statistics in Biopharmaceutical

Citation details: Volume 13, Number 1, pages 113-118

DOI: https://doi.org/10.1080/19466315.2020.1736140

This article refers to the W-square test, which is an asymptotic unconditional test for demonstrating noninferiority in terms of binomial proportions. This is entirely based on Wittes and Wallenstein’s (1987) paper, “The Power of the Mantel-Haenszel Test”.

The link to the original paper is given below.

https://www.tandfonline.com/eprint/GJFHHYAACF9JQZQE7ZM7/full?target=10.1080/19466315.2020.1736140

The power function defined in Section 5 (p. 116) of Koti’s (2021) W-square test, which is given below $$\psi =P\left(M_U>C_{\alpha }|{\delta }_i=0\text{ for all }i\right)$$

was wrong and rigid for a noninferiority clinical trial. The corrected power function definition (with new symbol $\Psi$) is $$\Psi (\mathit{\epsilon })=P\left(M_U>C_{\alpha }|K_A:{\pi }_{i1}-{\pi }_{i2}\ge -{\mathit{\epsilon }}_i,\text{ for all }i,\text{ and}{\pi }_{i1}-{\pi }_{i2}>\hspace{1em}-{\mathit{\epsilon }}_i,\text{ for some }i\right),$$

where $0<{\mathit{\epsilon }}_i\le {\delta }_i<1$, for all i from 1 to T. That is, $$\Psi \left(\mathit{\epsilon }\right)=\Phi \left[\left\{{\mu }_{\mathit{ALT}}\sqrt{W_{\mathit{NULL}}}-\left(z_{1-\alpha }{\sigma }_{\mathit{NULL}}+{\mu }_{\mathit{NULL}}\right)\sqrt{W_{\mathit{ALT}}}\right\}/({\sigma }_{\mathit{ALT}}\times \sqrt{W_{\mathit{NULL}}})\right],$$

where the finite sample moments with subscript NULL are obtained by substituting ${\pi }_{i1}-{\pi }_{i2}=-{\delta }_i$, and those with subscript ALT are obtained using ${\pi }_{i1}-{\pi }_{i2}=-{\mathit{\epsilon }}_i$ for all i from 1 to T.

Consequently, the corrected required sample size N is easily obtained: $$\begin{array}{c}N={\left[z_{1-\beta }({\sigma }_{\mathit{ALT}}\times \sqrt{W_{\mathit{NULL}}})+z_{1-\alpha }{\sigma }_{\mathit{NULL}}\sqrt{W_{\mathit{ALT}}}\right]}^2/\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}{\left[\sum {\lambda }_i{\rho }_i\left(1-{\rho }_i\right){\delta }_i\sqrt{W_{\mathit{ALT}}}+\sum {\lambda }_i{\rho }_i\left(1-{\rho }_i\right){\mathit{\epsilon }}_i\sqrt{W_{\mathit{NULL}}}\right]}^2.\end{array}$$

Of course, the sample size details provided in Koti’s (2021) Section 6 should be ignored.

Additional information

Funding

The author(s) reported there is no funding associated with the work featured in this article.