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Two new methods for non-inferiority testing of the ratio in matched-pair setting

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Abstract

Non-inferiority of one treatment to another based on ratio is a common issue in medical research. Nam and Blackwelder derived a Fieller-type statistic from constrained maximum likelihood estimation of nuisance parameters. Although it improved that of Lachenbruch and Lynch, its size still exceeds the claimed nominal level alpha for some cases. This is in accordance with Sidik who has found that the asymptotic tests are inaccurate in testing a hypothesis of a non-zero difference between marginal probabilities. In this paper, we extend Sidik’s exact tests to non-inferiority in ratio, and propose two new methods—one based on the standard \(p\) value and the other from restricted Bayesian estimation using an approximate \(p\) value. Our tests are both based on only one point of the two-dimension nuisance parameter space for accuracy improvement and computational purposes. The sizes and powers of our tests are considered. Simulation results suggest that our tests can control the type I error rates well with competitive powers while the Fieller-type statistic cannot for some cases be investigated.

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Correspondence to Hua Jin.

Appendices

Appendices

1.1 Appendix 1: Proof of (9)

We want to prove that the exact \(p\) value of the test based on the statistic \(Z_\mathrm{NB} (x_{11} ,x_{01} ,x_{10} )\)is

$$\begin{aligned} p_\mathrm{NB}&= \mathop {\sup }\limits _{S(p_{11} ,p_{10} )} \sum \limits _{(u,v,w)\in Rz} {\frac{n!}{u!v!w!(n-u-v-w)!}p_{11}^u \left( \frac{p_{11} +p_{10} }{\phi _0 }-p_{11} \right) ^v} \\&\times \, p_{10}^w \left( 1-p_{10} -\frac{p_{11} +p_{10} }{\phi _0 } \right) ^{n-u-v-w} \end{aligned}$$

where \(Rz=\{(u,v,w):u,v,w\in N,u+v+w\le n,\;Z_\mathrm{NB} (u,v,w)\ge Z_\mathrm{NB} (x_{11} ,x_{01} ,x_{10} )\}\)

$$\begin{aligned} S(p_{11} ,p_{10} )&= \{(p_{11} ,p_{10} ):0\le p_{11} \le \phi _0,\\&\quad 0\le p_{10} \le \phi _0 /(1+\phi _0 ),p_{11} +(1+\phi _0 )p_{10} \le \phi _0 \}. \end{aligned}$$

In fact, from the definition in (8), we have

$$\begin{aligned} p_\mathrm{NB}&= \mathop {\sup }\limits _{H_0 } P\{Z_\mathrm{NB} (X_{11} ,X_{01} ,X_{10} )\ge Z_\mathrm{NB} (x_{11} ,x_{01} ,x_{10} )\} \\&= \mathop {\sup }\limits _{H_0 } \sum \limits _{(u,v,w)\in Rz} {\frac{n!}{u!v!w!(n - u - v - w)!}} p_{11}^u p_{01}^v p_{10}^w\\&\times \,(1 - p_{11} - p_{01} - p_{10} )^{n-u-v-w} \end{aligned}$$

Given \(X_{01} =x_{01} \) and \(X_{11} =x_{11} \), the variable of \(X_{10} \) has a binomial distribution:

$$\begin{aligned}&b\left( x_{10} :n-x_{01} -x_{11} ,\frac{p_{10} }{1-p_{01} -p_{11} }\right) \\&\quad =\frac{(n-x_{01} -x_{11} )!}{x_{10} ![(n-x_{01} -x_{11} )-x_{10} ]!} \left( \frac{p_{10} }{1-p_{01} -p_{11} }\right) ^{x_{10} }\\&\quad \quad \times \,\left( 1-\frac{p_{10} }{1-p_{01} -p_{11} }\right) ^{(n-x_{01} -x_{11} )-x_{10} } \end{aligned}$$

By the same way, given any two elements, the third element will have a binomial distribution. According to Casella and Berger (1990), it can be concluded that the family of binomial distributions of \(X_{10} \) given \(X_{01} =x_{01} \) and \(X_{11} =x_{11} \) is stochastically increasing in \(p_{10} \) for any fixed \(p_{01} \) and \(p_{11} \). On the other hand, it is easy to see that \(Rz(x_{11} ,x_{01} ,x_{10} )\) is a Barnard convex set in the sense of Sidik (2003) when fix an element. In fact, fix \(x_{10} =w\), if \((u,v)\in Rz(x_{11} ,x_{01} ,w)\), and \({u}'\le u\), \({v}'\ge v\), then \({u}'-{v}'\le u-v\) and \(({u}',{v}')\in Rz(x_{11} ,x_{01} ,w)\). Therefore, it follows from Theorem 1 of Sidik (2003) that, if \(p_{01} \ge \frac{p_{11} +p_{10} }{\phi _0 }-p_{11} \), then

$$\begin{aligned}&P\{Z_\mathrm{NB} (X_{11} ,X_{01} ,X_{10} )\ge Z_\mathrm{NB} (x_{11} ,x_{01} ,x_{10} )\left| {p_{11} ,p_{01} ,p_{10} }\}\right. \\&\quad \le P\left\{ Z_\mathrm{NB} (X_{11} ,X_{01} ,X_{10} )\ge Z_\mathrm{NB} (x_{11} ,x_{01} ,x_{10} )| {p_{11} ,\frac{p_{11} +p_{10} }{\phi _0 }-p_{11} ,p_{10} } \right\} \end{aligned}$$

Further, it follows from \(p_{01} +p_{10} +p_{11} \le 1\) that \(p_{11} +(1+\phi _0 )p_{10} \le \phi _0 \), \(0\le p_{11} \le \phi _0 \) and \(0\le p_{10} \le \frac{\phi _0 }{1+\phi _0 }\). Then

$$\begin{aligned} p_\mathrm{NB}&= \mathop {\sup }\limits _{H_0 } P\{Z_\mathrm{NB} (X_{11} ,X_{01} ,X_{10} )\ge Z_\mathrm{NB} (x_{11} ,x_{01} ,x_{10} )\} \\&= {\mathop {\mathop {\mathop {\sup }\limits _{p_{11} ,p_{10} ,p_{01} \ge 0 }}\limits _{p_{11} +p_{10} +p_{01} \le 1 }}\limits _{p_1 \le \phi _0 p_0 }} P\left\{ Z_\mathrm{NB} (X_{11} ,X_{01} ,X_{10} )\ge Z_\mathrm{NB} (x_{11} ,x_{01} ,x_{10} )\vert p_{11} ,p_{01} ,p_{10} \right\} \\&= {\mathop {\mathop {\sum }\limits _{p_{11} ,p_{10} \ge 0 }}\limits _{p_{11} +(1+\phi _0 )p_{10} \le \phi _0 }} P\left\{ Z_\mathrm{NB} (X_{11} ,X_{01} ,X_{10} ) \ge Z_\mathrm{NB} (x_{11} ,x_{01} ,x_{10} )\vert p_{11} ,\frac{p_{11} +p_{10} }{\phi _0 }- p_{11} ,p_{10} \right\} \\ \end{aligned}$$

Thus the \(p\) value can be defined on the boundary of \(H_0 \). That is to say,

$$\begin{aligned} p_\mathrm{NB}&= \mathop {\sup }\limits _{S(p_{11} ,p_{10} )} P\left\{ Z_\mathrm{NB} (X_{11} ,X_{01} ,X_{10} ) \ge Z_\mathrm{NB} (x_{11} ,x_{01} ,x_{10} )\vert p_{11} ,\frac{p_{11} + p_{10} }{\phi _0 } - p_{11} ,p_{10} \right\} \\&= \mathop {\sup }\limits _{S(p_{11} ,p_{10} )} \sum \limits _{(u,v,w)\in Rz} {\frac{n!}{u!v!w!(n-u-v-w)!}p_{11}^u \left( \frac{p_{11} +p_{10} }{\phi _0 }-p_{11} \right) ^v }\\&\times \, p_{10}^w \left( 1-p_{10} -\frac{p_{11} +p_{10} }{\phi _0 }\right) ^{n-u-v-w} \end{aligned}$$

The proof is complete.

1.2 Appendix 2: Proof of (15)

According to (10), (11), (12) and (13), we have

$$\begin{aligned} \tilde{\theta }_{10}&= \int _0^{\frac{\phi _0 }{1+\phi _0 }} {\theta _{10} \cdot \pi _{\theta _{10} } (\theta _{10} \vert x)} d\theta _{10} \\&= \frac{\int _0^{\frac{\phi _0 }{1+\phi _0 }} {\int _0^{\phi _0 -(1+\phi _0 )\theta _{10} } {[\theta _{11} (1-\phi _0 )+\theta _{10} ]^{x_{01} }} \theta _{11} ^{x_{11} }\theta _{10} ^{x_{10} }(\phi _0 -\phi _0 \theta _{10} -\theta _{11} -\theta _{10} )^{n-x_{01} -x_{11} -x_{10} }d\theta _{11} d\theta _{10} } }{\int _0^{\frac{\phi _0 }{1+\phi _0 }} {\int _0^{\phi _0 -(1+\phi _0 )\theta _{10} } {[\theta _{11} (1-\phi _0 )+\theta _{10} ]^{x_{01} }} } \theta _{11} ^{x_{11} }\theta _{10} ^{x_{10} }(\phi _0 -\phi _0 \theta _{10} -\theta _{11} -\theta _{10} )^{n-x_{01} -x_{11} -x_{10} }d\theta _{11} d\theta _{10} } \end{aligned}$$

Let \(\theta _{11} =[\phi _0 -(1+\phi _0 )\theta _{10} ]t\), then \(\phi _0 -\phi _0 \theta _{10} -\theta _{11} -\theta _{10} =[\phi _0 -(1+\phi _0 )\theta _{10} ](1-t)\), and we have

$$\begin{aligned} \tilde{\theta }_{10}&= \frac{\int _0^{\frac{\phi _0 }{1+\phi _0 }} {\int _0^1 {\sum _{i=0}^{x_{01} } {\left( {{\begin{array}{c} {x_{01} } \\ i \\ \end{array} }}\right) } } } \theta _{10} ^{x_{10} +i+1}[\phi _0 -(1+\phi _0 )\theta _{10} ]^{n-x_{10} -i+1}t^{x_{11} +x_{01} -i}(1-\phi _0 )^{x_{01} -i}(1-t)^{n-x_{01} -x_{11} -x_{10} }dtd\theta _{10} }{\int _0^{\frac{\phi _0 }{1+\phi _0 }} {\int _0^1 {\sum _{i=0}^{x_{01} } {\left( {{\begin{array}{c} {x_{01} } \\ i \\ \end{array} }}\right) } } } \theta _{10} ^{x_{10} +i}[\phi _0 -(1+\phi _0 )\theta _{10} ]^{n-x_{10} -i+1}t^{x_{11} +x_{01} -i}(1-\phi _0 )^{x_{01} -i}(1-t)^{n-x_{01} -x_{11} -x_{10} }dtd\theta _{10} } \\&= \frac{\int _0^{\frac{\phi _0 }{1+\phi _0 }} {\sum _{i=0}^{x_{01} } {\left( {{\begin{array}{c} {x_{01} } \\ i \\ \end{array} }}\right) } } \theta _{10} ^{x_{10} +i+1}[\phi _0 -(1+\phi _0 )\theta _{10} ]^{n-x_{10} -i+1}(1-\phi _0 )^{x_{01} -i}B(x_{11} +x_{01} -i+1,n-x_{10} -x_{11} -x_{01} +1)d\theta _{10} }{\int _0^{\frac{\phi _0 }{1+\phi _0 }} {\sum _{i=0}^{x_{01} } {\left( {{\begin{array}{c} {x_{01} } \\ i \\ \end{array} }}\right) } } \theta _{10} ^{x_{10} +i}[\phi _0 -(1+\phi _0 )\theta _{10} ]^{n-x_{10} -i+1}(1-\phi _0 )^{x_{01} -i}B(x_{11} +x_{01} -i+1,n-x_{10} -x_{11} -x_{01} +1)d\theta _{10} } \end{aligned}$$

let \(\theta _{10} =\frac{\phi _0 }{1+\phi _0 }s\), then \([\phi _0 -(1+\phi _0 )\theta _{10} ]=\phi _0 (1-s)\), and therefore

$$\begin{aligned} \tilde{\theta }_{10} =\phi _0 \sum \limits _{i=0}^{x_{01} } {a_i b_i /\sum \limits _{i=0}^{x_{01} } {b_i } } \end{aligned}$$

where \(a_i =\frac{x_{10} +x_{01} -i+2}{n+3} \frac{1}{1+\phi _0 }\),

$$\begin{aligned} b_i&= \left( {{\begin{array}{cc} {x_{01} }\\ i \\ \end{array} }}\right) (1+\phi _0 )^{i-x_{01} }(1-\phi _0 )^i B(x_{10} +x_{01} -i+1,n-x_{01} -x_{10} +i+2)\nonumber \\&\times B(x_{11} +i+1,n-x_{01} -x_{10} -x_{11} +1). \end{aligned}$$

Similarly, we can prove that

$$\begin{aligned} \tilde{\theta }_{11} =\phi _0 \sum \limits _{i=0}^{x_{01} } {c_i d_i /\sum \limits _{i=0}^{x_{01} } {d_i } } \end{aligned}$$

where \(c_i =\frac{x_{11} +i+2}{n+3}\),

$$\begin{aligned} d_i&= \left( {{\begin{array}{cc} {x_{01} } \\ i \\ \end{array} }}\right) (1+\phi _0 )^{i-x_{01} }(1-\phi _0 )^i B(x_{11} +i+1,n-x_{11} -i+2)\nonumber \\&\times B(x_{01} +x_{10} -i+1,n-x_{01} -x_{10} -x_{11} +1). \end{aligned}$$

The proof of (15) is complete.

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Jin, H., Feng, X., Chen, M. et al. Two new methods for non-inferiority testing of the ratio in matched-pair setting. TEST 23, 691–707 (2014). https://doi.org/10.1007/s11749-014-0374-6

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