Hypostis testing
Binomial Non-inferiority Two-Sample Testing
Let \(\pi_c\) and \(\pi_t\) denote the response rates for the control and experimental treatments, respectively. Higher response rate means better performance. Let \(\delta = \pi_t - \pi_c\). We have
\[ H_0: \delta \leq \delta_0 \quad vs \quad H_1: \delta > \delta_0 \] where \(\delta_0 < 0\) is called non-inferiority margin. In other word,
\[ H_0: \pi_t \leq \pi_c + \delta_0 \quad vs \quad H_1: \pi_t > \pi_c + \delta_0 \] The test statistic can be defined by
\[ T = \frac{\hat{\pi}_t - \hat{\pi}_c - \delta_0}{\sqrt{\frac{\tilde{\pi}_t(1-\tilde{\pi}_t)}{n_t} + \frac{\tilde{\pi}_c(1-\tilde{\pi}_c)}{n_c}}} \] where \(\tilde{\pi}_t\) and \(\tilde{\pi}_c\) are the restricted maximum likelihood estimates (REML) of \(\pi_t\) and \(\pi_c\). From Miettinen and Nurminen (1985), \(\tilde{\pi}_t\) and \(\tilde{\pi}_c\) can be obtained by solving the third degree likelihood equation:
\[ \sum_{k=0}^3 L_k\tilde{\pi}_c^k = 0 \] with \(\tilde{\pi}_t = \tilde{\pi}_c + \delta_0\),
\[ \begin{aligned} L_3 & = N = n_c + n_t \\ L_2 & = (n_t + 2n_c)\delta_0 - N - x_c - x_t \\ L_1 & = (n_c\delta_0 - N - 2x_c)\delta_0 + x_c + x_t \\ L_0 &= x_c\delta_0(1-\delta_0) \end{aligned} \]
Cochran–Mantel–Haenszel statistics
In statistics, the Cochran–Mantel–Haenszel test (CMH) is a test used in the analysis of stratified or matched categorical data. It allows an investigator to test the association between a binary predictor or treatment and a binary outcome such as case or control status while taking into account the stratification
| Treatment | Control | Row Total | |
|---|---|---|---|
| Event | \(x_{Aj}\) | \(x_{Bj}\) | \(x_{+j}\) |
| No Event | \(y_{Aj}\) | \(y_{Bj}\) | \(y_{-j}\) |
| Column Totals | \(n_{Aj}\) | \(n_{Bj}\) | \(N_j\) |
The risk ratio is \(\widehat{RR} = \frac{x_{A}/x_+}{y_A/y_-}\) and the odds ratio = \(\widehat{OR} = \frac{x_A/x_B}{y_A/y_B} = \frac{x_Ay_B}{x_By_A}\). The CMH estimate for a risk ratio is \[ \widehat{RR}_{cmh} = \frac{\sum_{j = 1}^K x_{Aj}(y_{Aj}+y_{Bj})/N_j}{\sum_{j = 1}^K y_{Aj}(x_{Aj} + x_{Bj})/N_j} \] The CMH odds-ratio estimate is \[ \widehat{OR}_{cmh} = \frac{\sum_{j = 1}^K x_{Aj}y_{Bj}/N_j}{ \sum_{j = 1}^K x_{Bj}y_{Aj}/N_j} \]
The null hypothesis is that there is no association between the treatment and the outcome, i.e. \(H_0: OR = 1\) and the alternative hypothesis is \(H_1: OR\neq 1\).
The test statistics is \[\begin{align}\label{CMHtest} \xi_{C M H}=\frac{\left[\sum_{j=1}^{K}\left(x_{Aj}-\frac{x_{+j} n_{Aj}}{N_{j}}\right)\right]^{2}}{\sum_{j=1}^{K} \frac{x_{+j} y_{-j} n_{Aj} n_{Bj}}{N_{j}^{2}\left(N_{j}-1\right)}} \sim \chi^2_1 \end{align}\]
i.e. \(\xi_{CMH} = [\sum_i (x_{Aj} - \mu_{Aj})]^2/\sum_i Var(x_{Aj})\) where \(E(x_{Aj}) = \frac{x_{+j} n_{Aj}}{N_{j}}\) and \(Var(x_{Aj}) = \frac{x_{+j} y_{-j} n_{Aj} n_{Bj}}{N_{j}^{2}\left(N_{j}-1\right)}\).
Miettinen & Nurminen Method
Notation:| Treatment | Control | Row Total | |
|---|---|---|---|
| Event | \(x_{Aj}\) | \(x_{Bj}\) | \(x_{+j}\) |
| No Event | \(y_{Aj}\) | \(y_{Bj}\) | \(y_{-j}\) |
| Column Totals | \(n_{Aj}\) | \(n_{Bj}\) | \(N_j\) |
The null hypothesis is a family of hypotheses: \(H(\delta) = \pi_{Aj} - \pi_{Bj} = \delta\) for \(j = 1, \ldots, K\) and the corresponding chi-square test statistics: \[ \xi^{MN}(\delta) = \frac{\left[\sum_{j = 1}^K W_j(r_{Aj} - r_{Bj} - \delta) \right]^2}{\sum_{j=1}^K W_j^2 \widetilde{V}_{r_{Aj} - r_{Bj}}} \sim \chi_1^2 \] where \[ \widetilde{V}_{r_{A j}-r_{B j}}=\left\{\frac{\widetilde{R}_{A j}\left(1-\widetilde{R}_{A j}\right)}{n_{A j}}+\frac{\widetilde{R}_{B j}\left(1-\widetilde{R}_{B j}\right)}{n_{B j}}\right\} \frac{N_{j}}{\left(N_{j}-1\right)} \] is the variance of \(d_j = r_{Aj} - r_{Bj}\) under \(H(\delta)\), \(\widetilde{R}_{Aj}\) and \(\widetilde{R}_{Bj}\) are the contrained maximum likelihood estimates of \(\pi_{Aj}\) and \(\pi_{Bj}\) so that \(\widetilde{R}_{Aj} - \widetilde{R}_{Bj} = \delta\) (see Miettinen & Nurminen Appendix 1), \(W_j\) is the M&N weight for stratum \(j\):
\[\begin{align}\label{MNweight} W_{j}=\left[\frac{\widetilde{R}_{A}^{*}\left(1-\widetilde{R}_{A}^{*}\right)}{\widetilde{R}_{B}^{*}\left(1-\widetilde{R}_{B}^{*}\right)} / n_{A j}+1 / n_{B j}\right]^{-1}, \quad j=1, \ldots, K \end{align}\]
and \(\widetilde{R}_A^*\) and \(\widetilde{R}_B^*\) are the weighted averages of the contrained MLEs under \(H(\delta)\), \[\begin{align}\label{wMLE} \widetilde{R}_i^* = \frac{\sum_{j=1}^K W_j\widetilde{R}_{ij}}{\sum_{j=1}^K W_j}, \quad i = A, B. \end{align}\]
We reject \(H(\delta)\) if and only if \(\xi^{MN}(\delta) \geq \chi^2_{1, 1- \alpha}\). The \(100(1-\alpha)\%\) confidence limits of \(\delta\) can be obtained by solving \[ \xi^{MN}(\widehat{\delta}_L) = \xi^{MN}(\widehat{\delta}_U) = \chi_{1, 1-\alpha}^2 \] We can also define \[ Z(\delta) = \frac{\sum_{j=1}^K W_j(r_{Aj} - r_{Bj} - \delta)}{\sqrt{\sum_{j=1}^K W_j^2 \widetilde{V}_{r_{Aj} - r_{Bj}}}} \] so that \(Z(\widehat{\delta}_L) = z_{1-\alpha/2}\) and \(Z(\widehat{\delta}_U) = - z_{1-\alpha/2}\).
We can use the following steps to solve for the M&N weights:Implementation
\(\hat\delta\) can be obtained through \[ \hat\delta = \frac{\sum_{j=1}^K W_j(r_{Aj} - r_{Bj})}{\sum_{j=1}^K W_j} \]
Note that the weights \(W_j\) depend on \(\delta\). Alternatively, we have \(X^2(\hat\delta) = 0\) so that we can find \(\hat\delta\) that minimizes \(\xi^{MN}(\delta)\). We can transform \(\delta \in (-1, 1)\) to \(\tan (\delta \pi/2) \in (-\infty, \infty)\). We start from \(\delta_1 = \min\{r_{Aj}-r_{Bj}: j = 1,\ldots, K\}\) and \(\delta_2 = \max\{r_{Aj}-r_{Bj}: j = 1,\ldots, K\}\) and use golden section search to find the value minimizing the objective function.
After the first step, we have \(\hat\delta_L \in (-1, \hat\delta)\) and \(\hat\delta_U \in (\hat\delta, 1)\). By solving \(\xi^{MN}(\delta) = \chi^2_{1, 1-\alpha/2}\), we can finish the second step.
CMH Weights for M&N Method
The M&N weights are \[ W_j^{MN}(\delta) \propto \left[\frac{\widetilde{R}_{A}^{*}\left(1-\widetilde{R}_{A}^{*}\right)}{\widetilde{R}_{B}^{*}\left(1-\widetilde{R}_{B}^{*}\right)} / n_{A j}+1 / n_{B j}\right]^{-1} \] and CMH weights are \[ W_j^{CMH} \propto \left[\frac{1}{n_{Aj}} + \frac{1}{n_{Bj}} \right]^{-1} \] Two cases that two types of weights are identical:Equivalence of M&N Tests and CMH Test When \(\delta = 0\)
The CMH test statistics is \[ \xi_{C M H}=\frac{\left[\sum_{j=1}^{K}\left(x_{Aj}-\frac{x_{+j} n_{Aj}}{N_{j}}\right)\right]^{2}}{\sum_{j=1}^{K} \frac{x_{+j} y_{-j} n_{Aj} n_{Bj}}{N_{j}^{2}\left(N_{j}-1\right)}} \sim \chi^2_1 \] Note that \[ x_{Aj}-\frac{x_{+j} n_{Aj}}{N_{j}} = \frac{n_{Aj}n_{Bj}}{N_j}(r_{Aj} - r_{Bj}) \] and \[ \begin{aligned} \frac{x_{+j} y_{-j} n_{Aj} n_{Bj}}{N_{j}^{2}\left(N_{j}-1\right)} & = \left(\frac{n_{Aj}n_{Bj}}{N_j}\right)^2\frac{x_{+j}y_{-j}}{N_j^2}\frac{n_{Aj}+n_{Bj}}{n_{Aj}n_{Bj}}\frac{N_j}{N_j-1}\\ & = \left(\frac{n_{Aj}n_{Bj}}{N_j}\right)^2\left\{\bar r_j(1-\bar r_j)\left(\frac{1}{n_{Aj}}+\frac{1}{n_{Bj}}\right) \right\}\frac{N_j}{N_j-1} \end{aligned} \] The CMH test statistics can be modified as \[ \begin{aligned} \xi_{C M H} & =\frac{\left[\sum_{j=1}^{K}\left(x_{Aj}-\frac{x_{+j} n_{Aj}}{N_{j}}\right)\right]^{2}}{\sum_{j=1}^{K} \frac{x_{+j} y_{-j} n_{Aj} n_{Bj}}{N_{j}^{2}\left(N_{j}-1\right)}} \\ & = \frac{\left[\sum_{j=1}^K W_j^{CMH}(r_{Aj} - r_{Bj})\right]^2}{\sum_{j=1}^K (W_j^{CMH})^2 V_{r_{Aj} - r_{Bj}}^{CMH}} \end{aligned} \] where \[ W_j^{CMH} = \frac{n_{Aj}n_{Bj}}{N_j} = \left[\frac{1}{n_{Aj}} + \frac{1}{n_{Bj}}\right]^{-1} \] and \[ V_{r_{Aj} - r_{Bj}}^{CMH} = \left\{\bar r_j(1-\bar r_j)\left(\frac{1}{n_{Aj}}+\frac{1}{n_{Bj}}\right) \right\}\frac{N_j}{N_j-1} \] When \(\delta = 0\), from the above discussion, under the null hypothesis \(H(0): \pi_{Aj} = \pi_{Bj}\). The constrained MLEs have \(\widetilde{R}_{Aj} = \widetilde{R}_{Bj}\). From (), we have \(\widetilde{R}_A^* = \widetilde{R}_B^*\) so that () becomes \[ W_j^{MN} = \left[\frac{1}{n_{Aj}} + \frac{1}{n_{Bj}}\right]^2 \] and \[ V_{r_{Aj} - r_{Bj}}^{MN} = \left\{\widetilde{R}_{Aj}(1-\widetilde{R}_{Aj})\left(\frac{1}{n_{Aj}}+\frac{1}{n_{Bj}}\right) \right\}\frac{N_j}{N_j-1} \]
Note that \(\widetilde{R}_{Aj} \neq \bar r_{j}\) when \(\delta = 0\). From Appendix 1, we have \[ L0 = 0, \space L1 = x_{+j}, \space L_2 = -N_j - x_{+j}, \space L_3 = N_j \] so that \[ \begin{aligned} p & = \pm \frac{\sqrt{N_j^2 + x_{+j}^2 - N_jx_{+j}}}{3N_j}\\ q & = \frac{-(N_j+x_{+j})(2N_j-x_{+j})(N_j-2x_{+j})}{54N_j^3} \end{aligned} \] Based on triple angle formula, \[ \cos(3a) = 4\cos^3 a - 3\cos a = 4 \left(\frac{\widetilde{R}_{Bj} + \frac{L_2}{3L_3}}{2p}\right)^3 - 3\frac{\widetilde{R}_{Bj} + \frac{L_2}{3L_3}}{2p} \] Solving this equation, we have \(\widetilde{R}_{Bj} = \bar r_j\) (the other two solutions are 1 and 0).
Because \(W_j^{CMH} = W_j^{MN}\) and \(V_{r_{Aj} - r_{Bj}}^{CMH} = V_{r_{Aj} - r_{Bj}}^{MN}\), we can say the two test statistics \(\xi_{CMH} = \xi_{MN}\) under \(\delta = 0\).
Appendix 1
Miettinen and Nurminen provided a closed-form solution for the MLEs of \(\widetilde{R}_{Aj}\) and \(\widetilde{R}_{Bj}\) where \[ \begin{aligned} \widetilde{R}_{Bj} &=2 p \cos (a)-L_{2} /\left(3 L_{3}\right) \\ \widetilde{R}_{Aj} & = \widetilde{R}_{Bj} + \delta\\ a &=(1 / 3)\left[\pi+\cos ^{-1}\left(q / p^{3}\right)\right] \\ p &=\pm\left[L_{2}^{2} /\left(3 L_{3}\right)^{2}-L_{1} /\left(3 L_{3}\right)\right]^{1 / 2} \\ q &=L_{2}^{3} /\left(3 L_{3}\right)^{3}-L_{1} L_{2} /\left(6 L_{3}^{2}\right)+L_{0} /\left(2 L_{3}\right) \\ L_{3} &= N_j \\ L_{2} & =\left(n_{Aj}+2 n_{Bj}\right)\delta- N_j- x_{+j}\\ L_{1} & =\left[n_{Bj}\delta-n_{Bj}-n_{Aj}-2 x_{Bj}\right]\delta+x_{+j} \\ L_{0} & =x_{Bj}\delta(1-\delta) \end{aligned} \]