Eight predictive powers with historical and interim data for futility and efficacy analysis

Figures & data

Table 1. The eight predictive powers with historical and interim data, their analytical expressions, the predictive distributions, the data used, and the references for the hypotheses $H_0:\delta \le {\delta }_0$ versus $H_1:\delta >{\delta }_0$.

No.	Predictive power	Analytical expression	Predictive distribution	Data used	References
1	$\begin{array}{c}{I}_1=\mathrm{C}\mathrm{P}\mathrm{P}=P\left(S_{\alpha ,{\delta }_0}^{C,d_2}\right|d_0)\\ ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P\left(S_{\alpha ,{\delta }_0}^{C,d_2}\right|\delta \left)\pi \right(\delta \left|d_0\right)\mathrm{d}\delta \end{array}$	$E_1$	$\pi \left(d_2\right|d_0)$	H	(6.4) in Spiegelhalter et al. (2004);
					(6) in O'Hagan et al. (2005);
					(2) in Chuang-Stein (2006)
2	$\begin{array}{c}{I}_2=\mathrm{C}\mathrm{I}\mathrm{P}\mathrm{P}=P\left(S_{\alpha ,{\delta }_0}^{C,d_2}\right|d_0,d_1)\\ ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P\left(S_{\alpha ,{\delta }_0}^{C,d_2}\right|\delta \left)\pi \right(\delta |d_0,d_1)\mathrm{d}\delta \end{array}$	$E_2$	$\pi \left(d_2\right|d_0,d_1)$	HI
3	$\begin{array}{c}{I}_3=\mathrm{C}\mathrm{C}\mathrm{P}\mathrm{P}=P\left(S_{\alpha ,{\delta }_0}^{C,d_1,d_2}\right|d_0)\\ ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P\left(S_{\alpha ,{\delta }_0}^{C,d_1,d_2}\right|\delta ,d_1\left)\pi \right(\delta \left|d_0\right)\mathrm{d}\delta \end{array}$	$E_3$	$\pi \left(d_2\right|d_0)$	HI
4	$\begin{array}{c}{I}_4=\mathrm{C}\mathrm{C}\mathrm{I}\mathrm{P}\mathrm{P}=P\left(S_{\alpha ,{\delta }_0}^{C,d_1,d_2}\right|d_0,d_1)\\ ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P\left(S_{\alpha ,{\delta }_0}^{C,d_1,d_2}\right|\delta ,d_1\left)\pi \right(\delta |d_0,d_1)\mathrm{d}\delta \end{array}$	$E_4$	$\pi \left(d_2\right|d_0,d_1)$	HI${}^2$	(6.15) in Spiegelhalter et al. (2004)
5	$\begin{array}{c}{I}_5=\mathrm{B}\mathrm{P}\mathrm{P}=P\left(S_{\alpha ,{\delta }_0}^{B,d_0,d_2}\right|d_0)\\ ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P\left(S_{\alpha ,{\delta }_0}^{B,d_0,d_2}\right|\delta ,d_0\left)\pi \right(\delta \left|d_0\right)\mathrm{d}\delta \end{array}$	$E_5$	$\pi \left(d_2\right|d_0)$	H${}^2$	(6.7) in Spiegelhalter et al. (2004)
6	$\begin{array}{c}{I}_6=\mathrm{B}\mathrm{I}\mathrm{P}\mathrm{P}=P\left(S_{\alpha ,{\delta }_0}^{B,d_0,d_2}\right|d_0,d_1)\\ ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P\left(S_{\alpha ,{\delta }_0}^{B,d_0,d_2}\right|\delta ,d_0\left)\pi \right(\delta |d_0,d_1)\mathrm{d}\delta \end{array}$	$E_6$	$\pi \left(d_2\right|d_0,d_1)$	H${}^2$I
7	$\begin{array}{c}{I}_7=\mathrm{B}\mathrm{C}\mathrm{P}\mathrm{P}=P\left(S_{\alpha ,{\delta }_0}^{B,d_0,d_1,d_2}\right|d_0)\\ ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P\left(S_{\alpha ,{\delta }_0}^{B,d_0,d_1,d_2}\right|\delta ,d_0,d_1\left)\pi \right(\delta \left|d_0\right)\mathrm{d}\delta \end{array}$	$E_7$	$\pi \left(d_2\right|d_0)$	H${}^2$I
8	$\begin{array}{c}{I}_8=\mathrm{B}\mathrm{C}\mathrm{I}\mathrm{P}\mathrm{P}=P\left(S_{\alpha ,{\delta }_0}^{B,d_0,d_1,d_2}\right|d_0,d_1)\\ ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P\left(S_{\alpha ,{\delta }_0}^{B,d_0,d_1,d_2}\right|\delta ,d_0,d_1\left)\pi \right(\delta |d_0,d_1)\mathrm{d}\delta \end{array}$	$E_8$	$\pi \left(d_2\right|d_0,d_1)$	H${}^2$I${}^2$	(6.18) in Spiegelhalter et al. (2004)

Table 2. The eight predictive powers with historical and interim data, their analytical expressions, the predictive distributions, and the data used for the reversed hypotheses $H_0:\delta \ge {\delta }_0$ versus $H_1:\delta <{\delta }_0$.

No.	Predictive power	Analytical expression	Predictive distribution	Data used
1	$\begin{array}{c}{I}_1^{-}={\mathrm{C}\mathrm{P}\mathrm{P}}^{-}=P\left(S_{\alpha ,{\delta }_0}^{C-,d_2}\right|d_0)\\ ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P\left(S_{\alpha ,{\delta }_0}^{C-,d_2}\right|\delta \left)\pi \right(\delta \left|d_0\right)\mathrm{d}\delta \end{array}$	$E_1^{-}$	$\pi \left(d_2\right|d_0)$	H
2	$\begin{array}{c}{I}_2^{-}={\mathrm{C}\mathrm{I}\mathrm{P}\mathrm{P}}^{-}=P\left(S_{\alpha ,{\delta }_0}^{C-,d_2}\right|d_0,d_1)\\ ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P\left(S_{\alpha ,{\delta }_0}^{C-,d_2}\right|\delta \left)\pi \right(\delta |d_0,d_1)\mathrm{d}\delta \end{array}$	$E_2^{-}$	$\pi \left(d_2\right|d_0,d_1)$	HI
3	$\begin{array}{c}{I}_3^{-}={\mathrm{C}\mathrm{C}\mathrm{P}\mathrm{P}}^{-}=P\left(S_{\alpha ,{\delta }_0}^{C-,d_1,d_2}\right|d_0)\\ ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P\left(S_{\alpha ,{\delta }_0}^{C-,d_1,d_2}\right|\delta ,d_1\left)\pi \right(\delta \left|d_0\right)\mathrm{d}\delta \end{array}$	$E_3^{-}$	$\pi \left(d_2\right|d_0)$	HI
4	$\begin{array}{c}{I}_4^{-}={\mathrm{C}\mathrm{C}\mathrm{I}\mathrm{P}\mathrm{P}}^{-}=P\left(S_{\alpha ,{\delta }_0}^{C-,d_1,d_2}\right|d_0,d_1)\\ ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P\left(S_{\alpha ,{\delta }_0}^{C-,d_1,d_2}\right|\delta ,d_1\left)\pi \right(\delta |d_0,d_1)\mathrm{d}\delta \end{array}$	$E_4^{-}$	$\pi \left(d_2\right|d_0,d_1)$	HI${}^2$
5	$\begin{array}{c}{I}_5^{-}={\mathrm{B}\mathrm{P}\mathrm{P}}^{-}=P\left(S_{\alpha ,{\delta }_0}^{B-,d_0,d_2}\right|d_0)\\ ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P\left(S_{\alpha ,{\delta }_0}^{B-,d_0,d_2}\right|\delta ,d_0\left)\pi \right(\delta \left|d_0\right)\mathrm{d}\delta \end{array}$	$E_5^{-}$	$\pi \left(d_2\right|d_0)$	H${}^2$
6	$\begin{array}{c}{I}_6^{-}={\mathrm{B}\mathrm{I}\mathrm{P}\mathrm{P}}^{-}=P\left(S_{\alpha ,{\delta }_0}^{B-,d_0,d_2}\right|d_0,d_1)\\ ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P\left(S_{\alpha ,{\delta }_0}^{B-,d_0,d_2}\right|\delta ,d_0\left)\pi \right(\delta |d_0,d_1)\mathrm{d}\delta \end{array}$	$E_6^{-}$	$\pi \left(d_2\right|d_0,d_1)$	H${}^2$I
7	$\begin{array}{c}{I}_7^{-}={\mathrm{B}\mathrm{C}\mathrm{P}\mathrm{P}}^{-}=P\left(S_{\alpha ,{\delta }_0}^{B-,d_0,d_1,d_2}\right|d_0)\\ ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P\left(S_{\alpha ,{\delta }_0}^{B-,d_0,d_1,d_2}\right|\delta ,d_0,d_1\left)\pi \right(\delta \left|d_0\right)\mathrm{d}\delta \end{array}$	$E_7^{-}$	$\pi \left(d_2\right|d_0)$	H${}^2$I
8	$\begin{array}{c}{I}_8^{-}={\mathrm{B}\mathrm{C}\mathrm{I}\mathrm{P}\mathrm{P}}^{-}=P\left(S_{\alpha ,{\delta }_0}^{B-,d_0,d_1,d_2}\right|d_0,d_1)\\ ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P\left(S_{\alpha ,{\delta }_0}^{B-,d_0,d_1,d_2}\right|\delta ,d_0,d_1\left)\pi \right(\delta |d_0,d_1)\mathrm{d}\delta \end{array}$	$E_8^{-}$	$\pi \left(d_2\right|d_0,d_1)$	H${}^2$I${}^2$

Figure 1. The data structures of the historical data, interim data and future data.

Table 3. The eight predictive powers with historical and interim data of eventual conclusions for the B-14 trial after the first interim analysis in 1993. Two prior distributions are considered: a sceptical prior and an optimistic prior.

	Tamoxifen superior		Equivocal		Control superior
No.	Sceptical	Optimistic	Sceptical	Optimistic	Sceptical	Optimistic
1	0.156	0.656	0.687	0.336	0.156	0.008
2	0.015	0.077	0.760	0.857	0.225	0.066
3	0.011	0.161	0.781	0.821	0.208	0.017
4	0.000	0.003	0.610	0.846	0.389	0.151
5	0.120	0.771	0.761	0.228	0.120	0.001
6	0.005	0.195	0.869	0.803	0.126	0.002
7	0.005	0.321	0.852	0.678	0.142	0.001
8	0.000	0.017	0.724	0.972	0.276	0.011

Figure 2. The sensitivity analysis of $d_0$ on the eight predictive powers.

Figure 3. The sensitivity analysis of $m_0$ on the eight predictive powers.

Table 4. The increase–decrease characteristics of $I_i^{s-}$, $I_i^s$, $1-I_i^{s-}-I_i^s$, $I_i^{o-}$, $I_i^o$ and $1-I_i^{o-}-I_i^o$ for $i=1,\dots ,8$ observed from Figure 3.

No.	$I_i^{s-}$	$I_i^s$	$1-I_i^{s-}-I_i^s$	$I_i^{o-}$	$I_i^o$	$1-I_i^{o-}-I_i^o$
1	↓	↓	↑	↑	↓	$\nearrow \searrow$
2	—	↓	↑	↑	↓	$\nearrow \searrow$
3	↓	↓	↑	↓	↓	↑
4	—	↓	↑	↑	↓	↑
5	↓	↓	↑	↑	↓	$\nearrow \searrow$
6	—	↓	↑	↑	↓	$\nearrow \searrow$
7	↓	↓	↑	$\searrow \nearrow$	↓	$\nearrow \searrow$
8	—	↓	↑	↑	↓	$\nearrow \searrow$

Figure 4. The sensitivity analysis of $d_1$ on the eight predictive powers.

Figure 5. The sensitivity analysis of $m_1$ on the eight predictive powers.

Table 5. The increase–decrease characteristics of $I_i^{s-}$, $I_i^s$, $1-I_i^{s-}-I_i^s$, $I_i^{o-}$, $I_i^o$ and $1-I_i^{o-}-I_i^o$ for i = 2, 3, 4, 6, 7, 8 observed from Figure 5.

No.	$I_i^{s-}$	$I_i^s$	$1-I_i^{s-}-I_i^s$	$I_i^{o-}$	$I_i^o$	$1-I_i^{o-}-I_i^o$
2	↓	$\nearrow \searrow$	↑	↓	$\nearrow \searrow$	↑
3	↓	↑	$\nearrow \searrow$	↓	↑	$\nearrow \searrow$
4	↓	↑	$\nearrow \searrow$	↓	↑	$\nearrow \searrow$
6	↓	$\nearrow \searrow$	↑	↓	—	↑
7	↓	↑	$\nearrow \searrow$	↓	—	↑
8	↓	↑	$\nearrow \searrow$	↓	—	↑

Figure 6. The sensitivity analysis of $m_2$ on the eight predictive powers.

Table 6. The increase–decrease characteristics of $I_i^{s-}$, $I_i^s$, $1-I_i^{s-}-I_i^s$, $I_i^{o-}$, $I_i^o$ and $1-I_i^{o-}-I_i^o$ for $i=1,\dots ,8$ observed from Figure 6.

No.	$I_i^{s-}$	$I_i^s$	$1-I_i^{s-}-I_i^s$	$I_i^{o-}$	$I_i^o$	$1-I_i^{o-}-I_i^o$
1	↑	↑	↓	↑	—	↓
2	↑	$\nearrow \searrow$	↓	↑	$\nearrow \searrow$	↓
3	↑	↓	$\nearrow \searrow$	↑	↓	$\nearrow \searrow$
4	↑	↓	$\nearrow \searrow$	↑	↓	$\nearrow \searrow$
5	↑	↑	↓	↑	—	↓
6	↑	$\nearrow \searrow$	↓	↑	—	↓
7	↑	↓	$\nearrow \searrow$	↑	—	↓
8	↑	↓	$\nearrow \searrow$	↑	—	↓

Figure 7. The sensitivity analysis of t on the eight predictive powers.

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