The Imprecision Issues of Four Powers and Eight Predictive Powers with Historical and Interim Data

: Imprecision is commonly encountered with respect to powers and predictive powers in clinical trials. In this article, we investigate the imprecision issues of four powers (Classical Power, Classical Conditional Power, Bayesian Power, and Bayesian Conditional Power) and eight predictive powers. To begin with, we derive the probabilities of Control Superior (CS), Treatment Superior (TS), and Equivocal (E) of the four powers and the eight predictive powers, and evaluate the limits of the probabilities at point 0. Moreover, we conduct extensive numerical experiments to exemplify the imprecision issues of the four powers and the eight predictive powers. In the numerical experiments, ﬁrst, we compute the probabilities of CS, TS, and E for the four powers as functions of the sample size of the future data when the true treatment effect favors control, treatment, and equivocal, respectively. Second, we compute the probabilities of CS, TS, and E for the eight predictive powers as functions of the sample size of the future data under the sceptical prior and the optimistic prior, respectively. Finally, we carry out a real data example to show the prominence of the methods.


Introduction
As we understand, the power with extremely distinct values at different treatment effects (for example, an observed treatment effect in the interim analysis or a treatment effect under the alternative hypothesis) may cause difficulty for interpretation. However, the predictive power, which is the prior expectation of the power and averaged over the prior distribution for the unknown true treatment effect, is better than the power in giving a favorable indication of the probability that the trial will demonstrate a positive or statistically significant outcome. The predictive power has been studied intensively in the literature [1][2][3][4][5]. In addition, the predictive power is also known as assurance [6][7][8], Probability Of Success (POS) [9][10][11], Average Success Probability (ASP) [12,13], or Contemplated Average Success Probability (CASP) [14].
The predictive power is an average power with respect to some prior, that is, where δ is the true treatment effect of the early phase and Phase III trials. As described in [5], there are eight predictive powers with historical and interim data, because we have four choices for power(δ), including the Classical Power (CP) which does not use any data, the Classical Conditional Power (CCP), which uses the interim data once, the Bayesian In Section 4, we will perform some numerical experiments for a real data example. Section 5 provides some conclusions and discussions.

The Four Powers and the Eight Predictive Powers, and Their Limits at 0
In this section, after some preliminary results, we will derive the probabilities of CS, TS, and E of the four powers and the eight predictive powers, and evaluate their limits at point 0 (m 2 → 0 + ).

Preliminary
In this subsection, we will give some preliminary results. The four powers, the eight predictive powers, and the data structures of the historical data, interim data, and future data are described in Figure 1. On the basis of Figure 1 in [5], we have added the four powers in the two plots. From Figure 1, we see that the time when CP and BP are calculated in the upper plot is at the end of early phase and before the start of Phase III. At that time, only the first and fifth predictive powers can be calculated. Moreover, the time when CCP and BCP are calculated in the lower plot is at the interim of the Phase III trial. At the interim, there are six predictive powers; that is, the second, third, fourth, sixth, seventh, and eighth predictive powers. As described in [5], in the figure, H means historical data, I means interim data, and F means future data. The historical data could be the Phase II data, or the previous Phase III data, provided that the outcome variable and patient populations are the same between the historical data and the new Phase III data. Furthermore, the historical data could also be fictitious data corresponding to a sceptical or an optimistic prior, and in this case, d 0 and m 0 of the historical data are calculated to satisfy the requirements of the sceptical or optimistic prior. Notice that d 0 , d 1 , and d 2 are the observed treatment differences in the treatment group and the control (or placebo) group means of the historical data, interim data, and future data, respectively, and m 0 , m 1 , and m 2 are the per group numbers of patients of the historical data, interim data, and future data, respectively.

Early Phase
Phase III Let us use the normal models given in [5]. For CP, BP, I 1 , and I 5 , the model and prior are given by which implies where δ is the unknown true treatment effect, and σ 2 is a common known variance. The data structure of (1) is depicted in the upper plot of Figure 1. For CCP, BCP, I 2 , I 3 , I 4 , I 6 , I 7 , and I 8 , the model and prior are given by which leads to The data structure of (2) is depicted in the lower plot of Figure 1. We will consider the hypotheses This kind of hypothesis arises when we assume that a larger value in the population mean of the normal distribution means improvement in disease condition. Hence, a positive value of δ means better. Moreover, we are also interested in the reversed hypotheses This kind of hypothesis arises when we assume that a smaller value in the population mean of the normal distribution means improvement in disease condition. Hence, a negative value of δ means a better condition.
There are three conclusions, namely, CS, TS, and E. In terms of the δ values, assume that where TC stands for True Condition. For example, let h 1 and h 2 be the hazard rates corresponding to treatment and control, respectively. Hence, where HR is the Hazard Ratio. In terms of confidence or credible intervals of δ, we have CS D ⇔ the 1 − α confidence or credible interval of δ with lower limit [L, ∞) lying wholly above 0, TS D ⇔ the 1 − α confidence or credible interval of δ with upper limit (−∞, U] lying wholly below 0, E D ⇔ the 1 − 2α confidence or credible interval of δ with lower and upper limits [L, U] including 0, where D stands for Decision.

The CP, and the First and Second Predictive Powers
In this subsection, we will calculate the confidence intervals of δ with lower, upper, and both limits for the Classical Power (CP). Moreover, we will derive the probabilities of CS, TS, and E of the CP and the first and second predictive powers, and evaluate their limits at point 0.
Let us first calculate the confidence intervals of δ with lower, upper, and both limits for the CP. The CP is the probability of the classical rejection region with d 2 , given a value for δ, that is, CP = P S C,d 2 α,0 |δ . It is easy to show that the 1 − α confidence interval of δ with lower limit for the CP is Now, let us consider the CP − , which is the probability of the classical rejection region with d 2 , given a value for δ, that is, CP − = P S C−,d 2 α,0 |δ . It is easy to show that the 1 − α confidence interval of δ with upper limit for the CP is Moreover, the 1 − 2α confidence interval of δ with lower and upper limits for the CP is It is worth noting that the confidence level related to the lower or upper limit for the CP is 1 − α, while the confidence level related to both limits for the CP is 1 − 2α.
The expressions of the probabilities P(CS:CP), P(TS:CP), P(E:CP), P(CS:I 1 ), P(TS:I 1 ), P(E:I 1 ), P(CS:I 2 ), P(TS:I 2 ), and P(E:I 2 ) are given in Supplementary Materials. Note that the CS, TS, and E in the probabilities are for decisions. However, the subscripts D are omitted to lighten notations.
The main theoretical contributions of this article are summarized in the following four propositions, whose proofs are given in Supplementary Materials.
The limits at point 0 of the probabilities of CS, TS, and E of the CP and the first and second predictive powers are summarized in the following proposition.

The CCP, and the Third and Fourth Predictive Powers
In this subsection, we will calculate the confidence intervals of δ with lower, upper, and both limits for the Classical Conditional Power (CCP). Moreover, we will derive the probabilities of CS, TS, and E of the CCP and the third and fourth predictive powers, and evaluate their limits at point 0.
Let us first calculate the confidence intervals of δ with lower, upper, and both limits for the CCP. The CCP is the probability of the classical rejection region with d 1 and d 2 , given values of δ and the interim result d 1 , that is, CCP = P S C,d 1 ,d 2 It is easy to show that the 1 − α confidence interval of δ with a lower limit for the CCP is Now let us consider the CCP − , which is the probability of the classical rejection region with d 1 and d 2 , given values of δ and the interim result d 1 , that is, CCP − = P S C−,d 1 ,d 2 α,0 |δ, d 1 . It is easy to show that the 1 − α confidence interval of δ with the upper limit for the CCP is −∞, Moreover, the 1 − 2α confidence interval of δ with lower and upper limits for the CCP is It is worth noting that the confidence level related to the lower or upper limit for the CCP is 1 − α, while the confidence level related to both limits for the CCP is 1 − 2α.
The graph of the three limits (8)-(10) by one plot is depicted in Web Figure S1. Alternatively, the graph of the three limits (8)-(10) by three plots is depicted in Figure 2.  Figure 2. The graph of the three limits (8)-(10) by three plots.

The BP, and the Fifth and Sixth Predictive Powers
In this subsection, we will calculate the credible intervals of δ with lower, upper, and both limits for the Bayesian Power (BP). Moreover, we will derive the probabilities of CS, TS, and E of the BP and the fifth and sixth predictive powers, and evaluate their limits at point 0.
Let us first calculate the credible intervals of δ with lower, upper, and both limits for the BP. The BP is the probability of the Bayesian rejection region with d 0 , d 2 , given values of δ and the historical result d 0 , that is, It is easy to show that the 1 − α credible interval of δ with a lower limit for the BP is Now, let us consider the BP − , which is the probability of the Bayesian rejection region with d 0 , d 2 , given values of δ and historical result d 0 , that is, It is easy to show that the 1 − α credible interval of δ with an upper limit for the BP is Moreover, the 1 − 2α credible interval of δ with lower and upper limits for the BP is It is worth noting that the credible level related to the lower or upper limit for the BP is 1 − α, while the credible level related to both limits for the BP is 1 − 2α.
The limits at point 0 of the probabilities of CS, TS, and E of the BP and the fifth and sixth predictive powers are summarized in the following proposition.

The BCP, and the Seventh and Eighth Predictive Powers
In this subsection, we will calculate the credible intervals of δ with lower, upper, and both limits for the Bayesian Conditional Power (BCP). Moreover, we will derive the probabilities of CS, TS, and E of the BCP and the seventh and eighth predictive powers, and evaluate their limits at point 0.
Let us first calculate the credible intervals of δ with lower, upper, and both limits for the BCP. The BCP is the probability of the Bayesian rejection region with d 0 , d 1 , d 2 , It is easy to show that the 1 − α credible interval of δ with lower limit for the BCP is Now, let us consider the BCP − , which is the probability of the Bayesian rejection region with d 0 , d 1 , d 2 , It is easy to show that the 1 − α credible interval of δ with the upper limit for the BCP is Moreover, the 1 − 2α credible interval of δ with lower and upper limits for the BCP is It is worth noting that the credible level related to the lower or upper limit for the BCP is 1 − α, while the credible level related to both limits for the BCP is 1 − 2α. The

Numerical Experiments
In this section, we will conduct extensive numerical experiments to exemplify the imprecision issues of the four powers and the eight predictive powers. First, we will compute the probabilities of CS, TS, and E for the four powers as functions of m 2 when δ = 1, which favors control, δ = −1, which favors treatment, and δ = 0, which favors equivocal, respectively. Second, we will compute the probabilities of CS, TS, and E for the eight predictive powers as functions of m 2 under the sceptical prior and the optimistic prior, respectively.
As was used in [5,15], we assume that is calculated to ensure that an optimistic prior was centered on a 40% hazard reduction and a 5% chance of a negative effect (i.e., HR > 1), equivalent on the log(HR) scale to a normal prior with mean µ o = log(0.6) ≈ −0.51 and standard deviation 0.31 (σ = √ 2, m r 0 ≈ 41.4). The sceptical prior was adopted as a normal distribution with the same standard deviation as the optimistic prior, but centered on µ s = 0. The superscripts "r" in m r 0 , m r 1 , m r 2 , d r 1 , and t r are added to indicate that they are from the real data.
The four powers and the eight predictive powers as functions of the parameters and the data used are summarized in Table 1. From the table, we observe the following facts.
(1) The CP has relations to I 1 and I 2 , the CCP has relations to I 3 and I 4 , the BP has relations to I 5 and I 6 , and the BCP has relations to I 7 and I 8 .
(2) All four powers and eight predictive powers are functions of α, σ, and m 2 .
(3) The four powers are functions of δ, while the eight predictive powers are not functions of δ. (4) For the data used column, as described in [5], H means that the historical data are used, and I means that the interim data are used. HI means that the historical data are used once and that the interim data are also used once. HI 2 means that the historical data are used once and that the interim data are used twice. H 2 means that the historical data are used twice. H 2 I means that the historical data are used twice and that the interim data are used once. H 2 I 2 means that the historical data are used twice and that the interim data are also used twice. Note that CP does not use the historical data nor the interim data, and thus, 0 is used to indicate this fact. (5) For the eight predictive powers, I 1 and I 5 only use the historical data, while for the other predictive powers, they use both the historical data and the interim data.
The probabilities of CS, TS, and E for CP, CCP, BP, and BCP as functions of m 2 when δ = 1, which favors control, are plotted in Figure 3. From the figure, we observe the following facts.    Therefore, when m 2 is small (imprecision), the P(E:BCP_o) is large and it is hard to discriminate between CS and TS. Moreover, P(CS:BCP_o) is an increasing function of m 2 , P(TS:BCP_o) is almost 0, and P(E:BCP_o) is a decreasing function of m 2 . The three probabilities (P(CS:BCP_o), P(TS:BCP_o), and P(E:BCP_o)) sum to 1. (7) The central two plots are for BP, with the left plot being a sceptical prior and the right plot being an optimistic prior. The two plots display similar patterns of increasing and decreasing characteristics. The sceptical prior favors control, and thus P(CS:BP_s) is larger than P(CS:BP_o). The optimistic prior favors treatment; however, the two probabilities (P(TS:BP_s) and P(TS:BP_o)) for TS are almost 0, forcing P(E:BP_o)tobe larger than P(E:BP_s). (8) The lower two plots are for BCP, with the left plot being a sceptical prior and the right plot being an optimistic prior. The two plots display different patterns of increasing and decreasing characteristics. The sceptical prior favors control, and thus P(CS:BCP_s) is larger than P(CS:BCP_o). The optimistic prior favors treatment; however, the two probabilities (P(TS:BCP_s) and P(TS:BCP_o)) for TS are almost 0, forcing P(E:BCP_o)tobe larger than P(E:BCP_s). The sceptical prior favors control and the interim data (d r 1 = 0.435) also favors control, and thus, P(CS:BCP_s) is large. The optimistic prior favors treatment, but the interim data favors control, and thus, P(E:BCP_o) is large. The probabilities of CS, TS, and E for CP, CCP, BP, and BCP as functions of m 2 when δ = 1, which favors control, are summarized in Web Table S1. This table reports the numerical values of the probabilities in Figure 3.
The probabilities of CS, TS, and E for CP, CCP, BP, and BCP as functions of m 2 when δ = −1, which favors treatment, are plotted in Figure 4. From the figure, we observe the following facts.  (1) Compared to Figure 3, since δ = 1 favors control and δ = −1 favors treatment, the probabilities of CS in Figure 4 are smaller than those in Figure 3, while the probabilities of TS in Figure 4 are larger than those in Figure 3. (2) The central two plots are for BP, with the left plot being a sceptical prior and the right plot being an optimistic prior. The two plots display similar patterns of increasing and decreasing characteristics. The optimistic prior favors treatment, and thus, P(TS:BP_o) is larger than P(TS:BP_s). The sceptical prior favors control; however, the two proba-bilities (P(CS:BP_o) and P(CS:BP_s)) for CS are almost 0, forcing P(E:BP_s)tobe larger than P(E:BP_o). (3) The lower two plots are for BCP, with the left plot being a sceptical prior and the right plot being an optimistic prior. The two plots display different patterns of increasing and decreasing characteristics. The sceptical prior favors control, and thus, P(CS:BCP_s) is larger than P(CS:BCP_o). The optimistic prior favors treatment, and thus, P(TS:BCP_o) is larger than P(TS:BCP_s). The sceptical prior favors control and the interim data (d r 1 = 0.435) also favors control, and thus, P(CS:BCP_s) is large. The optimistic prior favors treatment but the interim data favors control, and thus, P(E:BCP_o) is large.
The probabilities of CS, TS, and E for CP, CCP, BP, and BCP as functions of m 2 when δ = −1 which favors treatment, are summarized in Web Table S2. This table reports the numerical values of the probabilities in Figure 4.
The probabilities of CS, TS, and E for CP, CCP, BP, and BCP as functions of m 2 when δ = 0, which favors equivocal, are plotted in Figure 5. From the figure, we observe the following facts.
(1) Compared to Figures 3 and 4, since δ = 0 favors equivocal, the probabilities of equivocal in Figure 5 are larger than those in Figures 3 and 4 for CP, BP_s, BP_o, and BCP_o. It is interesting to note that for P(E:CCP) in Figure 4, the interim data favors control and δ = −1 favors treatment, and thus, P(E:CCP) in Figure 4 is large.
Similarly, for P(E:BCP_s) in Figure 4, the interim data favors control, the sceptical prior favors control, and δ = −1 favors treatment, and thus, P(E:BCP_s) in Figure 4 is large. Nevertheless, the P(E:CCP) and P(E:BCP_s) in Figure 5 are large, since δ = 0 favors equivocal in this figure. The probabilities of CS, TS, and E for CP, CCP, BP, and BCP as functions of m 2 when δ = 0, which favors equivocal, are summarized in Web Table S3. This table reports  The probabilities of CS, TS, and E for the eight predictive powers as functions of m 2 under the sceptical prior are plotted in Web Figure S2 and summarized in Web Table S4.
The probabilities of CS, TS, and E for the eight predictive powers as functions of m 2 under the optimistic prior are plotted in Web Figure S3 and summarized in Web Table S5.

A Real Data Example
In this section, we perform some numerical experiments on a real data example. First, we will compute the probabilities of CS, TS, and E for CP, CCP, BP_s, BP_o, BCP_s, and BCP_o when δ = 1 (favors control), −1 (favors treatment), and 0 (favors equivocal), respectively. Second, we will compute the probabilities of CS, TS, and E for the eight predictive powers under either a sceptical or optimistic prior.
Long-term tamoxifen therapy is used to prevent the recurrence of breast cancer (see [5]; Example 6.7 in [15,16]). The aim of this study is to estimate disease-free survival benefit from tamoxifen over placebo, in patients who already have had 5 years of taking tamoxifen without a recurrence. It implies that patients were randomized to either continuation with placebo vs. continuation with tamoxifen therapy after having survived recurrence-free under tamoxifen for 5 years. So as to detect a 40% reduction in annual risk associated with tamoxifen (hazard ratio = 0.6), with 85% power and a one-sided tail area of 5%, 115 events were required. With summary using the approximate hazard ratio analysis, the proportional hazards regression model is the statistical model. If there are O T events on treatment, and O C events on control, then d 1 = 2(O T − O C )/m 1 is an approximate estimate of the log(hazard ratio) δ, with mean δ and variance 4/m 1 , as shown in [17]. There are two prior distributions that are used. An optimistic prior was centered on a 40% hazard reduction and a 5% chance of a negative effect (i.e., HR > 1), equivalent on the log(HR) scale to a normal prior with mean µ o = log(0.6) = −0.51 and standard deviation 0.31 (σ = √ 2, m 0 = 41.4). It is useful to note that in [15], the variance is σ 2 = 4, while in this article, the variance is 2σ 2 = 4, and thus, σ = √ 2 in this article. In addition, a sceptical prior was adopted with the same standard deviation as the optimistic prior, but centered on µ s = 0. The estimated log(HR) after the first interim analysis in 1993 is d 1 = 0.435; at that time, m 1 = 46 events have been observed, and a further m 2 = 115 − 46 = 69 events are to be observed. The chosen significance level α is 0.025. Therefore, we have The probabilities of CS, TS, and E for CP, CCP, BP_s, BP_o, BCP_s, and BCP_o when δ = 1 (favors control), −1 (favors treatment), and 0 (favors equivocal) are summarized in Table 2. From the table, we observe the following facts.
(1) In each row, the sum of the three probabilities should be equal to 1. However, in some circumstances, the sum is equal to 0.999, due to the rounding error. (2) When δ = 1, which favors control, the probabilities of CS are large for CP, CCP, BP_s, BP_o, BCP_s, and BCP_o; when δ = −1, which favors treatment, the probabilities of TS are large for CP, CCP, BP_s, BP_o, BCP_s, and BCP_o; and when δ = 0, which favors equivocal, the probabilities of E are large for CP, CCP, BP_s, BP_o, BCP_s, and BCP_o. In conclusion, for a given true condition (CS, TS, or E), the probabilities of that condition (CS, TS, or E) are large for CP, CCP, BP_s, BP_o, BCP_s, and BCP_o. (3) CP, BP_s, BP_o, and BCP_o have imprecision issues, while CCP and BCP_s do not have imprecision issues (see . However, for all the powers, the probabilities of E are large only when δ = 0, which favors equivocal. That is to say, when δ = 1 (favors control) and δ = −1 (favors treatment), the probabilities of E are not large for all the powers (an exception is the probability of E, which is equal to 0.501 for BCP_s when δ = −1). Therefore, whether the probabilities of E are large are affected by the δ values, but are not affected by whether the power has an imprecision issue.
The probabilities of CS, TS, and E for the eight predictive powers under the sceptical or optimistic prior are summarized in Table 3. From the table, we observe the following facts.
(1) In each row, the sum of the three probabilities corresponding to the sceptical prior (or the optimistic prior) should be equal to 1. However, in some cases, the sum is equal to 0.999, due to the rounding error. (2) For the eight predictive powers, I 1 _s, I 2 _s, I 5 _s, I 6 _s, I 1 _o, I 2 _o, I 5 _o, I 6 _o, I 7 _o, and I 8 _o have imprecision issues; however, I 3 _s, I 4 _s, I 7 _s, I 8 _s, I 3 _o, and I 4 _o do not have imprecision issues (see Web Figures S2 and S3). Moreover, the probabilities of E are large for all the eight predictive powers under the sceptical or optimistic prior except I 1 _o and I 5 _o. It is common that having an imprecision issue combines with P(E) large; for instance, I 1 _s, I 2 _s, I 5 _s, I 6 _s, I 2 _o, I 6 _o, I 7 _o, and I 8 _o. However, not having an imprecision issue may combine with P(E) large; for example, I 3 _s, I 4 _s, I 7 _s, I 8 _s, I 3 _o, and I 4 _o. Moreover, having an imprecision issue may combine with P(E) small; for instance, I 1 _o and I 5 _o. Therefore, whether the probabilities of E are large is not affected by whether the predictive power has an imprecision issue. (3) Note that the numerical values of this table are the same as those of Table 3 in [5].
However, in this table, we have discussed that the relationship between the imprecision issue of the predictive power and the probability of E large, and the relationship is described in item (2). (4) At the interim, the trial is stopped for futility, because the probabilities of TS are small for all the six predictive powers at the interim (I 2 , I 3 , I 4 , I 6 , I 7 , and I 8 ). However, it is not because the probabilities of CS are large, but because the probabilities of E are large. In a word, at the interim, the trial neither favors treatment nor control, but favors equivocal.

Conclusions and Discussions
Some conclusions and discussions are provided below.
• We have derived the probabilities of CS, TS, and E of the four powers and the eight predictive powers, and have evaluated the limits of the probabilities at point 0. Moreover, we have conducted extensive numerical experiments to exemplify the imprecision issues of the four powers and the eight predictive powers. In the numerical experiments, first, we have computed the probabilities of CS, TS, and E for the four powers as functions of m 2 when the true treatment effect favors control, treatment, and equivocal, respectively. Second, we have computed the probabilities of CS, TS, and E for the eight predictive powers as functions of m 2 under the sceptical prior and the optimistic prior, respectively. Finally, we have carried out a real data example to show the prominence of the methods. • For the four powers and the eight predictive powers with the parameters specified in (19), some have imprecision issues, but the others do not have these issues. More precisely, the CP, BP_s, BP_o, BCP_o, I 1 _s, I 2 _s, I 5 _s, I 6 _s, I 1 _o, I 2 _o, I 5 _o, I 6 _o, I 7 _o, and I 8 _o have imprecision issues. However, the CCP, BCP_s, I 3 _s, I 4 _s, I 7 _s, I 8 _s, I 3 _o, and I 4 _o do not have imprecision issues. • In fact, each of the four powers and the eight predictive powers has an opportunity to encounter the imprecision issue as long as the parameter values are chosen appropriately. Firstly, from (6), we see that the CP, I 1 , and I 2 will certainly have imprecision issues. Moreover, from (10), we see that if −Z α σ √ 2/s < d 1 < Z α σ √ 2/s, then the CCP, I 3 , and I 4 will have imprecision issues. Furthermore, from (14), we see that if −Z α σ 2/m 0 < d 0 < Z α σ 2/m 0 , then the BP, I 5 , and I 6 will have imprecision issues. Finally, from (18), we see that if −Z α σ 2(m 0 + s) < m 0 d 0 + sd 1 < Z α σ 2(m 0 + s), then the BCP, I 7 , and I 8 will have imprecision issues. • For simplicity, let P(CS) = P(CS:i), P(TS) = P(TS:i), and P(E) = P(E:i), Supplementary Materials: The following supporting information can be downloaded at: www.mdpi. com/xxx/s1, Figure S1: The graph of the three limits (10), (11), and (12) by one plot; Figure S2: The probabilities of CS, TS, and E for the 8 predictive powers as functions of m 2 under the sceptical prior; Figure S3: The probabilities of CS, TS, and E for the 8 predictive powers as functions of m 2 under the optimistic prior; Table S1: The probabilities of CS, TS, and E for CP, CCP, BP, and BCP as functions of m 2 when δ = 1 which favors control; Table S2: The probabilities of CS, TS, and E for CP, CCP, BP, and BCP as functions of m 2 when δ = −1 which favors treatment; Table S3: The probabilities of CS, TS, and E for CP, CCP, BP, and BCP as functions of m 2 when δ = 0 which favors equivocal. Table S4: The probabilities of CS, TS, and E for the 8 predictive powers as functions of m 2 under the sceptical prior.  Informed Consent Statement: Not applicable.

Data Availability Statement:
The authors confirm that the data supporting the findings of this study are available within the article.