# Conditional power, predictive power and probability of success in clinical
trials with continuous, binary and time-to-event endpoints

Madan G Kundu1 Sandipan Samanta2 Shoubhik Mondal3
1Daiichi-Sankyo Inc. (DSI), Basking Ridge, NJ, USA
2QIAGEN GmbH, Hilden, Germany
3Boehringer Ingelheim, CT, USA Corresponding author: Madan G. Kundu,
<EMAIL_ADDRESS>article reflects the views of the authors only and should
not be construed to represent the views or policies of their affiliated
organizations.

###### Abstract

Assessment of study success using conditional power (CP), the predictive power
of success (PPoS) and probability of success (PoS) is becoming increasingly
common for resource optimization and adaption of trials in clinical
investigation. Determination of these measures is often a non-trivial
mathematical task. Further, the terminologies used across the literature are
not consistent, and there is no consolidated presentation on this. Lastly,
certain types of trials received more attention where others (e.g., single-arm
trial with time-to-event endpoints) were completely ignored. We attempted to
fill these gaps. This paper first provides a detailed derivation of CP, PPoS
and PoS in a general setting with normally distributed test statistics and
normal prior. Subsequently, expressions for these measures are obtained for
continuous, binary, and time-to-event endpoints in single-arm and two-arm
trial settings. We have discussed both clinical success and trial success.
Importantly, we have derived the expressions for CP, PPoS and PoS in a single-
arm trial with a time-to-event endpoint that was never addressed in the
literature to our knowledge. In that discussion, we have also shown that
commonly recommended $1/d$ consistently under-estimates the variance of
log(median) and alternative expression for variance was derived. We have also
presented the PPoS calculation for the binomial endpoint with a beta prior.
Examples are given along with the comparison of CP and PPoS. Expressions
presented in this paper are implemented in `LongCART` package in `R`. An `R`
shiny app is also available at https://ppos.herokuapp.com/.

Keywords: B-value, Beta prior, Clinical success, Conditional power, Normal
prior, Predictive power of success (PPoS), Prior distribution, Probability of
success (PoS), Trial success.

##  1 Introduction

The need to determine the probability of “study success” may arise at various
stages of drug development. For example, the goal of such an exercise could be
making a go or no go decision at the beginning of a prospective trial based on
data from the earlier phase. It can also be used to monitor an ongoing
clinical trial to answer various questions: should the trial continue or
should it stop? Is the current sample size sufficient? Do we need any adaption
to the trial? The term “success” or “study success” is often understood in the
context of achieving a pre-specified threshold for p-value (e.g., two-sided
0.05 or one-sided 0.025) at the end of the trial. In this paper, we would call
it “trial success” to differentiate it from “Clinical success”. “Clinical
success” is defined as the observed treatment effect size exceeding some
threshold value that is often clinically meaningful [1, 2]. This paper focuses
on the determination of the conditional power (CP), predictive power of
success (PPoS) and probability of success (PoS) to assess the chance of “trial
success” and “clinical success”.

The CP, PPoS and PoS along with power are often used statistical tools to
quantify the chance of success (either “trial success” or “clinical success”).
Of these, power and PoS are calculated at the beginning of the trial, whereas
CP and PPoS are determined after observing the interim results. All these
measures attempt to determine pr(success) (i.e., chance of success) based on
the prior belief of effect size at the beginning of the trial, or the observed
value of effect size in the first part of the trial, or the combination of
both. At the beginning of a trial, pr(success) is determined solely based on
the prior belief. On the other hand, calculation of pr(success) at the interim
analysis has to take following things into consideration: (a) available
interim results, (b) limited uncertainty in the trial data arising from the
post-interim part, and (c) the decision to use or not the prior information.
Further, power and CP are frequentist tools, whereas PoS and PPoS follow the
Bayesian paradigm. The difference of Bayesian approach over the frequentist
approach is the way available knowledge on effect size is summarized: Bayesian
measures summarize this knowledge as distribution of effect whereas
frequentist measure makes the best guess about the effect size as a single
value, and thereby frequentist measures may not be a good indicator of
pr(success) [3, 4, 5, 6]. For this reason, PoS is also viewed as average
‘power’ over the prior distribution of $\theta$ [5, 6] whereas PPoS is the
average CP over the predictive distribution of $\theta$. Further, in the
Bayesian approach, both prior knowledge and available interim results can be
used together in assessing pr(success); however, all available knowledge must
be summarized into a single value in the frequentist approach.

CP is defined as pr(success) given the interim result and assuming a fixed
effect size for the remainder of the trial. Halperin et al. [7] first proposed
the use of CP in monitoring a long-term clinical trial. They proposed to
calculate two CPs for trial success given the current results: (a) assuming
the fixed effect size for the remainder of the trial as specified under the
null hypothesis ($H_{0}$), and (b) assuming the fixed effect size as expected
at the beginning of the trial under the alternative hypothesis ($H_{1}$). Lan
and Wittes [8] generalized the calculation of CP using the B-value (see
Section 2). Further, Lachin [9] has shown that any formal stopping boundaries
based on the CP (along with the type I and II error probabilities) can be
expressed using B value in a study with interim futility analysis. The use of
PPoS in a clinical trial can be traced back to Choi, Smith and Becker [10]:
they employed the predictive distribution of proportion using a beta prior to
obtaining the “desired probability” of trial success. However, it was
Spiegelhalter, Freedman and Blackburn [11] who first proposed a general
Bayesian framework to obtain unconditional power by averaging the conditional
probability of success over the current opinion of the treatment effect based
on the results observed in the first part of the trial. They termed this
unconditional power as the “predictive probability of rejecting the $H_{0}$.
In the remainder of this paper, this quantity is referred to as PPoS. For a
prospective trial, Spiegelhalter and Freedman [3] first suggested calculating
“average power” as the “overall predictive probability of obtaining a
significant result”, where the so-called statistical power of the trial is
averaged with respect over the prior distribution of belief about the possible
effect size. To distinguish it from the PPoS based on interim data, we would
call it PoS. The frameworks of [10, 11, 3] apply the Bayesian methodology in
monitoring clinical trials where analyses were carried out using conventional
frequentist techniques, and this framework was later referred to as ‘hybrid
classical-Bayesian’ [12]. Lan, Hu and Proschan [13] have discussed the
relationship between the CP and PPoS. The majority of the earlier works (e.g.,
[7, 10, 11]) considered a two-arm trial with a binary endpoint for
illustration. However, these frameworks are illustrated for the continuous
endpoints (e.g., see [14]) and the survival endpoints (e.g., see [15]) as
well.

Calculation of pr(success) (i.e., CP, PPoS and PoS) is often ‘non-trivial’
mathematical task [3]. To make it more difficult, the literature in this area
is severely suffered by the inconsistent use of terminologies for these
concepts. For example, PPoS has been referred as ‘predictive power’ [1, 13],
‘Bayesian predictive power (BPP)’ [16], ‘predictive probability of statistical
significance’ [2] and ‘probability of study success’ [17] in the literature.
On the other hand, the PoS in the literature is also referred in the
literature as ‘average success probability [5, 6], ‘assurance’ [4], and
‘expected power’ [6]. Further, certain areas (e.g., trials with binary
endpoint) received more attention compared to the rest. For example, we could
not find any literature discussing these probability measures in a single-arm
trial with the time-to-event endpoint. Despite the wide popularity of these
measures, the current literature still lacks the concise presentation of these
measures under a general framework of hypothesis testing. The present work
attempts to fill all these gaps.

In this paper, we focus on CP, PPoS with or without prior and PoS. We first
derive general expressions for these pr(success) measures for normally
distributed test statistics with normal prior (Section 3) under a unified
framework of hypothesis testing (Section 2). Subsequently, we present the
expressions for these measures in single-arm and two-arm trials with
continuous, binary and time-to-event endpoints separately in Section 4. For
two-arm trials, examples are presented along with the comparison between CP
and PPoS, and assessments of the impact of the prior distribution on
predictive distribution of effect size and PPoS. Importantly, we have derived
the expressions for CP, PPoS and PoS in a single-arm trial with the time-to-
event endpoint in Section 4.5 that was never addressed in the literature to
our knowledge. In that discussion, we have also shown that commonly
recommended approximated variance of $1/d$ (e.g. see [18]) consistently under-
estimates the variance of log(median) (see Figure 3), and we have derived
alternative expression for variance. Expressions of PPoS for the binary
endpoint with beta prior are presented in Section 5 with example.
Implementation in R through LongCART package [19] and R shiny app
(https://ppos.herokuapp.com/) are discussed in Section 6 and illustrated in
Appendix 2.

##  2 Preliminaries and notations

Table 1: Summary of notations used in Section 2 and Section 3 in deriving general expression of CP, PPoS and PoS. $t$ | fraction of information at interim $(0<t<1)$.
---|---
$\theta$ | true treatment effect.
$\hat{\theta}(t)$ | observed estimate of $\theta$ at the interim analysis with information $t$
$\hat{\theta}(1-t)$ | projected value of $\theta$ from post-interim data with remaining information of $1-t$
$k$ | SE of $\hat{\theta}(t)$ at final analysis ($t=1$)
$\tilde{k}$ | projected SE in the trial
$Z(t)$ | Z-test statistic based on interim data.
$Z(1-t)$ | Z-test statistic based on post-interim data.
$B(t)$ | B-values; $B(t)=Z(t)\cdot\sqrt{t}$
$c(t)$ | rejection boundary at analysis with information $t$
$\Phi(\cdot)$ | cumulative distribution function of a standard normal variate
$\theta^{\prime}$ | projected value of $\theta$ from post-interim data (excludes data upto interim).
$\theta_{0},\sigma_{0}$, | mean and SD of a normal prior for $\theta$
$\psi$ | $\sigma_{0}^{2}/(\sigma_{0}^{2}+k^{2}/t)$
$\theta_{\rm min}$ | threshold for “clinical success”
$\gamma$ |  $\gamma=c(1)$ for “trial success” and $\gamma=\frac{\theta_{\rm min}}{k}$ for “clinical success”.

Notations presented in this Section are listed in Table 1. We consider the
following general form of hypothesis testing in a clinical trial:

$H_{0}:\theta=0\qquad{\rm vs.}\qquad H_{1}:\theta>0$

where $\theta$ is a parameter of interest. For example, it could be either
mean or proportion in a given population or their difference between two
populations or log HR. We also assume results from the interim analysis
performed after the accrual of $t$ amounts of “information” ($0\leq t\leq 1$)
are available. At any given time of analysis, the information $t$ equals the
proportion of interim subjects ($n$, say) to the maximum number of planned
subjects ($N$, say) for continuous and binary endpoints, or proportion of
observed events at interim analysis ($d$, say) to the maximum planned events
($D$, say) for time-to-event endpoints. Let’s $\hat{\theta}(t)$ be the
estimate of $\theta$ at the interim analysis with corresponding ndard error
(SE) as $SE[\hat{\theta}(t)]=k\cdot\frac{1}{\sqrt{t}}$. Note that, $k$ is the
SE of the estimate at the final analysis and does not depend on $t$. For
example, $k=\sigma/\sqrt{N}$ in trials with continuous endpoint (where
$\sigma$ is the standard deviation (SD)), and $k\approx 2/\sqrt{D}$ in trials
with time-to-event endpoint, respectively. With this, for both Z-test and log-
rank test, test statistic $Z(t)$ can be expressed as

$Z(t)=\frac{\hat{\theta}(t)}{SE[\hat{\theta}(t)]}=\frac{\hat{\theta}(t)}{k}\cdot\sqrt{t}$

and,

${\rm Reject}\;\;H_{0},\;\;\rm{if}\;\;Z(t)>c(t)$

where $c(t)$ is the rejection boundary. $c(t)$ must be identified in advance
and should be such that it preserves overall type I error of $\alpha$. In a
single look design without any interim analysis, $c(1)=\Phi(1-\alpha)$, where
$\Phi(\cdot)$ denotes the cumulative distribution function of a standard
normal variate. In a multiple looks design with one or more interim analyses,
$c(1)$ must be determined according to the appropriate alpha spending function
(e.g. [20]).

Since, $E\left[Z(t)\right]=(\theta/k)\cdot\sqrt{t}$, the information growth in
$Z(t)$ is proportional to $\sqrt{t}$. Following Lan and Wittes [8], the
B-values are defined as follows:

$B(t)=Z(t)\cdot\sqrt{t}=\frac{\hat{\theta}(t)}{k}\cdot t$ (2.1)

with $B(0)=0$ at the trial initiation and $B(1)=Z(1)$ at the end of the trial.
Further,

${\rm Cov}[B(t),B(s)]=min(s,t)\qquad{\rm and}\qquad{\rm Var}[B(t)]=t$ (2.2)

$B(t)$ has following advantages over $Z(t)$: (a) information growth in $B(t)$
is proportional to $t$ as $E[B(t)]=(\theta/k)\cdot t$, and (b) $B(t)$ has
independent increments, implying $B(1)-B(t)$, is independent of the $B(t)$.
Because of these two advantages, it is often easier to work with $B(t)$
compared to $Z(t)$.

With interim data available at information time $t$, the uncertainty is now
restricted to the results from the post-interim data (i.e., data contributing
to remaining information of $(1-t)$). As $B(1)-B(t)$ is independent of $B(t)$,
we can decompose $B(1)$ as follows

$B(1)=B(t)+[B(1)-B(t)]$

Based on Eq. (2.1), it translates to

$Z(1)=\sqrt{t}\cdot Z(t)+\sqrt{1-t}\cdot Z(1-t)$ (2.3)

where $Z(1-t)$ is the test statistic using the post-interim data only (i.e.,
data accrued after information $t$) and is defined as follows:

$Z(1-t)=\frac{\hat{\theta}(1-t)}{SE\left[\hat{\theta}(1-t)\right]}=\frac{\hat{\theta}(1-t)}{k/\sqrt{1-t}}=\frac{\hat{\theta}(1-t)\cdot\sqrt{1-t}}{k}$
(2.4)

where, $\hat{\theta}(1-t)$ is the estimate of $\theta$ based on post-interim
data only. From Eq. (2.3), we have

$\hat{\theta}(1)=t\cdot\hat{\theta}(t)+(1-t)\cdot\hat{\theta}(1-t)$ (2.5)

Clearly, $Z(1-t)$ and $\hat{\theta}(1-t)$ are independent from $Z(t)$ and
$\hat{\theta}(t)$. Further, after the interim analysis, $Z(t)$ and
$\hat{\theta}(t)$ are fixed and known, but $Z(1-t)$ and $\hat{\theta}(1-t)$
are unknown and random. We call out “Trial success” at the time of final
analyses if $Z(1)>c(1)$. Based on Eq. (2.3) and Eq. (2.4), this translates to

$\mbox{ Trial success, if
}\qquad\hat{\theta}(1-t)>\frac{k}{1-t}\cdot\left[c(1)-\sqrt{t}\cdot
Z(t)\right]$

Further, we call out “clinical success” at the time of final analyses if
$\hat{\theta}(1)>\theta_{\rm min}$. Based on Eq. (2.5), this translates to

$\mbox{ Clinical success, if
}\qquad\hat{\theta}(1-t)>\frac{k}{1-t}\cdot\left[\frac{\theta_{\rm
min}}{k}-\sqrt{t}\cdot Z(t)\right]$

Note the similarity in the definition of the “Trial success” and “Clinical
success” criteria. By replacing $c(1)$ with $\frac{\theta_{\rm min}}{k}$ in
the “Trial success” criterion, we can obtain the “Clinical success” criterion.
Therefore, we define the general criteria of “success” as follows:

$\mbox{ Success, if
}\qquad\hat{\theta}(1-t)>\frac{k}{1-t}\cdot\left[\gamma-\sqrt{t}\cdot
Z(t)\right]$ (2.6)

where $\gamma=c(1)$ for “Trial success” and $\gamma=\frac{\theta_{\rm
min}}{k}$ for “Clinical success”.

##  3 Conditional power, Predictive power, Probability of success

In this section, we first explain the concept of CP, PPoS and PoS.
Subsequently, the generic expressions for these measures are derived using
normal approximation. Calculation of PPoS and PoS using Beta-Binomial
distribution are presented in Section 5.

###  3.1 Conditional power (CP) based on interim results

CP is the probability that the final study result will be statistically
significant (or clinically successful), given the data observed thus far and a
specific assumption about the pattern of the data to be observed in the
remainder of the study, such as assuming the original design effect, or the
effect estimated from the current data, or under the null hypothesis [9].
Projecting the estimate of $\theta$ from the post-interim period to be
$\theta^{\prime}$, the estimate of $\theta$ in the post-interim data would be
distributed as

$\hat{\theta}(1-t)\sim\mbox{Normal}\left[\theta^{\prime},\frac{k^{2}}{1-t}\right]$

Therefore, the CP is (from Eq (2.6))

$\mbox{CP}(t|\theta^{\prime})=1-\Phi\left(\frac{\frac{k}{1-t}\cdot\left[\gamma-\sqrt{t}\cdot
Z(t)\right]-\theta^{\prime}}{\frac{k}{\sqrt{1-t}}}\right)=\Phi\left(\frac{1}{\sqrt{1-t}}\left[\frac{\hat{\theta}(t)}{k}\left\\{t+(1-t)\frac{\theta^{\prime}}{\hat{\theta}(t)}\right\\}-\gamma\right]\right)$
(3.1)

It is very intuitive and common to replace $\theta^{\prime}$ by
$\hat{\theta}(t)$ in calculation of the CP. In that case, the expression of CP
reduces to (e.g., see [13])

$\mbox{CP}(t|\hat{\theta}(t))=\Phi\left(\frac{1}{\sqrt{1-t}}\left[\frac{\hat{\theta}(t)}{k}-\gamma\right]\right)=\Phi\left(\frac{1}{\sqrt{1-t}}\left[\frac{Z(t)}{\sqrt{t}}-\gamma\right]\right)$
(3.2)

This is the CP when the post-interim trend expected to follow the interim
trend.

###  3.2 Predictive power of success (PPoS) based on interim results

CP depends on the specified treatment effect in the post-interim data, and
therefore, calculation of CP can be arbitrary. An alternative would be to
obtain PPoS as averaged CP over the predictive distribution of
$\hat{\theta}(1-t)$ at interim. The use of prior in the calculation of PPoS
optional. We have discussed PPoS below both with and without prior.

Suppose the prior knowledge about $\theta$ can be summarized using the
following prior distribution:

$\theta\sim{\rm Normal}\left[\theta_{0},\sigma_{0}^{2}\right]$ (3.3)

With this prior, the posterior distribution of $\theta$ is

$\theta|\hat{\theta}(t)\sim{\rm
Normal}\left[\psi\cdot\hat{\theta}(t)+(1-\psi)\theta_{0},\;\;\;\psi\cdot
k^{2}/t\right]$

where, $\psi=\dfrac{\sigma_{0}^{2}}{\sigma_{0}^{2}+k^{2}/t}$ is the proportion
of the contribution of interim data. Accordingly, the predictive distribution
of $\hat{\theta}(1-t)$ is as follows:

$\hat{\theta}(1-t)|\hat{\theta}(t)\sim{\rm
Normal}\left[\psi\cdot\hat{\theta}(t)+(1-\psi)\theta_{0},\;\;\;k^{2}\left(\frac{1}{1-t}+\psi\cdot\frac{1}{t}\right)\right]$

We can now use the predictive distribution of $\hat{\theta}(1-t)$ and Eq.
(2.6) to derive the PPoS as follows

$\displaystyle\mbox{PPoS}(t|\mbox{interim},\mbox{prior})$
$\displaystyle=1-\Phi\left(\frac{\frac{k}{1-t}\left[\gamma-\sqrt{t}\cdot
Z(t)\right]-\psi\cdot\hat{\theta}(t)-(1-\psi)\theta_{0}}{k\sqrt{1/(1-t)+\psi/t}}\right)$

This is the PPoS given the interim results and the prior distribution. Without
the prior distribution, PPoS can be derived as a special case of Eq
(LABEL:eq:ppos) by setting $\psi=1$ which implies 100% contribution of interim
data to the predictive distribution of $\hat{\theta}(1-t)$. Therefore, after
some simple algebraic manipulation, PPoS without the prior distribution is
obtained as follows (e.g., see [13, 21]):

$\mbox{PPoS}(t|\mbox{interim})=\Phi\left(\frac{1}{\sqrt{1-t}}\left[\frac{Z(t)}{\sqrt{t}}-\gamma\right]\cdot\sqrt{t}\right)$
(3.5)

Expressions of the PPoS in Eq. (3.5) and the CP in Eq. (3.2) are very similar
except the additional $\sqrt{t}$ inside $\Phi(\cdot)$ for PPoS. It’s simple
consequence is that CP$>$PPoS for $CP>0.5$ and CP$<$PPoS for $CP<0.5$ [13]
(see Figures 1, 2 and 4). That is, the CP is less extreme than the PPoS. For
example, the stopping rule based on the PPoS will always make it harder to
stop a trial compared to the CP.

###  3.3 Probability of success (PoS) of a prospective trial at the design
stage

The concept of PoS is very similar to PPoS: PPoS is averaged CP over the
predictive distribution, whereas PoS is averaged power over the predictive
distribution. Unlike PPoS, PoS is calculated at the beginning of the trial,
and hence it only relies on the prior distribution of $\theta$. As mentioned
before, the PoS has also been referred to as ‘assurance’ [4], and ‘expected
power’ or ‘average success of probability’ [6].

With the prior distribution specified in Eq. (3.3) and expecting the SE in the
trial to be $\tilde{k}$, the predictive distribution of $\hat{\theta}(1)$ is

$\hat{\theta}(1)|\theta\sim{\rm
Normal}\left[\theta_{0},\;\;\;\sigma_{0}^{2}+var[\hat{\theta}(1)]\right]\equiv{\rm
Normal}\left[\theta_{0},\;\;\;\sigma_{0}^{2}+\tilde{k}^{2}\right]$

In that case, the PoS given the prior distribution would be

$\mbox{PoS}=Pr[Z(1)>\gamma|\theta]=Pr[\hat{\theta}(1)>\tilde{k}\cdot\gamma|\theta]=\Phi\left(\frac{\theta_{0}-\tilde{k}\cdot\gamma}{\sqrt{\sigma_{0}^{2}+\tilde{k}^{2}}}\right)$
(3.6)

##  4 Expressions of CP, PPoS and PoS by type of endpoints

Table 2: Summary of notations and expressions of CP, PPoS and PoS with
normally distributed test statistics and normal priors presented in Section 4

| Continuous endpoint | Binary endpoint | Time-to-event endpoint
---|---|---|---
| single-arm | two-arm | single-arm | two-arm | single-arm | two-arm
Hypothesis and notations
Hypothesis | $H_{0}:\mu=\mu_{1}$ | $H_{0}:\mu_{T}-\mu_{C}=\Delta_{1}$ | $H_{0}:\Pi=\Pi_{1}$ | $H_{0}:\Pi_{T}-\Pi_{C}=\Delta_{1}$ | $H_{0}:M=M_{1}$ | $H_{0}:\Delta=\Delta_{1}$
| vs. | vs. | vs. | vs. | vs. | vs.
| $H_{1}:\mu>\mu_{1}$ | $H_{1}:\mu_{T}-\mu_{C}>\Delta_{1}$ | $H_{1}:\Pi>\Pi_{1}$ | $H_{1}:\Pi_{T}-\Pi_{C}>\Delta_{1}$ | $H_{1}:M>M_{1}$ | $H_{1}:\Delta<\Delta_{1}$
Timing of analyses |  |  |  |  |  |
$-$ interim analysis |  $n$ subjects |  $n$ subjects |  $n$ subjects |  $n$ subjects |  $d$ events |  $d$ events
$-$ final analysis |  $N$ subjects |  $N$ subjects |  $N$ subjects |  $N$ subjects |  $D$ events |  $D$ events
Allocation ratio | - | $a$ | - | $a$ | - | $a$
SD1 | $\sigma$ | $\sigma$ | $\sigma=\sqrt{\Pi(1-\Pi)}$ |  $\sigma$, 2see footnote | - | -
Parameter of interest | mean ($\mu$) | mean diff ($\Delta$) | proportion ($\Pi$) | prop diff ($\Delta$) | median ($M$) | HR($\Delta$)
|  | $\Delta=\mu_{T}-\mu_{C}$ |  | $\Delta=\Pi_{T}-\Pi_{C}$ |  |
|  |  $\mu_{T}$: mean in Arm T |  |  $\Pi_{T}$: prop. in Arm T |  |
|  |  $\mu_{C}$: mean in Arm C |  |  $\Pi_{C}$: prop. in Arm C |  |
$-$value under $H_{0}$ | $\mu_{1}$ | $\Delta_{1}$ | $\Pi_{1}$ | $\Delta_{1}$ | $M_{1}$ | $\Delta_{1}$
$-$interim estimate | $\bar{x}_{n}$ | $\delta_{n}=\bar{x}_{T,n}-\bar{x}_{C,n}$ | $p_{n}$ | $\delta_{n}=p_{T,n}-p_{C,n}$ | $m_{d}$ | $\delta_{d}$
|  |  $\bar{x}_{T,n}$: estimate of $\mu_{T}$ |  |  $p_{T,n}$: estimate of $\Pi_{T}$ |  |
|  |  $\bar{x}_{C,n}$: estimate of $\mu_{C}$ |  |  $p_{C,n}$: estimate of $\Pi_{C}$ |  |
$-$projected value for |  |  |  |  |  |
post-interim data | $\mu^{\prime}$ | $\Delta^{\prime}$ | $\Pi^{\prime}$ | $\Delta^{\prime}$ | $M^{\prime}$ | $\Delta^{\prime}$
$-$projected value in |  |  |  |  |  |
the entire trial | $\tilde{\mu}$ | $\tilde{\Delta}$ | $\tilde{\Pi}$ | $\tilde{\Delta}$ | $\tilde{M}$ | $\tilde{\Delta}$
$-$prior distribution | $\mu\sim$ | $\Delta\sim$ | $\Pi\sim$ | $\Delta\sim$ | $\log{M}\sim$ | $\log{\Delta}\sim$
| $N(\mu_{0},\sigma_{0}^{2})$ | $N(\Delta_{0},\sigma_{0}^{2})$ | $N(\Pi_{0},\sigma_{0}^{2})$ | $N(\Delta_{0},\sigma_{0}^{2})$ | $N(\log{M_{0}},\sigma_{0}^{2})$ | $N(\log{\Delta_{0}},\sigma_{0}^{2})$
Estimated (pooled) | $s_{n}$ | $s_{n}$ | $s_{n}$ | $s_{n}$ | - | -
SD at interim |  |  |  |  |  |
Projected (pooled) | $\tilde{\sigma}$ | $\tilde{\sigma}$ | $\tilde{\sigma}=\sqrt{\tilde{\Pi}(1-\tilde{\Pi})}$ | $\tilde{\sigma}$ | - | -
SD at beginning |  |  |  |  |  |
At the beginning of trial
$-$probability of |  |  |  |  |  |
success (PoS) | Eq. (4.5) | Eq. (4.11) | Eq. (4.17) | Eq. (4.11)3 | Eq. (4.24) | Eq. (4.30)
At the interim analysis
$-$Conditional power |  |  |  |  |  |
(CP) with specified | Eq. (4.1) | Eq. (4.7) | Eq. (4.13) | Eq. (4.7)3 | Eq. (4.20) | Eq. (4.26)
trend |  |  |  |  |  |
$-$Conditional power |  |  |  |  |  |
(CP) with interim | Eq. (4.2) | Eq. (4.8) | Eq. (4.14) | Eq. (4.8)3 | Eq. (4.21) | Eq. (4.27)
trend |  |  |  |  |  |
$-$predictive power |  |  |  |  |  |
of success (PPoS) | Eq. (4.3) | Eq. (4.9) | Eq. (4.15) | Eq. (4.9)3 | Eq. (4.22) | Eq. (4.28)
without prior |  |  |  |  |  |
$-$predictive power |  |  |  |  |  |
of success (PPoS) | Eq. (4.4) | Eq. (4.10) | Eq. (4.16) | Eq. (4.10)3 | Eq. (4.23) | Eq. (4.29)
with prior |  |  |  |  |  |
$-$ Subscripts $T$ and $C$ indicates measurements in treatment (T) and control
(C) arms, respectively.
$-$ 1SD: Standard deviation. For two arm trials it is the pooled SD.
$-$ 2SD in two-arm trial with binary endpoint:
$\sigma=\sqrt{\dfrac{a}{a+1}\left\\{\dfrac{\Pi_{T}(1-\Pi_{T})}{a}+\Pi_{C}(1-\Pi_{C})\right\\}}$
$-$ 3The expression for PoS, CP and PPoS are identical for two-arm trials with
binary and continuous endpoints, although the
quantities (e.g., $\delta_{n}$, $s_{n}$ etc.) included in the expression are
to be obtained differently.

Expressions of CP, PPoS and PoS for continuous, binary and survival endpoints
in a single-arm and two-arm trials are derived separately in this section
based on the general expressions presented in the previous section with normal
approximation. Summary of notations and expressions of CP, PPoS and PoS
discussed in the Section are presented are presented in Table 2. For the two-
arm trial, the allocation ratio (treatment arm to control arm) is denoted as
$a:1$. We denote $r^{2}=(a+1)^{2}/a$. The expressions for single-arm trial can
be obtained directly from the corresponding expressions from the two-arm trial
by specifying $r=1$. Intuition for setting $r=1$ is simple: the single-arm
design can be thought of as $1:0$ allocation ratio (instead of $a:1$) in which
case, $r=(1+0)/\sqrt{1}=1$. For clarity, we present the expressions for both
single-arm and two-arm scenario separately.

We also would like to remind that here the expressions are presented for
general success criteria as presented in Eq. (2.6). One needs to set
$\gamma=c(1)$ for “Trial success” and $\gamma=\frac{\theta_{\rm min}}{k}$ for
“Clinical success”. We have illustrated the calculation of CP, PPoS and PoS
based on published clinical trial results for two-arm trials and also compared
the behaviour of CP and PPoS.

###  4.1 Continuous endpoint, single-arm trial

We start with the single-arm trial continuous endpoint. Denote the population
mean as $\mu$ and the maximum sample size in the trial as $N$. We test the
following hypotheses:

$H_{0}:\mu=\mu_{1}\qquad{\rm vs.}\qquad H_{1}:\mu>\mu_{1}$

Here, $\theta=\mu-\mu_{1}$. At interim analysis with sample size $n$, the
estimate of $\theta$ is $\hat{\theta}(t)=\overline{x}_{n}-\mu_{1}$ where
$\overline{x}_{n}$ is the sample mean at interim. The corresponding test
statistic is

$Z(t)=\frac{\overline{x}_{n}-\mu_{1}}{s_{n}/\sqrt{n}}=\frac{(\overline{x}_{n}-\mu_{1})\sqrt{n}}{s_{n}}$

where $s_{n}$ is the estimate of SD ($\sigma$) at interim analysis. Further,
in this case, $t=n/N$ and $k=s_{n}/\sqrt{N}$.

Conditional power (CP): The CP with the future trend of the sample mean as
$\mu^{\prime}$ is

$\Phi\left(\frac{1}{s_{n}}\sqrt{\frac{N}{N-n}}\left[\frac{1}{\sqrt{N}}\\{n(\bar{x}_{n}-\mu_{1})+(N-n)(\mu^{\prime}-\mu_{1})\\}-s_{n}\cdot\gamma\right]\right)$
(4.1)

If we assume that the current trend observed through interim analysis
continues to hold for future data as well (i.e.,
$\mu^{\prime}=\overline{x}_{n}$), then the expression of the CP reduces to

$\Phi\left(\frac{1}{s_{n}}\sqrt{\frac{N}{N-n}}\left[(\bar{x}_{n}-\mu_{1})\sqrt{N}-s_{n}\cdot\gamma\right]\right)$
(4.2)

Predictive power of success (PPoS): The PPoS solely based on the interim
information can be expressed as

$\Phi\left(\frac{1}{s_{n}}\sqrt{\frac{n}{N-n}}\left[(\bar{x}_{n}-\mu_{1})\sqrt{N}-s_{n}\cdot\gamma\right]\right)$
(4.3)

Incorporating prior information specified in Eq. (4.6), expression of PPoS can
be refined as

$\Phi\left(\frac{1}{s_{n}}\sqrt{\frac{n}{N-n}}\frac{\left[(1-\psi)\\{n(\bar{x}_{n}-\mu_{1})+(N-n)(\mu_{0}-\mu_{1})\\}/\sqrt{N}+\psi(\bar{x}_{n}-\mu_{1})\sqrt{N}-s_{n}\gamma\right]}{\sqrt{\psi+(1-\psi)n/N}}\right)$
(4.4)

where, $\psi=n\sigma_{0}^{2}/(n\sigma_{0}^{2}+s_{n}^{2})$.

Probability of success (PoS): The PoS of a prospective clinical trial with $N$
subjects and prior information specified in Eq. (4.6) can be expressed as

$\Phi\left(\frac{\sqrt{N}\cdot(\mu_{0}-\mu_{1})-\tilde{\sigma}\cdot\gamma}{\sqrt{N\cdot\sigma_{0}^{2}+\tilde{\sigma}^{2}}}\right)$
(4.5)

where $\tilde{\sigma}$ is the projected SD and
$\tilde{k}=\tilde{\sigma}/\sqrt{N}$ is the projected SE in the trial. For the
calculation of PPoS with prior distribution in Eq. (4.4) and PoS in Eq. (4.5),
following prior for $\mu$ was used

$\mu\sim{\rm Normal}\left[\mu_{0},\sigma_{0}^{2}\right]$ (4.6)

###  4.2 Continuous endpoint, two-arm trial

Now consider a two-arm trial comparing treatment (T) with control (C) arm with
population means as $\mu_{T}$ and $\mu_{C}$, respectively. Denote the maximum
total sample size as $N$. We test

$H_{0}:\mu_{T}-\mu_{C}=\Delta_{1}\qquad{\rm vs.}\qquad
H_{1}:\mu_{T}-\mu_{C}>\Delta_{1}$

We have, $\theta=\mu_{T}-\mu_{C}-\Delta_{1}$. At interim analysis with total
sample size $n$, $\hat{\theta}(t)=\delta_{n}-\Delta_{1}$ where
$\delta_{n}=\bar{x}_{T,n}-\bar{x}_{C,n}$, and $\bar{x}_{T,n}$ and
$\bar{x}_{C,n}$ are the sample means at interim in treatment and control arms,
respectively. The corresponding test statistic is

$Z(t)=\frac{\delta_{n}-\Delta_{1}}{r\cdot
s_{n}/\sqrt{n}}=\frac{(\delta_{n}-\Delta_{1})\sqrt{n}}{r\cdot s_{n}}$

where $s_{n}$ is the estimate of pooled SD ($\sigma$) at interim analysis.
Further, $t=n/N$ and $k=r\cdot s_{n}/\sqrt{N}$.

Conditional power (CP): The CP with estimated mean difference from post-
interim data as $\Delta^{\prime}$ is

$\Phi\left(\frac{1}{r\cdot
s_{n}}\sqrt{\frac{N}{N-n}}\left[\frac{1}{\sqrt{N}}\\{n(\delta_{n}-\Delta_{1})+(N-n)(\Delta^{\prime}-\Delta_{1})\\}-r\cdot
s_{n}\cdot\gamma\right]\right)$ (4.7)

Assuming the interim trend continues to hold for the future data (i.e.,
$\Delta^{\prime}=\delta_{n}$), the CP reduces to

$\Phi\left(\frac{1}{r\cdot
s_{n}}\sqrt{\frac{N}{N-n}}\left[(\delta_{n}-\Delta_{1})\sqrt{N}-r\cdot
s_{n}\cdot\gamma\right]\right)$ (4.8)

Predictive Power of success (PPoS): The PPoS without prior distribution can be
expressed as

$\Phi\left(\frac{1}{r\cdot
s_{n}}\sqrt{\frac{n}{N-n}}\left[(\delta_{n}-\Delta_{1})\sqrt{N}-r\cdot
s_{n}\cdot\gamma\right]\right)$ (4.9)

The PPoS incorporating prior information specified in Eq. (4.12) can be
expressed as

$\Phi\left(\frac{1}{rs_{n}}\sqrt{\frac{n}{N-n}}\frac{\left[(1-\psi)\\{n(\delta_{n}-\Delta_{1})+(N-n)(\Delta_{0}-\Delta_{1})\\}/\sqrt{N}+\psi(\delta_{n}-\Delta_{1})\sqrt{N}-r\cdot
s_{n}\gamma\right]}{\sqrt{\psi+(1-\psi)n/N}}\right)$ (4.10)

where, $\psi=n\sigma_{0}^{2}/(n\sigma_{0}^{2}+r^{2}s_{n}^{2})$.

Probability of success (PoS): The PoS of a prospective clinical trial with $N$
subjects and prior information specified in Eq. (4.12) can be expressed as
(e.g. see [4])

$\Phi\left(\frac{\sqrt{N}\cdot(\Delta_{0}-\Delta_{1})-r\cdot\tilde{\sigma}\cdot\gamma}{\sqrt{N\cdot\sigma_{0}^{2}+r^{2}\cdot\tilde{\sigma}^{2}}}\right)$
(4.11)

where $\tilde{\sigma}$ is the projected pooled SD and
$\tilde{k}=r\cdot\tilde{\sigma}/\sqrt{N}$ is the projected SE in the trial.
For PPoS with prior distribution (Eq. (4.10)) and PoS (Eq. (4.11)), following
prior was used

$\mu_{T}-\mu_{C}\sim{\rm Normal}\left[\Delta_{0},\sigma_{0}^{2}\right]$ (4.12)

Example 1: In the pragmatic, unblinded, non-inferiority CODA trial [22], 1552
subjects (=$N$) with appendicitis were equally randomized to receive either
antibiotics or to undergo appendectomy. The primary outcome was 30-day health
status, as assessed with the European Quality of Life–5 Dimensions (EQ-5D)
questionnaire (scores range from 0 to 1, with higher scores indicating better
health status; non-inferiority margin, 0.05 points). For this illustration, we
imagine to have an interim analysis at the sample size of 776 (=$n$).
According to O’Brien alpha spending function, the rejection boundaries for Z
test statistic are 2.96 and 1.97 (=$c(1)$) at interim and final analyses,
respectively.

Figure 1: Left: Predictive distributions of $\Delta=\mu_{T}-\mu_{C}$ and
Right: plots of the CP and PPoS for trial success against $\delta_{n}$
(interim estimate of $\Delta$) for Example 1. Horizontal and vertical
reference lines in the left panel correspond to 50% power and observed value
of $\delta_{n}=-0.025$, respectively.

We are testing following hypotheses: $H_{0}:\mu_{T}-\mu_{C}\leq-0.05$ vs.
$H_{1}:\mu_{T}-\mu_{C}>-0.05$. Therefore, $\Delta_{1}=-0.05$. Further, the
external information are summarized as following prior distribution

$\mu_{T}-\mu_{C}\sim{\rm
Normal}\left[\Delta_{0}=0,\;\;\sigma_{0}^{2}=(0.02)^{2}\right]$

Treatment allocation ratio is 1:1; therefore, $r^{2}=(1+1)^{2}/2=4$. First, we
illustrate the calculation of the PoS at the design stage: If the projected
pooled SD in the trial was 0.12 ($=\tilde{\sigma}$), the PoS for trial success
($\gamma=c(1)=1.97$) would be
$\Phi\left(\frac{\sqrt{1552}(0-(-0.05))-(2)(0.12)(1.97)}{\sqrt{(1552)(0.02)^{2}+(4)(0.12)^{2}}}\right)=0.965$
(see Eq. (4.11)).

For the calculation of CP and PPoS, let’s consider following interim results:
mean difference of -0.025 (=$\delta_{n}$) points with SD as 0.16 (=$s_{n}$).
Assuming the interim trend to be continued to the remaining part of trial as
well, based on Eq. (4.8), the conditional power for trial success
($\gamma=c(1)=1.97$) is 0.941 and PPoS would be 0.866 (see Eq. (4.10)). Now,
expecting -0.030 mean difference from post-interim data (i.e.,
$\Delta^{\prime}=-0.030$), the conditional power would be 0.871. Further, the
PPoS for trial success given the interim results and prior distribution is
0.944 (see Eq. (4.9)).

The predictive distributions of $\Delta=\mu_{T}-\mu_{C}$ with and without
prior distribution, and the CP and PPoS values for trial success against
$\delta_{n}$ (i.e., interim estimate of mean difference) are plotted in Figure
1. As mentioned in Section 3.2, we can verify that CP$>$PPoS for $CP>0.5$ and
CP$<$PPoS for $CP<0.5$, when prior information was not incorporated in PPoS.
Both the predictive distribution and PPoS were improved due to use of
optimistic prior.

###  4.3 Binary endpoint, single-arm trial

Let’s $\Pi$ denotes the population proportion in a single-arm trial with the
binary endpoint. The maximum sample size in the study is $N$. We test the
following set of hypotheses:

$H_{0}:\Pi=\Pi_{1}\qquad{\rm vs.}\qquad H_{1}:\Pi>\Pi_{1}$

We have, $\theta=\Pi-\Pi_{1}$. At interim analysis with sample size $n$,
$\hat{\theta}(t)=p_{n}-\Delta_{1}$ with $p_{n}$ being the sample proportion.
The corresponding test statistic is

$Z(t)=\frac{p_{n}-\Pi_{1}}{s_{n}/\sqrt{n}}=\frac{(p_{n}-\Pi_{1})\sqrt{n}}{s_{n}}$

where, $s_{n}=\sqrt{p_{n}(1-p_{n})}$ is the estimate of SD ($\sigma$).
Further, $t=n/N$ and $k=s_{n}/\sqrt{N}={\rm SE}(p_{n})\cdot\sqrt{t}$.
Expressions of the CP, PPoS and PoS in this case can be obtained from the
corresponding expressions in single-arm trial with continuous endpoint by
replacing $\bar{x}_{n}$ with $p_{n}$, $\mu_{1}$ with $\Pi_{1}$, and
$\mu^{\prime}$ with $\Pi^{\prime}$.

Conditional power (CP): The CP with projected proportion of $\Pi^{\prime}$ for
post-interim data is

$\Phi\left(\frac{1}{s_{n}}\sqrt{\frac{N}{N-n}}\left[\frac{1}{\sqrt{N}}\\{n(p_{n}-\Pi_{1})+(N-n)(\Pi^{\prime}-\Pi_{1})\\}-s_{n}\cdot\gamma\right]\right)$
(4.13)

If the interim trend continues to hold for the future data (i.e.,
$\Pi^{\prime}=p_{n}$), the CP reduces to

$\Phi\left(\frac{1}{s_{n}}\sqrt{\frac{N}{N-n}}\left[(p_{n}-\Pi_{1})\sqrt{N}-s_{n}\cdot\gamma\right]\right)$
(4.14)

Predictive power of success (PPoS): The PPoS based on the interim information
can be expressed as

$\Phi\left(\frac{1}{s_{n}}\sqrt{\frac{n}{N-n}}\left[(p_{n}-\Pi_{1})\sqrt{N}-s_{n}\cdot\gamma\right]\right)$
(4.15)

Incorporating prior information specified in Eq. (4.18), expression of the
PPoS can be refined as

$\Phi\left(\frac{1}{s_{n}}\sqrt{\frac{n}{N-n}}\frac{\left[(1-\psi)\\{n(p_{n}-\Pi_{1})+(N-n)(\Pi_{0}-\Pi_{1})\\}/\sqrt{N}+\psi(p_{n}-\Pi_{1})\sqrt{N}-s_{n}\gamma\right]}{\sqrt{\psi+(1-\psi)n/N}}\right)$
(4.16)

where, $\psi=n\sigma_{0}^{2}/(n\sigma_{0}^{2}+Ns_{n}^{2})$.

Probability of success (PoS): The PoS of a prospective clinical trial with $N$
subjects and prior information specified in Eq. (4.18) can be expressed as

$\Phi\left(\frac{\sqrt{N}\cdot(\Pi_{0}-\Pi_{1})-\tilde{\sigma}\cdot\gamma}{\sqrt{N\cdot\sigma_{0}^{2}+\tilde{\sigma}^{2}}}\right)$
(4.17)

where, $\tilde{\sigma}=\sqrt{\tilde{\Pi}(1-\tilde{\Pi})}$ is the projected SD
and $\tilde{k}=\tilde{\sigma}/\sqrt{N}$ is the projected SE in the trial with
$\tilde{\Pi}$ being the projected proportion in the trial. Following prior was
used for the PPoS with prior distribution in Eq. (4.16) and PoS in Eq. (4.17)

$\Pi\sim{\rm Normal}\left[\Pi_{0},\sigma_{0}^{2}=\sigma_{0}^{2}\right]$ (4.18)

###  4.4 Binary endpoint, two-arm trial

Consider a two-arm trial comparing treatment (T) with control (C) arm with
population proportions as $\Pi_{T}$ and $\Pi_{C}$, respectively. Denote the
maximum total sample size as $N$. We test

$H_{0}:\Pi_{T}-\Pi_{C}=\Delta_{1}\qquad{\rm vs.}\qquad
H_{1}:\Pi_{T}-\Pi_{C}>\Delta_{1}$

Here, $\theta=\Pi_{T}-\Pi_{C}-\Delta_{1}$. At interim analysis with total
sample size $n$, $\hat{\theta}(t)=\delta_{n}-\Delta_{1}$ where
$\delta_{n}=p_{T,n}-p_{C,n}$ is the difference between the estimated
proportions ($p_{T,n}$ and $p_{C,n}$) in the two arms.

$SE(\delta_{n})=\sqrt{\dfrac{p_{T,n}(1-p_{T,n})}{a\cdot
n/(1+a)}+\dfrac{p_{C,n}(1-p_{C,n})}{n/(1+a)}}=r\cdot s_{n}/\sqrt{n}$

where
$s_{n}^{2}=\dfrac{a}{a+1}\left\\{\dfrac{p_{T,n}(1-p_{T,n})}{a}+p_{C,n}(1-p_{C,n})\right\\}$
is the estimate of pooled SD ($\sigma$). Therefore, the corresponding test
statistic is

$Z(t)=\frac{\delta_{n}-\Delta_{1}}{r\cdot
s_{n}/\sqrt{n}}=\frac{(\delta_{n}-\Delta_{1})\sqrt{n}}{r\cdot s_{n}}$

Further, $t=n/N$ and $k=r\cdot s_{n}/\sqrt{N}={\rm
SE}(\delta_{n})\cdot\sqrt{t}$. Expressions of the CP, PPoS and PoS for two-arm
trial with binary endpoint are same with that of continuous case in Section
4.2. That is,

  * •

Eq. (4.7) for the CP with the projected difference from post-interim data as
$\Delta^{\prime}$.

  * •

Eq. (4.8) for the CP with the projected difference similar to that observed at
interim analysis.

  * •

Eq. (4.9) for the PPoS without prior distribution.

  * •

Eq. (4.10) for the PPoS with prior distribution with
$\psi=n\sigma_{0}^{2}/(n\sigma_{0}^{2}+Ns_{n}^{2})$.

  * •

Eq. (4.11) for the PoS with projected SD as
$\tilde{\sigma}=\sqrt{\dfrac{a}{a+1}\left\\{\dfrac{\tilde{\Pi}_{T}(1-\tilde{\Pi}_{T})}{a}+\tilde{\Pi}_{C}(1-\tilde{\Pi}_{C})\right\\}}$
and the projected SE as $\tilde{k}=r\cdot\tilde{\sigma}/\sqrt{N}$, where
$\tilde{\Pi}_{T}$ and $\tilde{\Pi}_{C}$ are the projected proportions in the
trial.

For the PPoS with prior distribution and PoS, following prior for
$\Pi_{T}-\Pi_{C}$ was considered

$\Pi_{T}-\Pi_{C}\sim{\rm
Normal}\left[\Delta_{0},\sigma_{0}^{2}=\sigma_{0}^{2}\right]$ (4.19)

Example 2: Fenaux et al. [23] reported the trial results of placebo-
controlled, phase 3 trial evaluating the effect of Luspatercept in patients
with lower-risk myelodysplastic syndromes. The primary endpoint was the
proportion of patients with transfusion independence for eight weeks or longer
during weeks 1 through 24. A total sample size of 210 patients (=$N$) with 2:1
treatment allocation ratio would give the study 90% power to detect
differences between response rates of 0.30 in the luspatercept arm and 0.10 in
the placebo arm with the one-sided alpha of 0.025 and 10% dropout rate. For
this illustration, we add an interim analysis at the sample size of 158
(=$n$). Further, we assume the clinically meaningful difference is 15%
(=$\theta_{\min}$). According to O’Brien alpha spending function, the
rejection boundaries for the Z test statistic are 2.34 and 2.012 (=$c(1)$) at
interim and final analyses, respectively.

Figure 2: (Left) Predictive distributions of $\Delta=\Pi_{T}-\Pi_{C}$, and
(Right) plots of the CP and PPoS for trial success against $\delta_{n}$
(interim estimate of $\Delta$) for Example 2. Horizontal and vertical
reference lines in the left panel correspond to 50% power and observed value
of $\delta_{n}=0.157$, respectively.

In this case, we are statistically testing the following hypotheses:
$H_{0}:\Pi_{T}-\Pi_{C}\leq 0$ against $H_{1}:\Pi_{T}-\Pi_{C}>0$. Therefore,
$\Delta_{1}=0$. We consider the following prior distribution for
$\Pi_{T}-\Pi_{C}$

$\Pi_{T}-\Pi_{C}\sim{\rm
Normal}\left[\Delta_{0}=0.20,\;\;\sigma_{0}^{2}=0.06\right]$

The allocation ratio is 2:1, resulting in $r^{2}=(2+1)^{2}/2=4.5$. At the
final analysis, the SD and $\mbox{SE}(\delta_{n})$ are projected as
$\tilde{\sigma}=\sqrt{(2/3)\cdot(0.30*0.70/2+0.10*0.90)}=0.361$ and
$\tilde{k}=\sqrt{4.5}(0.361)/\sqrt{210}=0.053$, respectively. Therefore, the
PoS for trial success ($\gamma=c(1)=2.012$) and clinical success
($\gamma=\theta_{\min}/\tilde{k}=0.15/0.053=2.83$) at the design stage are
$\Phi\left(\frac{\sqrt{210}(0.20-0)-\sqrt{4.5}(0.361)(2.012)}{\sqrt{(210)(0.06)+(4.5)(0.361)^{2}}}\right)=0.645$
and
$\Phi\left(\frac{\sqrt{210}(0.20-0)-\sqrt{4.5}(0.361)(2.83)}{\sqrt{(210)(0.06)+(4.5)(0.361)^{2}}}\right)=0.578$,
respectively (see Eq. (4.11)).

Now consider the following interim results: 37.9% of the patients responded in
the luspatercept arm ($n_{T}=105$) compared to 22.2% in the placebo arm
($n_{C}=53$). Therefore, $\delta_{n}=0.379-0.222=0.157$ with SE as
$\sqrt{0.379*0.621/105+0.222*0.778/53}=0.074$ and
$s_{n}=SE\cdot\sqrt{n}/r=(0.074)(\sqrt{158})/\sqrt{4.5}=0.4385$. Further,
$k=0.074\cdot\sqrt{0.75}=0.064$.Projecting the observed difference in
proportion from the post-interim data as 0.20 (=$\Delta^{\prime}$), the CP for
trial success ($\gamma=c(1)=2.012$) and clinical success
($\gamma=0.15/0.064=2.34$) are 0.884 and 0.709, respectively (Eq. (4.7)).
Assuming the interim trend for remaining part of the trial, the CP for trial
success and clinical success are 0.804 and 0.587, respectively (Eq. (4.8)).
The PPoS for trial success and clinical success based on interim results and
prior knowledge are 0.782 and 0.586, respectively (Eq. (4.9)). If we leave out
the prior distribution, the PPoS for trial success and clinical success are
0.772 and 0.575, respectively (Eq. (4.10)).

Plots of the predictive distributions of $\Delta=\Pi_{T}-\Pi_{C}$, and the CP
and PPoS against $\delta_{n}$ (i.e., interim estimate of difference in
proportion) are plotted presented in Figure 2. We can confirm that CP$>$PPoS
for $CP>0.5$ and CP$<$PPoS for $CP<0.5$ regardless the type of success (i.e.,
trial success or clinical success) when prior information are not incorporated
in the PPoS.

###  4.5 Survival endpoint, single-arm trial

Let’s consider a study with a single treatment arm and time-to-event endpoint.
Denote the population median as $M$ and the maximum number of events in the
study as $D$. We test the following hypotheses:

$H_{0}:M=M_{1}\qquad{\rm vs.}\qquad H_{1}:M>M_{1}$

Here, $\theta=\log{M}-\log{M_{1}}$. At interim analysis with $d$ ($<D$)
events, the estimate of $\theta$ is $\hat{\theta}(t)=\log{m_{d}}-\log{M_{1}}$
where $m_{d}$ is the estimated median at interim. Assuming that $m_{d}$
normally distributed (e.g., see [18, 24]), and with
$\mbox{var}(\log{m_{d}})=\xi^{2}/d$, a test can by constructed as follows:

$Z(t)=\frac{\log{m_{d}}-\log{M_{1}}}{\xi/\sqrt{d}}=\frac{\log{(m_{d}/M_{1})}\cdot\sqrt{d}}{\xi}$

Under the exponential time-to-event distribution,
$\mbox{var}(\log{m_{d}})=1/d$ (i.e., $\xi=1$) if $m_{d}$ is maximum likelihood
estimate (MLE) (see Appendix 1a), or
$\mbox{var}(\log{m_{d}})=(\log{2})^{-2}/d$ (i.e., $\xi=(\log{2})^{-1}=1.443$)
if $m_{d}$ is the plain sample median (see Appendix 1b). Further,
$\mbox{var}(\log{m_{d}})=(\log{2})^{-2}\cdot\beta^{-2}/d$ (i.e.,
$\xi=(\log{2})^{-1}\cdot\beta^{-1}=1.443\cdot\beta^{-1}$) if $m_{d}$ is the
plain sample median from a Weibull time-to-event distribution with $\beta$
shape parameter (see Appendix 1b). For other estimator (e.g., smallest time
with at least 50% Kaplan-Meier (KM) estimate of survival probability),
$\xi^{2}$ is simply the ratio of the variance of the log of estimated median
to that of MLE. We have plotted the empirical SE of log(KM estimate of median)
for various event sizes (ranging between 20 and 60) and sample sizes (0%, 30%
and 50% more than event size) in Figure 3. Empirical SE were obtained as
follows: (a) 5000 datasets were simulated, (b) in each of the simulated
datasets, event times were generated from exponential distribution with median
of 12 months and KM estimates for median were obtained, and (c) SD of log(KM
estimate of median) were obtained. The results suggest that $1/\sqrt{d}$
under-estimates the empirical SE. In comparison, $(\log{2})^{-1}/\sqrt{d}$
almost coincides with the empirical SE for $N=d$. However, as $N$ becomes
greater than $d$, $(\log{2})^{-1}/\sqrt{d}$ tends to over-estimate the SE as
the follow-up times of the censored subjects attributes to the increased
precision of KM estimates which was not factored into the derivation of
$(\log{2})^{-1}/\sqrt{d}$. In any case, $t=d/D$ and $k=\xi/\sqrt{D}$.

Figure 3: Comparison of empirical SE of log(KM estimate of median) with the
$1/d$ with $1/\log{2}\cdot 1/d$ as presented in Section 4.5

Conditional power (CP): The CP with the projected median from post-interim
data as $M^{\prime}$ is

$\Phi\left(\frac{1}{\xi}\sqrt{\frac{D}{D-d}}\left[\frac{1}{\sqrt{D}}\left\\{d\cdot\log{\frac{m_{d}}{M_{1}}}+(D-d)\cdot\log{\frac{M^{\prime}}{M_{1}}}\right\\}-\xi\cdot\gamma\right]\right)$
(4.20)

Projecting the future trend to be identical with the interim trend (i.e.,
$M^{\prime}=m_{d}$), the expression of the CP simplifies to

$\Phi\left(\frac{1}{\xi}\sqrt{\frac{D}{D-d}}\left[\sqrt{D}\cdot\log{\frac{m_{d}}{M_{1}}}-\xi\cdot\gamma\right]\right)$
(4.21)

Predictive power of success (PPoS): The PPoS solely based on the interim data
can be expressed as

$\Phi\left(\frac{1}{\xi}\sqrt{\frac{d}{D-d}}\left[\sqrt{D}\cdot\log{\frac{m_{d}}{M_{1}}}-\xi\cdot\gamma\right]\right)$
(4.22)

Now, incorporating prior information specified in Eq. (4.25), expression of
the PPoS is refined as

$\Phi\left(\frac{1}{\xi}\sqrt{\frac{d}{D-d}}\frac{\left[(1-\psi)\\{\frac{d}{\sqrt{D}}\cdot\log{\frac{m_{d}}{M_{1}}}+\frac{D-d}{\sqrt{D}}\cdot\log{\frac{M_{0}}{M_{1}}}\\}+\psi\cdot\sqrt{D}\cdot\log{\frac{m_{d}}{M_{1}}}-\xi\cdot\gamma\right]}{\sqrt{\psi+(1-\psi)\frac{d}{D}}}\right)$
(4.23)

where, $\psi=d\cdot\sigma_{0}^{2}/(d\cdot\sigma_{0}^{2}+\xi^{2})$.

Probability of success (PoS): The PoS of a prospective trial with $D$ events
(therefore, $\tilde{k}=\xi/\sqrt{D}$) and the prior information specified in
Eq. (4.25) can be expressed as follows:

$\Phi\left(\frac{\sqrt{D}\cdot\log{(M_{0}/M_{1})}-\xi\cdot\gamma}{\sqrt{D\cdot\sigma_{0}^{2}+\xi^{2}}}\right)$
(4.24)

For the calculation of the PPoS and PoS with prior distribution, the following
prior for $M$ was used

$\log{M}\sim{\rm Normal}\left[\log{M_{0}},\sigma_{0}^{2}\right]$ (4.25)

###  4.6 Survival endpoint, two-arm trial

Denoting the treatment to control HR as $\Delta$, in a two-arm clinical trial
with time-to-event endpoint and the maximum target number of events as $D$, we
test the following hypotheses:

$H_{0}:\Delta=\Delta_{1}\qquad{\rm vs.}\qquad H_{1}:\Delta<\Delta_{1}$

Here, $\theta=\log{(\Delta_{1}/\Delta)}$. At interim analysis with the total
$d$ events, $\hat{\theta}(t)=\log{(\Delta_{1}/\delta_{d})}$ where $\delta_{d}$
is the estimated HR. The corresponding log-rank statistic for trend test is
approximately equivalent to

$Z(t)=\frac{\log{(\Delta_{1}/\delta_{d})}}{r}\sqrt{d}$

Further, in this case, $t=d/D$ and $k=r/\sqrt{D}$.

Conditional power (CP): With the projected HR from future data as
$\Delta^{\prime}$, the expression of CP is

$\Phi\left(\frac{1}{r}\sqrt{\frac{D}{D-d}}\left[\frac{d}{\sqrt{D}}\log{\frac{\Delta_{1}}{\delta_{d}}}+\frac{D-d}{\sqrt{D}}\log{\frac{\Delta_{1}}{\Delta^{\prime}}}-r\cdot\gamma\right]\right)$
(4.26)

If the interim trend continues to hold in post-interim data (i.e.,
$\Delta^{\prime}=\delta_{d}$), the CP reduces to

$\Phi\left(\frac{1}{r}\sqrt{\frac{D}{D-d}}\left[\sqrt{D}\cdot\log{\frac{\Delta_{1}}{\delta_{d}}}-r\cdot\gamma\right]\right)$
(4.27)

Predictive power of success (PPoS): The PPoS solely based on the interim data
can be expressed as

$\Phi\left(\frac{1}{r}\sqrt{\frac{d}{D-d}}\left[\sqrt{D}\cdot\log{\frac{\Delta_{1}}{\delta_{d}}}-r\cdot\gamma\right]\right)$
(4.28)

Incorporating the prior information specified in Eq. (4.31), revised
expression of PPoS is [15]

$\Phi\left(\frac{1}{r}\sqrt{\frac{d}{D-d}}\frac{\left[(1-\psi)\\{\frac{d}{\sqrt{D}}\cdot\log{\frac{\Delta_{1}}{\delta_{d}}}+\frac{D-d}{\sqrt{D}}\cdot\log{\frac{\Delta_{1}}{\Delta_{0}}}\\}+\psi\cdot\sqrt{D}\cdot\log{\frac{\Delta_{1}}{\delta_{d}}}-r\cdot\gamma\right]}{\sqrt{\psi+(1-\psi)\frac{d}{D}}}\right)$
(4.29)

where, $\psi=\dfrac{d\cdot\sigma_{0}^{2}}{d\cdot\sigma_{0}^{2}+r^{2}}$.

Probability of success (PoS): The PoS of a prospective trial with $D$ events
(hence, $\tilde{k}=r/\sqrt{D}$) and the prior information specified in Eq.
(4.31) is (e.g., see [17]):

$\Phi\left(\frac{\sqrt{D}\cdot\log{(\Delta_{1}/\Delta_{0})}-r\cdot\gamma}{\sqrt{D\cdot\sigma_{0}^{2}+r^{2}}}\right)$
(4.30)

For the calculation of the PPoS and PoS with prior distribution, the following
prior for $\Delta$ was used

$\log{\Delta}\sim{\rm Normal}\left[\log{\Delta_{0}},\sigma_{0}^{2}\right]$
(4.31)

Example 3: In the INTELLANCE-I trial on glioblastoma patients comparing
investigational drug depatuxizumab mafodotin, total of 639 subjects (=$N$)
were enrolled with 1:1 allocation ratio [25]. The primary endpoint in the
study was overall survival. The target number of events at the final analysis
was 441 (=$D$) and an interim analysis was planned with 332 events. The trial
used a weighted log-rank test, however, here we illustrate assuming standard
log-rank test. Therefore, expected SE of $\log{HR}$ at final analysis is
$k=2/\sqrt{441}=0.0952$. The rejection boundaries for the Z test (i.e., trend
test) statistic are 2.340 and 2.012 at interim and final analyses,
respectively. For clinical success, we assume $\mbox{HR}\leq 0.80$
(=$\Delta_{\min}$) resulting in $\gamma=-\log{(0.80)}/0.0952=2.344$.

Figure 4: Predictive distribution of HR (i.e., $\Delta$) and the relationship
of CP and PPoS for trial and clinical success with $\delta_{d}$ (i.e., interim
estimate of HR) based on the Example 3. Horizontal and vertical lines in the
left panel correspond to 50% power and observed HR of 0.82 (=$\delta_{d}$),
respectively.

We test following hypotheses: $H_{0}:{\rm HR}=1$ vs. $H_{1}:{\rm HR}<1$ (i.e.,
$\Delta_{1}=1$). The phase 2 trial on recurrent disease [26] reported HR of
0.71 (=$\Delta_{0}$) with 133 events resulting in prior distribution

$\log{\Delta}\sim{\rm
Normal}\left[\log{0.71},\sigma_{0}^{2}=(2/\sqrt{133})^{2}=(0.173)^{2}\right]$

Due to 1:1 allocation ratio, $r^{2}=(1+1)^{2}/2=4$. Thus, the PoS for trial
success ($\gamma=1.96$) and clinical success ($\gamma=2.344$) are
$\Phi\left(\frac{\sqrt{441}\log{(1/0.71)}-(2)(1.96)}{\sqrt{(441)(0.173)^{2}+(2)^{2}}}\right)=0.785$
and
$\Phi\left(\frac{\sqrt{441}\log{(1/0.71)}-(2)(2.344)}{\sqrt{(441)(0.173)^{2}+(2)^{2}}}\right)=0.727$,
respectively, at the design stage of a single-look trial without interim
analysis (Eq. (4.30)).

Consider the following interim results: estimated HR of 0.82 (=$\delta_{d}$)
with 346 events (=$d$). Note that, $k=2/\sqrt{441}=0.0952$. Assuming the
projected HR from the post-interim data as 0.75 (=$\Delta^{\prime}$) as
assumed at the design stage, the CP for trial success ($\gamma=2.012$) and
clinical success ($\gamma=2.344$) are 0.722 and 0.451, respectively (see Eq.
(4.26)). However, if we assume that the interim trend continues to the
remaining part of the trial, the CP for trial success and clinical success are
0.561 and 0.288, respectively (see Eq. (4.27)). The PPoS for trial success and
clinical success solely based on the interim results are 0.554 and 0.310,
respectively (see Eq. (4.28)). If we incorporate prior distribution, the PPoS
for trial success and clinical success are 0.625 and 0.370, respectively (see
Eq. (4.29)).

In Figure 4, the predictive distribution of HR are plotted in the left panel,
and the CP and PPoS against the interim HR estimate are plotted in the right
panel. For both trial success and clinical success, CP$>$PPoS for $CP>0.5$ and
CP$<$PPoS for $CP<0.5$ when prior information is ignored.

##  5 Binary endpoint with beta prior

So far we have discussed normally distributed test statistics and normal
priors. Here, we discuss the derivation of PPoS in trials with binary endpoint
with beta prior. We have considered the same notations and hypothesis testings
presented in Section 4.3 and Section 4.4.

###  5.1 single-arm trial

Consider a prior distribution of probability of response $\Pi$ as ${\rm
Beta}(a,b)$. Denoting the observed number of response as $x_{n}$ from $n$
subjects at interim analyses, the posterior distribution of $\Pi$ is

$\Pi\;|\;x_{n}\sim{\rm Beta}(x_{n}+a,n-x_{n}+b)$

Let, $Y$ be the number of the observed response from remaining $N-n$ subjects.
The predictive distribution of $Y$ is (e.g., see [10, 27])

$\mbox{Pr}(Y=y|x_{n})={N-n\choose y}\frac{{\rm
B}(x_{n}+y+a,\;N-x_{n}-y+b)}{{\rm B}(x_{n}+a,\;n-x_{n}+b)}\qquad
y=0,1,\cdots,N-n$

where, ${\rm B}(u,v)=\dfrac{(u-1)!(v-1)!}{(u+v-1)!}$ is the beta function.
Thus, the PPoS would be

$\sum_{y=0}^{N-n}{I({\rm success}|x_{n}+y,N)\cdot{\rm Pr}(Y=y|x_{n})}$ (5.1)

where $I(\cdot)$ is the indicator function for trial success (e.g., based on
approximate Z test or exact binomial test) or clinical success (i.e.,
estimated proportion exceeds certain threshold value) is met.

###  5.2 two-arm trial

We assume following priors for the proportions in treatment (T) and control
(C) arms: $\Pi_{T}\sim{\rm Beta}(a_{T},b_{T})$ and $\Pi_{C}\sim{\rm
Beta}(a_{C},b_{C})$. At the interim analysis, $x_{T}$ of $n_{T}$ subjects in
the treatment arm and $x_{C}$ of $n_{C}$ subjects in the control arm are
responded. The posterior distributions are [2]

$\Pi_{T}\;|\;x_{T}\sim{\rm Beta}(x_{T}+a_{T},n_{T}-x_{T}+b_{T})$
$\Pi_{C}\;|\;x_{C}\sim{\rm Beta}(x_{C}+a_{C},n_{C}-x_{C}+b_{C})$

Let, $Y_{T}$ and $Y_{C}$ be the number of responders from remaining
$N_{T}-n_{T}$ and $N_{C}-n_{C}$ subjects, respectively. The predictive
distribution of $Y_{T}$ and $Y_{C}$ are (e.g., see [27])

${\rm Pr}(Y_{T}=y_{T}|x_{T})={N_{T}-n_{T}\choose y_{T}}\frac{{\rm
B}(x_{T}+y_{T}+a_{T},\;N_{T}-x_{T}-y_{T}+b_{T})}{{\rm
B}(x_{T}+a_{T},\;n_{T}-x_{T}+b_{T})}\qquad y_{T}=0,1,\cdots,N_{T}-n_{T}$ ${\rm
Pr}(Y_{C}=y_{C}|x_{C})={N_{C}-n_{C}\choose y_{C}}\frac{{\rm
B}(x_{C}+y_{C}+a_{C},\;N_{C}-x_{C}-y_{C}+b_{C})}{{\rm
B}(x_{C}+a_{C},\;n_{C}-x_{C}+b_{C})}\qquad y_{C}=0,1,\cdots,N_{C}-n_{C}$

Thus, the PPoS at the end of the trial is

$\sum_{y_{T}=0}^{N_{T}-n_{T}}\sum_{y_{C}=0}^{N_{C}-n_{C}}{I({\rm
success}|x_{T}+y_{T},x_{C}+y_{C},N_{T},N_{C})\cdot{\rm
Pr}(Y_{T}=y_{T}|x_{T})\cdot{\rm Pr}(Y_{C}=y_{C}|x_{C})}$ (5.2)

where $I(\cdot)$ is the indicator function for success criteria which could be
either trial success (e.g., based on approximate Z test or Fisher’s exact
test) or clinical success indicating the observed difference in proportion
exceeds the certain clinically meaningful value.

Example 4: This example is inspired by the example given in Johns and Andersen
[27]. Consider a clinical trial to demonstrate that the relapse rate in
patients treated in the experimental treatment arm is less than the control
arm’s response rate. It was planned to enrol 340 patients in each arm. The
interim analysis was planned after 170 patients in each arm completed
treatment. Non-informative uniform priors were assumed for the two relapse
rate: $\Pi_{T}\sim{\rm Beta}(a_{T}=1,b_{T}=1)$ and $\Pi_{C}\sim{\rm
Beta}(a_{C}=1,b_{C}=1)$.

In this case, we are statistically testing the following hypotheses:
$H_{0}:\Pi_{T}-\Pi_{C}\leq 0$ against $H_{1}:\Pi_{T}-\Pi_{C}>0$. Suppose we
observed following results at interim analysis: (a) in the treatment arm, 155
(=$n_{T}$) out of 170 patients responded with 13 (=$x_{T}$) subsequent
relapses, and (b) in the control arm, 152 (=$n_{C}$) out of 169 patients
responded with 21 (=$x_{C}$) subsequent relapses. Subsequently, additional
$340-170=170$ patients (=$N_{T}-n_{T}$) and $340-169=171$ patients
(=$N_{C}-n_{C}$) to be enrolled in the treatment arm and control arm,
respectively. With this information, the PPoS for trial success based on a Z
test at one sided 0.025 level is 0.536 (see Eq. (5.2)).

##  6 Software implementation

Figure 5: R shiny app for calculation of CP, PPoS and PoS

The expressions of CP, PPoS and PoS presented in this paper are implemented in
`LongCART` package in `R`: (a) `PoS()` to calculate PoS at the design stage,
(b) `succ_ia()` to calculate CP and PPoS, and (c) `succ_ia_betabinom_one()`
and `succ_ia_betabinom_two()` to calculate PPoS for binary endpoint with beta
prior. Examples of these functions are provided in the Appendix 2. A user-
friendly R shiny app is also available at https://ppos.herokuapp.com/ to
calculate these measures. A screenshot of this shiny app is presented in
Figure 5.

##  7 Discussion

In this paper, expressions for various measures of the probability of success
are presented by type of endpoints. The discussion in this paper is restricted
to the normally distributed test statistics along with normal prior, and beta
prior for binomial distributions. For other distributions, the relevant
expressions can be obtained using the general framework presented in Section 2
or simulation based methods such as Bayesian clinical trial simulation (BCTS)
[4, 17] may be used. Nevertheless, a natural question arises which pr(success)
measure one should prefer. Often PPoS is preferred over CP for the following
reasons: (1) these have better predictive interpretation, (2) unlike
frequentist counterpart, the knowledge on $\theta$ (the parameter of interest)
is used as distribution, whereas in frequentist calculation, we assume that
the value of $\theta$ is known without any uncertainty, and (3) unlike
frequentist paradigm, the prior information can be incorporated in the
Bayesian paradigm. As evident from Figure 1, Figure 2 and Figure 4, the CP is
more aggressive than the PPoS and hence use of the CP increases the chance of
early stopping for futility or efficacy. Lachin [9] has shown that futility
termination may markedly decrease the power in direct proportion to the
probability of stopping for futility. Therefore, PPoS seems to be more useful
while monitoring a trial for early termination.

Dallow and Fina [21] discussed the disadvantages of PoS and PPoS, especially
PPoS can lead to much larger sample sizes than the CP during sample size re-
estimation. On the other hand, the effect of varying prior distribution of
$\theta$ on predictive power in the context of futility monitoring is
discussed by Dmitrienko and Wang [1], and in general by Rufibach, Burger and
Abt [16]. In summary, they have proposed to use aggressive prior for futility
monitoring as the use of non-informative may increase the early termination
rate. Tang [15] suggested using the upper limit of PPoS in futility
monitoring. The effect of prior on PPoS in the context of the binomial
endpoint is discussed by Johns and Andersen [27].

One might consider the predictive power of clinical success (PPoCS) in
monitoring for early stopping as well; however, in general, its use should be
discouraged. Saville et al. [2] have pointed out that the PPoS with respect to
trial success (referred to as ’predictive probabilities’) are naturally
appealing for monitoring a clinical trial as (a) the PPoS directly addresses
the question of whether the study is going to be a success at the end, (b) and
the PPoS often changes drastically with the accrual of more data whereas the
PPoCS (referred as ’posterior probabilities’) may remain nearly identical.
Further, we also would like to point out the potential misuse of PPoS for
survival endpoints with delayed treatment effects. In that case, the use of
futility criteria for early stopping based on PPoS or CP may be misleading. In
these cases, the futility criteria, if any, must be determined through
exhaustive evaluation of operating characteristics.

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## Appendix

### Appendix 1: Variance of median estimates in a single-arm trial with time-
to-event endpoint

#### Appendix 1a: Variance of MLE of median with exponential time-to-event
distribution

For an exponential time-to-event distribution with density
$f(x)=\lambda\exp{(-\lambda x)}$ and $d$ events, the asymptotic variance of
$\hat{\lambda}$ (MLE of $\lambda$) is estimated by $\hat{\lambda}^{2}/d$, so
$1/d$ estimates the variance of $\log{\hat{\lambda}}$ [18]. The MLE of median
can be expressed as $m_{d}=\log{2}/\hat{\lambda}$. Therefore, the variance of
$\log{m_{d}}$ is

$\mbox{var}[\log{m_{d}}]=\mbox{var}[\log{(\log{2})}-\log{\hat{\lambda}}]=\mbox{var}[\log{\hat{\lambda}}]=1/d$

#### Appendix 1b: Variance of observed sample median with Weibull and
exponential time-to-event distribution

Consider a Weibull time-to-event distribution with density $f(x)=\beta\lambda
x^{\beta-1}\exp{(-\lambda x^{\beta-1})}$. The corresponding cumulative
distribution function (CDF) is $F(x)=1-\exp{(-\lambda x^{\beta-1})}$. Equating
this CDF with 0.5, we obtain the median value as
$M=(\log{2}/\lambda)^{1/\beta}$. The density at $M$ is

$f(M)=\frac{1}{2}\cdot\beta\cdot\lambda\cdot\left(\frac{\log{2}}{\lambda}\right)^{\frac{\beta-1}{\beta}}$

If $m_{d}$ is the sample median based on $d$ events (and, entirely ignoring
the censored subjects), estimated variance of $m_{d}$ is [28]

$\mbox{var}(m_{d})=\frac{1}{4\cdot d\cdot
f(M)}=\frac{1}{d}\cdot\frac{1}{\beta^{2}}\cdot\frac{1}{\lambda^{2}}\cdot\left(\frac{\lambda}{\log{2}}\right)^{2\cdot\frac{\beta-1}{\beta}}$

Consequently, the variance of $\log{m_{d}}$ is

$\mbox{var}[\log{m_{d}}]=\left(\frac{1}{m_{d}}\right)^{2}\mbox{var}(m_{d})=\frac{1}{d}\cdot\frac{1}{\beta^{2}}\cdot\left(\frac{1}{\log{2}}\right)^{2}$

For $\beta=1$ (i.e., exponential underlying distribution),
$\mbox{var}[\log{m_{d}}]=d^{-1}\cdot(\log{2})^{-2}$.

#### Appendix 1c: R codes used to compare empirical SD of log(KM estimate of
median) with the $1/d$ with $1/\log{2}\cdot 1/d$ in Figure 3

⬇

library(survival)

sim.fn<- function(N, D, med, M, ltfu.rate=0.05){

lambda.event<- log(2)/med

lambda.censor<- 1/(1/ltfu.rate - 1)*lambda.event

event.t<- rexp(n = N*M, rate = lambda.event)

censor.t<- rexp(n = N*M, rate = lambda.censor)

replicate<- kronecker(1:M, rep(1,N))

dat<- data.frame(replicate, event.t, censor.t)

dat<- transform(dat, fup=pmin(event.t, censor.t),
event=I(event.t<=censor.t)*1)

dat.event<- subset(dat, event==1)

dat.event.sort<- dat.event[order(dat.event$replicate, dat.event$fup),]

myfn<- function(x, d) x[min(length(x),d)] #myfn(1:20, 10)

max.event.t<- tapply(dat.event.sort$fup, dat.event$replicate, FUN=myfn, d=D)

dat$max.event.t<- kronecker(max.event.t, rep(1,N))

dat<- transform(dat, fup=pmin(fup, max.event.t),
event2=I(fup<=max.event.t)*event)

fit <- survfit(Surv(fup, event2) ~ strata(replicate), data = dat)

medians.km<- summary(fit)$table[,”median”]

names(medians.km)<- NULL

sd.obs.median<- sd(log(medians.km))

sd.theory1<- 1/sqrt(D)

sd.theory2<- 1/(log(2))*1/sqrt(D)

ret<- c(N, D, med, sd.obs.median, sd.theory1, sd.theory2, ltfu.rate, M)

ret

}

#— Example: sim.fn(N=round(80*1.30), D=80, med=12, M=5000, ltfu.rate=0.000005)

### Appendix 2: Example R codes

#### Appendix 2a: Calculation of PoS with normal approximation

⬇

#— Example 1 in the paper (continuous endpoint)

PoS(type=”cont”, nsamples=2, null.value=-0.05, alternative=”greater”,

N=1552, a=1,

succ.crit=”trial”, Z.crit.final=1.97,

se.exp=0.12*sqrt(1/776 + 1/776),

meandiff.prior=0, sd.prior=0.02)

#— Example 3 in the paper (survival endpoint)

PoS(type=”surv”, nsamples=2, null.value=1, alternative=”less”,

D=441,

succ.crit=”trial”, Z.crit.final=1.96,

hr.prior=0.71, D.prior=133)

PoS(type=”surv”, nsamples=2, null.value=1, alternative=”less”,

D=441,

succ.crit=”clinical”, clin.succ.threshold =0.8,

hr.prior=0.71, D.prior=133)

#### Appendix 2b: Calculation of CP and PPoS with normal approximation

⬇

#— Example 1 in the paper (continuous endpoint)

succ_ia(type=”cont”, nsamples=2, null.value=-0.05, alternative=”greater”,

N=1552, n=776, a=1,

meandiff.ia=-0.025, sd.ia=0.16,

succ.crit=”trial”, Z.crit.final=1.97,

meandiff.exp=-0.030,

meandiff.prior=0, sd.prior=0.02)

#— Example 2 in the paper (binary endpoint)

p1<- 0.379; p2<- 0.222

n1<- 105; n2<- 53

#– Trial success

succ_ia(type=”bin”, nsamples=2, null.value=0, alternative=”greater”,

N=210, n=158, a=2,

propdiff.ia=p1-p2,

stderr.ia=sqrt(p1*(1-p1)/n1 + p2*(1-p2)/n2),

succ.crit=”trial”, Z.crit.final=2.012,

propdiff.exp=0.20,

propdiff.prior=0.20, sd.prior=sqrt(0.06))

#– Clinical success

succ_ia(type=”bin”, nsamples=2, null.value=0, alternative=”greater”,

N=210, n=158, a=2,

propdiff.ia=p1-p2,

stderr.ia=sqrt(p1*(1-p1)/n1 + p2*(1-p2)/n2),

succ.crit=”clinical”, clin.succ.threshold=0.15,

propdiff.exp=0.20,

propdiff.prior=0.20, sd.prior=sqrt(0.06))

#— Example 3 in the paper (survival endpoint)

#— Trial success

succ_ia(type=”surv”, nsamples=2, null.value=1, alternative=”less”,

D=441, d=346, a=1,

hr.ia=0.82,

succ.crit=”trial”, Z.crit.final=2.012,

hr.exp=0.75,

hr.prior=0.71, D.prior=133)

#— clinical success

succ_ia(type=”surv”, nsamples=2, null.value=1, alternative=”less”,

D=441, d=346, a=1,

hr.ia=0.82,

succ.crit=”clinical”, clin.succ.threshold=0.80,

hr.exp=0.75,

hr.prior=0.71, D.prior=133)

#### Appendix 2c: Calculation of PPoS for binary endpoint with beta-binomial
approximation

⬇

#— Trial success

succ_ia_betabinom_two( N.trt=155+170, N.con=152+171,

n.trt=155, x.trt=13, n.con=152, x.con=21,

alternative=”less”, test=”z”,

succ.crit = ”trial”, Z.crit.final = 1.96,

a.trt = 1, b.trt=1, a.con=1, b.con=1)

#— clinical success

succ_ia_betabinom_two( N.trt=155+170, N.con=152+171,

n.trt=155, x.trt=13, n.con=152, x.con=21,

alternative=”less”, test=”fisher”,

succ.crit = ”trial”, Z.crit.final = 1.96,

a.trt = 1, b.trt=1, a.con=1, b.con=1)