Sizing clinical trials when comparing bivariate time-to-event outcomes

Abstract

Clinical trials with multiple primary time-to-event outcomes are common. Use of multiple endpoints creates challenges in the evaluation of power and the calculation of sample size during trial design particularly for time-to-event outcomes. We present methods for calculating the power and sample size for randomized superiority clinical trials with two correlated time-to-event outcomes. We do this for independent and dependent censoring for three censoring scenarios: (i) the two events are non-fatal; (ii) when one event is fatal (semi-competing risk); and (iii) when both are fatal (competing risk). We derive the bivariate log-rank test in all three censoring scenarios and investigate the behavior of power and the required sample sizes. Separate evaluations are conducted for two inferential goals, evaluation of whether the test intervention is superior to the control on (1) all of the endpoints (multiple co-primary) or (2) at least one endpoint (multiple primary).

1. Introduction

The use of two time-to-event outcomes as primary endpoints has become common in clinical trials evaluating interventions in many disease areas. For example, co-infection/comorbidity trials may utilize primary endpoints to evaluate multiple comorbities, for example, a trial evaluating therapies to treat Kaposi’s sarcoma in HIV-infected individuals may have the time to Kaposi’s sarcoma progression and the time to HIV virologic failure, as primary endpoints. In new anti-cancer drug trials, the most commonly used primary endpoint is overall survival (OS) defined as the time from randomization until death from any cause. However, OS in general requires long follow-up periods after disease progression leading to long and expensive studies. Therefore, in addition to OS, many clinical trials include progression-free survival (PFS) as a primary endpoint, defined as the time from randomization to the first of tumor progression or death. Meanwhile in trials aimed at evaluating treatments to reduce a specific type of mortality, the two endpoints of time to disease-specific mortality and all-cause mortality are primary endpoints. In the first example of co-infection/comorbidity trials, both events are non-fatal (the ‘both non-fatal case’ where neither event-time is censored by the other event). However, in the oncology example, one event is fatal (death) potentially censoring the other non-fatal event (the ‘one fatal case’). Here, we use the term ‘fatal’ to describe an event that censors future events of interest. This is referred as to ‘semi-competing risk problem’, first introduced by [1]. In the last example, both events are fatal as each event-time may be censored by the other event (the ‘both fatal case’).

We developed methods for sizing clinical trials with two primary time-to-event outcomes under a time-dependent correlation structure of three bivariate exponential distributions, where (i) both events are non-fatal [2,3]. In this paper, we discuss the log-rank test-based method using the normal approximation for power and sample size calculations in clinical trials with two time-to-event outcomes, to accommodate two additional situations: (ii) when one event is fatal and (iii) when both are fatal. We also evaluate composite endpoints as a strategy. Hence, six scenarios are evaluated as classified by the three censoring schemes (both non-fatal, one fatal, and both fatal) for the two time-to-event outcomes and whether a composite endpoint is used. Table I illustrates examples of the six scenarios.

Table I.

Classification of event types by the composite and non-composite examples (six scenarios)

Type	Non-composite examples	Composite examples
Both non-fatal events	HIV trial Time to virologic failure Time to discontinuation of treatment due to toxicity Cardiovascular trial: Time to revascularization Time to major bleeding	HIV trial Time to virologic failure or to discontinuation of treatment due to toxicity
One fatal event (one non-fatal event)	Oncology trial Time to progression (TTP) Time to all cause death (OS: overall survival) Cardiovascular trial Time to cardiovascular hospital admissions Time to all cause death (OS)	Oncology trial Time to progression or all cause death (PFS: progression free survival) Time to all cause death (OS) Cardiovascular trial Time to myocardial infarction Time to major adverse cardiac events (MACE) (acute myocardial infarction, ischaemic stroke, coronary arterial occlusion, death)
Both fatal events	Cardiovascular trial Time to cardiac-specific death Time to the other cause death	Cardiovascular trial Time to cardiac-specific death Time to all cause death (OS)

We consider two inferential goals for clinical trials with multiple endpoints: (1) ‘multiple co-primary endpoints’ (where the trial is designed to evaluate if the intervention is superior to the control on all of the endpoints) and (2) ‘multiple primary endpoints’ (or ‘alternative primary endpoints’, where the trial is designed to evaluate if the intervention is superior to the control on at least one endpoint [4–6]). When considering two or more endpoints as co-primary, no adjustment is needed to control the Type I error rate if the hypothesis associated with each endpoint is evaluated at the same significance level as that required for all of the objectives. However, the Type II error rate increases as the number of endpoints being evaluated increases. Thus, design adjustments are needed to maintain the overall power. In contrast, when designing the trial to evaluate an effect on at least one of the endpoints, an adjustment is needed to control the Type I error rate because the Type I error rate increases as the number of endpoints being evaluated increases.

The paper is structured as follows: in Section 2, we describe the dependence measure and censoring schemes, and then discuss the correlation structure of the bivariate log-rank statistic. In Sections 3 and 4, we provide the power for comparing two groups with respect to two time-to-event outcomes as co-primary or multiple primary and describe methods for calculating the sample size. We also investigate the behaviors of power and the required sample sizes with a real example. In Section 5, we summarize our findings.

2. Censoring schemes, dependency, and correlation

2.1. Notation and framework

Consider a randomized clinical trial designed to compare two interventions with total N participants being recruited and randomized. Suppose that r⁽²⁾N participants are assigned to the test intervention group and r⁽¹⁾N participants to the control intervention group (r⁽¹⁾ + r⁽²⁾ = 1). Patients are then followed to evaluate bivariate survival times for two endpoints. Let $T_{ik}^{\ast }$ and C_ik be the underlying continuous survival time and potential censoring time of the kth primary endpoint for the ith participant (k = 1, 2; i = 1, … , N). Assume C_i = C_i1 = C_i2, because C_i1 and C_i2 are usually the same time. Hence, we observe the bivariate time-to-event data of ${\{(T_{i1},T_{i2},{\mathrm{\Delta }}_{i1},{\mathrm{\Delta }}_{i2},g_i)\}}_{i=1}^N$, where T_ik and Δ_ik are the ith observable survival time and right-censoring indicator for the kth primary endpoint, respectively, and g_i is the group index j (j = 2 if the ith participant belongs to the test, and j = 1 otherwise). For example, typically we see $T_{ik}=min\left(T_{ik}^{\ast },C_{ik}\right)$ and ${\mathrm{\Delta }}_{ik}=𝟙\left(T_{ik}^{\ast }\le C_i\right)$ (𝟙 (·) is the index function). The information of (T_ik, Δ_ik) is represented by the counting process 𝒩_ik(t) = 𝟙 (T_ik ≤ t, Δ_ik = 1) and the at-risk process 𝒴_ik(t) = 𝟙 (T_ik ≥ t).

Denote the marginal hazard function and its cumulative function for $T_{ij}^{\ast }$ in the group j by

$${\lambda }_k^{(j)}(t)=\underset{\mathrm{d}t\to 0}{lim}\frac{1}{\mathrm{d}t}Pr\left(t\le T_{ik}^{\ast }<t+\mathrm{d}t\mid t\le T_{ik}^{\ast },g_i=j\right)\text{and}{\mathrm{\Lambda }}_k^{(j)}={\int }_0^t{\lambda }_k^{(j)}(s)\mathrm{d}s.$$

Let ψ_k(t) be the hazard ratio (HR) ${\lambda }_k^{(2)}(t)/{\lambda }_k^{(1)}(t)$ between the two intervention groups. To test the single hypothesis ‘H_0k : ψ_k(t) = 1 for all t’ restricted to the kth endpoint (k = 1, 2), the standardized log-rank statistic

$$Z_k^{(0)}=U_k(\tau )/\sqrt{{\widehat{V}}_{kk}^{(0)}(\tau )}$$ (1)

can be applied to the univariate data set of ${\{(T_{ik},{\mathrm{\Delta }}_{ik},g_i)\}}_{i=1}^N$, where τ is the maximum observed follow-up time, U_k(t) is the log-rank process

$$U_k(t)=\sqrt{N}{\int }_0^t{\widehat{H}}_k(s)\left\{\mathrm{d}{\widehat{\mathrm{\Lambda }}}_k^{(2)}(s)-\mathrm{d}{\widehat{\mathrm{\Lambda }}}_k^{(1)}(s)\right\},k=1,2,$$

${\widehat{\mathrm{\Lambda }}}_k^{(j)}(t)={\int }_0^{t}d{\mathcal{N}}_k^{(j)}(s)/{\mathcal{Y}}_k^{(j)}(s)$ is the Nelson–Aalen estimator of ${\mathrm{\Lambda }}_k^{(j)}(t),{\widehat{V}}_{kk}^{(0)}(t)$ is the conditional variance of U_k(t) under the null hypothesis H_0k,

$${\widehat{V}}_{kk}^{(0)}(t)=N{\int }_0^t{\widehat{H}}_k{(s)}^2\left\{1-\frac{\mathrm{d}{\mathcal{N}}_k(s)-1}{{\mathcal{Y}}_k(s)-1}\right\}\left\{\frac{\mathrm{d}{\widehat{\mathrm{\Lambda }}}_k^{(1)}(s)}{{\mathcal{Y}}_k^{(2)}(s)}+\frac{\mathrm{d}{\widehat{\mathrm{\Lambda }}}_k^{(2)}(s)}{{\mathcal{Y}}_k^{(1)}(s)}\right\},$$

where ${\widehat{H}}_k(t)=N^{-1}{\mathcal{Y}}_k^{(1)}(t){\mathcal{Y}}_k^{(2)}(t)/{\mathcal{Y}}_k(t),{\mathcal{Y}}_k(t)={\mathcal{Y}}_k^{(1)}(t)+{\mathcal{Y}}_k^{(2)}(t),{\mathcal{N}}_k(t)={\mathcal{N}}_k^{(1)}(t)+{\mathcal{N}}_k^{(2)}(t),{\mathcal{Y}}_k^{(j)}(t)={\sum }_{i=1}^N𝟙(g_i=j){\mathcal{Y}}_{ik}(t)$ and ${\mathcal{N}}_k^{(j)}(t)={\sum }_{i=1}^N𝟙(g_i=j){\mathcal{N}}_{ik}(t)$ are the at-risk and counting processes for the kth endpoint of individuals belonging to the group j.

2.2. Censoring schemes and dependence measures for two time-to-event outcomes

In the trial where the bivariate time-to-event data of ${\{(T_{i1},T_{i2},{\mathrm{\Delta }}_{i1},{\mathrm{\Delta }}_{i2},g_i)\}}_{i=1}^N$ is observed, various situations are considered. First, consider the censoring scheme. [3] discuss the simplest case where both events are non-fatal. We consider the two other situations, that is, where one event is fatal but another is non-fatal (one fatal), and when both events are fatal (both fatal). We also briefly describe the measure of dependence between the two time-to-event outcomes.

Examples of the three censoring scenarios

1. Both non-fatal outcomes: In HIV clinical trials, if $T_{i1}^{\ast }$ is the time to infant HIV infection and $T_{i2}^{\ast }$ is the time to infant Hepatitis B infection, neither event-time is censored by the other event. If the subjects do not experience both events, then the events are censored at the same time (e.g., by the end of the study or patient drop-out) in the end of follow-up period. So that we have $T_{i1}=min(T_{i1}^{\ast },C_i)$ and $T_{i2}=min(T_{i2}^{\ast },C_i)$.

2. One fatal outcome: In oncology trials, if $T_{i1}^{\ast }$ is the time-to-progression (TTP: defined as the time from randomization until objective tumor progression and does not include deaths) and $T_{i2}^{\ast }$ is the OS (time to all cause death),one endpoint (TTP) is censored by the other endpoint (OS) being completely observed. So that, we have $T_{i1}=min(T_{i1}^{\ast },T_{i2}^{\ast },C_i)$ and $T_{i2}=min(T_{i2}^{\ast },C_i)$. As T_i2 is a competing risk for T_i1 but T_i1 is not for T_i2 (T_i1 ≤ T_i2), this situation describes a semi-competing risk discussed by [1] (see [7] for further discussions). If there is a non-zero correlation between the endpoints (i.e., dependent censoring), the situation requires modifying the standard log-rank test to account for the dependent censoring. We may be able to avoid this problems by creating a composite endpoint.

3. Both fatal outcomes: In trials aimed at reducing a specific type of mortality, if $T_{i1}^{\ast }$ is the time to disease-specific mortality and $T_{i2}^{\ast }$ is the time to other-cause mortalities, each event may be censored by the other event. This situation is called a competing risk. Here, we have $T_{i1}=min(T_{i1}^{\ast },T_{i2}^{\ast },C_i)$, and $T_{i2}=min(T_{i1}^{\ast },T_{i2}^{\ast },C_i)$.

Definition for the composite endpoint and handling censoring

We provide the definition for the composite endpoint in the scenarios with two time-to-event outcomes. See Table I for examples of the composite endpoints, such as PFS composing (combining) the TTP and OS or major adverse cardiac events, where the difference between composing and not composing the endpoints is based on handling rule for censoring in each endpoint. That is, we can consider handling censoring in two ways in our time-to-event context. The first way is as usual to use ${\mathrm{\Delta }}_{ik}=𝟙(T_{ik}=T_{ik}^{\ast })$, k = 1, 2, which is the non-composite setting in the context of this paper. The second one is to handle the censoring indicators by ${\mathrm{\Delta }}_{i1}=𝟙(T_{i1}=min(T_{i1}^{\ast },T_{i2}^{\ast }))$ and ${\mathrm{\Delta }}_{i2}=𝟙(T_{i2}=T_{i2}^{\ast })$, which is the definition of the composite setting in this paper. For an example, consider the situation where $T_{i1}^{\ast }$ and $T_{i2}^{\ast }$ are the TTP and OS endpoints, respectively. Then, (T_i1, Δ_i1) is the ith observation for the TTP endpoint in the former handling censoring (the TTP outcome), while (T_i1, Δ_i1) is that for the PFS endpoint in the latter (the PFS outcome). Note that the difference between the TTP and PFS outcomes is brought by handling of censoring in this paper, but T_i1 is the observable time with the same length and notation in both TTP and PFS endpoints. Alternatively, because one usually define the PFS endpoint as ${\stackrel{\sim }{T}}_{i1}^{\ast }=min(T_{i1}^{\ast },T_{i2}^{\ast })$, the former handling of censoring for the composite endpoint ${\stackrel{\sim }{T}}_{i1}^{\ast }$ leads to the PFS outcome. However, to avoid confusion in derivations, we discuss the case of composite setting based on the handling rule for censoring without notations to define the composite endpoint such as ${\stackrel{\sim }{T}}_{i1}^{\ast }$. Also, in this paper, the observations for the 1st endpoint, (T_i1, Δ_i1), are used in either non-composite or composite setting. On the other hand, those for the second endpoint, (T_i2, Δ_i2), are consistently considered as the non-composite setting.

Dependence measure

Two times $T_{i1}^{\ast }$ and $T_{i2}^{\ast }$ may be correlated. We consider a correlation structure between $T_{i1}^{\ast }$ and $T_{i2}^{\ast }$. Let $S^{(j)}(t,s)=Pr(t<T_{i1}^{\ast },s<T_{i2}^{\ast }\mid g_i=j)$ and $S_k^{(j)}(t)=Pr(t<T_{ik}^{\ast }\mid g_i=j)(k=1,2)$ (k = 1, 2) be the joint survival and marginal survival functions for the bivariate survival data ${\{(T_{i1}^{\ast },T_{i2}^{\ast })\}}_{i=1}^n$ in the group j, respectively. We consider the correlation between the two cumulative hazard variates [8] defined by

$${\rho }^{(j)}=\text{corr}\left[{\mathrm{\Lambda }}_1^{(j)}(T_{i1}^{\ast }),{\mathrm{\Lambda }}_2^{(j)}(T_{i2}^{\ast })\right]={\int }_0^{\infty }{\int }_0^{\infty }{S}^{(j)}(t,s)\mathrm{d}{\mathrm{\Lambda }}_1^{(j)}(t)\mathrm{d}{\mathrm{\Lambda }}_2^{(j)}(s)-1.$$

If the marginal of the bivariate survival data are exponential, ρ^(j) is the same as the correlation coefficient of the raw data ${\{(T_{i1}^{\ast },T_{i2}^{\ast })\}}_{i=1}^n$ [3]. In order to generate the joint survival function S^(j)(t, s) from the $S_1^{(j)}(t)$ and $S_2^{(j)}(s)$, we prepare a copula function 𝒞 (·), which gives

$$S^{(j)}(t,s)=\mathcal{C}(S_1^{(j)}(t),S_2^{(j)}(s);{\theta }^{(j)}),$$

where the association parameter θ^(j) included in 𝒞 (·) is a one-to-one function of ρ^(j).

2.3. Bivariate structure of two log-rank test statistics

Consider applying the log-rank statistic (1) to the data ${\{(T_{ik},{\mathrm{\Delta }}_{ik},g_i)\}}_{i=1}^N$ for each endpoint (k = 1, 2) extracted from the bivariate time-to-event data ${\{(T_{i1},T_{i2},{\mathrm{\Delta }}_{i1},{\mathrm{\Delta }}_{i2},g_i)\}}_{i=1}^N$ obtained under the aforementioned situation. Then, the pair ${\left(Z_1^{(0)},Z_2^{(0)}\right)}^{\mathrm{T}}$ of the standardized log-rank statistics is approximately bivariate normally distributed with mean vector $\sqrt{N}\mathit{\delta }$ and variance–covariance matrix Σ when N is sufficiently large (see [3]), where

$$\mathit{\delta }=\left(\begin{array}{c}{\delta }_1\\ {\delta }_2\end{array}\right)=\left(\begin{array}{c}\frac{{\mu }_1(\tau )}{\sqrt{V_{11}^{(0)}(\tau )}}\\ \frac{{\mu }_2(\tau )}{\sqrt{V_{22}^{(0)}(\tau )}}\end{array}\right)\text{and}\mathbf{\sum }=\left(\begin{array}{cc}\frac{V_{11}(\tau )}{V_{11}^{(0)}(\tau )}& \frac{V_{12}(\tau )}{\sqrt{V_{11}^{(0)}(\tau )V_{22}^{(0)}(\tau )}}\\ \frac{V_{12}(\tau )}{\sqrt{V_{11}^{(0)}(\tau )V_{22}^{(0)}(\tau )}}& \frac{V_{22}(\tau )}{V_{22}^{(0)}(\tau )}\end{array}\right),$$

μ_k(t) and $V_{kk}^{(0)}(t)$ are asymptotic forms of N^−1/2U_k(t) and ${\widehat{V}}_{kk}^{(0)}(t)$, respectively, and V_kk(t), k = 1, 2 and V₁₂(t) are the asymptotic variances and covariance of U₁(t) and U₂(t), respectively. These elements can be written as, in all censoring scenarios with/without the composite setting,

$$\begin{array}{ll}\hfill {\mu }_k(\tau )& ={\int }_0^{\tau }{H}_k(t)\left\{\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_k^{(2)}(t)-d{\stackrel{\sim }{\mathrm{\Lambda }}}_k^{(1)}(t)\right\},k=1,2\hfill \\ \hfill V_{kk}^{(0)}(\tau )& ={\int }_0^{\tau }{H}_k{(t)}^2\left\{\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_k^{(1)}(t)/r^{(2)}{y}_k^{(2)}(t)+\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_k^{(2)}(t)/r^{(1)}{y}_k^{(1)}(t)\right\},k=1,2\hfill \\ \hfill V_{kk}(\tau )& ={\int }_0^{\tau }{H}_k{(t)}^2\left\{\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_k^{(1)}(t)/r^{(1)}{y}_k^{(1)}(t)+\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_k^{(2)}(t)/r^{(2)}{y}_k^{(2)}(t)\right\},k=1,2\hfill \\ \hfill V_{12}(\tau )& ={\int }_0^{\tau }{\int }_0^{\tau }{H}_1(t)H_2(s)G(t\vee s)\left\{\frac{\mathrm{d}{A}^{(1)}(t,s)}{r^{(1)}{y}_1^{(1)}(t)y_2^{(1)}(s)}+\frac{\mathrm{d}{A}^{(2)}(t,s)}{r^{(2)}{y}_1^{(2)}(t)y_2^{(2)}(s)}\right\},\hfill \end{array}$$ (2)

where H_k(t) is the asymptotic form of Ĥ_k(t), $y_k^{(j)}(t)=\mathrm{E}[{\mathcal{Y}}_{ik}(t)\mid g_i=j],\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_k^{(j)}(t)=\mathrm{E}[\mathrm{d}{\mathcal{N}}_{ik}(t)\mid {\mathcal{Y}}_{ik}(t)=1,g_i=j]$,

$$G(t\vee s)\mathrm{d}{A}^{(j)}(t,s)=\mathrm{E}[\mathrm{d}{M}_{i1}(t)\mathrm{d}{M}_{i2}(s)\mid g_i=j],j=1,2$$

is a covariance function for a martingale variation of $d{M}_{ik}(t)=\mathrm{d}{\mathcal{N}}_{ik}(t)-{\mathcal{Y}}_{ik}(t)\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_k^{(j)}(t)$, G(t) is the survival function of censoring time C_i, and t ∨ s represents max(t, s). However, note that the forms of H_k(t), $y_k^{(j)}(t),\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_k^{(j)}(t)$, and dA^(j)(t, s) are different among the censoring scenarios as provided thereafter. Some details of the derivations are in Appendix A. In advance, let Λ^(j)(t, s) = −log S^(j)(t, s) be the joint cumulative hazard function in group j. Using this notation, note that we can write $\mathrm{d}{\mathrm{\Lambda }}_1^{(j)}(t)={\mathrm{\Lambda }}^{(j)}(\mathrm{d}t,0)$ and $\mathrm{d}{\mathrm{\Lambda }}_2^{(j)}(s)={\mathrm{\Lambda }}^{(j)}(0,\mathrm{d}s)$.

• (i)
  Both non-fatal outcomes: In this situation, the setting of the censoring indicators is consistently ${\mathrm{\Delta }}_{ik}=𝟙(T_{ik}^{\ast }=T_{ik})=𝟙(T_{ik}^{\ast }\le C_i)$. The forms of H_k(t), $y_k^{(j)}(t),d{\stackrel{\sim }{\mathrm{\Lambda }}}_k^{(j)}(t)$ and dA^(j)(t,s) are given by [3]. See Table II for the details. In particular, ${\stackrel{\sim }{\mathrm{\Lambda }}}_k^{(j)}(t)={\mathrm{\Lambda }}_k^{(j)}(t)$ and

  $$\mathrm{d}{A}^{(j)}(t,s)=S^{(j)}(\mathrm{d}t,\mathrm{d}s)+S^{(j)}(t,\mathrm{d}s)\mathrm{d}{\mathrm{\Lambda }}_1^{(j)}(t)+S^{(j)}(\mathrm{d}t,s)\mathrm{d}{\mathrm{\Lambda }}_2^{(j)}(s)+S^{(j)}(t,s)\mathrm{d}{\mathrm{\Lambda }}_1^{(j)}(t)\mathrm{d}{\mathrm{\Lambda }}_2^{(j)}(s).$$
• (ii)One fatal outcome: Assume that $T_{i1}^{\ast }$ is the time to non-fatal event while $T_{i2}^{\ast }$ is the time to fatal event, so that $T_{i1}=min(T_{i1}^{\ast },T_{i2}^{\ast },C_i)$ and $T_{i2}=min(T_{i2}^{\ast },C_i)$. For example, in oncology trials, the TTP and OS endpoints are defined by non-fatal event time $T_{i1}^{\ast }$ and fatal one $T_{i2}^{\ast }$, respectively, while the PFS endpoint is defined by $min(T_{i1}^{\ast },T_{i2}^{\ast })$ as the composite. The log-rank test is applied to survival outcomes obtained using their endpoints. The applications to two sets of bivariate data defined by the TTP and OS endpoints and by the PFS and OS endpoints are discussed in the following non-composite and composite settings, respectively. See Appendix A for the details provided below.

Table II.

Elements of U_k(τ), $V_{kk}^{(0)}(\tau )$, V_kk(τ), and V₁₂(τ) for the three censoring scenarios under the non-composite setting. t ∨ s = max(t, s).

Elements		Both non-fatal case	One fatal case	Both fatal case
	k
$\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_k^{(j)}(t)$	1		Λ(j)(dt, t)
	2	$\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_k^{(j)}(t)$		Λ (j)(t, dt)
$y_k^{(j)}(t)$	1		S(j)(t, t)G(t)
	2	$S_k^{(j)}(t)G(t)$
Hk(t)	1		$G(t)\frac{r^{(1)}{r}^{(2)}{S}^{(1)}(t,t)S^{(2)}(t,t)}{r^{(1)}{S}^{(1)}(t,t)+r^{(2)}{S}^{(2)}(t,t)}$
	2	$G(t)\frac{r^{(1)}{r}^{(2)}{S}_k^{(1)}(t)S_k^{(2)}(t)}{r^{(1)}{S}_k^{(1)}(t)+r^{(2)}{S}_k^{(2)}(t)}$
dA(j)(t, s)		$\begin{array}{l}{S}^{(j)}(\mathrm{d}t,\mathrm{d}s)\\ +S^{(j)}(t,\mathrm{d}s)\mathrm{d}{\mathrm{\Lambda }}_1^{(j)}(t)\\ +S^{(j)}(\mathrm{d}t,s)\mathrm{d}{\mathrm{\Lambda }}_2^{(j)}(s)\\ +S^{(j)}(t,s)\mathrm{d}{\mathrm{\Lambda }}_1^{(j)}(t)\mathrm{d}{\mathrm{\Lambda }}_2^{(j)}(s)\end{array}$	$\begin{array}{l}𝟙(t\le s)S^{(j)}(\mathrm{d}t,\mathrm{d}s)\\ +𝟙(t\le s)S^{(j)}(t,\mathrm{d}s){\mathrm{\Lambda }}^{(j)}(\mathrm{d}t,t)\\ +S^{(j)}(\mathrm{d}t,t\vee s)\mathrm{d}{\mathrm{\Lambda }}_2^{(j)}(s)\\ +S^{(j)}(t,t\vee s){\mathrm{\Lambda }}^{(j)}(\mathrm{d}t,t)\mathrm{d}{\mathrm{\Lambda }}_2^{(j)}(s)\end{array}$	0

Non-composite case

Consider the non-composite setting, that is, set ${\mathrm{\Delta }}_{ik}=𝟙(T_{ik}^{\ast }=T_{ik})$, k = 1, 2 for the censoring indicators, where non-fatal $T_{i1}^{\ast }$ is censored by observing fatal $T_{i2}^{\ast }$ earlier than $T_{i1}^{\ast }$. The forms of H_k(t), $y_k^{(j)}(t)$, and $\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_k^{(j)}(t)$, j = 1, 2 are that, for the non-fatal endpoint,

$$H_1(t)=G(t)\frac{r^{(1)}{r}^{(2)}{S}^{(1)}(t,t)S^{(2)}(t,t)}{r^{(1)}{S}^{(1)}(t,t)+r^{(2)}{S}^{(2)}(t,t)},y_1^{(j)}(t)=G(t)S^{(j)}(t,t)\text{and}\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_1^{(j)}(t)={\mathrm{\Lambda }}^{(j)}(\mathrm{d}t,t),$$

depending on the correlation between $T_{i1}^{\ast }$ and $T_{i2}^{\ast }$, while, for the fatal endpoint,

$$H_2(t)=G(t)\frac{r^{(1)}{r}^{(2)}{S}_2^{(1)}(t)S_2^{(2)}(t)}{r^{(1)}{S}_2^{(1)}(t)+r^{(2)}{S}_2^{(2)}(t)},y_2^{(j)}(t)=G(t)S_2^{(j)}(t)\text{and}\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_2^{(j)}(t)=\mathrm{d}{\mathrm{\Lambda }}_2^{(j)}(t)$$

being the same as those of situation (i). Also, dA^(j)(t, s) related to the covariance of the martingale variation is

$$\begin{array}{l}\mathrm{d}{A}^{(j)}(t,s)=𝟙(t\le s)S^{(j)}(\mathrm{d}t,\mathrm{d}s)+𝟙(t\le s)S^{(j)}(t,\mathrm{d}s){\mathrm{\Lambda }}^{(j)}(\mathrm{d}t,t)\\ +S^{(j)}(\mathrm{d}t,t\vee s)\mathrm{d}{\mathrm{\Lambda }}_2^{(j)}(s)+S^{(j)}(t,t\vee s){\mathrm{\Lambda }}^{(j)}(\mathrm{d}t,t)\mathrm{d}{\mathrm{\Lambda }}_2^{(j)}(s).\end{array}$$ (3a)

Composite case

In the composite setting, set ${\mathrm{\Delta }}_{i1}=𝟙(T_{i1}=min(T_{i1}^{\ast },T_{i2}^{\ast }))$ and ${\mathrm{\Delta }}_{i2}=𝟙(T_{i2}=T_{i2}^{\ast })$ for the censoring indicators. The forms of H_k(t), $y_k^{(j)}(t)$ and $\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_2^{(j)}(t)$ are the same as those in the non-composite case, while $\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_1^{(j)}(t)$ and dA^(j)(t, s) are changed from the non-composite case, where the intensity information on the second endpoint (fatal event) is added into them (k = 1, 2, j = 1, 2). That is, in this case, we have

$$\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_1^{(j)}(t)={\mathrm{\Lambda }}^{(j)}(\mathrm{d}t,t)+{\mathrm{\Lambda }}^{(j)}(t,\mathrm{d}t)$$

and

$$\begin{array}{l}\mathrm{d}{A}^{(j)}(t,s)=𝟙(t\le s)S^{(j)}(\mathrm{d}t,\mathrm{d}s)-𝟙(t=s)S^{(j)}(t,\mathrm{d}s)\\ +𝟙(t\le s)S^{(j)}(t,\mathrm{d}s)\{{\mathrm{\Lambda }}^{(j)}(\mathrm{d}t,t)+{\mathrm{\Lambda }}^{(j)}(t,\mathrm{d}t)\}\\ +\{S^{(j)}(\mathrm{d}t,t\vee s)+𝟙(t\le s)S^{(j)}(\mathrm{d}t,t)\}\mathrm{d}{\mathrm{\Lambda }}_2^{(j)}(s)\\ +S^{(j)}(t,t\vee s)\{{\mathrm{\Lambda }}^{(j)}(\mathrm{d}t,t)+{\mathrm{\Lambda }}^{(j)}(t,\mathrm{d}t)\}\mathrm{d}{\mathrm{\Lambda }}_2^{(j)}(s).\end{array}$$ (3b)

• (iii)Both fatal outcomes: Both $T_{i1}^{\ast }$ and $T_{i2}^{\ast }$ are the times to fatal events, so that we observe $T_{i1}=min\left(T_{i1}^{\ast },T_{i2}^{\ast },C_i\right)$ and $T_{i2}=min\left(T_{i1}^{\ast },T_{i2}^{\ast },C_i\right)$. See Appendix A for the details provided in the subsequent section.

Non-composite case

Assume the non-composite setting, that is, ${\mathrm{\Delta }}_{ik}=𝟙(T_{ik}^{\ast }=T_{ik})$, k = 1, 2. The forms of H_k(t), $y_k^{(j)}(t)$, and $\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_k^{(j)}(t)$, k = 1, 2, j = 1, 2 are similar to those for the 2nd endpoint in situation (ii), because, if either of T_i1 or T_i2 is completely observed, the other is always censored. Inthis case, we can derive

$$\begin{array}{ll}\hfill H_k(t)& =G(t)\frac{r^{(1)}{r}^{(2)}{S}^{(1)}(t,t)S^{(2)}(t,t)}{r^{(1)}{S}^{(1)}(t,t)+r^{(2)}{S}^{(2)}(t,t)},y_k^{(j)}(t)=G(t)S^{(j)}(t,t),k=1,2\hfill \\ \hfill \mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_1^{(j)}(t)& ={\mathrm{\Lambda }}^{(j)}(\mathrm{d}t,t),\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_2^{(j)}(t)={\mathrm{\Lambda }}^{(j)}(t,\mathrm{d}t)\hfill \end{array}$$

and

$$\mathrm{d}{A}^{(j)}(t,s)=0.$$ (4a)

Composite case

Here, set ${\mathrm{\Delta }}_{i1}=𝟙(T_{i1}=min(T_{i1}^{\ast },T_{i2}^{\ast }))$ and ${\mathrm{\Delta }}_{i2}=𝟙(T_{i2}=T_{i2}^{\ast })$ for the composite setting. In this case, only $\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_1^{(j)}(t)$ and dA^(j)(t, s), j = 1, 2 are changed from the non-composite version. Hence, we have

$$\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_1^{(j)}(t)={\mathrm{\Lambda }}^{(j)}(\mathrm{d}t,t)+{\mathrm{\Lambda }}^{(j)}(t,\mathrm{d}t)$$

and

$$\begin{array}{l}\mathrm{d}{A}^{(j)}(t,s)=-𝟙(t=s)S^{(j)}(t\vee s,\mathrm{d}s)+𝟙(t\le s)S^{(j)}(s,\mathrm{d}s){\mathrm{\Lambda }}^{(j)}(t,\mathrm{d}t)\\ +𝟙(t\ge s)S^{(j)}(t,\mathrm{d}t){\mathrm{\Lambda }}^{(j)}(s,\mathrm{d}s)+S^{(j)}(t\vee s,t\vee s){\mathrm{\Lambda }}^{(j)}(t,\mathrm{d}t){\mathrm{\Lambda }}^{(j)}(s,\mathrm{d}s).\end{array}$$ (4b)

The forms of H_k(t), $y_k^{(j)}(t)$, and $\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_2^{(j)}(t)$, k, j = 1, 2 are the same as those in the non-composite setting.

Table II summarizes the statistics regarding $y_k^{(j)}(t),\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_k^{(j)}(t)$, H_k(t), and dA^(j)(t, s) among the three censoring scenarios under the non-composite setting. The table clearly shows how these statistics change among the three scenarios. For all scenarios, the power is given by

$$1-\beta =Pr\left[(Z_1^{(0)},Z_2^{(0)})\in {\mathcal{Z}}^{(0)}\right]\approx \int {\int }_{(z_1,z_2)\in {\mathcal{Z}}^{(0)}}\varphi (z_1,z_2;\sqrt{N}\mathit{\delta },\mathbf{\sum })\mathrm{d}{z}_1\mathrm{d}{z}_2$$ (5)

because the hypothesis is rejected if the bivariate statistic ${\left(Z_1^{(0)},Z_2^{(0)}\right)}^{\mathrm{T}}$ takes the realized values on the area 𝒵⁽⁰⁾, where $\varphi (z_1,z_2;\sqrt{N}\mathit{\delta },\mathbf{\sum })$ is the bivariate normal density with mean vector $\sqrt{N}\mathit{\delta }$ and variance–covariance matrix Σ.

3. Co-primary endpoints

3.1. Hypothesis testing, power, and sample sizes

We are interested in testing the hypotheses on the HRs to evaluate a joint reduction of occurrence of events over time on both outcomes, that is, ${\mathrm{H}}_0^{\text{cp}}=\{{\mathrm{H}}_{01}:{\psi }_1(t)=1\text{or}{\mathrm{H}}_{02}:{\psi }_2(t)=1\text{for}\text{all}t\}$ versus ${\mathrm{H}}_1^{\text{cp}}=\{{\mathrm{H}}_{11}:{\psi }_1(t)<1\text{and}{\mathrm{H}}_{12}:{\psi }_2(t)<1\text{at}\text{some}t\}$. In all of the censoring scenarios, using the two log-rank statistics, $Z_k^{(0)}=U_k(\tau )/\sqrt{{\widehat{V}}_{kk}^{(0)}(\tau )}$, k = 1, 2 for this hypothesis testing, we are able to

$$\text{reject}\text{the}\text{null}\text{hypothesis}{\mathrm{H}}_0^{\text{cp}}\text{if}\text{and}\text{only}\text{if}{\mathrm{Z}}_1^{(0)}<-{\mathrm{z}}_{\alpha }\text{and}{\mathrm{Z}}_2^{(0)}<-{\mathrm{z}}_{\alpha }$$ (6)

at the prespecified significance level of α, where z_α is the 100(1 − α) percentile of N(0, 1). The overall Type I error associated with the null hypothesis ${\mathrm{H}}_0^{\text{cp}}$ is controlled by the maximum of the marginal Type I errors [5]. This means that, to investigate whether the overall Type I error is larger than the nominal level, it is enough to investigate whether the marginal ones are larger than the nominal level [2]. The behaviors of the Type I error for the univariate log-rank test are well known [9–11]. The one-sided test procedure (6) based on the asymptotic normality may inflate under some situations such as small sample size and/or unbalanced design (in particular, with r⁽¹⁾ > 0.5). We can improve the precision by correcting the critical value based on the sample size N and the allocation rate r⁽¹⁾ to control the marginal Type I errors. However, the overall Type I error on ${\mathrm{H}}_0^{\text{cp}}$ is, for example, the product of marginal ones for independent endpoints, and usually quite smaller than the maximum of marginal ones as long as ρ^(j) is not so high, so that the inflation problem seen in one-sided log-rank test may be moderately reduced in multiple co-primary problem.

In the procedure (6), the rejection region of ${\mathrm{H}}_0^{\text{cp}}$ is { $\left(Z_1^{(0)},Z_2^{(0)}\right):Z_1^{(0)}<-{\mathrm{z}}_{\alpha }$ and $Z_2^{(0)}<-{\mathrm{z}}_{\alpha }$}. In all of the scenarios, therefore, we have the power function for the joint reduction in both time-to-event outcomes given as

$$1-\beta =Pr\left[{\cap }_{k=1}^2\{Z_k^{(0)}<-{\mathrm{z}}_{\alpha }\}\mid {\mathrm{H}}_1^{\text{cp}}\right]$$

This overall power is referred to as ‘complete power’ [12] or ‘conjunctive power’ [13], which is simply calculated using the cumulative distribution function of the bivariate normal distribution. The power can be approximately calculated (under large samples) by

$$1-\beta \approx {\int }_{-\infty }^{{\mathrm{z}}_{\alpha }}{\int }_{-\infty }^{-{\mathrm{z}}_{\alpha }}\varphi \left(z_1,z_2;\sqrt{N}\mathit{\delta },\mathbf{\sum }\right)\mathrm{d}{z}_1\mathrm{d}{z}_2$$ (7)

Let N_cp be the minimum of total sample size N required for testing ${\mathrm{H}}_0^{\text{cp}}$ against ${\mathrm{H}}_1^{\text{cp}}$. Thus, N_cp is the smallest integer not less than satisfying the power (7) and given by

$$N_{\text{cp}}=\lceil {\left(K_{\beta }+z_{\alpha }{\sigma }_2^{(0)}/{\sigma }_2\right)}^2/{\delta }_2^2\rceil$$ (8)

where ⌈x⌉ is the smallest integer not less than x, ${\sigma }_k^{(0)}$ and σ_k(k = 1, 2) are the standard deviations $\sqrt{V_{kk}^{(0)}(\tau )}$ and $\sqrt{V_{kk}(\tau )}$, respectively, K_β is the solution of the integral equation

$$1-\beta ={\int }_{-\infty }^{\frac{{\delta }_1}{{\delta }_2}{K}_{\beta }+{\mathrm{z}}_{\alpha }\left\{\frac{{\delta }_1}{{\delta }_2}\frac{{\sigma }_2^{(0)}}{{\sigma }_2}-\frac{{\sigma }_1^{(0)}}{{\sigma }_1}\right\}}{\int }_{-\infty }^{K_{\beta }}\varphi (z_1,z_2;\mathbf{0},\mathbf{R})\mathrm{d}{z}_1\mathrm{d}{z}_2$$

and R is the following correlation matrix

$$\mathbf{R}=\left(\begin{array}{cc}1& V_{12}(\tau )/\sqrt{V_{11}(\tau )V_{22}(\tau )}\\ V_{12}(\tau )/\sqrt{V_{11}(\tau )V_{22}(\tau )}& 1\end{array}\right).$$

A grid search to find a value of N_cp often requires considerable computing time. The search proceeds by gradually increasing N_cp until the power (7) exceeds the desired power. Alternative methods to reduce the computational time are the Newton–Raphson algorithm for finding K_β in [14] or the basic linear interpolation algorithm in [2].

One may expect to calculate the required number of participants from that of events such as the Reference [15] procedure in univariate data. However, we encounter difficulty in finding such a procedure in the bivariate case [3]. One of the causes is that the relationship between numbers of events and participants is more complicated than the univariate case, because the patterns of observations (censored or not) increase twofold. Another is that it is difficult to let the difference between the correlation models (such as early or late dependency) on a restricted time-interval reflect using both numbers of events without censoring, because both numbers of events are usually considered under the uncensored model (or after long enough follow-up). Even so, the required numbers of events are useful to monitor a trial and they can be obtained using D_𝚤𝚥 = N_cp × P_𝚤𝚥, where P_𝚤𝚥 is the proportion of each observation pattern, $P_{𝚤𝚥}=Pr({\mathrm{\Delta }}_i^{(1)}=𝚤,{\mathrm{\Delta }}_i^{(2)}=𝚥)$, D_𝚤𝚥 is the expected number of events in each case where the first endpoint is observed (𝚤 = 1) or censored (𝚤 = 0) and the second is observed (𝚥 = 1) or not (𝚥 = 0). See Appendix B for details for calculation of P_𝚤𝚥 under the three censoring schemes.

3.2. Behavior of the sample size

We investigate the behavior of the sample size and power for detecting the joint reduction ${\mathrm{H}}_1^{\text{cp}}$ in bivariate time-to-event data under the three censoring schemes with the two time-dependent association structures, where the time-dependent association structures are asymmetric late (tail) dependency generated by the Clayton copula [16] and early (tail) dependency by the Gumbel copula [17], which have been widely used in practice. We generate bivariate time-to-event data by supposing that the marginals of $T_{i1}^{\ast }$ and $T_{i2}^{\ast }$ are exponential, and that C_i = U(0, τ_a) + τ_f (hence, τ = τ_a + τ_f), where τ_a and τ_f are the lengths of the entry period to the trial and follow-up period, respectively, and U(0, τ_a) denotes a uniform random number on (0, τ_a). The target power 1 − β = 0.8, the significance level α = 0.025, τ_a = 2, and τ_f = 3 are used, and all empirical powers are computed by Monte Carlo trials with 100,000 replications throughout.

Let N_sim be the simulation-based sample size required for testing ${\mathrm{H}}_0^{\text{cp}}$, and let N_k denote the minimum of total sample size required to test the single hypothesis H_0k for the kth endpoint. To avoid confusion in N_cp and N₁, we write them as $N_{\text{cp}}^{(\mathrm{c})}$ and $N_1^{(\mathrm{c})}$ if the first endpoint is replaced by the composite of the first and the second ones (composite setting), and as $N_{\text{cp}}^{(\text{nc})}$ and $N_1^{(\text{nc})}$ otherwise (non-composite setting). Let P̃_cp denote the empirical power (%) for detecting ${\mathrm{H}}_1^{\text{cp}}$ when the total number of participants is N_cp designed using the formula (8). See Section B.1 of Supporting Information for results other than ones provided later.

Formula performance and sample size behavior: one fatal case

Focusing on the one fatal case scenario, suppose that $T_{i1}^{\ast }$ and $T_{i2}^{\ast }$ are the TTP and OS endpoints, respectively. We evaluate the performance of the sample size formula (8) for practicality by comparing it with alternative solutions sizing based on univariate versions (N_TTP, N_PFS, and N_OS) and simulation (N_sim) under the common α and β, where $N_{\text{TTP}}(=N_1^{(\text{nc})}),N_{\text{PFS}}(=N_1^{(\mathrm{c})})$, and N_OS (= N₂) are N_k when the kth endpoint is the TTP, PFS, and OS. In the one fatal case, the bivariate survival data (T_i1, Δ_i1) and (T_i2, Δ_i2) are of the TTP and OS outcomes under the non-composite setting, but they are of the PFS and OS outcomes under the composite setting.

Table III displays the required total sample sizes N_cp, N_sim, N₁, and N₂ with the empirical power P̃_cp under ${\psi }_{\text{TTP}}^{-1}=1.7,S_{\text{TTP}}^{(1)}(\tau )=0.3$ and $S_{\text{OS}}^{(1)}(\tau )=0.4$, when ${\psi }_{\text{OS}}^{-1}$ and ρ^(k) vary for the combinations of ${\psi }_{\text{OS}}^{-1}=1.3$, 1.5, 1.7 and ρ⁽¹⁾ = ρ⁽²⁾ = 0, 0.3, 0.5 and 0.8. Note that ψ_TTP (= ψ₁) and ψ_OS (=ψ₂) are the HRs for the TTP and OS, respectively. Similarly, $S_{\text{TTP}}^{(1)}(\tau )(=S_1^{(1)}(\tau ))$ and $S_{\text{OS}}^{(1)}(\tau )(=S_2^{(1)}(\tau ))$ are the τ-time survival rates for the TTP and OS.

Table III.

The case of one fatal outcome (ii) (semi-competing risk): total numbers of participants N_cp calculated from (8), the corresponding empirical powers P̃_cp (%) and alternative sizing solutions N_sim, N_TTP and N_OS under ${\psi }_{\text{TTP}}^{-1}=1.7,S_{\text{TTP}}^{(1)}(\tau )=0.3$ and $S_{\text{OS}}^{(1)}(\tau )=0.4$.

Dependence			Composite setting					Non-composite setting
Structure	${\psi }_{\text{OS}}^{-1}$	ρ(k)	Nsim	$N_{\text{cp}}^{(\mathrm{c})}$	P̃cp	NPFS	NOS	Nsim	$N_{\text{cp}}^{(\text{nc})}$	P̃cp	NTTP	NOS
Late (Clayton copula)	1.3	0.0	968	972	80.1	262	972	966	972	80.1	288	972
	1.3	0.3	968	972	80.2	312	972	972	974	80.2	314	972
	1.3	0.5	968	972	80.2	348	972	970	976	80.3	328	972
	1.3	0.8	968	972	79.8	418	972	966	972	80.3	324	972
	1.5	0.0	430	436	80.5	196	432	474	480	80.9	284	432
	1.5	0.3	436	440	80.6	232	432	492	496	80.9	320	432
	1.5	0.5	440	444	80.6	260	432	500	506	80.8	342	432
	1.5	0.8	450	456	80.5	316	432	504	510	80.9	362	432
	1.7	0.0	272	278	81.2	160	266	354	360	81.3	280	266
	1.7	0.3	280	286	80.9	190	266	380	388	81.0	324	266
	1.7	0.5	288	294	81.0	214	266	402	408	81.1	354	266
	1.7	0.8	308	314	80.9	262	266	426	432	80.9	394	266
Early (Gumbel copula)	1.3	0.0	968	972	80.2	262	972	968	972	80.1	288	972
	1.3	0.3	968	972	80.3	282	972	966	972	80.2	262	972
	1.3	0.5	968	972	80.3	294	972	970	972	80.1	232	972
	1.3	0.8	970	972	80.1	306	972	968	972	80.2	158	972
	1.5	0.0	432	436	80.4	196	432	474	480	80.9	284	432
	1.5	0.3	430	434	80.5	212	432	464	472	80.7	278	432
	1.5	0.5	430	434	80.5	224	432	456	462	80.5	268	432
	1.5	0.8	428	432	80.3	238	432	432	438	80.8	220	432
	1.7	0.0	272	278	81.0	160	266	352	360	81.2	280	266
	1.7	0.3	274	278	80.9	176	266	356	364	81.2	294	266
	1.7	0.5	274	278	80.8	186	266	358	362	81.0	300	266
	1.7	0.8	274	278	80.8	206	266	336	342	80.9	290	266

The sample sizes N_cp calculated from the formula (8) usually provide slightly conservative results compared with N_sim, and the corresponding empirical powers P̃_cp are preferable, that is, P̃_cp are slightly larger than the target powers (although P̃_cp tends to be further away from the target as the HR ${\psi }_2^{-1}$ increases larger than 1). Times to compute N_sim are usually much longer than ones for N_cp, which becomes greater as the effect size is smaller. Hence, the formula (8) reduces the cost greatly, regardless of the effect size, and it is also useful as an initial value to search N_sim. Also, $N_{\text{cp}}^{(\mathrm{c})}$ (N_cp under the composite setting) is smaller than $N_{\text{cp}}^{(\text{nc})}$ (N_cp under the non-composite setting) in all cases. As the absolute value of $log{\psi }_{\text{OS}}^{-1}$ decreases below $\mid log{\psi }_{\text{TTP}}^{-1}\mid$, the value of N_cp approaches N_OS and diverges from N_PFS and N_TTP. If both HRs are approximately equal, ${\psi }_{\text{TTP}}^{-1}\approx {\psi }_{\text{OS}}^{-1}$, then N_cp is slightly larger than the max(N₁, N₂) (the ratios of N_cp/max(N₁, N₂) are at most about 1.29 in Table III). Further, when comparing late dependency (Clayton copula) and early one (Gumbel copula), if ${\psi }_{\text{TTP}}^{-1}$ is relatively close to ${\psi }_{\text{OS}}^{-1},N_{\text{cp}}^{(\text{nc})}$ increases proportionally to ρ^(j) in the late dependency, while $N_{\text{cp}}^{(\text{nc})}$ decreases or does not change as ρ^(j) varies in the early dependency. Similar tendencies but more moderate variation is observed for $N_{\text{cp}}^{(\mathrm{c})}$. That is, a high value of ρ^(j) makes the two log-rank statistics correlated, but higher ρ^(j) increases the censoring rate, so that the higher ρ^(j) does not contribute very much to the reduction of N_cp compared with the sample size when ρ^(j) = 0. Hence, N_cp increases in the late dependency, because many observations are censored before the dependent effect works.

Sample size behavior under correlated both fatal outcomes

Consider the both fatal outcomes scenario such that $T_{i1}^{\ast }$ is the time to disease-specific mortality and $T_{i2}^{\ast }$ is the time to other-cause mortalities. If there are such fatal endpoints, it is often assumed that their endpoints are uncorrelated for purposes of ease. But the assumption of no correlation is often unjustified scientifically. We investigate how the required sample sizes $N_{\text{cp}}^{(\mathrm{c})},N_{\text{cp}}^{(\text{nc})},N_1^{(\mathrm{c})},N_1^{(\text{nc})}$, and N₂ behave when the fatal endpoints are correlated. We compute the sample sizes using formula (8) and provide results in Table IV, displaying N_cp, N₁, and N₂ with the empirical power P̃_cp under ${\psi }_2^{-1}=1.7,S_1^{(1)}(\tau )=S_2^{(1)}(\tau )=0.5$, when ${\psi }_2^{-1}$ and ρ^(k) vary for the combinations of ${\psi }_2^{-1}=1.5$, 1.7 and ρ⁽¹⁾ = ρ⁽²⁾ = 0, 0.3, 0.5, and 0.8.

Table IV.

The case of both fatal outcomes (iii) (Competing risk): total numbers of participants N_cp calculated from (8), the corresponding empirical powers P̃_cp (%) and alternative sizing solutions N_sim, N₁ and N₂ under ${\psi }_2^{-1}=1.7$ and $S_1^{(1)}(\tau )=S_2^{(1)}(\tau )=0.5$.

Dependence			Composite setting				Non-composite setting
Structure	${\psi }_1^{-1}$	ρ(k)	$N_{\text{cp}}^{(\mathrm{c})}$	P̃cp	$N_1^{(\mathrm{c})}$	N2	$N_{\text{cp}}^{(\text{nc})}$	P̃cp	$N_1^{(\text{nc})}$	N2
Late (Clayton copula)	1.5	0	418	80.7	254	404	710	80.7	648	404
	1.5	0.3	474	80.8	294	456	834	80.6	774	456
	1.5	0.5	518	80.4	328	498	944	80.5	886	498
	1.5	0.8	626	80.7	414	592	1258	80.2	1211	592
	1.7	0	404	80.8	200	400	528	81.0	400	400
	1.7	0.3	466	80.8	232	462	608	81.0	462	462
	1.7	0.5	518	80.6	258	514	676	80.8	514	514
	1.7	0.8	652	80.7	324	646	850	80.7	646	646
Early (Gumbel copula)	1.5	0	418	80.8	254	404	710	80.8	648	404
	1.5	0.3	444	81.0	288	422	854	80.7	814	422
	1.5	0.5	452	80.9	312	422	1010	80.5	992	422
	1.5	0.8	436	81.0	360	364	1840	80.6	1840	364
	1.7	0	404	80.7	200	400	528	81.1	400	400
	1.7	0.3	458	80.8	228	454	598	81.3	454	454
	1.7	0.5	496	80.8	246	492	648	81.1	492	492
	1.7	0.8	566	81.0	282	562	740	81.2	562	562

The sample sizes N_cp from (8) usually provide slightly more conservative results from the perspective that the corresponding empirical powers P̃_cp are slightly larger than the target powers, similarly to the one fatal case. The value of $N_{\text{cp}}^{(\mathrm{c})}$ is smaller than $N_{\text{cp}}^{(\text{nc})}$ in any case, and $N_{\text{cp}}^{(\mathrm{c})},N_{\text{cp}}^{(\text{nc})},N_1^{(\mathrm{c})},N_1^{(\text{nc})}$ and N₂ usually increase proportionally to ρ^(j) in both of late and early dependencies, but $N_{\text{cp}}^{(\mathrm{c})}$ and N₂ decrease inversely only when $({\psi }_1^{-1},{\psi }_2^{-1})=(1.5,1.7)$, ρ^(j) = 0.8 and the Gumbel copula. That is, the stronger dependency makes the censoring rate increase more than the one fatal case, and hence, the reduction effect of N_cp based on correlated log-rank statistics are not obtained. On the other hand, the composite endpoint strategy using $N_1^{(\mathrm{c})}$ is the most reasonable in terms of smaller sample sizing. In particular, if ${\psi }_1^{-1}\approx {\psi }_2^{-1}$, we have $N_{\text{cp}}^{(\mathrm{c})}\approx N_2\approx 2N_1^{(\mathrm{c})}$.

3.3. Illustration: the ICON7 study

We illustrate the sample size methods with an example. Consider ‘A Randomized, Two-Arm, Multi-Centre Gynaecologic Cancer Inter Group Trial of Adding Bevacizumab to Standard Chemotherapy (Carboplatin and Paclitaxel) in Patients With Epithelial Ovarian Cancer’ (ICON7) [18]. The study was designed to investigate the addition of bevacizumab to standard chemotherapy for first-line treatment of woman with ovarian cancer. The primary endpoints of interest were the PFS and OS. The protocol stated that the number of PFS events, 684 was needed to detect a 28% change in the PFS from a median value of 18 months in the control group to 23 months in the bevacizumab group (i.e., ${\psi }_{\text{PFS}}^{-1}\doteq 1.28$), with the power of 90% at the significance level of 5% (a two-sided log-rank test), while the number of OS events, 715 was required to detect a 23% improvement in the OS from a median value of 43 months in the control group to 53 months in the bevacizumab group (i.e., ψ_OS = 1.23), with the power of 80% at a significance level of 5% (two-sided test). The protocol sample size of 1520 patients was determined for the reasons that the required numbers of PFS and OS events were expected to occur until 36 and 60 months after the first randomized, respectively, assuming constant recruitment over 24 months, and some elements of uncertainty are considered.

We set 60 months PFS and OS rates in the control group as $S_{\text{PFS}}^{(1)}(60)=0.1$ and $S_{\text{OS}}^{(1)}(60)=0.4$ based on the aforementioned information, which is taking account of hazard rates ${\lambda }_{\text{PFS}}^{(1)}=-\frac{1}{18}log(0.5)$ and ${\lambda }_{\text{OS}}^{(1)}=-\frac{1}{43}log(0.5)$ from the protocol and the exponential assumptions, and its uncertainty. Also, we derive the HR for TTP, because our calculation is based on the TTP and OS rather than the PFS and OS. That is, the HR and survival rate for TTP can be computed as ${\psi }_{\text{TTP}}^{-1}={\lambda }_{\text{TTP}}^{(1)}/{\lambda }_{\text{TTP}}^{(2)}\doteq 1.31$ and $S_{\text{TTP}}^{(1)}(60)\doteq 0.25$ from the information on medians based on

$$e^{-{\lambda }_{\text{TTP}}^{(j)}t}=S_{\text{TTP}}^{(j)}(t)=S_{\text{PFS}}^{(j)}(t)/S_{\text{OS}}^{(j)}(t)=e^{-{\lambda }_{\text{PFS}}^{(j)}t}/e^{-{\lambda }_{\text{OS}}^{(j)}t},j=1,2$$

from the independence and exponential assumptions. These values of the HR and $S_{\text{TTP}}^{(1)}$ are used for an illustration although the independence assumption is suspect.

Table V shows the total sample size N_cp, and the empirical powers P̃_cp, P̃_TTP, P̃_PFS, and P̃_OS (%) to evaluate the joint and single hypotheses ( ${\mathrm{H}}_1^{\text{cp}}$ and H_1j) of TTP, PFS, and OS given N_cp, respectively, under ${\psi }_{\text{TTP}}^{-1}=1.31$ and ${\psi }_{\text{OS}}^{-1}=1.23$. Alternative sizing of NTTP, NPFS, and NOS gives the sample sizes required to test the single hypotheses of the TTP, PFS, and OS, respectively. The sample sizes were calculated to evaluate the joint reduction in both time-to-event outcomes of the TTP and OS or the PFS and OS with the target power of 80% at the significance level of 2.5%, based on the assumptions of the ICON7 study, assuming τ_a = 24 months accrual duration and additional τ_f = 36 months follow-up. The empirical power P̃_PFS is larger than the target power of 90% to test the PFS written in the protocol, because the analysis times in the protocol design (τ_f = 12) and ours (τ_f = 36) are different. Similarly, note that N_PFS is calculated under the target power of 80% smaller than the power 90% in the protocol. Also, the expected numbers of bivariate events are listed using the notation corresponding to D_𝚤𝚥 = N_cp × P_𝚤𝚥 in which N_cp is replaced by $N_{\text{cp}}^{(\mathrm{c})}$ or $N_{\text{cp}}^{(\text{nc})}$, and P_𝚤𝚥 is the probability ( $P_{𝚤𝚥}^{\text{TTP}}$: see Appendix B) calculated under the non-composite setting. Note that, even in the composite setting with the PFS and OS, D_𝚤𝚥 represents the number that either or both of the TTP and OS are observed. In addition, the correlation is assumed to be common between the two groups, that is, ρ⁽¹⁾ = ρ⁽²⁾.

Table V.

Total sample size N_cp and the empirical power for detecting the joint reduction in the TTP or PFS and the OS, where N_cp is designed with the target power 80% at α = 0.025, based on the assumption of ICON7 study, ${\psi }_{\text{TTP}}^{-1}=1.31,{\psi }_{\text{OS}}^{-1}=1.23$, τ_a = 24 and τ_f = 36 with $S_{\text{TTP}}^{(1)}(60)=0.25$ and $S_{\text{OS}}^{(1)}(60)=0.4$.

Dependence		Composite setting				Expected number of (TTP,OS)
Structure	ρ(k)	$N_{\text{cp}}^{(c)}$	P̃cp (P̃PFS, P̃OS)	NPFS	NOS	D11	D10	D01	D00
Late (Clayton copula)	0.0	1510	80.1 (99.0,80.3)	658	1498	240	476	487	306
	0.3	1528	80.0 (97.4,80.7)	789	1498	275	429	461	363
	0.5	1544	80.0 (96.0,81.1)	881	1498	302	395	441	405
	0.8	1564	80.0 (93.4,81.7)	1024	1498	370	316	383	495
Early (Gumbel copula)	0.0	1510	80.3 (99.0,80.6)	658	1498	240	476	487	306
	0.3	1508	80.1 (98.5,80.3)	700	1498	315	384	411	398
	0.5	1506	80.2 (98.2,80.4)	726	1498	374	327	351	454
	0.8	1498	80.0 (97.9,80.1)	744	1498	510	243	211	533
Dependence		Non-composite setting				Expected number of (TTP,OS)
Structure	ρ (k)	$N_{\text{cp}}^{(nc)}$	P̃cp (P̃TTP, P̃OS)	NTTP	NOS	D11	D10	D01	D00
Late (Clayton copula)	0.0	1628	80.3 (96.3,83.3)	913	1498	259	514	525	330
	0.3	1674	80.2 (94.9,84.2)	1023	1498	301	470	505	397
	0.5	1658	80.2 (94.1,84.7)	1076	1498	332	434	484	445
	0.8	1658	80.3 (94.1,84.1)	1056	1498	392	335	406	525
Early (Gumbel copula)	0.0	1628	80.2 (96.4,83.3)	913	1498	259	514	525	330
	0.3	1594	80.1 (96.7,82.3)	878	1498	333	406	435	420
	0.5	1562	80.2 (97.0,81.8)	830	1498	388	339	364	471
	0.8	1510	80.0 (98.6,80.2)	695	1498	514	245	213	538

P̃_TTP, P̃_PFS, and P̃_OS are 100 times empirical powers (%) when single hypotheses on the TTP, PFS and OS are tested, given total sample sizes N_cp.

In calculating the sample size, we assume that one event is fatal because the TTP may be censored by the OS of fatal event. If the association between the TTP and OS is late-time dependent, then the total sample sizes $N_{\text{cp}}^{(\mathrm{c})}$ required to test the PFS and OS jointly with common correlation between the two groups, ρ^(k) = 0.0, 0.3, 0.5, and 0.8 are 1510, 1528, 1544, and 1564, respectively. The sample size increases monotonically to ρ^(k) = 0.8 from 0: the difference between the smallest and largest sample sizes may seem relatively large, although the ratio is only about 1.04. If the association is early-time dependent, then the total sample sizes $N_{\text{cp}}^{(\mathrm{c})}$ required under ρ^(k) = 0.0, 0.3, 0.5 and 0.8 are 1510, 1508, 1506, and 1498, respectively. The sample size decreases with increasing correlation, but the reduction rate from the largest sample size given by ρ^(k) = 0.0 is quite small. Also, comparing the composite and non-composite endpoints, $N_{\text{cp}}^{(\mathrm{c})}$ is smaller than the $N_{\text{cp}}^{(\text{nc})}$ required to test the TTP and OS jointly, but the difference is slight. The expected numbers of bivariate events, D_𝚤𝚥 provide useful information in monitoring process of the trial.

4. Multiple primary endpoints

4.1. Hypothesis testing, power, and sample sizes

We discuss calculating the required sample size for trials with multiple primary endpoints. For simplicity, we will consider the problem on the power and sample size calculation using the simplest procedures, the (weighted) Bonferroni procedure as well known and widely used. The other procedures for controlling the Type I error rate are available [6, 19, 20].

The weighted Bonferroni procedure allocates the Type I error rate α between the endpoints with weight ω, that is, α₁ = ωα for the first endpoint and α₂ = (1−ω)α for the second endpoint. We are then interested in testing ${\mathrm{H}}_0^{\text{mp}}=\{{\mathrm{H}}_{01}:{\psi }_1(t)=1\text{and}{\mathrm{H}}_{02}:{\psi }_2(t)=1\}\text{for}\text{all}t\}$ versus ${\mathrm{H}}_1^{\text{mp}}=\{{\mathrm{H}}_{11}:{\psi }_1(t)<1\text{or}{\mathrm{H}}_{12}:{\psi }_2(t)<1\text{at}\text{some}t\}$ at the (overall) significance level of 3, based on the log-rank test statistics $Z_1^{(0)}$ and $Z_2^{(0)}$ given in Section 2. The testing procedure is to

$$\text{reject}\text{the}\text{null}\text{hypothesis}{\mathrm{H}}_0^{\text{mp}}\text{if}\text{and}\text{only}\text{if}{Z}_1^{(0)}<-{\mathrm{z}}_{{\alpha }_1}\text{or}{\mathrm{Z}}_2^{(0)}<-{\mathrm{z}}_{{\alpha }_2},$$ (9)

where z_α1 and z_α2 are the 100(1 − α₁) and 100(1 − α₂) percentiles of N(0, 1), respectively. Therefore, because the rejection region of ${\mathrm{H}}_0^{\text{mp}}$ is $\left\{\left(Z_1^{(0)},Z_2^{(0)}\right):Z_1^{(0)}<-{\mathrm{z}}_{{\alpha }_1}\text{or}{Z}_2^{(0)}<-{\mathrm{z}}_{{\alpha }_2}\right\}$, we have the power function for a reduction in either of the time-to-event outcomes with the weighted Bonferroni procedure,

$$1-\beta =Pr\left[{\cup }_{k=1}^2\{Z_k^{(0)}<-{\mathrm{z}}_{{\alpha }_k}\}\mid {\mathrm{H}}_1^{\text{mp}}\right]=1-Pr\left[{\cap }_{k=1}^2\{Z_k^{(0)}\ge -{\mathrm{z}}_{{\alpha }_k}\}\mid {\mathrm{H}}_1^{\text{mp}}\right]$$

This overall power is referred to as ‘minimal power’[12] or ‘disjunctive power’[13]. Similarly to Section 3.1, for all of the three censoring scenarios, the power based on the function (5) can be approximated by

$$1-\beta \approx 1-{\int }_{-z_{{\alpha }_1}}^{\infty }{\int }_{-z_{{\alpha }_2}}^{\infty }\varphi \left(z_1,z_2;\sqrt{N}\mathit{\delta },\mathbf{\sum }\right)\mathrm{d}{z}_1\mathrm{d}{z}_2.$$ (10)

Letting N_mp be the minimum of total sample size N required for testing ${\mathrm{H}}_0^{\text{mp}}$ against ${\mathrm{H}}_1^{\text{mp}}$, the formula for N_mp is given by

$$N_{\text{mp}}=\lceil {\left(L_{\beta }-z_{{\alpha }_2}{\sigma }_2^{(0)}/{\sigma }_2\right)}^2/{\delta }_2^2\rceil ,$$ (11)

where L_β is the solution of the integral equation

$$\beta ={\int }_{-\infty }^{\frac{{\delta }_1}{{\delta }_2}{L}_{\beta }+\left\{{\mathrm{z}}_{{\alpha }_1}\frac{{\sigma }_1^{(0)}}{{\sigma }_1}-{\mathrm{z}}_{{\alpha }_2}\frac{{\delta }_1}{{\delta }_2}\frac{{\sigma }_2^{(0)}}{{\sigma }_2}\right\}}{\int }_{-\infty }^{L_{\beta }}\varphi (z_1,z_2;\mathbf{0},\mathbf{R})\mathrm{d}{z}_1\mathrm{d}{z}_2$$

and ${\sigma }_k^{(0)}$, σ_k, and R are the same as the definitions given in Section 3.1. Similarly to the methods to obtain N_cp, we can use a search method of L_β such as the Newton–Raphson algorithm [14] or the basic linear interpolation algorithm [2] in order to compute N_mp, which generally take shorter computing time than direct search that let N_mp increase sequentially until (10) exceeds the desired power. Also, similarly to Section 3.1, the required numbers of events that are useful to monitor a trial can be obtained by D_𝚤𝚥 = N_mp × P_𝚤𝚥 (see Appendix B for details regarding calculation of P_𝚤𝚥 under the three censoring scenarios).

The overall Type I error associated with the null ${\mathrm{H}}_0^{\text{mp}}$ is controlled by the sum of marginal Type I errors. Similarly to Section 3, the inflation problem of the overall Type I error in the procedure (6) based on the asymptotic normal approximation is parallel to that of the marginal ones. The overall significance level α₁ + α₂ on ${\mathrm{H}}_0^{\text{mp}}$ is quite close to the level α₁ + α₂ − α₁α₂ considered when two endpoints are independent, so that the influence on marginal inflations may be practically larger in multiple primary problem than in multiple co-primary one. However, we can control the errors by correcting the critical values based on the sample size N and the allocation rate r⁽¹⁾, using simulation or theoretical result well known in the univariate log-rank statistic [9–11]. The practical need for sample size formula is a balanced weighting of the precision and cost, and then the use of the formula based on the asymptotic normal approximation is left in each stage of practice.

4.2. Illustration: one fatal case

We illustrate the behavior of the sample size and power for detecting at least one reduction ${\mathrm{H}}_1^{\text{mp}}$ in bivariate time-to-event data focusing on the one fatal outcome scenario with the two time-dependent association structures. Similarly to Section 3.2, we generate bivariate time-to-event data and perform Monte Carlo trials. The target power 1 − β = 0.8, the significance level α = 0.025, τ_a = 2, and τ_f = 3 are used. The notations of N_EP, ψ_EP, and $S_{\text{EP}}^{(j)}(t)$ for EP = TTP, PFS, and OS are the same as those in Section 3. Also, N_mp is written as $N_{\text{mp}}^{(\text{nc})}$ when the first endpoint is the TTP, or as $N_{\text{mp}}^{(\mathrm{c})}$ when the first endpoint is the PFS. Let P̃_mp denote the empirical power (%) for detecting ${\mathrm{H}}_1^{\text{mp}}$ under N_mp participants designed using the formula (11). Further, we select ω = ω₀ as an optimal value of ω to consider a variable weighting strategy for the weighted Bonferroni procedure, that is, ω₀ is ω that gives a minimum of N_mp over ω = 0, 0.05, …, 0.95, 1 (ω₀ = argmin_{ω∈{0,0.05,…,0.95,1}}N_mp). One may consider the other testing procedures such as the fixed-sequence procedure when ω₀ = 0 or ω₀ = 1 is suggested.

Table VI displays the required total sample sizes N_mp (given ω = ω₀), the alternative sizing solutions N_PFS, N_TTP, and N_OS, the selected Bonferroni weight ω₀ and the empirical power P̃_mp under ${\psi }_{\text{TTP}}^{-1}=1.7,S_{\text{TTP}}^{(1)}(\tau )=0.3$, and $S_{\text{OS}}^{(1)}(\tau )=0.5$ when ${\psi }_{\text{OS}}^{-1}$ and ρ^(k) vary in the combinations of ${\psi }_{\text{OS}}^{-1}=1.3$, 1.5, 1.7 and ρ⁽¹⁾ = ρ⁽²⁾ = 0, 0.3, 0.5, 0.8.

Table VI.

The case of one fatal outcome (ii) (semi-competing risk): total numbers of participants N_mp calculated from (11), the corresponding empirical powers P̃_mp (%) and alternative sizing solutions N_PFS, N_TTP, and N_OS under ${\psi }_{\text{TTP}}^{-1}=1.7,S_{\text{TTP}}^{(1)}(\tau )=0.3$ and $S_{\text{OS}}^{(1)}(\tau )=0.5$.

Dependence			Composite setting					Non-composite setting
structure	${\psi }_{\text{OS}}^{-1}$	ρ(k)	ω 0	$N_{\text{mp}}^{(\mathrm{c})}$	P̃mp	NPFS	NOS	ω0	$N_{\text{mp}}^{(\text{nc})}$	P̃mp	NTTP	NOS
Late (Clayton copula)	1.3	0.0	1.00	254	80.7	254	1194	0.95	268	80.8	270	1194
	1.3	0.3	1.00	284	80.5	284	1194	0.95	288	81.0	288	1194
	1.3	0.5	1.00	322	80.9	322	1194	0.95	296	80.5	298	1194
	1.3	0.8	1.00	366	80.8	366	1194	1.00	288	80.7	288	1194
	1.5	0.0	1.00	200	80.9	200	532	0.80	242	80.2	266	532
	1.5	0.3	1.00	232	80.7	232	532	0.80	266	80.4	292	532
	1.5	0.5	1.00	256	80.5	256	532	0.80	282	80.5	306	532
	1.5	0.8	1.00	300	80.9	300	532	0.80	296	80.5	312	532
	1.7	0.0	1.00	168	81.1	168	328	0.55	208	80.4	264	328
	1.7	0.3	1.00	196	80.9	196	328	0.55	228	80.7	296	328
	1.7	0.5	0.95	218	80.6	218	328	0.55	242	80.7	314	328
	1.7	0.8	0.95	260	80.8	262	328	0.50	264	80.7	332	328
Early (Gumbel copula)	1.3	0.0	1.00	254	80.7	254	1194	0.95	268	80.9	270	1194
	1.3	0.3	1.00	266	80.6	266	1194	1.00	246	81.0	246	1194
	1.3	0.5	1.00	272	80.6	272	1194	1.00	222	80.8	222	1194
	1.3	0.8	1.00	258	80.3	258	1194	1.00	174	81.0	174	1194
	1.5	0.0	1.00	200	80.7	200	532	0.80	242	80.5	266	532
	1.5	0.3	1.00	216	81.0	216	532	0.90	250	80.5	260	532
	1.5	0.5	1.00	224	80.7	224	532	0.95	246	80.6	248	532
	1.5	0.8	1.00	230	80.9	230	532	1.00	214	81.0	214	532
	1.7	0.0	1.00	168	81.3	168	328	0.55	208	80.4	264	328
	1.7	0.3	1.00	186	81.0	186	328	0.60	228	80.3	272	328
	1.7	0.5	1.00	196	80.7	196	328	0.65	240	80.3	280	328
	1.7	0.8	1.00	214	80.7	214	328	1.00	248	81.0	248	328

The sample sizes N_mp obtained from the formula (11) consistently provide conservative results because the empirical powers P̃_mp are slightly larger than the target powers. Although $N_{\text{mp}}^{(\mathrm{c})}<N_{\text{mp}}^{(\text{nc})}$ in many cases, we observe $N_{\text{mp}}^{(\mathrm{c})}>N_{\text{mp}}^{(\text{nc})}$ in some cases where N_TTP is relatively close to N_PFS or smaller than N_PFS, which occurs when ρ^(j) is relatively large and the effect size of OS (absolute value of $log{\psi }_{\text{OS}}^{-1}$) is smaller than that of TTP. Also, if the effect size of OS is smaller than that of TTP, then N_TTP ≪ N_OS or N_PFS ≪ N_OS occurs, so that we have $N_{\text{mp}}^{(\text{nc})}\approx N_{\text{TTP}}$ or $N_{\text{mp}}^{(\mathrm{c})}\approx N_{\text{PFS}}$ accompanied with the value of ω₀ close to 1. When N_TTP (or N_PFS) is closer to N_OS, ω₀ takes a value between 0 and 1.

Further, we add a summary about the tendency how ω₀-value ranges from 0 to 1 in Section B.2 of Supporting Information, based on the underlying results by varying S_TTP(τ) and ρ^(k) leaving the other factors fixed, whose parts are provided in Figures B.4, B.5, B.6, and B.7. We provide a guideline based on the summary about ω₀-value: generally, the correlation ρ^(k) and the dependence structure are unknown, so that these should be examined using meta analysis and/or data from a pilot study, similarly to consideration for the HRs of effect sizes. Although it may be desirable to consider a maximum sample size considering such unknown factors, it is necessary to balance on a degree of uncertainty and the cost. When the TTP and OS are used as two primary endpoints, ω₀ ranges from 0 to 1, but ω₀ is approximately 0.5 when N_TTP/N_OS is close to 1. When the PFS and OS are used as two primary endpoints, the ω₀-value is usually close to 1 under early dependency and under a late dependency with not high ρ^(k), if ${\psi }_{\text{PFS}}^{-1}\ge {\psi }_{\text{OS}}^{-1}$. The value of ω₀ may be away from 0 and 1 otherwise. These considerations about ω₀-value suggest that the use of PFS endpoint is one of the reasonable strategies in the multiple primary problem for the TTP and OS if ${\psi }_{\text{TTP}}^{-1}\ge {\psi }_{\text{OS}}^{-1}$, although the PFS endpoint may be used without consideration in practice. Hence, it is useful to incorporate ω₀ in the sample size calculation proposed for multiple primary endpoints.

4.3. Behavior of the sample size as a function of the correlation

We extend the study to the both non-fatal and both fatal cases. We use the same setting as in Section 4.2 (1 − β = 0.8, α = 0.025, τ_a = 2, and τ_f = 3), while the notations of N_k, ψ_k, and $S_k^{(j)}(t)$ for the kth endpoint are used because of the consistent expression of the results from the three situations. Similarly to Section 4.2, we select a minimum of N_mp over ω = 0, 0.05, …, 0.95, 1 via a strategy for the weighted Bonferroni procedure, $N_{\text{mp}}^{(\mathrm{c})}$ is N_mp calculated under the composite setting, and $N_{\text{mp}}^{(\text{nc})}$ is N_mp calculated under the non-composite setting.

Figure 1 shows the required total sample sizes N_mp as a function of ρ⁽¹⁾ = ρ⁽²⁾ = 0, 0.05, …, 0.95 when ${\psi }_2^{-1}$ varies from 1.3, 1.5 and 1.7 under ${\psi }_1^{-1}=1.7,S_1^{(1)}(\tau )=0.3$ and the early dependency (Gumbel copula). The 12 plots are arranged from the left for $S_2^{(1)}(\tau )=0.4$, 0.5 and 0.6, and from the top for the four scenarios: the both non-fatal case without the composite setting ( $N_{\text{cp}}^{(\text{nc})}$), the one fatal case without and with the composite setting ( $N_{\text{mp}}^{(\text{nc})}$ and $N_{\text{mp}}^{(\mathrm{c})}$), and the both fatal case with the composite setting ( $N_{\text{mp}}^{(\mathrm{c})}$). The plots for the late dependency case are provided in Figure B.3 in Section B.1 of Supporting Information (generated under the Clayton copula (late dependency) using the same conditions as Figure 1 except the copula model).

Figure 1.

Behavior of the total sample sizes N_mp as a function of the correlation ρ^(j) for ${\psi }_2^{-1}=1.3$, 1.5 and 1.7, arranged from the left for $S_2^{(1)}(\tau )=0.4$, 0.5, 0.6 and from the top following to $N_{\text{mp}}^{(\text{nc})}$ from the both non-fatal case, $N_{\text{mp}}^{(\text{nc})}$ and $N_{\text{mp}}^{(\mathrm{c})}$ of one fatal case and $N_{\text{mp}}^{(\mathrm{c})}$ of both fatal case, given 1 − β = 0.8, α = 0.025, τ_a = 2, τ_f = 3, ${\psi }_1^{-1}=1.7,S_1^{(1)}(\tau )=0.3$ and early dependency

As ρ^(j) increases, the behavior of $N_{\text{mp}}^{(\text{nc})}$ is monotone increasing up to a constant value (univariate sample size) in the both non-fatal case but is more complicated in the one fatal case. In the one fatal case, there is a complicated interaction between the effect size and the correlation, where a smaller sample size is required with increasing effect size and decreasing correlation, while the censoring rate for non-fatal events increases with increasing correlation. Consider the case of ρ^(j) = 1 as a standard, because the sample size N_mp required in bivariate primary endpoints reduces to that of the single endpoint in the optimal weighting Bonferroni strategy when the correlations ρ^(j) are 1. In the one fatal case with the composite setting, the complicated behavior is moderated, and $N_{\text{mp}}^{(\mathrm{c})}$ tends to decrease as the correlation decreases, but $N_{\text{mp}}^{(\mathrm{c})}$ is sometimes larger than $N_{\text{mp}}^{(\text{nc})}$. The behavior of $N_{\text{mp}}^{(\mathrm{c})}$ in the both fatal case with the composite setting is similar to that of the one fatal case.

5. Summary

Utilizing multiple endpoints in clinical trials may provide the opportunity for characterizing intervention’s multidimensional effects but also creates challenges in design and analysis of clinical trials. Specifically controlling the Types I and II error rates is non-trivial when the multiple primary endpoints are potentially correlated. When designing the trial to detect effects for all of the endpoints, no adjustment is needed to control the Type I error. However, the Type II error increases as the number of endpoints being evaluated increases. In contrast, when designing the trial to detect an effect for at least one of the endpoints, then an adjustment is needed to control the Type I error.

We describe an approach to the evaluation of power and sample size for comparing the effect of two interventions in superiority clinical trials with two time-to-event outcomes for multiple co-primary and multiple primary cases. Designing clinical trials with multiple time-to-event outcomes is more complex compared with endpoints with the other scales, requiring censoring scheme challenges and time-dependent associations among outcomes. We consider the three censoring scenarios based on the types of outcomes: (i) both outcomes are non-fatal; (ii) one outcome is fatal; and (iii) both outcomes are fatal, and we evaluate their composite and non-composite settings. We discuss the two time-dependent association structures: the asymmetric late time-dependency generated by the Clayton copula and the early time-dependency generated by the Gumbel copula. Our findings are summarized as follows.

• In the co-primary endpoint situation, if the two time-to-event outcomes are non-fatal, then the required sample size $N_{\text{cp}}^{(\text{nc})}$ decreases with increasing correlations ρ^(j), in both of the late and early time-dependencies, except for the case where one HR is larger than the other. When the correlations are zero, $N_{\text{cp}}^{(\text{nc})}$ is the largest. However, when one or both outcomes are non-fatal, the behaviors of required sample sizes $N_{\text{cp}}^{(\text{nc})}$ and $N_{\text{cp}}^{(\mathrm{c})}$ have complicated shapes owing to an interaction between the correlations and effect sizes based on the HRs. Zero correlation does not provide the largest required sample size. Thus, careful consideration is required in practice.

• In the multiple primary endpoint situation based on the optimal weighting Bonferroni strategy, a standard situation occurs when the correlations ρ^(j) are one, corresponding to the single endpoint case. Considering the two non-fatal case, the sample size $N_{\text{mp}}^{(\text{nc})}$ does not vary or increases monotonously up to a constant (univariate sample size) as a function of the correlations, in both of the late and early time-dependencies. However, when one or both outcomes are non-fatal, the behaviors of the required sample sizes $N_{\text{mp}}^{(\text{nc})}$ and $N_{\text{mp}}^{(\mathrm{c})}$ are complex with an interaction between the correlations and the effect size ratios. Thus, the proposed formula is useful for determining the sample size in practice, noting, when the sample size is small, for example, < 100, one may have to modify a critical value based on the normal approximation.

• Unlike the both non-fatal case, larger correlations do not increase the statistical power in the one fatal or both fatal cases. The reason for this is that higher correlation leads to increased censoring. Hence, the standard log-rank statistic must be corrected by incorporating information from informative censoring. We focus on providing a foundation for designing clinical trials with two time-to-event outcomes. Informative censoring is a topic for future work.

When designing a clinical trial with the proposed methods, one needs parameter estimates using the available methods ([7, 21–25, 27, 28]). But specification for the joint distribution is challenging as data to base selection is often limited during trial design. One conservatively alternative is to select the largest sample size of all of the correlation and joint distribution combinations, and stop the clinical trial when the appropriate number of events required for each outcome is observed. Another option is to use group-sequential designs. This may lead to potentially fewer patients than the fixed-sample designs when evidence is overwhelming and thus offers efficiency but introduces the other challenges. Information for the endpoints may not accrue at the same rate and require different information times.

As discussed in [29] and [26], the Type I error rate of the log-rank test for each endpoint may be inflated in small sample sizes or with unequally sized intervention groups. When this occurs in the co-primary endpoint situation, our simulation studies suggest that the overall Type I error associated with the null hypothesis ${\mathrm{H}}_0^{\text{cp}}$ is not larger than the target significance level, except when the correlation is very high (i.e., close to one), even though the marginal Type I error rate is inflated. On the other hand with multiple primary endpoints, the overall Type I error associated with the null hypothesis ${\mathrm{H}}_0^{\text{mp}}$ is larger than the target significance level, particularly when the correlation is small. In these cases, we may consider more direct ways of calculating sample size without using a normal approximation such as the methods in [29] and [26].

APPEENDIX A. Asymptotic forms in the bivariate log-rank statistic

We provide the details of asymptotic forms of the bivariate log-rank statistic discussed in Section 2.3. See [3] for the case when both endpoints are non-fatal. We obtain the covariance process between two incremental differences $\mathrm{d}{M}_{ik}(t)=\mathrm{d}{\mathcal{N}}_{ik}(t)-{\mathcal{Y}}_{ik}(t)\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_k^{(g_i)}(t)$, k = 1, 2, where M_i1(t) and M_i2(t) are martingale processes relative to the filtration generated by the history prior to time t under the null hypothesis. This corresponds to calculating the expectation of

$$\begin{array}{l}\mathrm{d}{M}_{i1}(t)\mathrm{d}{M}_{i2}(s)=\mathrm{d}{\mathcal{N}}_{i1}(t)\mathrm{d}{\mathcal{N}}_{i2}(s)-{\mathcal{Y}}_{i1}(t)\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_1^{(g_i)}(t)\mathrm{d}{\mathcal{N}}_{i2}(s)\\ -{\mathcal{Y}}_{i2}(s)\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_2^{(g_i)}(s)\mathrm{d}{\mathcal{N}}_{i1}(t)+{\mathcal{Y}}_{i1}(t){\mathcal{Y}}_{i2}(s)\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_1^{(g_i)}(t)\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_2^{(g_i)}(s).\end{array}$$ (A.1)

One fatal case

Consider the non-composite setting with the censoring indicator ${\mathrm{\Delta }}_{ik}=𝟙\left(T_{ik}=T_{ik}^{\ast }\right)$, k = 1, 2. Because $T_{i1}=min\left(T_{i1}^{\ast },T_{i2}^{\ast },C_i\right)$ and $T_{i2}=min\left(T_{i2}^{\ast },C_i\right)$, the expectations of the ith at-risk processes given g_i = j, that is, $y_k^{(j)}(t)$, k = 1, 2 are

$$y_1^{(j)}(t)=\mathrm{E}[{\mathcal{Y}}_{i1}(t)\mid g_i=j]=Pr\left(T_{i1}^{\ast }\ge t,T_{i2}^{\ast }\ge t,C_i\ge t\mid g_i=j\right)=S^{(j)}(t,t)G(t)$$

for the non-fatal endpoint and

$$y_2^{(j)}(t)=\mathrm{E}[{\mathcal{Y}}_{i2}(t)\mid g_i=j]=Pr(T_{i2}^{\ast }\ge t,C_i\ge t\mid g_i=j)=S_2^{(j)}(t)G(t)$$

for the fatal endpoint (the latter form is identical to that of both non-fatal case). So each element in Ĥ_k(t) converges to the corresponding expectation almost surely by Glivenko–Cantelli’s theorem under some regular conditions, so that we have the asymptotic forms of Ĥ_k(t), k = 1, 2 as

$$H_k(t)=N^{-1}\frac{\mathrm{E}\left[{\mathcal{Y}}_k^{(1)}(t)\right]\mathrm{E}\left[{\mathcal{Y}}_k^{(2)}(t)\right]}{\mathrm{E}\left[{\mathcal{Y}}_k^{(1)}(t)\right]+\mathrm{E}\left[{\mathcal{Y}}_k^{(2)}(t)\right]}=\{\begin{array}{ll}G(t)\frac{r^{(1)}{r}^{(2)}{S}^{(1)}(t,t)S^{(2)}(t,t)}{r^{(1)}{S}^{(1)}(t,t)+r^{(2)}{S}^{(2)}(t,t)},\hfill & k=1,\hfill \\ G(t)\frac{r^{(1)}{r}^{(2)}{S}_2^{(1)}(t)S_2^{(2)}(t)}{r^{(1)}{S}_2^{(1)}(t)+r^{(2)}{S}_2^{(2)}(t)},\hfill & k=2.\hfill \end{array}$$

Similarly, the conditional expectations of the ith counting processes given 𝒴_ik(t) = 1 are

$$\begin{array}{l}\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_1^{(g_i)}(t)=\mathrm{E}[\mathrm{d}{\mathcal{N}}_{i1}(t)\mid {\mathcal{Y}}_{i1}(t)=1.g_i]\\ =\frac{Pr(t\le T_{i1}^{\ast }<t+\mathrm{d}t,t\le T_{i2}^{\ast },t\le C_i\mid g_i)}{Pr({\mathcal{Y}}_{i1}(t)=1\mid g_i)}=-\frac{S^{(g_i)}(\mathrm{d}t,t)}{S^{(g_i)}(t,t)}={\mathrm{\Lambda }}^{(g_i)}(\mathrm{d}t,t)\end{array}$$

for the non-fatal endpoint and $\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_2^{(g_i)}(t)=\mathrm{E}[\mathrm{d}{\mathcal{N}}_{i2}(t)\mid {\mathcal{Y}}_{i2}(t)=1,g_i]=\mathrm{d}{\mathrm{\Lambda }}_2^{(g_i)}(t)$ for the fatal endpoint. Hence, the expectation of (A.1) is

$$\begin{array}{l}\mathrm{E}[\mathrm{d}{M}_{i1}(t)\mathrm{d}{M}_{i2}(s)\mid g_i]=Pr({\mathcal{Y}}_{i1}(t){\mathcal{Y}}_{i2}(s)=1\mid g_i)\mathrm{E}[\mathrm{d}{M}_{i1}(t)\mathrm{d}{M}_{i2}(s)\mid {\mathcal{Y}}_{i1}(t){\mathcal{Y}}_{i2}(s)=1,g_i]\\ =G(t\vee s)\{𝟙(t\le s)S^{(g_i)}(\mathrm{d}t,\mathrm{d}s)+𝟙(t\le s)S^{(g_i)}(t,\mathrm{d}s){\mathrm{\Lambda }}^{(g_i)}(\mathrm{d}t,t)+S^{(g_i)}(\mathrm{d}t,t\vee s)\mathrm{d}{\mathrm{\Lambda }}_2^{(g_i)}(s)+S^{(g_i)}(t,t\vee s){\mathrm{\Lambda }}^{(g_i)}(\mathrm{d}t,t)\mathrm{d}{\mathrm{\Lambda }}_2^{(g_i)}(s)\},\end{array}$$ (A.2)

because, noting that T_i1 ≤ T_i2 always holds, we have

$$\begin{array}{ll}\hfill Pr({\mathcal{Y}}_{i1}(t){\mathcal{Y}}_{i2}(s)=1\mid g_i)& =Pr(t\le T_{i1}^{\ast },t\vee s\le T_{i2}^{\ast },t\vee s\le C_i\mid g_i)=G(t\vee s)S^{(g_i)}(t,t\vee s),\hfill \\ \hfill \mathrm{E}[\mathrm{d}{\mathcal{N}}_{i1}(t)\mathrm{d}{\mathcal{N}}_{i2}(s)\mid {\mathcal{Y}}_{i1}(t){\mathcal{Y}}_{i2}(s)=1,g_i]& =\frac{Pr(t\le T_{i1}^{\ast }<t+\mathrm{d}t,s\le T_{i2}^{\ast }<s+\mathrm{d}s\mid g_i)}{Pr(t\le T_{i1}^{\ast },t\vee s\le T_{i2}^{\ast }\mid g_i)}=𝟙(t\le s)\frac{S^{(g_i)}(\mathrm{d}t,\mathrm{d}s)}{S^{(g_i)}(t,t\vee s)},\hfill \\ \hfill \mathrm{E}[{\mathcal{Y}}_{i1}(t)\mathrm{d}{\mathcal{N}}_{i2}(s)\mid {\mathcal{Y}}_{i1}(t){\mathcal{Y}}_{i2}(s)=1,g_i]& =\frac{Pr(t\le T_{i1}^{\ast },s\le T_{i2}^{\ast }<s+\mathrm{d}s,t\vee s\le T_{i2}^{\ast }\mid g_i)}{Pr(t\le T_{i1}^{\ast },t\vee s\le T_{i2}^{\ast }\mid g_i)}=-𝟙(t\le s)\frac{S^{(g_i)}(t,\mathrm{d}s)}{S^{(g_i)}(t,t,\vee s)}\text{and}\hfill \\ \hfill \mathrm{E}[\mathrm{d}{\mathcal{N}}_{i1}(t){\mathcal{Y}}_{i2}(s)\mid {\mathcal{Y}}_{i1}(t){\mathcal{Y}}_{i2}(s)=1,g_i]& =\frac{Pr(t\le T_{i1}^{\ast }<t+\mathrm{d}t,t\vee s\le T_{i2}^{\ast }\mid g_i)}{Pr(t\le T_{i1}^{\ast },t\vee s\le T_{i2}^{\ast }\mid g_i)}=-\frac{S^{(g_i)}(\mathrm{d}t,t\vee s)}{S^{(g_i)}(t,t\vee s)}.\hfill \end{array}$$

Therefore, we have the form of dA^(j)(t, s) = G(t ∨ s)⁻¹E[dM_i1(t)dM_i2(s) | g_i = j] given in (3a) of Section 2.3.

Next, assume the composite setting ${\mathrm{\Delta }}_{i1}=𝟙(T_{i1}=min(T_{i1}^{\ast },T_{i2}^{\ast }))$ and ${\mathrm{\Delta }}_{i2}=𝟙(T_{i2}=T_{i2}^{\ast })$. That is, the definition of d𝒩_i1(t) is only changed from the non-composite version. Because observing the 2nd endpoint is composed in the ith counting processes 𝒩_i1(t) for the first endpoint, the conditional expectations of d𝒩_i1(t) given 𝒴_i1(t) = 1 is

$$\begin{array}{l}\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_1^{(g_i)}(t)=\mathrm{E}[\mathrm{d}{\mathcal{N}}_{i1}(t)\mid {\mathcal{Y}}_{i1}(t)=1,g_i]=\frac{{\sum }_{k=1,2}Pr(t\le T_{ik}^{\ast }<t+\mathrm{d}t,t\le T_{i{k}^{\prime }}^{\ast },t\le C_i\mid g_i)}{Pr({\mathcal{Y}}_{i1}=1\mid g_i)}\\ ={\mathrm{\Lambda }}^{(g_i)}(\mathrm{d}t,t)+{\mathrm{\Lambda }}^{(g_i)}(t,\mathrm{d}t),\end{array}$$

where k′ = 3−k. So, the terms of $\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_1^{(g_i)}={\mathrm{\Lambda }}^{(g_i)}(\mathrm{d}t,t)$ in (A.2) are replaced by Λ^(g_i)(dt, t)+Λ^(g_i)(t, dt) in this composite setting. Similarly, the other corrections in (A.2) are of terms of the conditional expectations including d𝒩_i1(t), and derived as

$$\begin{array}{ll}\hfill \mathrm{E}[\mathrm{d}{\mathcal{N}}_{i1}(t)\mathrm{d}{\mathcal{N}}_{i2}(s)\mid {\mathcal{Y}}_{i1}(t){\mathcal{Y}}_{i2}(s)=1,g_i]& =\frac{𝟙(t\le s)S^{(g_i)}(\mathrm{d}t,\mathrm{d}s)-𝟙(t=s)S^{(g_i)}(t,\mathrm{d}s)}{S^{(g_i)}(t,t\vee s)}\text{and}\hfill \\ \hfill \mathrm{E}[\mathrm{d}{\mathcal{N}}_{i1}(t){\mathcal{Y}}_{i2}(s)\mid {\mathcal{Y}}_{i1}(t){\mathcal{Y}}_{i2}(s)=1,g_i]& =-\frac{S^{(g_i)}(\mathrm{d}t,t\vee s)+𝟙(t\le s)S^{(g_i)}(\mathrm{d}t,t\vee s)}{S^{(g_i)}(t,t\vee s)}.\hfill \end{array}$$

We achieve (3b) applying these results into the expectation of (A.1). In this composite setting, we can see that the intensity information on the 2nd endpoint (death event) is added into $\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_1^{(j)}(t)$ and dA^(j)(t, s) compared with the non-composite version.

Both fatal case

We consider the non-composite setting, ${\mathrm{\Delta }}_{ik}=𝟙(T_{ik}=T_{ik}^{\ast })$, k = 1, 2 and omit the composite setting for simplicity. Following $T_{i1}=min\left(T_{i1}^{\ast },T_{i2}^{\ast },C_i\right)$ and $T_{i2}=min\left(T_{i1}^{\ast },T_{i2}^{\ast },C_i\right)$, the expectations of the ith at-risk processes given g_i = j are the same forms as

$$y_k^{(j)}(t)=\mathrm{E}[{\mathcal{Y}}_{ik}(t)\mid g_i=j]=Pr\left(t\le T_{i1}^{\ast },t\le T_{i2}^{\ast },t\le C_i\mid g_i\right)=S^{(j)}(t,t)G(t),$$

for both endpoints, identical to that obtained for the one fatal case. Hence, by Glivenko–Cantelli’s theorem under some regular conditions, the asymptotic forms of Ĥ_k(t), k = 1, 2 are

$$H_k(t)=N^{-1}\frac{\mathrm{E}\left[{\mathcal{Y}}_k^{(1)}(t)\right]\mathrm{E}\left[{\mathcal{Y}}_k^{(2)}(t)\right]}{\mathrm{E}\left[{\mathcal{Y}}_k^{(1)}(t)\right]+\mathrm{E}\left[{\mathcal{Y}}_k^{(2)}(t)\right]}=G(t)\frac{r^{(1)}{r}^{(2)}{S}^{(1)}(t,t)S^{(2)}(t,t)}{r^{(1)}{S}^{(1)}(t,t)+r^{(2)}{S}^{(2)}(t,t)}.$$

The conditional expectations of the ith counting processes given 𝒴_ik(t) = 1 are

$$\begin{array}{l}\mathrm{d}{\stackrel{\sim }{\mathrm{\Lambda }}}_k^{(g_i)}(t)=\mathrm{E}[\mathrm{d}{\mathcal{N}}_{ik}(t)\mid {\mathcal{Y}}_{ik}(t)=1,g_i]=\frac{Pr\left(t\le T_{ik}^{\ast }<t+\mathrm{d},t\le T_{i{k}^{\prime }}^{\ast },t\le C_i\mid g_i\right)}{Pr({\mathcal{Y}}_{ik}(t)=1\mid g_i)}\\ =\{\begin{array}{ll}\frac{S^{(g_i)}(\mathrm{d}t,t)}{S^{(g_i)}(t,t)}\hfill & ={\mathrm{\Lambda }}^{(g_i)}(\mathrm{d}t,t),k=1\hfill \\ \frac{S^{(g_i)}(t,\mathrm{d}t)}{S^{(g_i)}(t,t)}\hfill & ={\mathrm{\Lambda }}^{(g_i)}(t,\mathrm{d}t),k=2\hfill \end{array}\end{array}$$

similar to the result for the non-fatal endpoint in the one fatal case, where k′ = 3 − k. Hence, the expectation of (A.1) can be derived as

$$\begin{array}{l}\mathrm{E}[\mathrm{d}{M}_{i1}(t)\mathrm{d}{M}_{i2}(s)\mid g_i]=G(t\vee s)\{𝟙(t=s)S^{(g_i)}(\mathrm{d}t,\mathrm{d}s)+𝟙(t\le s)S^{(g_i)}(t\vee s,\mathrm{d}s){\mathrm{\Lambda }}^{(g_i)}(\mathrm{d}t,t)\\ +𝟙(t\ge s)S^{(g_i)}(\mathrm{d}t,t\vee s)\mathrm{d}{\mathrm{\Lambda }}_2^{(g_i)}(s)+S^{(g_i)}(t\vee s,t\vee s){\mathrm{\Lambda }}^{(g_i)}(\mathrm{d}t,t){\mathrm{\Lambda }}^{(g_i)}(s,\mathrm{d}s)\},\end{array}$$ (A.3)

because

$$\begin{array}{ll}\hfill Pr({\mathcal{Y}}_{i1}(t){\mathcal{Y}}_{i2}(s)=1\mid g_i)& =Pr(t\vee s\le T_{i1}^{\ast },t\vee s\le T_{i2}^{\ast },t\vee s\le C_i\mid g_i)=G(t\vee s)S^{(g_i)}(t\vee s,t\vee s),\hfill \\ \hfill \mathrm{E}[\mathrm{d}{\mathcal{N}}_{i1}(t)\mathrm{d}{\mathcal{N}}_{i2}(s)\mid {\mathcal{Y}}_{i1}(t){\mathcal{Y}}_{i2}(s)=1,g_i]& =\frac{Pr\left(t\le T_{i1}^{\ast }<t+\mathrm{d}t,s\le T_{i2}^{\ast }<s+\mathrm{d}s,t\vee s\le T_{i1}^{\ast },t\vee s\le T_{i2}^{\ast }\mid g_i\right)}{Pr\left(t\vee s\le T_{i1}^{\ast },t\vee s\le T_{i2}^{\ast }\mid g_i\right)}\hfill \\ \hfill & =𝟙(t=s)\frac{S^{(g_i)}(\mathrm{d}t,\mathrm{d}s)}{S^{(g_i)}(t\vee s,t\vee s)},\hfill \\ \hfill \mathrm{E}[{\mathcal{Y}}_{i1}(t)\mathrm{d}{\mathcal{N}}_{i2}(s)\mid {\mathcal{Y}}_{i1}(t){\mathcal{Y}}_{i2}(s)=1,g_i]& =\frac{Pr\left(t\le T_{i1}^{\ast },s\le T_{i2}^{\ast }<s+\mathrm{d}s,t\vee s\le T_{i1}^{\ast },t\vee s\le T_{i2}^{\ast }\mid g_i\right)}{Pr\left(t\vee s\le T_{i1}^{\ast },t\vee s\le T_{i2}^{\ast }\mid g_i\right)}\hfill \\ \hfill & =-𝟙(t\le s)\frac{S^{(g_i)}(t\vee s,\mathrm{d}s)}{S^{(g_i)}(t\vee s,t\vee s)},\hfill \\ \hfill \mathrm{E}[\mathrm{d}{\mathcal{N}}_{i1}(t){\mathcal{Y}}_{i2}(s)\mid {\mathcal{Y}}_{i1}(t){\mathcal{Y}}_{i2}(s)=1,g_i]& =\frac{Pr\left(t\le T_{i1}^{\ast }<t+\mathrm{d}t,s\le T_{i2}^{\ast },t\vee s\le T_{i1}^{\ast },t\vee s\le T_{i2}^{\ast }\mid g_i\right)}{Pr\left(t\vee s\le T_{i1}^{\ast },t\vee s\le T_{i2}^{\ast }\mid g_i\right)}\hfill \\ \hfill & =-𝟙(t\ge s)\frac{S^{(g_i)}(\mathrm{d}t,t\vee s)}{S^{(g_i)}(t\vee s,t\vee s)}.\hfill \end{array}$$

Now, let us add an assumption natural for continuous time-to-event data that we cannot observe $T_{i1}^{\ast }$ and $T_{i2}^{\ast }$ simultaneously in both fatal event times. Then, the factors that occur at only t = s, such as 𝟙(t = s) S^(j)(dt, ds), do not contribute to the integral of the form ${\int }_0^{\tau }{\int }_0^{\tau }(·)\mathrm{d}{A}^{(j)}(t,s)$, so that we can ignore 𝟙(t = s)S^(j)(dt, ds) in (A.3) as the zero term. We can also apply the relations S^(j)(s, ds) = −S^(j)(s, s)Λ^(j)(s, ds) and S^(j)(dt, t) = −S^(j)(t, t)Λ^(j)(dt, t) into (A.3) further. Hence, (A.3) becomes to zero under continuous time-to-event data. As a result, the form of dA^(j)(t, s) = G(t ∨ s)⁻¹E[dM_i1(t)dM_i2(s) | g_i = j] is obtained as (4a) in Section 2.3.

APPENDIX B. Probability formula for the observed two endpoints

We provide the formula of $P_{ab}=Pr({\mathrm{\Delta }}_i^{(1)}=a,{\mathrm{\Delta }}_i^{(2)}=b)$ for the observation ( ${\mathrm{\Delta }}_i^{(1)},{\mathrm{\Delta }}_i^{(2)}$) of the four patterns. We may monitor a trial based on the number of events obtained from the required sample size using the probability formula. Let $P_{ab}^{(j)}=Pr({\mathrm{\Delta }}_i^{(1)}=a,{\mathrm{\Delta }}_i^{(2)}=b\mid g_i=j)$, a, b = 0, 1, j = 1, 2. Using this notation, we can write

$$P_{ab}=r^{(1)}{P}_{ab}^{(1)}+r^{(2)}{P}_{ab}^{(2)}$$

and then, for example, in the one fatal case, we can obtain $P_{11}^{(j)}=Q_1^{(j)},P_{10}^{(j)}=R_2^{(j)}-Q_1^{(j)},P_{01}^{(j)}=R_1^{(j)}$, and $P_{00}^{(j)}=1-R_1^{(j)}-R_2^{(j)}$, j = 1, 2, where

$$\begin{array}{ll}{Q}_1^{(j)}\hfill & ={\int }_0^{\infty }\left({\int }_0^s{f}^{(j)}(t,s)dt\right)G(s)ds,Q_2^{(j)}={\int }_0^{\infty }\left({\int }_0^t{f}^{(j)}(t,s)ds\right)G(t)dt,\hfill \\ R_1^{(j)}\hfill & ={\int }_0^{\infty }\left({\int }_s^{\infty }{f}^{(j)}(t,s)dt\right)G(s)ds,R_2^{(j)}={\int }_0^{\infty }\left({\int }_t^{\infty }{f}^{(j)}(t,s)ds\right)G(t)dt,\hfill \end{array}$$

$f^{(j)}(x,y)\left(=\frac{d^2}{\mathit{dxdy}}{S}^{(j)}(x,y)\right)$ is the density function of $\left(T_{i1}^{\ast },T_{i2}^{\ast }\right)$ for the ith participant assigned as g_i = j. The changes of $P_{ab}^{(j)}$ in the other censoring scenarios under the non-composite setting are summarized as Table BI including the one fatal case. For an example with the composite setting, let $P_{ab}^{\text{TTP}}$ and $P_{ab}^{\text{PFS}}$ be P_ab when the first endpoint corresponds to the TTP and the PFS, respectively, and the second endpoint is consistently of the OS. Then we have $P_{11}^{\text{PFS}}=P_{11}^{\text{TTP}}+P_{01}^{\text{TTP}},P_{10}^{\text{PFS}}=P_{10}^{\text{TTP}},P_{00}^{\text{PFS}}=P_{00}^{\text{TTP}}$, and $P_{01}^{\text{PFS}}=0$ It would be enough for us to consider P_ab under the non-composite setting as the probability formula, because it includes more information than P_ab under the composite setting.

Table BI.

The probability formula on observing ( ${\mathrm{\Delta }}_i^{(1)},{\mathrm{\Delta }}_i^{(2)}$) in the three censoring scenarios under the non-composite setting

$P_{ab}^{(j)}$	Both non-fatal case (Non-competing model)	One fatal case (Semi-competing model)	Both fatal case (Full-competing model)
$P_{11}^{(j)}$	$Q_1^{(j)}+Q_2^{(j)}$	$Q_1^{(j)}$	0
$P_{10}^{(j)}$	$R_2^{(j)}-Q_1^{(j)}$	$R_2^{(j)}-Q_1^{(j)}$	$R_2^{(j)}$
$P_{01}^{(j)}$	$R_1^{(j)}-Q_2^{(j)}$	$R_1^{(j)}$	$R_1^{(j)}$
$P_{00}^{(j)}$	$1-R_1^{(j)}-R_2^{(j)}$	$1-R_1^{(j)}-R_2^{(j)}$	$1-R_1^{(j)}-R_2^{(j)}$