Adaptive two-stage seamless sequential design for clinical trials

ABSTRACT

We propose an adaptive sequential testing procedure for the selection and testing of multiple treatment options, such as dose/regimen, different drugs, sub-populations, endpoints, or a mixture of them in a seamlessly combined phase II/III trial. The selection is to be made at the end of phase 2 stage. Unlike in many of the published literature, the selection rule is not required to be to “select the best”, and does not need to be pre-specified, which provides flexibility and allows the trial investigators to use any efficacy and safety information/criteria, or surrogate or intermediate endpoint to make the selection. Sample size and power calculations are provided. The calculations have been confirmed to be accurate by simulations. Interim analysis can be performed after the selection, sample size can be modified if the observed efficacy deviates from the assumed. Inference after the trial, including p-value, median unbiased point estimate and confidence intervals, are provided. By applying a dominance theorem, the procedure can be applied to normal, binary, Poisson, negative binomial distributed endpoints and time-to-event endpoints, and a mixture of these distributions (in trials involving endpoint selection).

KEYWORDS:

• Brownian motion approximation
• seamless combination
• adaptive two stage sequential design
• treatment selection

1. Introduction

Drug development is a lengthy and expensive process. It usually involves phase 2 and phase 3 trials. A phase 2 trial is generally of exploratory nature, in which a number of treatment options (such as treatment [e.g., dose/regimen], patient population, endpoints) can be investigated. One of the options with the most potential to be successful will be selected to proceed to a phase 3 trial. Conducting a clinical trial requires a sequence of activities including protocol development, seeking regulatory approvals, Institutional Review Board (IRB) approvals, budgeting, budget negotiation with contract service providers (commonly referred as vendors), site negotiation and initiation, and enrolment ramp up (Due to the lengthy process of site initiations, enrollment is typically very slow at the beginning of the trial). Each of the activities can be time and resource consuming. In addition, when phase 2 and 3 trials are conducted separately, there is usually a time gap of at least several months between the completion of the phase 2 trial and the planning and starting of the phase 3 trial. Combining phase 2 and phase 3 into one trial can reduce two sequences of activities into one and eliminate the time gap between the phase 2 and phase 3 trials, and the drug development process can be shortened and more efficient. Trial designs that combine phase 2 and phase 3 trials are commonly referred to as seamless phase II/III designs.

In a seamless design, at the end of phase 2 stage, an experiment treatment option must be chosen to proceed to phase 3 stage to be compared with the control. The main statistical challenges in such trials include: i) Family-wise error (FWE) control due to the selection procedure at the end of phase 2 stage. This selection is a multiple comparison procedure in nature and proper procedure should be included to control FWE at intended level. An optimal control is such that the FWE never exceeds the nominal error level but can be equal to the nominal level under certain conditions. A conservative control is such that the FWE never reaches the nominal level under any situation. The Bonferroni method is an example of conservative control. Conservative control will lead to reduced power. Hence, methods that achieve optimal or less conservative control are more preferable; ii) Because the trial includes phase 2 and phase 3 stages, it is desirable to combine the data from the two stages and analyze the data in an efficient way. But how to combine the data has been a main challenge since the data from phase 2 is not normally distributed because of the selection; iii) Sample size and power calculation (as always, such calculations involve assumptions on the effect size) are important to trial planning. The multiplicity issue and complicated methods for combining phase 2 and 3 data can complicate sample size and power calculations. iv) The assumptions for sample size calculation could be inaccurate (due to uncertainty about effect sizes). Thus, adaptive measures, such as sample size re-estimation (SSR), may be desirable to achieve adequate power. To obtain a proper new sample size, a method for calculating conditional power will be necessary; v) Inference must be made after the completion of the trial. Hence, p-values, estimates of the effect sizes, confidence intervals are desirable. The multiplicity issue from phase 2, also leads to challenges to finding unbiased estimates or exact inference. vi) When adaptations are made to the trial, inferences will be more complicated, but their relevance and importance is not reduced; vii) The usual group sequential design (GSD) testing comparing one new treatment arm and a control relies critically on the recursive algorithm of Armitage et al. (1969). Computations with multiple comparison adjustment and testing in a sequential procedure will be more complicated than a usual group sequential. In this article, we propose a procedure that addresses all of the above-mentioned statistical issues.

2. Background and features of proposed method

2.1. Background

Stallard and Todd (2010) summarized the methods into three categories: the group-sequential approach proposed by Stallard and Todd (2003), the combination test approach proposed by Bauer and Kieser (1999) (also Bretz et al. 2006), and the adaptive Dunnett method of Koenig et al. (2008). Later, several other methods were published that uses partition (e.g., Sugitani et al. (2013)), or the graph-based testing procedure (Glimm et al. 2018; Klinglmueller et al. 2014). Most methods require that the treatment option with the best observed outcome be selected (i.e., play-the-winner) at the end of phase 2 stage. There have been no methods for sample size and power calculations.

The published methods use various means to control the FWE, such as the Bonferroni method, or the closed testing procedure (Marcus R, Eric P, Gabriel. 1976) (e.g., Stallard and Todd 2003; Bauer and Kieser 1999; Bretz et al. 2006; Glimm et al. 2018); or the Dunnett method testing procedure (Koenig et al. 2008), or the partition method (Sugitani, Hamasaki, and Hamada 2013); or the graph-based testing procedure (Klinglmueller et al. 2014).

More recently, Chen et al. (2018) proposed a 2-in-1 adaptive phase 2/3 design for oncology trials. The focus of the design is to allow a decision to expand an ongoing Phase 2 trial into a Phase 3 trial, with fixed sample sizes for both the phase 2 and 3 trials. The phase 2 trial in this design does not involve multiple comparisons, and thus the design does not involve multiplicity issue due to selection, it also does not involve sample size re-estimation for the phase 3 trial. This design was later expanded in two directions: i) including dose selection (e.g., Jin and Zhang 2022; Zhang et al. 2022, 2023) in phase 2 and without sample size re-estimation for phase 3. The type I error (or FWE) control due to selection is achieved either by partitioning the $\mathit{\alpha }$ (Zhang et al. 2022, 2023), or by a graphical method (Jin and Zhang 2022). ii) including sample size re-estimation for phase 3 but without dose selection for phase 2 (Li et al. 0000). The type I error (or FWE) control with sample size re-estimation is achieved by the selection of a $C_{\mathit{min}}$.

2.2. Features of proposed method and notations

The majority of the methods in the literature are based on the Wald statistics. Our method is based on the score function (e.g., Jennison and Turnbull 1997). Unlike other methods that uses the score function (e.g. Stallard and Todd 2003), in our method the score function from phase 2 is directly added to the score function from phase 3 to form a combined score function. The combined score function is not normally distributed (because the score function from phase 2 is not normally distributed due to the selection process), but is a one-dimensional Markov process, and its mathematical properties can be precisely described.

Like Stallard and Todd (2003), we discuss a seamless design in which only one treatment option will be selected to continue into the phase 3 stage to compare with the control. A more general case in which several treatment options may continue into the phase 3 stage (e.g., Stallard and Friede 2008) will be discussed elsewhere in the context of adaptive sequential designs for multiple comparisons.

The control of the FWE is through a dominance theorem (section 3.3) on the Markov process. By utilizing the dominance theorem, our method does not require play-the-winner; hence, the selection does not need to depend solely on the primary outcome, but all information on efficacy and safety can be used for selection. The selection rule does not need to be pre-specified. Any emerging factors that were not foreseen at trial planning can be included in the selection. With such flexibility, it is possible to select options with the best combination of efficacy (not necessarily the best among all options) and safety profile. The selection can also be based on surrogate or intermediate endpoints. FWE control is conservative if the selection does not follow play-the-winner, but is precise if play -the-winner is followed and the endpoint is normally distributed. In this sense, the FWE control is optimal.

Our method includes a conservative point estimate if the null hypothesis is rejected at the end of phase 2. If the trial entered the phase 3 stage, then a median unbiased point estimate, an exact upper confidence limit, are available. We also include a conservative lower confidence limit that is consistent with the hypothesis test in the sense that the lower limit excludes zero if and only if the null hypothesis is rejected. In addition, our method includes sample size and power calculations that match exactly with simulations, confirming the accuracy of these calculations. Finally, our method includes sample size re-estimation in the phase 3 stage to increase power if the observed effect size at the interim analyses is smaller than the assumed one.

2.3. Comparison with methods in the literature

Our method is based on the Markov process properties of the combined score function. No method in literature utilizes the properties of Markov process. However, the presentation in this article does not involve Markov process theory, and our presentation does not require the readers to be familiar with Markov process theory. The dominance theorem (section 3.3.1), together with Slepian’s lemma (Huffer 1986; Slepian 1962), is fundamental to our approach. Its application assumes that the correlation coefficients between the score function statistics at the end of the phase 2 stage are non-negative (see section 3.3.2 for the discussion of this assumption). This assumption is usually satisfied in practice. For example, in trials involving selection among different doses of the same drug, the endpoints for all the doses are the same, and they are expected to trend in the same direction, hence the correlations are expected to be non-negative. In trials involving comparisons in different sub-populations, the endpoints for all the sub-populations are the same, and they are expected to trend in the same direction, with non-negative correlations. Multiple efficacy endpoints are usually assessed in any clinical trial. Suppose that all endpoints are standardized such that positive mean value indicates better efficacy. Then in most cases, it’s unlikely (although not entirely impossible) that an experiment drug will have positive efficacy in one endpoint but negative with another, hence, the assumption of non-negative correlation is reasonable in trials involving endpoint selection as well. However, if there is a possibility that some of the endpoints are negatively correlated, our method may not be appropriate.

A major difference between our method and those in the literature is how the data from the phase 2 and 3 stages are combined. All published methods (e.g., see Stallard and Todd 2010) use a multiplicity adjustment procedure such as the Bonferroni method, or the closed testing procedure, or the Dunnett’s procedure, or other multiple testing procedure (e.g., Bauer et al. 2015; Klinglmueller et al. 2014; Sugitani et al. 2013) on the phase 2 data and obtain a p-value. Then the p-value (usually the smallest) is converted into a normally distributed variable through the inverse normal transformation. Using the smallest p-value corresponds to the play-the-winner rule. This normalization conversion has been a necessary step in all published methods. With this conversion, data from both phases 2 and 3 are normally distributed and can be added together. However, the precise description of phase 2 data is lost in the inverse normal conversion, and it’s not possible to use the combined data for any purpose of parameter estimation (i.e., either point estimate or confidence limits). This conversion is also conservative since basically all multiplicity adjustment methods control the FWE conservatively. Our method preserves the precise description of the phase 2 data, since the score function from phase 2 data is directly added (without any conservative data conversion) to the score function from phase 3 data to form a Markov process. FWE control with our method is optimal (i.e., it is conservative in general, but is precise under the play-the-winner rule and when the endpoint is normally distributed). The combined data (as a Markov process) can be used for estimating the lower confidence limit of the efficacy for the selected option. Under the play-the-winner rule, all calculations on sample size, power, critical boundaries and selection probabilities are accurate for normally distributed endpoints, and can be confirmed by simulations (e.g., simulations in Figure 4 confirmed calculated results in Figure 3). Further, group sequential design and adaptive sequential design can also be described using the score function, based on the Markov process properties. With the commonly shared Markov process properties, we incorporated results from Gao et al. (2008) to propose sample size re-estimation with exact type I error control, and also incorporated results from Gao et al. (2013) to propose optimally exact point estimate and confidence interval (i.e., the point estimate is median unbiased if the trial proceeds into phase 3, and conservative if the trial stops at the end of phase 2; We derive exact upper confidence interval limit as well as a lower confidence interval limit that is consistent with hypothesis testing).

The dominance theorem provides rigorous theoretical justification for flexible selection rule. In many scenarios (e.g., when the selection is based on the primary endpoint, and there is no safety concern), it is most logical to select the option with the best efficacy, or play the winner. However, there may be situations in which the winner is not necessarily the most desirable. For example, in a trial with dose selection, the dose with the smallest p-value may be the highest dose, which may be related to higher rates of adverse events. A lower dose with acceptable effect size and lower rates of adverse events (better safety profile) may be more desirable. Another example is a trial in which a “surrogate or intermediate endpoint” (U.S. 2019) is used for the selection. Due to randomness, the treatment option with the best observed surrogate or intermediate endpoint may not be the same option with the best observed primary outcome endpoint, which precludes the use of play-the-winner rule. Hence, a method that does not require “play the winner” can provide some flexibility.

By using the dominance theorem, the seamless sequential design is completely parallel to the traditional group sequential/adaptive sequential design that compares a test treatment with a control, with critical boundaries, sample size and power calculation, interim analysis in exactly the same way as in Gao, Ware, Mehta (2008), and final analysis with point estimate and two-sided confidence interval obtained similarly to Gao, Liu, Mehta (2013). If the strategy of play-the-winner is used and the endpoint is normally distributed, then the critical boundaries control the type I error (FWE) exactly (in the strong sense), and the sample size and power calculations are exact. Otherwise, the FWE control is slightly conservative, per the dominance theorem and Slepian’s lemma (Huffer 1986; Slepian 1962).

We note that the correlation coefficients in a multi-variate normal distribution were critical in both Chen et al. (2018) and our approach: Chen et al. (2018) used the correlation coefficients in a bivariate normal distribution to control type I error for expanding a phase 2 trial into a phase 3 trial, and Slepian’s lemma (for multi-variate normal distribution) was critical to our dominance theorem.

Due to the application of the dominance theorem and Slepian’s lemma (Huffer 1986; Slepian 1962), there is an important difference between Stallard and Todd (2003) and our method in that we don’t make any assumption on the covariance matrix while Stallard and Todd (2003) assumes that the off- diagonal coefficients of the covariance matrix are ½ which holds only when the endpoint is normally distributed. Thus, our procedure can be applied to more general cases, such as when the endpoint is a non-normally distributed variable (e.g., survival analysis), and for different therapies in which the covariance coefficient may be unknown.

The proposed is broad in the sense that it can be applied to any designs that have been discussed in the literature, including those designs for oncology trials, such as those by (Chen et al. 2018; Jin and Zhang 2022; Zhang et al. 2022, 2023, Li et al. 2022).

We refer to our method as the two-stage seamless sequential design/adaptive two-stage seamless sequential design, TSSSD/ATSSSD. A seamless phase II/III design is an adaptive design. The FDA guidance on adaptive designs (2019) summarized the major potential advantages of adaptive designs: a) Statistical efficiency; b) Ethical considerations; c) Improved understanding of drug effects; d) Acceptability to stakeholders. In the same guidance, the FDA (2019) recommended that three statistical principles should be satisfied: a) Controlling the Chance of Erroneous Conclusions (type I error control); b) Estimating Treatment Effects (valid inference, including point estimate, two-sided confidence interval, and p-value); c) Trial Planning (including sample size/power calculation); (The FDA guidance (2019) also included a fourth principle, which is operational: d) Maintaining Trial Conduct and Integrity). Our method, the TSSSD/TSSASD, satisfies all the three statistical principles in the FDA guidance (2019). The TSSSD/ATSSSD is supported by the DACT (Design and Analysis for Clinical trials) software at https://www.innovatiostat.com/software.html (The software is free for academic researchers and research institutions). Computation codes are available upon request.

3. Method

3.1. Score function and GSD

To demonstrate the similarities between GSD and TSSSD, we first describe the GSD. In a GSD, one experimental treatment is compared with a control. Let $\mathit{\theta }$ be the parameter of interest, larger $\mathit{\theta }$ indicates better efficacy. Let the null hypothesis be $H_0$: $\mathit{\theta }=0$, and the alternative hypothesis $H_a$: $\mathit{\theta }>0$. The GSD (O’Brien and Fleming 1979; Pocock 1977) was based on the Wald statistics. Interim analyses are planned, to be performed at information time points $t_1<\dots <t_K=\mathit{T}$. Critical boundaries $c_1,\dots ,c_K$ are chosen such that $H_0$ is rejected in favor of $H_a$ if the event ${\cup }_{i=1}^K\left(Z_i\ge c_i\right)$ is observed ($Z_i=\mathit{Z}\left(t_i\mathit{\hspace{0.17em}}\right)$ is the Wald statistics at $t_i$). Type I error is controlled by requiring $\mathit{P}\left({\cup }_{i=1}^K\left(Z_i\ge c_i\right)\right)\le \mathit{\alpha }/2$. Let $\hat{\theta }$ be an unbiased estimate, and $\mathit{s}.\mathit{e}.\left(\hat{\theta }\right)$the standard error. The estimated Fisher’s information time is $\mathit{t}={\left(s.e.\left(\hat{\theta }\right)\right)}^{-2}$. Then, $\mathit{Z}\left(\mathit{t}\right)=\hat{\theta }\sqrt{\mathit{t}}$ is the Wald statistics, and the score function is $\mathit{S}\left(\mathit{t}\right)=\hat{\theta }\mathit{t}$. Approximately, $\mathit{S}\left(\mathit{t}\right)\approx \mathit{W}\left(\mathit{t}\right)=\mathit{B}\left(\mathit{t}\right)+\mathit{\theta t}{ }_{}\mathit{N}\left(\mathit{\theta t},\mathit{t}\right)$ (Jennison and Turnbull 1997). Let $e_i=c_i\sqrt{t_i}$ and name the $e_i$ ‘s as the “exit” boundaries. The rules of a GSD can be equivalently stated as $\mathit{P}\left({\cup }_{i=1}^K\left(\mathit{S}\left(t_i\right)\ge e_i\right)\right)\le \mathit{\alpha }/2$.

3.2. The TSSSD

3.2.1. Data description

The score function in a trial that compares one experimental treatment with a control arm is approximately a Brownian motion (Jennison and Turnbull 1997). In a seamless design that compares several treatment options with a control, there are several score functions in the phase 2 stage, each corresponding to the comparison of one experimental treatment option with the control. Take for example a trial that tests multiple doses in phase 2 stage, other situations can be dealt with similarly. Suppose that there are $\mathit{M}$ dose groups and a control arm in the phase 2 stage. Let ${\theta }_m$ (m = 1,…, M) be the efficacy for each dose being considered, larger $\mathit{\theta }$ indicates better efficacy. Let $\vec{\theta }=\left({\theta }_1,\dots ,{\theta }_M\right)$. Let the related hypotheses for each dose comparison be: null hypothesis $H_{0,\mathit{m}}$: ${\theta }_m=0$, and one-sided alternative hypothesis $H_{\mathit{a},\mathit{m}}$: ${\theta }_m>0$. At the end-of-phase 2, one dose (e.g., the $\mathit{n}$-the dose) is selected to proceed to phase 3. The overall null hypothesis of the trial is the null hypothesis about the selected dose, that $H_{0,\mathit{n}}$: ${\theta }_n=0$, and one-sided alternative hypothesis $H_{\mathit{a},\mathit{n}}$: ${\theta }_n>0$. Each of the $\mathit{M}$ parameters ${\theta }_1,\dots ,{\theta }_M$, is associated with an estimate ${\hat{\theta }}_{\mathit{m}}^{\left(0\right)}$ and the standard error $\mathit{s}.\mathit{e}.\left({\hat{\theta }}_{\mathit{m}}^{\left(0\right)}\right)$, from the phase 2 data, $\mathit{m}=1,\dots ,\mathit{M}$. There are $\mathit{M}$ Wald statistic $Z_m^{\left(0\right)}={\hat{\theta }}_{\mathit{m}}^{\left(0\right)}/\mathit{s}.\mathit{e}.\left({\hat{\theta }}_{\mathit{m}}^{\left(0\right)}\right)$, and $\mathit{M}$ score functions $S_m^{\left(0\right)}={\hat{\theta }}_{\mathit{m}}^{\left(0\right)}/{\left[s.e.\left({\hat{\theta }}_{\mathit{m}}^{\left(0\right)}\right)\right]}^2$. The superscript ⁽⁰⁾ indicates that the statistics uses only data from the phase 2 stage. Statistics using only data from the phase 3 stage will be denoted with a superscript ⁽¹⁾, and statistics using both the phase 2 and 3 data will be denoted by the superscript ^(0,1). Let ${\hat{t}}_{\mathit{m}}^{\left(0\right)}={\left[s.e.\left({\hat{\theta }}_{\mathit{m}}^{\left(0\right)}\right)\right]}^2$ be the estimator for $t_m^{\left(0\right)}$, the information time for the $\mathit{m}$ -th comparison. Suppose that the n-th option is selected. Let $S_n^{\left(0,1\right)}=S_n^{\left(0\right)}+S_n^{\left(1\right)}$ be the combined score function of the selected treatment option vs. the control, where $S_n^{\left(0\right)}$ is from the phase 2 stage, and $S_n^{\left(1\right)}$ is from the phase 3 stage. $S_n^{\left(1\right)}$ is independent from all the phase 2 data and is approximately a Brownian motion. $S_n^{\left(0\right)}$ is normally distributed in isolation, but not normally distributed as the chosen one from $S_1^{\left(0\right)}$, … , $S_M^{\left(0\right)}$ due to the selection procedure (which is a multiple comparison procedure). However, $S_n^{\left(0,1\right)}$, as the sum of a random variable $S_n^{\left(0\right)}$ and a Brownian motion $S_n^{\left(1\right)}$, is a Markov process. We note that a Brownian motion is a special Markov process, but a Markov process is not necessarily a Brownian motion in general.

A fundamental property of Brownian motion is that it has independent increments. The increments of $S_n^{\left(0,1\right)}$ are the same as those of $S_n^{\left(1\right)}$(which is approximately a Brownian motion), and thus are independent. This property is essential to our method. Because our method is based on the score function and its Brownian motion properties, we refer this as the Brownian motion approximation (BMA) approach. The BMA approach is motivated by the thinking in the theory of Markov processes, but such thinking has been modified and expressed only in terms of multivariate normal distributions and only uses calculus and four basic (widely known) properties of Brownian motion: i) It is additive: $\mathit{B}\left(\mathit{t}\right)+\mathit{B}\left(\mathit{u}\right)=\mathit{B}\left(\mathit{t}+\mathit{u}\right)$; ii) It has independent increment; iii) $\mathit{cov}\left(\mathit{B}\left(t_i\right),\mathit{B}\left(t_j\right)\mathit{\hspace{0.17em}}\right)=\mathit{min}\left\{t_i,t_j\right\}$; iv) For $0<t_1<..<t_K$, $\left(\mathit{B}\left(t_1\right),\dots ,\mathit{B}\left(t_K\right)\right)$ has a joint multivariate normal distribution (e.g., Jennison and Turnbull 2000). No prior knowledge of Markov processes is required to follow our method.

3.2.2. Properties of the combined score function

Let the score function $S_m^{\left(0\right)}=S_m^{\left(0\right)}\left({\hat{\theta }}_{\mathit{m}}^{\left(0\right)}\mathit{\hspace{0.17em}}\right)\approx W_{m,t_m^{\left(0\right)}}^{\left(0\right)}=W_m^{\left(0\right)}\left(t_m^{\left(0\right)}\right)=B_{m,t_m^{\left(0\right)}}^{\left(0\right)}+{\theta }_m{t}_m^{\left(0\right)}\sim \mathit{N}\left({\theta }_m{t}_m^{\left(0\right)},t_m^{\left(0\right)}\right)$. Let ${\vec{W}}^{\left(0\right)}=\left(W_{1,t_1^{\left(0\right)}}^{\left(0\right)},\dots ,W_{M,t_M^{\left(0\right)}}^{\left(0\right)}\right)$ be the vector of score functions at the end of phase 2, and ${\vec{t}}^{\left(0\right)}=\left(t_1^{\left(0\right)},\dots ,t_M^{\left(0\right)}\right)$. Let $\mathit{\hspace{0.17em}}{\vec{t}}^{\left(0\right)}\circ \mathit{\hspace{0.17em}}\vec{\theta }=\left({\theta }_1{t}_1^{\left(0\right)},\dots ,{\theta }_M{t}_M^{\left(0\right)}\right)$. The joint density function for ${\vec{W}}^{\left(0\right)}=\left(W_{1,t_1^{\left(0\right)}}^{\left(0\right)},\dots ,W_{M,t_M^{\left(0\right)}}^{\left(0\right)}\right)$ is $\mathit{u}\left(\vec{\theta },\vec{x},\mathrm{\Sigma },{\vec{t}}^{\left(0\right)}\right)={\left(2\mathit{\pi }\right)}^{-\frac{M}{2}}{\left(\mathit{det}\left(\mathrm{\Sigma }\right)\right)}^{-\frac{1}{2}}\mathit{exp}\left(-\frac{{\left(\vec{x}-\mathit{\hspace{0.17em}}{\vec{t}}^{\left(0\right)}\circ \mathit{\hspace{0.17em}}\vec{\theta }\right)}^{\prime }{\mathrm{\Sigma }}^{-1}\left(\vec{x}-\mathit{\hspace{0.17em}}{\vec{t}}^{\left(0\right)}\circ \mathit{\hspace{0.17em}}\vec{\theta }\right)}{2}\right)$, where $\mathrm{\Sigma }$ =${\mathrm{\Sigma }}_{\mathit{\hspace{0.17em}}{\vec{t}}^{\left(0\right)}}=\left({\sigma }_{\mathit{ij}}\right)$, $\mathit{i},\mathit{j}=1,\dots ,$, with ${\sigma }_{\mathit{ij}}=\mathit{cov}\left(W_{i,t_i^{\left(0\right)}}^{\left(0\right)},W_{j,t_j^{\left(0\right)}}^{\left(0\right)}\right)$. The derivation of the covariance matrix for dose selection, sub-population enrichment has been discussed in Gao et al. (2014). For simplicity of notations, we assume that $t_1^{\left(0\right)}=\dots =t_M^{\left(0\right)}=t^{\left(0\right)}$. Unequal $t_m^{\left(0\right)}$ ‘s can be dealt with similarly. At the end of phase 2, an option (e.g., with corresponding $W_{n,t_n^{\left(0\right)}}^{\left(0\right)}$, which is a component of ${\vec{W}}^{\left(0\right)}$) is selected to proceed to phase 3. As a result of the selection, the data from phase 2 is $X_0=W_{n,t_n^{\left(0\right)}}^{\left(0\right)}$. This $\mathit{n}$ is dependent on the selection, hence a random number, and $X_0$ does not have the same distribution as $W_{m,t_m^{\left(0\right)}}^{\left(0\right)}$, for any fixed $\mathit{m}=1,\dots ,\mathit{M}$, and $X_0$ is not normally distributed. For example, under the play-the-winner rule, $X_0=\mathit{max}\left\{W_{1,t_1^{\left(0\right)}}^{\left(0\right)},\dots ,W_{M,t_M^{\left(0\right)}}^{\left(0\right)}\right\}$, $\mathit{n}$ is such that $W_{n,t_n^{\left(0\right)}}^{\left(0\right)}=\mathit{max}\left\{W_{1,t_1^{\left(0\right)}}^{\left(0\right)},\dots ,W_{M,t_M^{\left(0\right)}}^{\left(0\right)}\right\}$ and the distribution of $X_0$ is $\mathit{P}\left(X_0\le \mathit{x}\right)=\mathit{P}\left({\cap }_{m=1}^M\left(W_{1,t_m^{\left(0\right)}}^{\left(0\right)}\le \mathit{x}\right)\right)={\int }_{-\mathrm{\infty }}^x\dots {\int }_{-\mathrm{\infty }}^x\mathit{u}\left(\vec{\theta },\vec{x},\mathrm{\Sigma },{\vec{t}}^{\left(0\right)}\right)\mathit{d}{x}_1\dots \mathit{d}{x}_M.$ The distribution can be precisely calculated if $\mathrm{\Sigma }$ is known (e.g., if phase 2 stage involves dose selection with a common control, then the off-diagonal elements = 0.5, and the diagonal elements = 1). The cases in which $\mathrm{\Sigma }$ are unknown are discussed in section 3.3. On the other hand, the score function using only the phase 3 data for the selected option (say the $\mathit{n}$ –th, denoted as $W_n\left(t^{\left(1\right)}\right)=\mathit{B}\left(t^{\left(1\right)}\right)+{\theta }_n{t}^{\left(1\right)}$), is still approximately a Brownian motion. The cumulative data from phase 2 and 3 will be represented as $\mathit{X}\left(t^{\left(1\right)}\right)=X_0\mathit{\hspace{0.17em}}+W_n\left(t^{\left(1\right)}\right)\mathit{\hspace{0.17em}}=W_{n,t_n^{\left(0\right)}}^{\left(0\right)}\mathit{\hspace{0.17em}}+W_n\left(t^{\left(1\right)}\right)=W_n\left(t_n^{\left(0\right)}+t^{\left(1\right)}\right)\mathit{\hspace{0.17em}}$. Accordingly, $\mathit{X}\left(\mathit{t}\right)=X_0\mathit{\hspace{0.17em}}+\mathit{W}\left(\mathit{t}\right)$, or $X_t=X_0+W_t=X_0+B_t+\mathit{\theta t}$ (where $\mathit{W}\left(\mathit{t}\right)=W_n\left(t^{\left(1\right)}\right)$ if the $\mathit{n}$ –the option is selected), is the sum of a random variable $X_0$ and a Brownian motion with a (random) drift ${\theta }_n$. To simplify the notation, we remove the subscript of the selected option.

3.2.3. Hypothesis test and type I error control

In a TSSSD, interim analyses may be performed on $\mathit{X}\left(\mathit{t}\right)$ at information time points $0<t^{\left(0\right)}<t_1^{\left(0,1\right)}<\dots <t_K^{\left(0,1\right)}$, where $t_i^{\left(0,1\right)}=t^{\left(0\right)}+t_i^{\left(1\right)}$, and $t_i^{\left(1\right)}$ is the information time in the phase 3 stage. Let $r^{\left(0\right)}=t^{\left(0\right)}/t_K^{\left(0,1\right)}$, $r_i^{\left(0,1\right)}=t_i^{\left(0,1\right)}/t_K^{\left(0,1\right)}$, $r_i^{\left(1\right)}=t_i^{\left(1\right)}/t_K^{\left(0,1\right)}$, and $s_i^{\left(1\right)}=t_i^{\left(1\right)}/t_K^{\left(1\right)}$. The null hypothesis $H_0$ is rejected if $\left(X_0\ge e_0^{\left(0\right)}\right)\cup \left({\cup }_{i=1}^M\left(\mathit{X}\left(t_i^{\left(0,1\right)}\right)\ge e_i^{\left(0,1\right)}\right)\right)$ is observed. In a TSSSD, sample size and power calculation, inferences such as $\mathit{p}$-value, confidence interval and estimates all require the calculation of probabilities of the form of $\mathit{P}\left(\left(X_0\ge e_0^{\left(0\right)}\right)\cup \left({\cup }_{i=1}^{I=1}\left(\mathit{X}\left(t_i^{\left(0,1\right)}\right)\ge e_i^{\left(0,1\right)}\right)\right)\cup \left(\mathit{X}\left(t_I^{\left(0,1\right)}\right)\ge x_I^{\left(0,1\right)}\right)\right)$. The details of the calculation are provided in the online supporting material. The exit boundaries$e_0^{\left(0\right)}$, $e_1^{\left(0,1\right)},\dots ,e_K^{\left(0,1\right)}$, or the critical boundaries $c_0^{\left(0\right)}=e_0^{\left(0\right)}/\sqrt{t^{\left(0\right)}}$, $c_i^{\left(0,1\right)}=e_i^{\left(0,1\right)}/\sqrt{t_i^{\left(0,1\right)}}$, i = 1, … ,K, can be chosen to satisfy$\mathit{P}{\left(\left(X_0\ge e_0^{\left(0\right)}\right)\cup \left({\cup }_{i=1}^K\left(\mathit{X}\left(t_i^{\left(0,1\right)}\right)\ge e_i^{\left(0,1\right)}\right)\right)\right)}_{\vec{\theta }=\vec{0}}\le \mathit{\alpha }/2$. Figure 1 provides a flowchart of the design and Figure 2 illustrates the critical boundaries.

Figure 1. Trial design flowchart.

Figure 2. Cumulative data $\mathit{X}\left(t^{\left(1\right)}\right)$ (using play the winner rule) and critical boundaries.

In Figure 2, $S_m^{\left(0\right)}$ is the score function for option m, and $X_0=S_n^{\left(0\right)}=\mathit{max}\left\{S_m^{\left(0\right)},\mathit{m}=1,\dots ,\mathit{M}\right\}$ (i.e., $X_0$ is obtained by play-the-winner rule).

3.2.4. O’Brien-Fleming boundary

The O’Brien-Fleming boundaries satisfy $e_0^{\left(0\right)}=\dots =e_K^{\left(0,1\right)}$, which is equivalent to $c_0^{\left(0\right)}\sqrt{r^{\left(0\right)}}=c_1^{\left(0,1\right)}\sqrt{r_1^{\left(0,1\right)}}\dots =c_K^{\left(0,1\right)}\sqrt{r_K^{\left(0,1\right)}}$). Let $\mathit{f}\left(\mathit{e}\right)=\mathit{P}\left(\left(X_0\ge \mathit{e}\right)\cup \left({\cup }_{i=1}^K\left(\mathit{X}\left(t_i^{\left(0,1\right)}\right)\ge \mathit{e}\right)\right)\right)$. Then $\mathit{f}\left(\mathit{e}\right)$ is a decreasing function of e. Let $e_{0.7\mathit{ex\alpha }/\alpha 2-0.5\mathrm{mu}0.7\mathrm{ex}2}=f^{-1}\left(\mathit{\alpha }/2\right)$. Then $c_0=e_{0.7\mathit{ex\alpha \alpha }2-0.5\mathit{mu}0.7\mathit{ex}2}/\sqrt{r^{\left(0\right)}}$, $c_i=e_{0.7\mathit{ex\alpha }/\alpha 2-0.5\mathrm{mu}0.7\mathrm{ex}2}/\sqrt{r_i^{\left(0,1\right)}}]$, i = 1, …, K are the desired O’Brien-Fleming boundaries (O’Brien and Fleming 1979). The details of calculation are provided in the online supporting material.

3.2.5. α-spending functions

Let $A_0=\left(X_0\ge e_0^{\left(0\right)}\right)$, $A_i=(X_0<e_0^{\left(0\right)})\cap \left({\cap }_{i=0}^K(\mathit{X}\left(t_j^{\left(0,1\right)}\right)<e_j^{\left(0,1\right)})\cap \left(\mathit{X}\left(t_i^{\left(0,1\right)}\right)\ge e_i^{\left(0,1\right)}\right)\right)$. $A_i$ is the event that the critical boundaries were not crossed at any interim analysis before $t_i^{\left(0,1\right)}$, and was crossed at $t_i^{\left(0,1\right)}$. Note that the events $A_i$ are mutually exclusive, or that $A_i\cap A_j=\mathrm{\varnothing }$ for $\mathit{i}\ne \mathit{j}$. Hence, $\mathit{P}\left\{{\cup }_{i=0}^K{A}_i\right\}={\sum }_{i=0}^K\mathit{P}\left(A_i\right)$. Let $\mathit{P}\left(A_i\right)={\alpha }_i$. Then ${\alpha }_i$ is the type I error “spent” at $t_i^{\left(0,1\right)}$. The only requirement on the ${\alpha }_i$’s is that ${\sum }_{i=0}^K{\alpha }_i=\mathit{\alpha }/2$. The $c_i$’s can be successively chosen to satisfy $\mathit{P}\left(A_i\right)={\alpha }_i$. The details of calculation are provided in the online supporting material.

3.2.6. Power calculation and sample size determination

Suppose that the “information fractions”, $r_i^{\left(0,1\right)}$, and the critical boundaries $c_0^{\left(0\right)}$, $c_1^{\left(0,1\right)},\dots ,c_K^{\left(0,1\right)}$, i = 1,…,K have been determined. Let the Fisher’s information time at the final analysis be denoted as $\mathit{T}=t_K^{\left(0,1\right)}$. Under the alternative hypothesis, $\vec{\theta }=\left({\theta }_1,\dots ,{\theta }_M\right)\ne \vec{0}$. The power is calculated as $[\$P_{\mathit{\hspace{0.17em}}\vec{\theta },\mathit{T}}=\mathit{P}{\left(\left(X_0\ge e_0^{\left(0\right)}\right)\left(\cup { }_{i=1}^K\left(\mathit{X}\left(t_i^{\left(0,1\right)}\right)\ge e_i^{\left(0,1\right)}\right)\right)\right)}_{\vec{\theta }}$. (The details for calculating $P_{\mathit{\hspace{0.17em}}\vec{\theta },\mathit{T}}$ are in the online supporting material). The power of a trial depends on the selection rule and the configuration $\vec{\theta }$. The desired power of the trial can be obtained by choosing an information time T that is large enough such that $P_{\mathit{\hspace{0.17em}}\vec{\theta },\mathit{T}}\ge 1-\mathit{\beta }$. Sample size $n_K$ can then be determined by the relationship that $n_K\approx \mathit{aT}$ for some distribution specific constant $\mathit{a}$ (e.g., Whitehead 1997)

3.3. General situations

The above discussed calculations of type I error, exit/critical boundaries, sample size/power assume that the covariance matrix is known, which is possible only in the special situation of dose selection with a common control, and the variable is normally distributed. In addition, the selection rule is strictly play- the-winner. We refer all other situations as the general situation, such as treatment selection with non-normally distributed variable (e.g., time-to-event), population selection, endpoint selection, or mixed selections (FDA guidance, adaptive designs, 2019). Further, the selection rule may not necessarily be play- the-winner as in the case when “surrogate or intermediate endpoint” (FDA. 2019) are used for the selection (e.g., when the primary endpoint requires long term follow-up, biomarkers and other surrogate endpoints may be used. Such situations are common in rare disease trials), or when the safety profile of the option with the best efficacy is not satisfactory, and the option with the next best efficacy needs to be selected. The FDA guidance (2019) noted that “It may be particularly difficult to estimate Type I error probability and other operating characteristics for designs that incorporate multiple adaptive features.” The FDA guidance (2019) hypothesized that, for the general situation, “it can be argued that assuming independence among multiple endpoints will provide an upper bound on the Type I error probability.” We provide a mathematically rigorous proof of this hypothesis by using the dominance theorem and Slepian’s lemma (Huffer 1986; Slepian 1962). We provide upper bound of type I error that does not exceed the desired level, conservative lower confidence limit and upper bound of p-value. These conservative statistics satisfy the three statistical principles in the FDA guidance (2019). These calculations are confirmed through simulations shown in later sections. Our approach also mathematically, and rigorously, confirms the hypothesis about the assumption of independence in the FDA guidance (2019).

3.3.1. The “Dominance” Theorem and Slepian’s lemma

As discussed in section 3.2.1, the combined score function has the form of $X_t=X_0+W_t=X_0+B_t+\mathit{\theta t}$. The dominance theorem establishes that the probabilities related to $X_t$ are largely determined by the distribution of $X_0$.

Definition:

Let $X_0$ and $Y_0$ be random variables, and that $\mathit{P}\left(Y_0\mathit{\hspace{0.17em}}\ge \mathrm{x}\right)\ge \mathit{P}\left(X_0\mathit{\hspace{0.17em}}\ge \mathrm{x}\right)$ for all $\mathit{x}$. Then $Y_0$ is said to dominate $X_0$.

Theorem

(The Dominance Theorem): Suppose that $Y_0$ dominates $X_0$. Let $X_t=\mathit{X}\left(\mathit{t}\right)=X_0+\mathit{B}\left(\mathit{t}\right)+\mathit{\theta t}$ and $Y_t=\mathit{Y}\left(\mathit{t}\right)=Y_0+\tilde{B}\left(\mathit{t}\right)+\mathit{\theta t}$, where $\mathit{B}\left(\mathit{t}\right)$ and $\tilde{B}\left(\mathit{t}\right)$ are both Brownian motions. Let $t_1<\dots <t_K$ be a sequence of time points and $u_0,u_1,\dots ,u_K$ be some exit boundaries. Then

$\mathit{P}\left\{\left(Y_0\ge u_0\right)\cup \left(\underset{\mathit{i}=1}{\overset{K}{\cup }}{Y}_{t_i}\ge u_i\right)\right\}\ge \mathit{P}\left\{\left(X_0\ge u_0\right)\cup \left(\underset{\mathit{i}=1}{\overset{K}{\cup }}{X}_{t_i}\ge u_i\right)\right\}.$

Lemma 1.

Let $X_t=\mathit{X}\left(\mathit{t}\right)=X_0+\mathit{B}\left(\mathit{t}\right)+\mathit{\theta t}$ and $Y_t=\mathit{Y}\left(\mathit{t}\right)=Y_0+\tilde{B}\left(\mathit{t}\right)+\mathit{\theta t}$, where $\mathit{B}\left(\mathit{t}\right)$ and $\tilde{B}\left(\mathit{t}\right)$ are both Brownian motions, and $Y_0\ge X_0$. Let $t_1<\dots <t_K$ be a sequence of time points and $e_0,e_1,\dots ,e_K$ be some exit boundaries. Then

$\mathit{P}\left(Y_0\ge e_0\right)\cup \cup \mathit{Ki}=1Y_{t_i}\ge e_i\mathit{P}\left(X_0\ge e_0\right)\cup \cup \mathit{Ki}=1X_{t_i}\ge e_i.$

Proof: Let $X_t=\mathit{X}\left(\mathit{t}\right)=X_0+\mathit{B}\left(\mathit{t}\right)+\mathit{\theta t}$ and ${\tilde{Y}}_{\mathit{t}}=\tilde{Y}\left(\mathit{t}\right)=Y_0+\mathit{B}\left(\mathit{t}\right)+\mathit{\theta t}$, where B(t) is a Brownian motion. Then $\tilde{Y}\left(\mathit{t}\right)\ge$ $\mathit{X}\left(\mathit{t}\right)$. For any $e_0,e_1,\dots ,e_K$, $\left({\tilde{Y}}_0\ge e_0\right)\supseteq \left(X_0\ge e_0\right)$, $\left({\tilde{Y}}_{t_i}\ge e_i\right)\supseteq \left(X_{t_i}\ge e_i\right)$, and $\left({\tilde{Y}}_0\ge e_0\right)\underset{\mathit{i}=1}{\overset{K}{\cup }}\left({\tilde{Y}}_{t_i}\ge e_i\right)\supseteq \left(X_0\ge e_0\right)\underset{\mathit{i}=1}{\overset{K}{\cup }}\left(X_{t_i}\ge e_i\right)$. Hence

$\mathit{P}\left({\tilde{Y}}_0\ge e_0\right)\cup \mathit{i}=1\mathit{K}\left({\tilde{Y}}_{t_i}\ge e_i\right)\ge \mathit{P}\left(X_0\ge e_0\right)\cup \mathit{i}=1\mathit{K}\left(X_{t_i}\ge e_i\right)$

Since $\tilde{Y}$(t) and Y(t) have the same distribution,

$\mathit{P}\left\{\left(Y_0\ge e_0\right)\underset{\mathit{i}=1}{\overset{K}{\cup }}\left(Y_{t_i}\ge e_i\right)\right\}=\mathit{P}\left\{\left({\tilde{Y}}_0\ge e_0\right)\underset{\mathit{i}=1}{\overset{K}{\cup }}\left({\tilde{Y}}_{t_i}\ge e_i\right)\right\}$. Hence,

$\mathit{P}\left(Y_0\ge e_0\right)\cup \mathit{i}=1\mathit{K}\left(Y_{t_i}\ge e_i\right)\ge \mathit{P}\left(X_0\ge e_0\right)\cup \mathit{i}=1\mathit{K}\left(X_{t_i}\ge e_i\right)$

The lemma is proved. ■

Proof of The Dominance Theorem: Let $\mathit{F}\left(\mathit{x}\right)=\mathit{P}\left(Y_0<\mathit{x}\right)$, and $\mathit{G}\left(\mathit{x}\right)=\mathit{P}\left(X_0<\mathit{x}\right)$. Since $\mathit{P}\left(Y_0\ge \mathit{x}\right)\ge \mathit{P}\left(X_0\ge \mathit{x}\right)$, then F(x) ≤ G(x). And $F^{-1}\left(\mathit{y}\right)\ge G^{-1}\left(\mathit{y}\right)$. Let 0 ≤ U ≤1 be a random variable with a uniform distribution. Let ${\tilde{Y}}_0=F^{-1}\left(\mathit{U}\right)$, and ${\tilde{X}}_0=G^{-1}\left(\mathit{U}\right)$. Then ${\tilde{Y}}_0\ge {\tilde{X}}_0$ and $\mathit{P}\left({\tilde{Y}}_0\ge \mathit{x}\right)=\mathit{P}\left(F^{-1}\left(\mathit{U}\right)\ge \mathit{x}\right)=\mathit{P}\left(\mathit{U}\ge \mathit{F}\left(\mathit{x}\right)\right)=1-\mathit{F}\left(\mathit{x}\right)=\mathit{P}\left(Y_0\ge \mathit{x}\right),\mathrm{and}\mathit{P}\left({\tilde{X}}_0\ge \mathit{x}\right)=\mathit{P}(G^{-1}\left(\mathit{U}\right)$ $\ge \mathit{x})=\mathit{P}\left(\mathit{U}\ge \mathit{G}\left(\mathit{x}\right)\right)=1-\mathit{G}\left(\mathit{x}\right)=\mathit{P}\left(X_0\ge \mathit{x}\right)$.

Let B(t) be a Brownian motion, $\tilde{X}$(t) =  ${\tilde{X}}_0$ + B(t) + θt, and $\tilde{Y}$(t) =  ${\tilde{Y}}_0$+ B(t) + θt. Then by Lemma 1,

$\mathit{P}\left({\tilde{Y}}_0\ge e_0\right)\cup \mathit{i}=1\mathit{K}\left({\tilde{Y}}_{t_i}\ge e_i\right)\ge \mathit{P}\left({\tilde{X}}_0\ge e_0\right)\cup \mathit{i}=1\mathit{K}\left({\tilde{X}}_{t_i}\ge e_i\right).$

Since $\tilde{X}$(t) and X(t) have the same distribution,

$\mathit{P}\left(X_0\ge e_0\right)\cup \mathit{i}=1\mathit{K}\left(X_{t_i}\ge e_i\right)=\mathit{P}\left({\tilde{X}}_0\ge e_0\right)\cup \mathit{i}=1\mathit{K}\left({\tilde{X}}_{t_i}\ge e_i\right).$

Since $\tilde{Y}$(t) and Y(t) have the same distribution,

$\mathit{P}\left(Y_0\ge e_0\right)\cup \mathit{i}=1\mathit{K}\left(Y_{t_i}\ge e_i\right)\mathit{P}\left({\tilde{Y}}_0\ge e_0\right)\cup \mathit{i}=1\mathit{K}\left({\tilde{Y}}_{t_i}\ge e_i\right).$

Therefore,

$\mathit{P}\left(Y_0\ge e_0\right)\cup \mathit{i}=1\mathit{K}\left(Y_{t_i}\ge e_i\right)\ge \mathit{P}\left(X_0\ge e_0\right)\cup \mathit{i}=1\mathit{K}\left(X_{t_i}\ge e_i\right)$

The theorem is proved. ■

Slepian’s lemma (Huffer 1986; Slepian 1962): Suppose that $\vec{X}=\left(X_1,\dots ,X_n\right)\in {\mathbb{R}}^{\mathit{n}}$ and $\vec{Y}=\left(Y_1,\dots ,Y_n\right)\in {\mathbb{R}}^{\mathit{n}}$ are Gaussian vectors and that $\mathit{E}\left(X_i\right)=\mathit{E}\left(Y_i\right)=0$, $\mathit{E}\left(X_i^2\right)=\mathit{E}\left(Y_i^2\right)$, i = 1, … ,n, and $\mathit{E}\left(Y_i{Y}_j\right)\le \mathit{E}\left(X_i{X}_j\right)$ for i ≠ j. Then the following inequality holds for all real numbers $u_1,\dots ,u_n$: $\mathit{P}\left\{\left(\underset{\mathit{i}=1}{\overset{n}{\cup }}{Y}_i>u_i\right)\right\}\ge \mathit{P}\left\{\left(\underset{\mathit{i}=1}{\overset{n}{\cup }}{X}_i>u_i\right)\right\}$.

3.3.2. Conservative exit/critical boundary selection

Suppose that it is known that $\mathit{P}\left(Y_0\mathit{\hspace{0.17em}}\ge \mathrm{x}\right)\ge \mathit{P}\left(X_0\mathit{\hspace{0.17em}}\ge \mathrm{x}\right)$ for all $\mathit{x}$, and that the distribution of $Y_0$ is known, but the distribution of $X_0$ is not. Per the dominance theorem, $\mathit{Y}\left(\mathit{t}\right)=\mathit{\hspace{0.17em}}{Y}_0+\mathit{B}\left(\mathit{t}\right)+\mathit{\theta t}$ is more likely to exit the same exit boundaries than $\mathit{X}\left(\mathit{t}\right)$. Then $\mathit{Y}\left(\mathit{t}\right)$can be used to select the exit/critical boundaries, and these boundaries can be applied to $\mathit{X}\left(\mathit{t}\right)$to control type I error. Consider $W_m^{\left(0\right)}=B_m^{\left(0\right)}+{\theta }_m{r}^{\left(0\right)}{ }_{}\mathit{N}\left({\theta }_m{r}^{\left(0\right)},r^{\left(0\right)}\right)$, m = 1,…,M, as the $X_i$‘s in Slepian’s lemma. Then $\mathit{E}\left(W_m^{\left(0\right)}{W}_l^{\left(0\right)}\right)=\mathit{E}\left(B_m^{\left(0\right)}{B}_l^{\left(0\right)}\right)$, and $\mathit{E}\left(X_i^2\right)=\mathit{E}\left(W_i^{\left(0\right)}{W}_i^{\left(0\right)}\right)=\mathit{E}\left(B_i^{\left(0\right)}{B}_i^{\left(0\right)}\right)=r^{\left(0\right)}$. Let $B_{m,ind}^{\left(0\right)}\left(r^{\left(0\right)}\right){ }_{}\mathit{N}\left({\theta }_m{r}^{\left(0\right)},r^{\left(0\right)}\right)$, $\mathit{m}=1,\dots ,\mathit{M}$ be independent Brownian motions, and $W_{m,ind}^{\left(0\right)}=B_{m,ind}^{\left(0\right)}+{\theta }_m{r}^{\left(0\right)}$. Let $W_{i,ind}^{\left(0\right)}\left(r^{\left(0\right)}\right)$ be the $Y_i$’s in Slepian’s lemma. Then $\mathit{E}\left(Y_i^2\right)=\mathit{E}\left(W_{i,ind}^{\left(0\right)}{W}_{i,ind}^{\left(0\right)}\right)=\mathit{E}\left(B_{i,ind}^{\left(0\right)}{B}_{i,ind}^{\left(0\right)}\right)=r^{\left(0\right)}=\mathit{E}\left(X_i^2\right)$. Further, $0=\mathit{E}\left(W_{m,ind}^{\left(0\right)}{W}_{l,ind}^{\left(0\right)}\right)=\mathit{E}\left(B_{m,ind}^{\left(0\right)}{B}_{l,ind}^{\left(0\right)}\right)=\mathit{E}\left(Y_m{Y}_l\right)$. For all potential options/comparisons to be considered in a two-stage trial, it is reasonable to assume that $\mathit{E}\left(B_m^{\left(0\right)}{B}_l^{\left(0\right)}\right)\ge 0$, or that the endpoints in the phase 2 stage are non-negatively correlated. This assumption is in agreement with the opinion that (FDA guidance, 2019) “Most secondary endpoints in clinical trials are correlated with the primary endpoint, often very highly correlated). This assumption is independent of the selection rule. Let $B_{m,ind}^{\left(0\right)}\left(r^{\left(0\right)}\right)$, $\mathit{m}=1,\dots ,\mathit{M}$ be independent Brownian motions, and $W_{m,ind}^{\left(0\right)}=B_{m,ind}^{\left(0\right)}+{\theta }_m{r}^{\left(0\right)}$. Let $B_{i,ind}^{\left(0\right)}\left(r^{\left(0\right)}\right)$ be the $Y_i$‘s in Slepian’s lemma. Then $0=\mathit{E}\left(B_{m,ind}^{\left(0\right)}{B}_{l,ind}^{\left(0\right)}\right)\le \mathit{E}\left(B_m^{\left(0\right)}{B}_l^{\left(0\right)}\right)$. Hence, $\mathit{E}\left(Y_m{Y}_l\right)\le 0\le \mathit{E}\left(B_m^{\left(0\right)}{B}_l^{\left(0\right)}\right)=\mathit{E}\left(X_m{X}_l\right)$, and the conditions for Slepian’s lemma are met. Thus, for all real numbers $u_1,\dots ,u_m$: $\mathit{P}\left\{\left(\underset{\mathit{m}=1}{\overset{M}{\cup }}{Y}_m>u_m\right)\right\}\ge \mathit{P}\left\{\left(\underset{\mathit{m}=1}{\overset{M}{\cup }}{X}_m>u_i\right)\right\}$, i.e., $\mathit{P}\left\{\left(\underset{\mathit{m}=1}{\overset{M}{\cup }}{W}_{m,ind}^{\left(0\right)}>u_m\right)\right\}\ge \mathit{P}\left\{\left(\underset{\mathit{m}=1}{\overset{M}{\cup }}{W}_m^{\left(0\right)}>u_m\right)\right\}$, this leads to $\mathit{P}\left(\mathit{max}\left(W_{m,ind}^{\left(0\right)},\dots ,W_{M,ind}^{\left(0\right)}\mathit{\hspace{0.17em}}\right)>\mathit{x}\right)\ge \mathit{P}\left(\mathit{max}\left(W_m^{\left(0\right)},\dots ,W_M^{\left(0\right)}\mathit{\hspace{0.17em}}\right)>\mathit{x}\right)$. Let $X_{0,\mathit{ind}}=\mathit{max}\left(W_{m,ind}^{\left(0\right)},\dots ,W_{M,ind}^{\left(0\right)}\mathit{\hspace{0.17em}}\right)$, $X_{0,\mathit{max}}=\mathit{max}\left(W_m^{\left(0\right)},\dots ,W_M^{\left(0\right)}\mathit{\hspace{0.17em}}\right)$. Let $X_0$ be the selected $W_n^{\left(0\right)}$ at the end of the phase 2 (regardless of the selection rule). Then $X_{0,\mathit{ind}}$ dominates $X_{0,\mathit{max}}$ per the above discussion., and $X_{0,\mathit{max}}$ dominates $X_0$ (regardless of the selection rule). Type I error (FWE) can be controlled in all situations (treatment selection of any distribution, population enrichment, and endpoint selection, or mixed selection) as follows: i) If the covariance matrix is known, but the selection rule is not strict, let $Y_0=X_{0,\mathit{max}}$ and apply the dominance theorem; ii) If the covariance matrix is unknown, and the selection rule is not strict, let $Y_0=X_{0,\mathit{ind}}$ and apply the dominance theorem. The critical boundaries $c_0^{\left(0\right)}$, $c_1^{\left(0,1\right)},\dots ,c_K^{\left(0,1\right)}$, for a sequential testing on X(t) can be determined by requiring that $\mathit{P}\left\{Y_0\ge c_0^{\left(0\right)}\sqrt{r^{\left(0\right)}}\mathit{\hspace{0.17em}}){\cup }_{i=1}^K\left(\mathit{Y}\left(r_i^{\left(0,1\right)}\right)\ge c_K^{\left(0,1\right)}\sqrt{r_i^{\left(0,1\right)}}\right)\right\}\le \mathit{\alpha }/2$. Then, regardless the selection rule, or the covariance matrix, P $\left\{\left(X_0\ge c_0^{\left(0\right)}\sqrt{r^{\left(0\right)}}\mathit{\hspace{0.17em}}\right)\cup ({\cup }_{i=1}^K\left(\mathit{X}\left(r_i^{\left(0,1\right)}\right)\ge c_i^{\left(0,1\right)}\sqrt{r_i^{\left(0,1\right)}}\right)\right\}\le \mathit{P}$ $\left\{Y_0\ge c_0^{\left(0\right)}\sqrt{r^{\left(0\right)}}\mathit{\hspace{0.17em}})\cup (\underset{\mathit{i}=1}{\overset{K}{\cup }}\left(\mathit{Y}\left(r_i^{\left(0,1\right)}\right)\ge c_i^{\left(0,1\right)}\sqrt{r_i^{\left(0,1\right)}}\right)\right\}\le \mathit{\alpha }/2$

3.4. Adaptive sequential testing (ATSSSD)

In a TSSSD, the power calculation is based on assumptions that may or may not reflect the true efficacy, and the comparison with the best efficacy may not get selected even if the rule is play-the-winner (Figures 3 and 5 provide examples for such probabilities). Hence, it may be desirable to perform sample size re-estimation at an interim analysis in the phase 3 stage (preferably using effect size estimated from phase 3 data only, which is unbiased). To facilitate the sample size change discussion we use the superscript ⁽²⁾ to indicate statistics, parameters after the sample size change in the phase 3 stage. For example, the Information times in the phase 3 stage after sample size change will be $t_i^{\left(2\right)}$, the combined phase 2 and 3 time points will be denoted $t_i^{\left(0,2\right)}$, the new critical and exit boundaries will be denoted as $c_i^{\left(0,2\right)}$, $e_i^{\left(0,2\right)}$, respectively. The exit boundaries $e_1^{\left(0,2\right)},\dots ,$ $e_{K_2}^{\left(0,2\right)}$ should maintain type I error control, the selection of which is mathematically identical to sample size modification for adaptive sequential designs (Gao et al. 2008), the only difference is the notations. Details of calculations are provided in the online supporting material.

Figure 3. Trial Design with two treatments and three looks, common control, and normal distribution – DACT.

3.4.1. Conditional power and conditional type I error

Suppose that $\mathit{X}\left(t_L^{\left(0,1\right)}\right)=x_L^{\left(0,1\right)}$ is observed at the L-th interim analysis. Let $\mathit{\theta }$ be the effect size of the selected comparison. The first step in sample size re-estimation is to calculate the conditional power with the planned sample size.

• Under the alternative hypothesis $H_a$: θ > 0. W(t) = B(t) +$\mathit{\theta t}\sim \mathit{N}\left(\mathit{\theta t},\mathit{t}\right)$. The conditional power at information times $t_{L+1}^{\left(0,1\right)}<\dots <t_{K_1}^{\left(0,1\right)}$ is $\mathit{P}\left(\mathit{\theta },e_{L+1}^{\left(0,1\right)},\dots ,e_{K_1}^{\left(0,1\right)}|\mathit{\hspace{0.17em}}\mathit{X}\left(t_L^{\left(0,1\right)}\right)=x_L^{\left(0,1\right)}\right)$

$=\mathit{P}{\left\{\underset{\mathit{i}=1}{\overset{K_1-\mathit{L}}{\cup }}\left(\mathit{X}\left(t_{L+i}^{\left(0,1\right)}\right)\ge e_{L+i}^{\left(0,1\right)}\right)|\mathit{X}\left(t_L^{\left(0,1\right)}\right)=x_L^{\left(0,1\right)}\right\}}_{\mathit{\theta }}$. Conditional type I error can be obtained by setting θ = 0, ${\alpha }_c^{\left(1\right)}=\mathit{P}{\left\{\underset{\mathit{i}=\mathit{L}+1}{\overset{K_1}{\cup }}\left(\mathit{X}\left(t_i^{\left(0,1\right)}\right)\ge e_i^{\left(0,1\right)}\right)|\mathit{X}\left(t_L^{\left(0,1\right)}\right)=x_L^{\left(0,1\right)}\right\}}_0$. If the conditional power is low or too high, then sample size can be either increased or decreased.

• Suppose that the sample size is to be modified with new information times $t_1^{\left(0,2\right)},\dots ,$ $t_{K_2}^{\left(0,2\right)}$, and new exit boundaries $e_1^{\left(0,2\right)},\dots ,$ $e_{K_2}^{\left(0,2\right)}$ chosen. The conditional power ($t_i^{\left(0,2\right)}$ = $t^{\left(0\right)}$ + $t_i^{\left(2\right)}$) is (similar to the above) $\mathit{P}\left(\mathit{\theta },e_1^{\left(0,2\right)},\dots ,{>}_{K_2}^{\left(0,2\right)}|\mathit{\hspace{0.17em}}\mathit{X}\left(t_L^{\left(0,1\right)}\right)=x_L^{\left(0,1\right)}\right)=$

${\left\{\underset{\mathit{i}=1}{\overset{K_2}{\cup }}\left(\mathit{X}\left(t_i^{\left(0,2\right)}\right)\ge e_i^{\left(0,2\right)}\right)|\mathit{X}\left(t_L^{\left(0,1\right)}\right)=x_L^{\left(0,1\right)}\right\}}_{\mathit{\theta }}$. This is an increasing function of $T_2=t_{K_2}^{\left(0,2\right)}$. Desired power (e.g., $1-\mathit{\beta }$) can be obtained by choosing an appropriately large $T_2$. The new conditional type I error is obtained by setting θ = 0, ${\alpha }_c^{\left(2\right)}=\mathit{P}{\left\{\underset{\mathit{i}=1}{\overset{K_2}{\cup }}\left(\mathit{X}\left(t_i^{\left(0,2\right)}\right)\ge e_i^{\left(0,2\right)}\right)|\mathit{X}\left(t_L^{\left(0,1\right)}\right)=x_L^{\left(0,1\right)}\right\}}_0$.

3.4.2. Adjusting the exit boundaries after sample size modification

To maintain the total type I error, it is sufficient to maintain the conditional type I error (same as in Gao, Ware, Mehta 2008), i.e. if the future exit boundaries $e_1^{\left(0,2\right)},\dots ,$ $e_{K_2}^{\left(0,2\right)}$ are chosen such that ${\alpha }_c^{\left(2\right)}={\alpha }_c^{\left(1\right)}$, then the overall type I error will be maintained. Details of the calculation are provided in the online supporting material.

3.5. Inference

3.5.1. Conservative lower confidence limit and $\mathit{P}$-value

Let $\vec{\theta }=\left({\theta }_1,\dots ,{\theta }_M\right)=\left(\mathit{\theta },\dots ,\mathit{\theta }\right)$, and denote it as ${\vec{\theta }}_{=}=\left(\mathit{\theta },\dots ,\mathit{\theta }\right)$. Suppose that the trial ended at $t_I^{\left(0,1\right)}$, with $\mathit{X}\left(t_I^{\left(0,1\right)}\right)=x_I^{\left(0,1\right)}=z_I^{\left(0,1\right)}\sqrt{t_I^{\left(0,1\right)}}$. Let $f_X\left(\mathit{\theta }\right)=\mathit{P}{\left\{\left(X_0\ge c_0^{\left(0\right)}\sqrt{t^{\left(0\right)}}\mathit{\hspace{0.17em}}\right)\cup (\underset{\mathit{i}=1}{\overset{\mathit{I}-1}{\cup }}\left(\mathit{X}\left(t_i^{\left(0,1\right)}\right)\ge c_i^{\left(0,1\right)}\sqrt{t_i^{\left(0,1\right)}}\right)\cup \left(\mathit{X}\left(t_I^{\left(0,1\right)}\right)\ge z_I^{\left(0,1\right)}\sqrt{t_I^{\left(0,1\right)}}\right)\right\}}_{{\vec{\theta }}_{=}}$, $f_X\left(\mathit{\theta }\right)=\mathit{P}{\left\{\left(Y_0\ge c_0^{\left(0\right)}\sqrt{t^{\left(0\right)}}\mathit{\hspace{0.17em}}\right)\cup (\underset{\mathit{i}=1}{\overset{\mathit{I}-1}{\cup }}\left(\mathit{Y}\left(t_i^{\left(0,1\right)}\right)\ge c_i^{\left(0,1\right)}\sqrt{t_i^{\left(0,1\right)}}\right)\cup \left(\mathit{Y}\left(t_I^{\left(0,1\right)}\right)\ge z_I^{\left(0,1\right)}\sqrt{t_I^{\left(0,1\right)}}\right)\right\}}_{{\vec{\theta }}_{=}}$Where $\mathit{Y}\left(\mathit{t}\right)=Y_0+\mathit{Y}\left(\mathit{t}\right)$, $Y_0=X_{0,\mathit{max}}$ if the covariance matrix is known, and $Y_0=X_{0,\mathit{ind}}$ if the covariance matrix is not known. Then both $f_X\left(\mathit{\theta }\right)$ and $f_Y\left(\mathit{\theta }\right)$ are increasing functions of $\mathit{\theta }$ and $f_X\left(\mathit{\theta }\right)\le f_Y\left(\mathit{\theta }\right)$ (per the dominance theorem). Hence for any $0<\mathit{\gamma }<1$, $f_Y^{-1}\left(\mathit{\gamma }\right)\le f_X^{-1}\left(\mathit{\gamma }\right)$. In particular, $f_Y^{-1}\left(\mathit{\alpha }/2\right)\le f_X^{-1}\left(\mathit{\alpha }/2\right)$. Therefore, ${\theta }_{\mathit{\alpha }/2,two-s\mathrm{tage}}={\theta }_{\mathit{\alpha }/2}=f_Y^{-1}\left(\mathit{\alpha }/2\right)$ is a conservative lower confidence limit, and $\mathit{p}=f_Y\left(0\right)$ is a conservative $\mathit{p}$- value. Further, $\mathit{p}<\mathit{\alpha }/2$ if and only if ${\theta }_{\mathit{\alpha }/2}>0$.

3.5.2. Unbiased point estimate and exact upper confidence limit

An unbiased estimate of θ = ${\theta }_{\mathit{selected}}={\theta }_n\mathit{\hspace{0.17em}}$ from phase 2 data (hence from combined phase 2/3 data) is not possible due to the bias from the selection (a multiple comparison) procedure. However, if only the phase 3 data is used, a median unbiased point estimate ${\hat{\theta }}_{\mathit{selected}}={\hat{\theta }}_{\mathit{phase}3}$ and exact upper confidence limit ${\theta }_{1-\mathit{\alpha }/2,p\mathrm{hase}3}$ can be obtained (Gao et al. 2013). Therefore, we propose to use ${\hat{\theta }}_{\mathit{phase}3}$ as the point estimate, and (${\theta }_{\mathit{\alpha }/2,two-s\mathrm{tage}}$, ${\theta }_{1-\mathit{\alpha }/2,p\mathrm{hase}3}$) as the conservative two-sided confidence interval. Mathematically, the method is exactly the same as in Gao, Liu, Mehta (2013) (see section 5.2 for explanations). To obtain ${\hat{\theta }}_{\mathit{phase}3}$, it is necessary to associate a separate group sequential design for phase 3. The design includes information time points for interim in the phase 3 stage are $t_1^{\left(1\right)}<\dots <t_{K_1}^{\left(1\right)},$ the pre-selected separate phase 3 stage exit/critical boundaries $e_1^{\left(1\right)},\dots ,e_{K_1}^{\left(1\right)},$ and $c_1^{\left(1\right)},\dots ,c_{K_1}^{\left(1\right)}$. The relative information fraction for phase 3 are $s_i^{\left(1\right)}=t_i^{\left(1\right)}/t_{K_1}^{\left(1\right)}$ (see Figures 1 and 3). Let

${\mathrm{\Sigma }}_{{\mathit{s}}^{\left(1\right)},\mathit{i}}=\left(\begin{array}{cc}\begin{array}{cc}{s}_1^{\left(1\right)}& s_1^{\left(1\right)}\\ s_1^{\left(1\right)}& s_2^{\left(1\right)}\end{array}& \begin{array}{cc}\dots & s_1^{\left(1\right)}\\ \dots & s_2^{\left(1\right)}\end{array}\\ \begin{array}{cc}\dots & \dots \\ s_1^{\left(1\right)}& s_2^{\left(1\right)}\end{array}& \begin{array}{cc}\dots & \dots \\ \dots & s_i^{\left(1\right)}\end{array}\end{array}\right)$

Suppose that the trial terminated at$\mathit{I}$ -th interim analysis in the phase 3 stage (information time $t_I^{\left(1\right)}$), with $\mathit{Z}\left(t_I^{\left(1\right)}\right)=z_I^{\left(1\right)}$. Let

$\mathit{g}\left(\mathit{\theta }\right)=1{-}_{-\mathrm{\infty }}^{c_1^{\left(1\right)}\sqrt{s_1^{\left(1\right)}}-\mathit{\theta }\sqrt{t_{K_1}^{\left(1\right)}}{s}_1^{\left(1\right)}}{\dots }_{-\mathrm{\infty }}{c_{I-1}^{\left(1\right)}\sqrt{s_{I-1}^{\left(1\right)}}-\mathit{\theta }\sqrt{t_{K_1}^{\left(1\right)}}{s}_{I-1}^{\left(1\right)}}_{-\mathrm{\infty }}^{z_I^{\left(1\right)}\sqrt{s_I^{\left(1\right)}}-\mathit{\theta }\sqrt{t_{K_1}^{\left(1\right)}}{s}_I^{\left(1\right)}}\mathit{u}\left(\vec{0},\vec{x},{\mathrm{\Sigma }}_{{\mathit{s}}^{\left(1\right)},\mathit{I}}\right)\mathit{d}{x}_1\dots \mathit{d}{x}_I.$

$\mathit{g}\left(\mathit{\theta }\right)$ is an increasing function of $\mathit{\theta }$. Let ${\hat{\theta }}_{\mathit{phase}3}=g^{-1}\left(0.5\right)$ be the point estimate, and ${\theta }_{1-\mathit{\alpha }/2,p\mathrm{hase}3}=g^{-1}\left(1-\mathit{\alpha }/2\right)$ be the upper limit of the (1- α) × 100% confidence interval data. Then ${\hat{\theta }}_{\mathit{phase}3}$ is a median unbiased estimator, and $\mathit{P}\left(\mathit{\theta }\ge {\theta }_{1-\mathit{\alpha }/2,p\mathrm{hase}3}\right)=\mathit{\alpha }/2$, i.e., the upper limit is exact (e.g., see Gao, Liu, Mehta 2013).

3.6. Inference with adaptation

3.6.1. Backward image

To obtain the statistics for inference, it is necessary to derive a backward image (see Gao, Liu, Mehta 2013, section 4) first. The backward image can be obtained in exactly the same way as in Gao et al. (2013) as follows (see section 5.2 for explanations): Suppose that the trial terminated at $I^{\left(2\right)}$-th interim analysis (information time $t_{{\mathit{I}}^{\left(2\right)}}^{\left(0,2\right)}$), with X($t_{{\mathit{I}}^{\left(2\right)}}^{\left(0,2\right)}$) = $x_{{\mathit{I}}^{\left(2\right)}}^{\left(0,2\right)}$. Then, there is a unique $J_{\theta }^{\left(0,1\right)},$ and a unique $x_{J_{\theta }^{\left(0,1\right)}}^{\left(0,1\right)}$ such that $\mathit{P}\left\{\underset{\mathit{i}=1}{\overset{{\mathit{I}}^{\left(2\right)}-1}{\cup }}\left(\mathit{X}\left(t_i^{\left(0,2\right)}\right)\ge e_i^{\left(0,2\right)}\right)\cup \left(\mathit{\hspace{0.17em}}\mathit{X}\left(t_{{\mathit{I}}^{\left(2\right)}}^{\left(0,2\right)}\right)\ge x_{{\mathit{I}}^{\left(2\right)}}^{\left(0,2\right)}\right)|\mathit{X}\left(t_L^{\left(0,1\right)}\right)=x_L^{\left(0,1\right)}\right\}=\mathit{P}\left\{\underset{i=L+1}{\overset{J_{\theta }^{\left(0,1\right)}-1}{\cup }}\left(\mathit{X}\left(t_i^{\left(0,1\right)}\right)\ge e_i^{\left(0,1\right)}\right)\cup \left(\mathit{X}\left(t_{J_{\theta }^{\left(0,1\right)}}^{\left(0,1\right)}\right)\ge x_{J_{\theta }^{\left(0,1\right)}}^{\left(0,1\right)}\right)|\mathit{X}\left(t_L^{\left(0,1\right)}\right)=x_L^{\left(0,1\right)}\right\}$. The pair $\left(t_{J_{\theta }^{\left(0,1\right)}}^{\left(0,1\right)},x_{J_{\theta }^{\left(0,1\right)}}^{\left(0,1\right)}\right)$ is the backward image of $\left(t_{{\mathit{I}}^{\left(2\right)}}^{\left(0,2\right)},x_{{\mathit{I}}^{\left(2\right)}}^{\left(0,2\right)}\right)$, given $\mathit{X}\left(t_L^{\left(0,1\right)}\right)=x_L^{\left(0,1\right)}$.

3.6.2. Lower confidence limit and p-value

$\mathrm{Let}{f}_X\left(\mathit{\theta }\right)=\mathit{P}\left(X_0\ge e_0^{\left(0\right)}\right)\cup \mathit{i}=1J_{\theta }^{\left(0,1\right)}-1\left(\mathit{X}\left(t_i^{\left(0,1\right)}\right)\ge e_i^{\left(0,1\right)}\right)\cup \left(\mathit{X}\left(t_{J_{\theta }^{\left(0,1\right)}}^{\left(0,1\right)}\right)\ge x_{J_{\theta }^{\left(0,1\right)}}^{\left(0,1\right)}\right).$

Let $\mathit{Y}\left(\mathit{t}\right)=Y_0+\mathit{Y}\left(\mathit{t}\right)$, where $Y_0$ is chosen as above and define $f_Y\left(\mathit{\theta }\right)$ similarly. Per earlier discussion regarding the application of the dominance theorem, for any $0<\mathit{\gamma }<1$, $f_Y^{-1}\left(\mathit{\gamma }\right)\le f_X^{-1}\left(\mathit{\gamma }\right)$. A conservative $\mathit{p}$-value and lower confidence bound can be obtained as: i) $\mathit{p}=f_Y\left(0\right)$, ${\theta }_{\mathit{\alpha }/2,two-s\mathrm{tage}}={\theta }_{\mathit{\alpha }/2}=f_Y^{-1}\left(\mathit{\alpha }/2\right)$. The p-value is consistent with ${\theta }_{\mathit{\alpha }/2},$ in the sense that $\mathit{p}\le \mathit{\alpha }/2$ if and only if ${\theta }_{\mathit{\alpha }/2}$ ≥0.

3.6.3. Unbiased point estimate and exact upper confidence limit

Unbiased point estimate and upper confidence interval can be obtained, in exactly the same way as in Gao, Liu, Mehta (2013, as follows (see section 5.2 for explanations): Let $t_i^{\left(2\right)}=t_i^{\left(0,2\right)}-t^{\left(0\right)}$ be the information time for the phase 3 stage. Let X($t_{{\mathit{I}}^{\left(2\right)}}^{\left(2\right)}$) = $x_{{\mathit{I}}^{\left(2\right)}}^{\left(2\right)}=x_{{\mathit{I}}^{\left(2\right)}}^{\left(0,2\right)}-x_L^{\left(0,1\right)}$ be the observed value for the phase 3 stage score function. Let $e_i^{\left(2\right)}=e_i^{\left(0,2\right)}-x_0$, where $x_0$ is the observed value of $X_0$. Then, there is a unique $J_{\theta }^{\left(1\right)}$ and a unique $x_{J_{\theta }^{\left(1\right)}}^{\left(1\right)}$ such that $\begin{array}{rl}& \mathit{P}\left\{\underset{\mathit{i}=1}{\overset{{\mathit{I}}^{\left(2\right)}-1}{\cup }}\left(\mathit{X}\left(t_i^{\left(2\right)}\right)\ge e_i^{\left(2\right)}\right)\cup \left(\mathit{X}\left(t_{{\mathit{I}}^{\left(2\right)}}^{\left(2\right)}\right)\ge x_{{\mathit{I}}^{\left(2\right)}}^{\left(2\right)}\right)|\mathit{X}\left(t_L^{\left(1\right)}\right)=x_L^{\left(0,1\right)}-x_0\right\}\\ & =\mathit{P}\left\{\underset{i=L+1}{\overset{J_{\theta }^{\left(1\right)}-1}{\cup }}\left(\mathit{X}\left(t_i^{\left(1\right)}\right)\ge e_i^{\left(1\right)}\right)\left(\mathit{X}\left(t_i^{\left(1\right)}\right)\ge e_i^{\left(1\right)}\right)\cup \left(\mathit{X}\left(t_{J_{\theta }^{\left(1\right)}}^{\left(1\right)}\right)\ge x_{J_{\theta }^{\left(1\right)}}^{\left(1\right)}\right)|\mathit{X}\left(t_L^{\left(1\right)}\right)=x_L^{\left(0,1\right)}-x_0\right\}\end{array}$. The pair $\left(t_{J_{\theta }^{\left(1\right)}}^{\left(1\right)},>x_{J_{\theta }^{\left(1\right)}}^{\left(1\right)}\right)$ is the phase 3 stage backward image of $\left(t_{{\mathit{I}}^{\left(2\right)}}^{\left(2\right)},x_{{\mathit{I}}^{\left(2\right)}}^{\left(2\right)}\right)$, given $\mathit{X}\left(t_L^{\left(1\right)}\right)=x_L^{\left(0,1\right)}-x_0$. Let $\mathit{Z}\left(t_{\hspace{0.17em}{J}_{\theta }^{\left(1\right)}}^{\left(1\right)}\right)=z_{\hspace{0.17em}{J}_{\theta }^{\left(1\right)}}^{\left(1\right)}=x_{J_{\theta }^{\left(1\right)}}^{\left(1\right)}/\sqrt{t_{J_{\theta }^{\left(1\right)}}^{\left(1\right)}}.\mathrm{Let}\mathit{g}\left(\mathit{\theta }\right)=1-{\int }_{-\mathrm{\infty }}^{c_1^{\left(1\right)}\sqrt{s_1^{\left(1\right)}}-\mathit{\theta }\sqrt{t_{K_1}^{\left(1\right)}}{s}_1^{\left(1\right)}}$

${\int }_{-\mathrm{\infty }}^{c_{J_{\theta }^{\left(1\right)}-1}^{\left(1\right)}\sqrt{s_{J_{\theta }^{\left(1\right)}-1}^{\left(1\right)}}-\mathit{\theta }\sqrt{t_{K_1}^{\left(1\right)}}{s}_{J_{\theta }^{\left(1\right)}-1}^{\left(1\right)}}{\int }_{-\mathrm{\infty }}^{z_{J_{\theta }^{\left(1\right)}}^{\left(1\right)}\sqrt{s_{J_{\theta }^{\left(1\right)}}^{\left(1\right)}}-\mathit{\theta }\sqrt{t_{K_1}^{\left(1\right)}}{s}_{J_{\theta }^{\left(1\right)}}^{\left(1\right)}}\mathit{u}\left(\vec{0},\vec{x},{\mathrm{\Sigma }}_{{\mathit{s}}^{\left(1\right)},J_{\theta }^{\left(1\right)}}\right)\mathit{d}{x}_1\dots \mathit{d}{x}_{J_{\theta }^{\left(1\right)}}$

$\mathit{g}\left(\mathit{\theta }\right)$ is an increasing function of $\mathit{\theta }$. Let ${\hat{\theta }}_{\mathit{phase}3}=g^{-1}\left(0.5\right)$ be the point estimate, and ${\theta }_{1-\mathit{\alpha }/2,p\mathrm{hase}3}=g^{-1}\left(1-\mathit{\alpha }/2\right)$ be the upper limit of the (1– α) × 100% confidence interval data. Then ${\hat{\theta }}_{\mathit{phase}3}$ is a median unbiased estimator, and $\mathit{P}\left(\mathit{\theta }\ge {\theta }_{1-\mathit{\alpha }/2,p\mathrm{hase}3}\right)=\mathit{\alpha }/2$, i.e., the upper limit is exact (e.g., see Gao, Liu, Mehta 2013).

3.6.4. Repeated sample size modification

If desired, similar to the situation in adaptive sequential design (e.g., Gao, Liu, Mehta 2008) sample size modifications can be repeated any number of times. The details are provided in the online supporting material.

3.7. Special case $\mathit{L}={\mathit{K}}_{\mathbf{1}}-\mathbf{1}$ and ${\mathit{K}}_{\mathbf{2}}=\mathbf{1}$

In general, the calculation of conditional probabilities requires numerical integrations. In a special, and likely the most common case, all statistics for sample size modification, such as conditional power, conditional type I error at interim analysis, new sample size, and new critical boundaries can all be expressed using closed form formulas. This is the case in which the sample size modification is performed at the penultimate interim analysis and there is no more interim analysis after the sample size change. Hence, $\mathit{L}=K_1-1$ and $K_2=1$. The calculations are exactly the same as in Gao, Ware, Mehta (2008). The backward image can also be calculated using a formula, exactly the same as in Gao et al. (2013). Details of calculations are provided in the online supporting material.

3.7.1. Sample size re-estimation

Suppose that $\mathit{X}\left(t_{K_1-1}^{\left(0,1\right)}\right)=x_{K_1-1}^{\left(0,1\right)}$ has been observed. Let $\hat{\theta }$ and $\mathit{s}.\mathit{e}.\left(\hat{\theta }\right)$ be the naïve point estimate and the standard error. Then $x_{K_1-1}^{\left(0,1\right)}=\hat{\theta }{\left[\mathit{\hspace{0.17em}}s.e.\left(\hat{\theta }\right)\right]}^{-2}$. The conditional power is $\mathrm{\Phi }\left(\hat{\theta }\sqrt{t_{K_1}^{\left(1\right)}-t_{K_1-1}^{\left(1\right)}}-\left(e_{K_1}^{\left(0,1\right)}-x_{K_1-1}^{\left(0,1\right)}\right)/\sqrt{t_{K_1}^{\left(1\right)}-t_{K_1-1}^{\left(1\right)}}\right)$. The required new information time that provides $\left(1-\mathit{\beta }\right)\times 100\mathrm{\%}$ conditional power is $t_1^{\left(0,2\right)}\ge {\left(\hat{\theta }\right)}^{-2}{\left(\left(e_{K_1}^{\left(0,1\right)}-x_{K_1-1}^{\left(0,1\right)}\right)/\sqrt{t_{K_1}^{\left(1\right)}-t_{K_1-1}^{\left(1\right)}}+{\mathrm{\Phi }}^{-1}\left(1-\mathit{\beta }\right)\right)}^2+t_{K_1-1}^{\left(0,1\right)}$. (Preferably, the $\hat{\theta }$ in this formula should be derived using the phase 3 data only). The new adjusted exit boundary is $e_1^{\left(0,2\right)}=\left\{\sqrt{t_1^{\left(0,2\right)}-t_{K_1-1}^{\left(0,1\right)}}/\sqrt{t_{K_1}^{\left(0,1\right)}-t_{K_1-1}^{\left(0,1\right)}}\right\}\left(e_{K_1}^{\left(0,1\right)}-x_{K_1-1}^{\left(0,1\right)}\right)+x_{K_1-1}^{\left(0,1\right)}$ and the new critical boundary is $c_1^{\left(0,2\right)}=e_1^{\left(0,2\right)}/\sqrt{t_1^{\left(0,2\right)}}$.

3.7.2. Backward image

In this case, the trial ends at $t_1^{\left(0,2\right)}$. Let the point estimate (using data from both stages) be ${\hat{\theta }}^{\left(0,1\right)}$, and the standard error be $\mathit{s}.\mathit{e}.\left({\hat{\theta }}^{\left(0,1\right)}\right)$. The final observation (using data from both stages) is $\mathit{X}\left(t_1^{\left(0,2\right)}\right)=x_1^{\left(0,2\right)}={\hat{\theta }}^{\left(0,1\right)}{\left[\mathit{\hspace{0.17em}}s.e.\left({\hat{\theta }}^{\left(0,1\right)}\right)\right]}^{-2}$. Also, $J_{\theta }^{\left(0,1\right)}=K_1$. For any $\mathit{\theta }$, the backward image is given as $x_{K_1,\mathit{\theta }}^{\left(0,1\right)}=\left\{\sqrt{t_{K_1}^{\left(0,1\right)}-t_{K_1-1}^{\left(0,1\right)}}/\sqrt{t_1^{\left(0,2\right)}-t_{K_1-1}^{\left(0,1\right)}}\right\}\left(x_1^{\left(0,2\right)}-x_{K_1-1}^{\left(0,1\right)}-\mathit{\theta }\left(t_1^{\left(0,2\right)}-t_{K_1-1}^{\left(0,1\right)}\right)\right)+x_{K_1-1}^{\left(0,1\right)}+\mathit{\theta }\left(t_{K_1}^{\left(0,1\right)}-t_{K_1-1}^{\left(0,1\right)}\right)$.

4. Examples and simulations

Probability calculations (e.g., critical boundaries, power, and sample size) and simulations require clearly pre-set rules. Play-the-winner is a clear rule, all calculations and simulations with this rule can be performed using the Markov process properties. If the rule is not play-the-winner, probability calculations may or may not be possible, depending on the complexity of the rules, even if the rule can be pre-set. For these reasons, probability calculations are provided only for play-the-winner rule in DACT. If a non-play-the winner rule is clearly specified, then customized simulations can be conducted to assess the probabilities and operating characteristics. There could be various reasons for not using the play-the-winner rule in practice, the most common reason may be the need for including safety profile (i.e., rate of adverse events) together with efficacy in trials involving dose selection. However, safety considerations often involve clinical judgement which can be difficult to pre-specify. For trials involving “surrogate or intermediate endpoint” (FDA. 2019), using play-the-winner on observations of the primary endpoint is precluded. Hence, only simulations using the play-the-winner rules are presented in this article. Type I error (FWE) control and conservativeness of probability calculations for non-play-the-winner rules are ensured by the dominance theorem.

If the play-the-winner rule is adopted and the distribution of the endpoint is normally distributed, then the FWE control is exact with the critical boundaries, and the power and sample size calculations are precise. If the selection rule is not play-the-winner, or if the endpoint is not normally distributed, then the FWE is conservatively controlled per the dominance theorem. For non-play-the-winner rules, due to the arbitrariness of the selection rule, the power and sample size can’t be precisely calculated, and the true power could be smaller than the actual power, and the sample size may have a power less than $1-\mathit{\beta }$. To ensure adequate power, sample size re-estimation may be conducted (and is recommended) in the phase 3 stage (where there is no bias due to selection).

4.1. Critical boundary selection and sample size calculation in trial design

In this section, two trial designs with three looks are presented. The first look is at the end of phase 2. One for a trial that compares two treatments with a common control and that the efficacy variable is normally distributed. The second trial design is for survival analysis endpoint. The covariance matrix for the first trial is known with off-diagonal covariance coefficients equal to 0.5, while that for the second trial is not. The critical boundaries for the second example are obtained by using the dominance theorem with a covariance matrix whose off-diagonal covariance coefficients equal to 0. These boundaries are therefore conservative, hence also applicable to any two comparisons, such as treatment selection with any non-normal distributions, or endpoint selection, or sub-population selection. In a TSSSD, several aspects of information are important: i) Critical boundaries for the combined phase II/III testing (for overall hypothesis testing, overall type I error (FWE) control, and lower confidence limit calculation); ii) Critical boundaries for phase 3 only (for unbiased point estimate) as a standalone trial; iii) Sample size per arm; iv) Probability of each test arm being observed as the best at end of phase 2 (whether the selection rule is to play the winner or not, understanding this probability can be helpful for trial planning); v) Power for each arm (this is the probability of the arm being the best at the end of phase 2, and rejecting the null hypothesis with this arm in either phase 2 or 3); vi) Conditional power for each arm if the arm showed best efficacy at the end of phase 2 (this is the probability of rejecting the null hypothesis conditional on a particular arm been selected for phase 3 testing). All these information are provided in the DACT output.

The information in Figure 3 involves many complicated calculations. It would be assuring if these calculations can be confirmed through simulations. In Figure 4, simulation results are presented for the design in Figure 3. The designing parameters (such as critical boundaries, and sample size) are as specified in Figure 3. The sample size for the control arm is 110. The simulation was repeated 100,000 times. The simulated probabilities (i.e., overall power, the probability of each treatment being the best at the end of phase 2, the power of each arm [i.e. the probability of being selected as the winner and rejecting the null hypothesis at the completion of the trial], and the conditional power for each arm after being selected as the best) match those calculated ones (within random error of simulations), confirming that the BMA approach mathematically precisely describes probability events of the seamless sequential designs.

Figure 4. Simulated probabilities-DACT.

It is noted that: i) The two-stage boundaries in Figure 5 are more conservative (higher) than that in Figure 3 for conservative type I error control, since the covariance matrix in Figure 5 (for trial 2) is not known, and the dominance theorem and Slepian’s lemma (Huffer 1986; Slepian 1962) are used; ii) The probability of the arm with lesser efficacy showing better efficacy is greater than zero (in both Figure 3 and 5), this is the probability of the “loser” being selected as the “winner”. The observed efficacy of the arm will very likely be biased. This suggests that unbiased estimate should not include phase 2 data (This is why there should be critical boundaries for phase 3 only).

Figure 5. Trial Design with two arms/regimens, survival analysis, three looks.

4.2. Type I error simulations (with sample size re-estimation)

To simulate type I error control, critical boundaries in Figure 3 is used for normal distribution and Figure 5 for survival analysis. Let the planned sample size be $N_p$, and the maximum sample size be $N_{\mathit{max}}$. Sample size is re-estimated at the second interim analysis. Let the futility threshold be ${\theta }_{\mathit{futi}}$. If the observed efficacy $\hat{\theta }$ is less than ${\theta }_{\mathit{futi}}$, the trial will continue to the planned sample size (immediate termination could be another option, but not used for the simulations here). If the conditional power is no less than 1 − β = 0.9, the trial will proceed to planned sample size. If the conditional power is less than 1 − β = 0.9, and the re-calculated sample size is less than $N_{\mathit{max}}$, then the trial will proceed to the new sample size. if the re-estimated sample size is greater than $N_{\mathit{max}}$, then the trial will proceed to planned sample size (proceed to $N_{\mathit{max}}$ could be another option). For the trial with normal distribution, ${\theta }_{\mathit{futi}}=0.05$, $N_p=100$ per arm, and $N_{\mathit{max}}=500$ per arm was used for the simulations. For the trial with survival analysis endpoint, ${\theta }_{\mathit{futi}}=-\mathit{log}\left(0.95\right)$ (i.e., the trial is futile if the observed hazards ratio is greater than 0.95), the planned sample size $N_p$ is 200 event, $N_{\mathit{max}}$ is 500 events. The simulation is repeated 100,000 times for normal distribution and 10,000 time for survival analysis. We note that the sample size modification rules in the simulations are chosen as an example for illustration, and the type I error control remains valid with any other simulation rules.

When (θ₁, θ₂) = (0,0) (or (HR₁, HR₂) = (1, 1)), any rejection of the null hypothesis results in a type I error. If θ₁ = 0 (or HR₁ = 1) but θ₂ > 0 (or HR₂ < 1), a rejection of the null hypothesis results in a type I error only when the dose associated with θ₁ = 0 (or HR₁ = 1) was selected at the end of the phase 2 stage. In Table 1, the selection rule is to play the winner, in which the treatment with the largest Wald statistics was chosen for phase 3 stage.

Table 1. Type I error simulations-play the winner.

Normal distribution, common control		Survival analysis
(θ1, θ2)	Type I error	(HR1, HR2)	Type I error
(0,0)	0.024721	(1,1)	0.0229
(0,0.2)	0.007555	(1,0.9)	0.0052
(0, 0.5)	0.001425	(1, 0.8)	0.002
(0,0.8)	6.5e-05	(1,0.7)	0.0002

In Table 2, three selection rules were simulated. One is to play the winner, another is to select the treatment with the second largest Wald statistics from phase 2 stage. Such a scenario could happen in a dose selection trial, and the dose with the larger Wald statistics was the higher dose but with higher toxicity, and the lower dose was chosen for lower toxicity. The third rule is random selection. This could happen when the selection was based on a surrogate or intermediate endpoint, whose order of efficacy does not exactly match that of the primary endpoint. By the dominance theorem, playing the winner will result in the highest rejection rate among all possible selection methods (see section 5.2 for explanations) under full null hypothesis of ${\theta }_i=0$ for all i. Hence, under the full null hypothesis, play the winner is associated with the highest type I error. Under partial null hypothesis, with some ${\theta }_i=0$ and other ${\theta }_i>0$, there is no dominance relationship of type I error between different selection rules, but all type I errors will be less than that under full null hypothesis (per the dominance theorem). Detailed explanations will be too tedious and won’t be presented here. However, proper type I error control is demonstrated through simulation results in Table 2.

Table 2. Type I error simulations: play the winner, selecting the 2^nd best, or random selection.

	Survival analysis
(HR1, HR2)	Type I error
	Play the winner	Select 2^nd best	Random selection
(1,1)	0.0224	0.0049	0.0141
(1,0.9)	0.0075	0.0041	0.007
(1, 0.8)	0.0054	0.0081	0.0064
(1,0.7)	0.0043	0.0097	0.0083

The simulations in Table 1 can be confirmed using the DACT software. The simulations in Table 2 require specific programming (included in the online support material).

4.3. Point estimates and confidence interval

Simulations are performed for dose selection studies. The selection rule for the simulations in Table 3 is for a two (active) dose (with one control) design in which the dose with the largest observed effect size (i.e. ${\hat{\theta }}_{\mathit{n}}^{\left(0\right)}$ = ${\hat{\theta }}_{\mathit{max}}^{\left(0\right)}$ = max{${\hat{\theta }}_1^{\left(0\right)},{\hat{\theta }}_2^{\left(0\right)}$}) will be selected at the end of the phase 2 stage. Point estimates and confidence interval coverage for ${\theta }_{\mathit{selected}}={\theta }_n\mathit{\hspace{0.17em}}$are presented. In a three-look design, one analysis is performed at the end of phase 2 stage, one interim analysis during the phase 3 stage, and one final analysis at the end of the trial. In our simulations, sample size re-estimation is performed after the second look in a three-look design.

Table 3. Bias and CI coverage, three looks, with sample size re-estimation at second look.

(θ1, θ2)	Coverage of 2 sided 95% CI (${\theta }_{\frac{\alpha }{2},\mathit{two}-\mathit{stage}}$, ${\theta }_{1-\frac{\alpha }{2},\mathit{phase}3}$)	Rate of ${\theta }_{\mathit{selected}}$ < ${\theta }_{\frac{\alpha }{2},\mathit{two}-\mathit{stage}}$	Rate of ${\theta }_{\mathit{selected}}$ > ${\theta }_{1-\frac{\alpha }{2},\mathit{phase}3}$	Median bias median(${\hat{\theta }}_{\mathit{phase}3}$ − ${\theta }_{\mathit{selected}}$)
(0,0)	0.95043	0.02454*	0.02503	−0.0001321661
(0,0.2)	0.95871	0.01536	0.02593	0.002318219
(0.1,0.2)	0.9603	0.01479	0.02491	0.003503076
(0.1,0.3)	0.96172	0.01313	0.02515	0.01666743
(0.15,0.43)	0.95143	0.02483	0.02374	−0.0234763

*The FWE was also exactly 0.02454, confirming the consistency of the test and the lower confidence limit.

The simulations confirm: i) The lower limit of the confidence interval is conservative, but consistent with the hypothesis testing; ii) The point estimate is median unbiased; iii) The upper confidence limit is exact.

4.4. A trial example

We use a hypothetical trial to demonstrate how to perform sample size re-estimation and final analysis. Suppose that an oncology trial is being conducted (see Figure 5 for the design), in which two treatments are compared to a common control. Suppose that the endpoint is PFS (Progression free survival), and the planned total number of events is 140. It is noted that the covariance matrix from phase 2 stage is not known. The selection rule does not require play- the-winner. Suppose that the trial is designed according to Figure 3 (for survival analysis, $\mathit{\theta }=-\mathit{log}\left(\mathit{HR}\right)$). Note that $K_1=2$ ($K_1$ is the number of looks in phase 3, total number of looks in the whole trial is $K_1+1$). Suppose that a sample size re-estimation is conducted at the second look in the TSSSD (first look in phase 3 stage). Hence, $\mathit{L}=1$ in our notations (however, the number 2 should be chosen when using DACT for the calculation, since it is the second interim analysis in the TSSSD). Suppose that the total number of events for the selected regimen and the control at the interim analysis is 80. Suppose that ${\hat{\theta }}^{\left(0,1\right)}=0.4$ ($\widehat{\mathit{HR}}=0.67$) and $\mathit{s}.\mathit{e}.({\hat{\theta }}^{\left(0,1\right)})=0.18$ (Using both phase 2 and 3 data), ${\hat{\theta }}^{\left(1\right)}=0.37$ ($\widehat{\mathit{HR}}=0.691$). $\mathit{s}.\mathit{e}.({\hat{\theta }}^{\left(1\right)})=0.2$ (Using phase 3 data only) have been observed. Accordingly, $z_1^{\left(0,1\right)}=0.4/0.18=$2.222. Since $z_1^{\left(0,1\right)}<c_1^{\left(0,1\right)}=$2.869285 at the interim analysis and the trial continues with a sample size re-estimation. We note that ${\hat{\theta }}^{\left(0,1\right)}$ can be biased due to the selection process at the end of phase 2, and ${\hat{\theta }}^{\left(1\right)}$ is unbiased since it is obtained using only phase 3 data. So the effect size is assumed to be $\hat{\theta }(1)=0.37$. Using DACT, the conditional power with the planned the sample size is 0.7716, and the new total number of events (from the remaining treatment and the control) required to achieve a new conditional power of 0.9 is $156$ (see Figure S1 in the online supporting material). The new critical boundary is $c_1^{\left(0,2\right)}=2.09$. Suppose that at the end of the trial with the new sample size, ${\hat{\theta }}^{\left(0,2\right)}=0.39$ and $\mathit{s}.\mathit{e}.({\hat{\theta }}^{\left(0,2\right)})=0.0923$ (using both phase 2 and 3 data), ${\hat{\theta }}^{\left(2\right)}=0.365$. $\mathit{s}.\mathit{e}.({\hat{\theta }}^{\left(2\right)})=0.1$ (Using phase 3 data only). Accordingly, $z_1^{\left(0,2\right)}=0.39/0.0923=4.2253\mathit{\hspace{0.17em}}$. Since $z_1^{\left(0,2\right)}>c_1^{\left(0,2\right)}$, the null hypothesis is rejected. The final p-value is $\mathit{p}=0.000176$.The point estimate and the conservative two-sided 95% confidence interval for $\mathit{\theta }=-\mathit{log}\left(\mathit{HR}\right)$ are 0.3657 (0.186661,0.568683) (readers can repeat these calculations using the DACT software).

5. Discussion

5.1. The role of the score function

The properties of the score function are well known (Jennison and Turnbull 1997). It is approximately a Brownian motion. Because of this property, it serves as a rigorous mathematical foundation for an ATSSSD procedure that satisfies the three statistical principles for adaptive designs (FDA guidance, 2019). By utilizing the dominance theorem and Slepian’s lemma, the procedure can be applied to a wide range of trials, including dose, treatment, end point selection, or a mixture selection, for any distributions. And it provides complete inference.

The Brownian motion properties of score function is based on large sample theory. Such properties may not hold well when the sample size is small and/or when the endpoint is non-Gaussian distributed. There is no clear threshold on sample size when the Brownian motion properties will hold. Simulations (which can be conducted using DACT) may be conducted on the operating characteristics of the design for non-Gaussian distributed endpoints, and/or when the sample size is not large.

5.2. Parallels and differences between seamless sequential and usual sequential designs

The adaptive seamless sequential design is fundamentally parallel to usual adaptive sequential design for phase 3 trials, but with some important differences.

Parallels:

• Both procedures are based on the score function on the accumulative data. The score function in a usual adaptive sequential design is $\mathit{W}\left(\mathit{t}\right)=\mathit{B}\left(\mathit{t}\right)+\mathit{\theta }{ }_{}\mathit{N}\left(\mathit{\theta t},\mathit{t}\right)$, where t is Fisher’s information time from the beginning of the trial. $\mathit{B}\left(\mathit{t}\right){ }_{}\mathit{N}\left(0,\mathit{t}\right)$ is a Brownian motion and a Markov process. Thus W(t) is a Brownian motion with drift θt. On the other hand, the accumulative data in a seamless sequential design is $\mathit{X}\left(\mathit{t}\right)=X_0+\mathit{W}\left(\mathit{t}\right)=X_0+\mathit{B}\left(\mathit{t}\right)+\mathit{\theta t}$, where $X_0$ is the random variable from the phase 2 stage, $\mathit{B}\left(\mathit{t}\right){ }_{}\mathit{N}\left(0,\mathit{t}\right)$), and t is Fisher’s information time from the beginning of the phase 3 stage. X(t) is a Markov process, but not a Brownian motion.

• Let $\vec{\theta }=\left({\theta }_1,\dots ,{\theta }_M\right)$ (for seamless design), $\vec{a}=\left(a_1,\dots ,a_K\right)$, $\vec{x}=\left(x_1,\dots ,x_K\right)$, and $\vec{t}=\left(t_1,\dots ,\mathit{t}\right)$. Both procedures are fundamentally based on an exit function of the form

  $\mathit{g}\left(\mathit{\theta },\vec{a},\vec{t}\right){=}_{-\mathrm{\infty }}^{a_1}{\dots }_{-\mathrm{\infty }}^{a_K}\mathit{u}\left(\mathit{\theta },\vec{x},\vec{t}\right)\mathit{d}{x}_1\dots \mathit{d}{x}_K$
  $\mathit{f}\left(\vec{\theta },\vec{a},\vec{t}\right)={\int }_{-\mathrm{\infty }}^{a_1}\dots {\int }_{-\mathrm{\infty }}^{a_K}\mathit{u}\left(\vec{\theta },\vec{x},\vec{t}\right)\mathit{d}{x}_1\dots \mathit{d}{x}_K$

where $\mathit{f}$(∙) is for the seamless design and g(∙) is for the usual sequential design. $\mathit{u}$(∙) is the joint density function of either $\vec{W}\left(\vec{t}\right)=\left(\mathit{W}\left(t_1\right),\dots ,\mathit{W}\left(t_K\right)\right)$ or $\vec{X}\left(\vec{t}\right)=\left(\mathit{X}\left(t_1\right),\dots ,\mathit{X}\left(t_K\right)\right)$). $\mathit{f}\left(\vec{\theta },\vec{a},\vec{t}\right)$ is the probability of $\vec{X}\left(\vec{t}\right)$ exiting the upper boundary $\vec{a}=\left(a_1,\dots ,a_K\right)$ at any $a_i$, and $\mathit{g}\left(\mathit{\theta },\vec{a},\vec{t}\right)$ is the exiting probability of $\vec{W}\left(\vec{t}\right)$. For any fixed $\vec{a}$ and $\vec{t}$, $\mathit{f}\left(\cdot ,\vec{a},\vec{t}\right)$ is an increasing function of each component of $\vec{\theta }=\left({\theta }_1,\dots ,{\theta }_M\right)$, $\mathit{g}\left(\cdot ,\vec{a},\vec{t}\right)$ is an increasing function of $\mathit{\theta }$. By setting $\vec{\theta }=\left({\theta }_1,\dots ,{\theta }_M\right)$ to be $\vec{\theta }=\left(\mathit{\theta },\dots ,\mathit{\theta }\right)$ in a seamless sequential design, this property is used to obtain ${\theta }_{\gamma }=f^{-1}\left(\mathit{\gamma }\right)$, ($g^{-1}\left(\mathit{\gamma }\right)$ for usual sequential design) for 0<$\mathit{\gamma }$ <1. ${\theta }_{0.5}$serves as the point estimate for $\mathit{\theta }$, and $\left({\theta }_{\mathit{\alpha }/2},{\theta }_{1-\mathit{\alpha }/2}\right)$ forms the (1−$\mathit{\alpha }$)×100% confidence interval for $\mathit{\theta }$. For any fixed $\mathit{\theta }$, $\mathit{f}\left(\vec{\theta },\cdot ,\vec{t}\right)$ or $\mathit{g}\left(\mathit{\theta },\cdot ,\vec{t}\right)$ is a decreasing function of each $a_i$. Setting $\mathit{\theta }$ = 0, or $\vec{\theta }$=(0, … ,0), this property is used to obtain critical boundaries for both sequential designs.

• Sample size re-estimation (SSR): the SSR for the seamless sequential design only involves conditional probabilities for phase 3, and only involves W(t), the tail of $\mathit{X}\left(\mathit{t}\right)=X_0+\mathit{W}\left(\mathit{t}\right)$. Similarly, the SSR of a usual adaptive design also involves W(t). Hence, the SSR for both adaptive sequential designs are exactly the same (e.g., Gao, Ware, Mehta 2008). The only difference is in notations.

Differences:

• The distribution for $\vec{W}\left(\vec{t}\right)$ is much simpler because W(t) is a Brownian motion. In a seamless design, the distribution of $X_0$ can be complex, depending on the configuration of $\vec{\theta }=\left({\theta }_1,\dots ,{\theta }_M\right)$, the correlation between the components of ${\vec{W}}^{\left(0\right)}=\left(W_{1,t_1^{\left(0\right)}}^{\left(0\right)},\dots ,W_{M,t_M^{\left(0\right)}}^{\left(0\right)}\right)$, the selection procedure in the phase 2 stage. In this article, a conservative approach is taken to make a conservative assumption that the selection is based on “play the winner” and obtain an $X_{0,\mathit{max}}=\left(W_{1,t_1^{\left(0\right)}}^{\left(0\right)},\dots ,W_{M,t_M^{\left(0\right)}}^{\left(0\right)}\right)$. This guarantees that ${\tilde{X}}_0\mathit{\hspace{0.17em}}\le X_{0,\mathit{max}}$ if ${\tilde{X}}_0$ is the result of any other selection procedure. $X_0$ may be influenced by unpredictable factors and hence may be unpredictable and unknown. Denote $f_{X_0}\left(\vec{\theta },\vec{a},\vec{t}\right)$ as the exit function for $\mathit{X}\left(\mathit{t}\right)=X_0+\mathit{W}\left(\mathit{t}\right)$. By the dominance theorem, $f_{{\tilde{X}}_0}\left(\vec{\theta },\vec{a},\vec{t}\right)\le f_{{\mathit{X}}_{0,\mathit{max}}}\left(\vec{\theta },\vec{a},\vec{t}\right)$. Hence, the critical boundaries obtained using $f_{{\mathit{X}}_{0,\mathit{max}}}\left(\vec{\theta },\vec{a},\vec{t}\right)$ are larger (and thus more conservative) than those obtained using $f_{{\tilde{X}}_0}\left(\vec{\theta },\vec{a},\vec{t}\right)$, ${\theta }_{\gamma }$’s obtained using $f_{{\mathit{X}}_{0,\mathit{max}}}\left(\vec{\theta },\vec{a},\vec{t}\right)$ are smaller (thus more conservative) than those obtained using $f_{{\tilde{X}}_0}\left(\vec{\theta },\vec{a},\vec{t}\right)$. The dominance theorem is the foundation of the seamless sequential design.

• In a usual sequential design that compares a test arm and a control arm, $\mathit{g}\left(\mathit{\theta },\vec{a},\vec{t}\right)$ is an increasing function of a single parameter θ. The estimates of ${\theta }_{\gamma }=g^{-1}\left(\mathit{\gamma }\right)$ is straightforward. In a seamless sequential design, $\mathit{f}\left(\vec{\theta },\vec{a},\vec{t}\right)$ is an increasing function of every component of $\vec{\theta }=\left({\theta }_1,\dots ,{\theta }_M\right)$. On condition that ${\theta }_n$ is the parameter for the selected option, then $\mathit{X}\left(\mathit{t}\right)=X_0+\mathit{W}\left(\mathit{t}\right)=X_0+\mathit{B}\left(\mathit{t}\right)+{\theta }_n\mathit{t}$. Note that $X_0$ depends on the configuration of $\vec{\theta }=\left({\theta }_1,\dots ,{\theta }_M\right)$ and the correlation between the components of ${\vec{W}}^{\left(0\right)}=\left(W_{1,t_1^{\left(0\right)}}^{\left(0\right)},\dots ,W_{M,t_M^{\left(0\right)}}^{\left(0\right)}\right)$, the vector of score functions at the end of phase 2. In practice, both the configuration and the correlations are unknown. If the phase 2 data were to be used in the estimation of ${\theta }_n$, then the estimation can’t be unbiased because of the unknowns. Hence, a conservative approach is taken in assuming that $\vec{\theta }=\left({\theta }_1,\dots ,{\theta }_M\right)=\left({\theta }_n,\dots ,{\theta }_n\right)=\left(\mathit{\theta },\dots ,\mathit{\theta }\right)$, and the components of ${\vec{W}}^{\left(0\right)}$ are independent. Thus $\mathit{f}\left(\vec{\theta },\vec{a},\vec{t}\right)=\mathit{f}\left(\mathit{\theta },\vec{a},\vec{t}\right)$ becomes a function of a single parameter θ. Per the dominance theorem, the estimator ${\theta }_{\gamma }$ is conservatively biased (i.e., median(${\theta }_{0.5}$)≤${\theta }_n$). An important feature of this estimator is consistency, such that ${\theta }_{\mathit{\alpha }/2}$ >0 if and only if the hypothesis test rejects the full null hypothesis of $\left({\theta }_1,\dots ,{\theta }_M\right)=\left(0,\dots ,0\right)$ at the one-sided α/2 level. To obtain an unbiased estimate of ${\theta }_n$, the phase 2 data must be excluded (section 3.5.2, 3.6.3).

• By the dominance theorem, conservative estimates of ${\theta }_{\gamma }$ and conservative p-values can be obtained using $\mathit{f}\left(\vec{\theta },\vec{a},\vec{t}\right)=\mathit{f}\left(\mathit{\theta },\vec{a},\vec{t}\right)$ either with sample size change (sections 3.6.2) or without sample size change (section 3.5.1). But unbiased estimates may be more desirable than conservative ones. Note that the tail of $\mathit{X}\left(\mathit{t}\right)=X_0+\mathit{W}\left(\mathit{t}\right)$ is W(t), which is exactly the Brownian motion in a usual sequential design. Hence, this tail can be used to obtain unbiased θ_y, either with sample size change (section 3.6.3), or without sample size change (section 3.5.2).

• When there is sample size change, a backward image is necessary for the estimates. The backward image depends only on the tail W(t). Hence the algorithm for obtaining the backward image is exactly as in a usual adaptive sequential design (Gao, Liu, Mehta 2013).

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Supplementary material

Supplemental data for this article can be accessed online at https://doi.org/10.1080/10543406.2024.2342518

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