Following our motivating example, we explore a seamless Phase II/III FAST design which includes 2 interim assessments and a final assessment. The primary aim of this work is to examine the impact of the timing for the interim assessments on final operating characteristics. The interim analyses will constitute the Phase II portion of the trial with the final assessment representing the Phase III portion of the trial. Sample sizes for the interim analysis and final operating characteristics will be determined via simulation.

To accommodate the clinical aims of the hypothesized clinical trial, we propose a trial design which allows for multiple questions of interest to be jointly assessed through a FAST design. Henceforth, we will reference each question of interest as a domain. In general, we let \(d = A,B,\ldots \Delta\) reference the domains of interest with \(\Delta\) representing the maximum number of domains. From our motivating trial, there are two domains of interest which we will reference as: \(A=\) Fluid and \(B=\) Mineralocorticoid. Within in each domain, we will note treatment assignment within domain as \(d_t\) where \(t=0,1,\ldots \tau _d\) and \(\tau _d\) represents the maximum number of treatment arms in domain d. Again utilizing our motivating trial as an example, there would then be three treatment arms in domain A, \(A_0=Saline\), \(A_1=Fluid_1\), and \(A_2=Fluid_2\), and two treatments in domain B, \(B_0=Control\) and \(B_1=Mineralocorticoid\). We further note r as the combination of treatment assignments across domains as a treatment regime with \(r=1,\ldots ,R= (\tau _1 \times \ldots \times \tau _\Delta )\). If allowing all combination of treatment assignments across domains, then there are 6 possible treatment regimes for our motivating example: \(r=1,\ldots 6\) with regimes being \(( (A_0,B_0),(A_1,B_0),(A_2,B_0),(A_0,B_1),(A_1,B_1),(A_2,B_1))\).

For the outcomes of interest, we will note outcomes by trial phase. Thus, let \(Y_\) represent the \(o^\) outcome of interest for the \(p^\) trial phase, where \(o=1,2,\ldots ,\theta\) and \(p=1,2,\ldots ,\rho\). From our motivating example, the outcomes utilized in the Phase II portion of the trial represent biomarkers assumed to be affected by treatment and the Phase III outcome of interest represents a functional outcome. We will assume two analyses within the Phase II portion of the trial and one in the Phase III portion of the trial. This is done to mirror the proposed design for the motivating trial. The Phase II portion of the trial will consist of 2 analyses: a feasibility analysis and an arm-dropping analysis.

The goal of the arm-dropping analysis is to determine which, if any, of the treatment arms (novel fluid therapies) can be discontinued from enrollment. We let \(\mu _^}\) and \(\mu _^}\) represent the average value for outcome \(Y_\) (bicarbonate) for subjects in arms 1 and 2 in domain A, respectively. Further, we let \(\mu _^}\) and \(\mu _^}\)represent the average value for outcome \(Y_\) (sodium) for subjects in arms 1 and 2 in domain A, respectively. Then, utilizing independent t-tests by outcome, the non-control arms will be compared by assessing two hypotheses:

$$\begin H_0:\mu _^}=\mu _^} \ vs.\ H_a:\mu _^} \ne \mu _^} \end$$

$$\begin H_0:\mu _^}=\mu _^} \ vs.\ H_a:\mu _^} \ne \mu _^} \end$$

The results of these hypothesis tests will be used to determine which arms will continue to enroll. If there is a statistically significant difference observed for outcome \(Y_\), the treatment arm in domain A with the highest average outcome \(Y_\) value will be retained. If there is a difference in outcome \(Y_\) between the treatment arms in domain A, then the arm with the lowest outcome \(Y_\) average will be retained. There is a possibility that neither arm will be dropped.

The goal of the feasibility analysis is to assess, utilizing one or more biomarkers, if the expected effect of treatment is being observed on the biomarker(s). While the comparison for this example focuses on responses in key biomarkers, any appropriate Phase II assessment could be included in this analysis. Treatment arms will be compared using an independent one-sided t-test. Letting \(\mu _^}\) represent the average value for outcome Y11 for subjects in arm \(A_0\) (the control arm for domain A) and \(\mu _}^}\) represent the average outcome 11 for the pooled treatment arms from domain A, we then propose a one-sided test, with hypothesis below:

$$\begin H_0:\mu _^} \ge \mu _}^} vs.H_a:\mu _^} \le \mu _}^} \end$$

If statistical significance is not obtained, then the assessment of treatment effect within domain A will terminate and subjects will no longer randomized to any of the treatments within domain A.

For this analysis, all subject data are utilized in the comparison. Pooling data within \(A_1\) and \(A_2\) could result in an assessment which is under-powered if there is a differential effect of treatment on \(Y_\). Given this pooling, in addition to the treatment effect, the impact of this pooling is likely affected by the timing and ordering of the arm-dropping and feasibility assessments. The impact of these trial characteristics and their impact on overall operating characteristics will be assessed via simulation.

For any trial where domain A proceeds beyond the Phase II portion of the trial, the final analysis will consist of an assessment comparing each treatment arm to control within domain. The final analysis will be conducted by constructing a generalized linear model where the outcome of interest is the Phase III outcome, assumed to be a binary outcome, and covariates in the model represent the treatment assignments for each subject. An appropriate correction for the multiple comparisons will be applied to \(\alpha\) so that type 1 error is controlled at a 5% level [7]. Due to the arm dropping analysis, there are three possible scenarios for the final analysis: where a treatment in domain A is dropped, where no treatment in domain A is dropped, where domain A is terminated.

In this scenario, while the majority of subjects will be randomized to one of the newer fluid arms or saline, a small proportion of subjects will be randomized to the dropped fluid arm prior to the arm dropping analysis. For the final analysis, subjects will be grouped as either having been randomized to saline or one of the newer fluid arms; that is, subjects randomized to either of the non-saline arms throughout the duration of the trial are pooled into a single fluid therapy arm for the final analysis.

Therefore, letting \(I(X_=1)\) represent an indicator variable for if subject i was randomized to saline, \(I(X_=1)\) represent an indicator variable for if subject i was randomized to mineralocorticoid, and \(p_i = P(Y_=1)\), we can then construct the primary analysis model as below.

$$\begin logit(p_i)=\beta _0+\beta _1 I(X_\ne 1)+\beta _2 I(X_=1) \end$$

To control the family-wise type I error in the strong sense at \(\alpha =0.05\), we will then assess for any significant treatment effect among the fluid and mineralocorticoid arms using a gatekeeping procedure:

$$\begin H_0:\beta _1=\beta _2=0 \ vs.\ H_a:\beta _1\ne 0 \text \beta _2\ne 0 \end$$

If the null hypothesis for the test above is rejected, implying that either the pooled fluid arm or the mineralocorticoid arm has a non-zero treatment effect relative to the saline/no mineralocorticoid arm, then individual tests comparing each arm to control will be completed.

$$\begin H_0:\beta _1=0 \ vs. \ H_a:\beta _1 \ne 0 \end$$

$$\begin H_0:\beta _2=0 \ vs. \ H_a:\beta _2 \ne 0 \end$$

As with the feasibility assessment in Phase II, it is worth noting that the comparisons above utilize the pooling approach so that all patient data inform the final analysis. This approach could reduce power if there is a different treatment effect within the pooled treatment arms; however, the impact of this effect is expected to vary with the timing of the assessments within Phase II. The relationship between the design parameters and the operating characteristics will be assessed in Phase III.

If no arm is dropped, then the final analysis will also proceed using a gatekeeping approach. The primary analysis model including fluid arm and mineralocorticoid arm will be constructed as below. Unlike the previous scenario, if neither of the newer fluid arms are dropping, then we will construct the following model where \(I(X_=1)\) and \(I(X_=1)\) represent indicator variables for if subject i was randomized to either of the newer fluids and \(I(X_=1)\) represents an indicator variable for if subject i was randomized to mineralocorticoid.

$$\begin logit(p_i )= & \beta _0+\beta _1 I(X_=1)+\beta _2 I(X_=1)\nonumber \\ & +\beta _3 I(X_=1) \end$$

To preserve \(\alpha\), a gatekeeping procedure will be used to assess for any significant treatment effect among the fluid and mineralocorticoid arms:

$$\begin H_:\beta _1=\beta _2=\beta _3= 0 \ vs.\ H_:\beta _1\ne 0\text \beta _2\ne 0\text \beta _3\ne 0 \end$$

If the null hypothesis for the test above is rejected, then all pairwise combinations will be tested as below.

$$\begin H_:\beta _1=\beta _2= 0 \ vs.\ H_:\beta _1 \ne 0 \text \beta _2 \ne 0 \end$$

$$\begin H_:\beta _1=\beta _3= 0 \ vs.\ H_:\beta _1 \ne 0\text \beta _3 \ne 0 \end$$

$$\begin H_:\beta _2=\beta _3= 0 \ vs.\ H_:\beta _2 \ne 0\text \beta _3 \ne 0 \end$$

Depending on which of the null hypotheses for the tests above are rejected, then individual tests comparing each arm to control will be completed.

$$\begin H_:\beta _1=0 \ vs.\ H_:\beta _1 \ne 0 \end$$

$$\begin H_:\beta _2=0 \ vs.\ H_:\beta _2 \ne 0 \end$$

$$\begin H_:\beta _3=0 \ vs.\ H_:\beta _3 \ne 0 \end$$

For example, the null hypothesis \(H_\) will only be assessed if \(H_\), \(H_\), and \(H_\) are all rejected in their respective tests.

In this scenario, after the feasibility assessment, randomization will cease within domain A. As such, for the final analysis will consider only the treatment assignments for domain B. For the final analysis in this scenario, subjects will be grouped as either having been randomized to mineralocorticoid or no mineralocorticoid. Therefore, let \(I(X_=1)\) represent an indicator variable for if subject i was randomized to mineralocorticoid, we can then construct the primary analysis model as below.

$$\begin logit(p_i)=\beta _0+\beta _1 I(X_=1) \end$$

Under this scenario, only a single parameter \(\beta _1\) is being assessed; therefore, no methods for the preservation of \(\alpha\) are required.