We will demonstrate the use of `gsMAMS`

package and
provide a detailed example for each type of outcome explaining the
output and every argument of all the functions.

For the continuous outcome, we will consider TAILoR trial, which is a phase II trial, and it compares three doses of telmisartan (20, 40, 80mg) with no intervention (control) for the reduction of insulin resistance in human immunodeficiency virus-positive patients receiving combination antiretroviral therapy. The primary outcome measure is a reduction in mean homeostasis model assessment of insulin resistance (HOMA-IR) score at 24 weeks. The standardized desirable and minimal effect sizes for efficacy are set as \(\delta^{(1)}\) = 0.545 for 80mg group and \(\delta^{(0)}\) = 0.178 for 20 and 40 mg groups, respectively, for the trial design. The sample size calculation is based on a one-sided type I error of 5% and a power of 90%. Based on the trial characteristics, we will design the trial for a two-stage design. The design parameters of the trial can be calculated using the design_cont function and the arguments in the function correspond to standardized effect size in ineffective arm(delta0} and effective arm(delta1), type I error(alpha), type II error(beta), total number of treatment arms(K) and the information time (0.5, 1) is denoted by frac argument in the function.

```
design_cont(delta0 = 0.178,delta1 = 0.545,alpha = 0.05,beta = 0.1,k=3,
frac = c(0.5,1))
#> $`Sample size`
#> Stage 1 Stage 2
#> Cumulative sample size for treatment group 40 79
#> Cumulative sample size for control group 40 79
#>
#> $`Maximum total sample size for the trial`
#> [1] 316
#>
#> $`Boundary values`
#> Stage 1 Stage 2
#> Lower bound 0.006 2.062
#> Upper bound 2.910 2.062
```

The design output shows the cumulative sample size for treatment and control groups at each stage. The SCPRT lower and upper boundaries are (0.006, 2.062) and (2.91, 2.062) respectively. Based on the design parameters, the first interim analysis can be conducted after the enrollment of 40 patients in the control arm. If the test statistic \(Z_{k,l}<0.006\), the \(k_{th}\) arm is rejected for futility at \(1^{st}\) stage and the trial continues with the remaining treatment arms and the control. If the test statistic 0.006 \(\le\) \(Z_{k,l}\) \(\le\) 2.91 for \(k\)=1,2,3 then the trial continues to the next stage and 39 patients are further enrolled per arm. If \(Z_{k,1} >2.91\) for some \(k\), the trial is terminated and the arm with maximum value of \(Z_{k,1}\), \(k\)=1,2,3 would be recommend for further study.

For FWER and Stagewise FWER:

The operating characteristics of the trial can be generated using the op_power_cont and op_fwer_cont functions for power under alternative hypothesis and FWER under global null hypothesis respectively. Most of the arguments in the function are similar to size and SCPRT functions with the exception of number of simulations(nsim) and seed number(seed).

```
op_fwer_cont(alpha=0.05,beta=0.1,p=3,frac=c(0.5,1),delta0=0.178,
delta1=0.545,nsim=10000,seed=10)
#> $FWER
#> [1] 0.05
#>
#> $`Stagewise FWER`
#> look1 look2
#> 0.0050 0.0466
#>
#> $`Stopping probability under null`
#> look1 look2
#> 0.2599 0.7401
#>
#> $`Probability of futility under null`
#> look1 look2
#> 0.2549 0.6949
#>
#> $`Average sample size used per arm under null`
#> [1] 61.645
```

Based on the simulation results, the type I error at the first interim analysis is 0.5% and at the second interim analysis, it is around 4.6%. Therefore, the overall type I error of the trial is close to 5%. The sample size required for the trial was 79 patients per arm but the trial used around an average of 62 subjects per arm. The stopping probability(probability of stopping the trial either due to futility or efficacy) should add up to 1 which is the case here and since this is under null configuration, the probability of futility(probability of stopping the trial when all the treatment arms becomes futile in the trial) should be around 95% and it holds true in this case.

For Power and Stagewise Power:

```
op_power_cont(alpha=0.05,beta=0.1,p=3,frac=c(0.5,1),delta0=0.178,
delta1=0.545,nsim=10000,seed=10)
#> $Power
#> [1] 0.893
#>
#> $`Stagewise Power`
#> look1 look2
#> 0.3126 0.5804
#>
#> $`Stopping probability under alternative`
#> look1 look2
#> 0.3258 0.6742
#>
#> $`Probability of futility under alternative`
#> look1 look2
#> 0.0035 0.0821
#>
#> $`Average sample size used per arm under alternative`
#> [1] 62.652
```

Based on the simulation results, the probability of success/power at the first stage is 31.26% and at the second stage is around 58.04%. Therefore, the overall power is approximately 90%. The sample size required for the trial was 79 patients per arm but the trial used around an average of 62 subjects per arm. The stopping probability should add up to 1 which is the case here and under alternate configuration, the probability of futility is approximately 8.5% which is less than 10% type II error. The reason is that the type II error comes from both failing to find any efficacious arm (futility) and finding the less efficacious arm as the most efficacious arm. The latter part was not included when the probability of futility was calculated.

For ordinal outcome, we will consider ASCLEPIOS trial, a phase II
trial for patients with stroke. The primary outcome response is the
patientâ€™s Barthel index assessed 90 days after randomization. This is an
ordered categorical score ranging from 0 (vegetative state) to 100
(complete recovery) in steps of 5, and relates to activities of daily
living that the patient is able to undertake. Following the ASCLEPIOS
study, we group the outcome categories of the score into six larger
categories. We will consider the treatment worthwhile if the odds ratio
between the effective and control arms is 3.06 and we set the null odds
ratio to be 1.32 which is the odds ratio between the ineffective and
control arms.

The sample size calculation is based on a one-sided FWER of 5% and a
power of 90%. Based on the trial characteristics, we will design the
trial for a three-stage design. The design parameters for a five-arm (K
= 4) trial can be calculated using design_ord function and the arguments
in the function correspond to probability of outcomes in control
group(prob), odds ratio of ineffective treatment group vs control(or0),
odds ratio of effective treatment group vs control(or) and the remaining
arguments are similar to the design_cont function for continuous
outcome.

```
design_ord(prob=c(0.075, 0.182, 0.319, 0.243, 0.015, 0.166),or=3.06,
or0=1.32,alpha=0.05,beta=0.1,k=4,frac = c(1/3,2/3,1))
#> $`Sample size`
#> Stage 1 Stage 2 Stage 3
#> Cumulative sample size for treatment group 21 42 62
#> Cumulative sample size for control group 21 42 62
#>
#> $`Maximum total sample size for the trial`
#> [1] 310
#>
#> $`Boundary values`
#> Stage 1 Stage 2 Stage 3
#> Lower bound -0.630 0.437 2.161
#> Upper bound 3.126 3.092 2.161
```

Based on the design parameters, the first interim analysis can be conducted after the enrollment of 21 patients(information time is 21/62 which is approximately 1/3 at this stage) in the control and treatment arms. If the test statistic \(Z_{k,l}<-0.63\), the \(k_{th}\) arm is rejected for futility at \(1^{st}\) stage and the trial continues with the remaining treatment arms and the control. If the test statistic -0.63 \(\le\) \(Z_{k,l}\) \(\le\) 3.126 for \(k\)=1,2,3,4 then the trial continues to the next stage and 21 patients are further enrolled per arm. If \(Z_{k,1} >3.126\) for some \(k\), the trial is terminated and the arm with maximum value of \(Z_{k,1}\), \(k\)=1,2,3,4 would be recommend for further study. Similar procedure is followed if the trial goes to second stage of interim analysis.

The operating characteristics can be generated using the functions op_fwer_ord and op_power_ord which are similar to that of continuous outcome.

For survival outcome, we will consider a MAMS trial with five arms (four treatment arms and a control arm, K=4) and two interim looks with balanced information time (0.5, 1). The null hazards ratio is 1 and the alternative hazards ratio is 0.65. The median survival time of control group is 20 months and the survival distribution is exponential without loss to follow-up. The sample size calculation is based on a one-sided type I error of 5% and a power of 90%.

The design parameters for a two-stage design can be calculated using the design_surv function and the arguments in the function correspond to median survival time of the control group(m0), hazard ratio of ineffective treatment vs control(HR0), hazard ratio of effective treatment vs control(HR1), accrual time(ta), follow-up time(tf), shape parameter of Weibull distribution(kappa), rate of loss to follow-up(eta)(assumed loss to follow-up follows an exponential distribution with rate parameter eta).

```
design_surv(m0=20,hr0=1, hr1=0.67032, ta=40,tf=20,alpha=0.05,beta=0.1,
k=4,kappa=1,eta=0,frac=c(0.5,1))
#> $`Sample size`
#> Stage 1 Stage 2
#> Cumulative number of events for combined treatment & control 164 328
#>
#> $`Maximum total number of events for the trial`
#> [1] 820
#>
#> $`Total number of subjects required for the trial`
#> [1] 1170
#>
#> $`Boundary values`
#> Stage 1 Stage 2
#> Lower bound 0.075 2.16
#> Upper bound 2.980 2.16
```

The design output shows the cumulative number of events for treatment and control groups combined at each stage along with the futility and efficacy boundaries. The total number of subjects required for the trial per arm is 234. Based on the design parameters, the first interim analysis can be conducted after the incidence of 164 events (information time is 164/328 which is 0.5 at this stage) which results from the aggregation of events in the control and treatment arms. If the test statistic \(Z_{k,l}<0.075\), the \(k_{th}\) arm is rejected for futility at \(1^{st}\) stage and the trial continues with the remaining treatment arms and the control. If the test statistic 0.075 \(\le\) \(Z_{k,l}\) \(\le\) 2.98 for \(k\)=1,2,3,4 then the trial continues to the next stage and we will wait for the incidence of additional 164 events to conduct the next interim analysis. If \(Z_{k,1} >2.98\) for some \(k\), the trial is terminated and the arm with maximum value of \(Z_{k,1}\), \(k\)=1,2,3,4 would be recommend for further study.

The operating characteristics of the trial can be generated using the op_power_surv and op_fwer_surv function.

For FWER and Stagewise FWER:

```
op_fwer_surv(m0=20,alpha=0.05,beta=0.1,p=4,frac=c(1/2,1),hr0=1,hr1=0.6703,
nsim=10000,ta=40,tf=20,kappa=1,eta=0,seed=12)
#> $FWER
#> [1] 0.05
#>
#> $`Stagewise FWER`
#> look1 look2
#> 0.0049 0.0460
#>
#> $`Stopping probability under null`
#> look1 look2
#> 0.2318 0.7682
#>
#> $`Probability of futility under null`
#> look1 look2
#> 0.2269 0.7234
#>
#> $`Average number of events happened per arm under null`
#> [1] 129.5902
#>
#> $`Average duration of trial(months)`
#> [1] 54.27148
```

Based on the simulation results, the type I error is around 0.4% at first interim analysis and around 4.6% at the second interim analysis. Therefore, the overall type I error is maintained at 5%. The stopping probability should be around 1 which is the case here and since this is under null configuration, probability of futility should be around 95% which holds true in this case. The average duration of trial is 54 months which is reasonable as the total duration of the trial is 60 months. The average number of events happened per arm is 130.

For Power and Stagewise Power :

```
op_power_surv(m0=20,alpha=0.05,beta=0.1,p=4,frac=c(1/2,1),hr0=1,hr1=.6703,
nsim=10000,ta=40,tf=20,kappa=1,eta=0,seed=12)
#> $Power
#> [1] 0.913
#>
#> $`Stagewise Power`
#> look1 look2
#> 0.3270 0.5863
#>
#> $`Stopping probability under alternative`
#> look1 look2
#> 0.3334 0.6666
#>
#> $`Probability of futility under alternative`
#> look1 look2
#> 0.0059 0.0800
#>
#> $`Average number of events happened per arm under alternative`
#> [1] 115.0483
#>
#> $`Average duration of trial(months)`
#> [1] 52.27229
```

Based on the simulation results, the probability of success/power at the first stage is 32.7% and at the second stage is around 58.63%. Therefore, the overall power is around 90%. The overall stopping probability should be around 1 which is the case here and the probability of futility is around 8.5% which is less than 10% type II error because of the same reason as mentioned in the continuous outcome. The average duration of trial is 52 months which is reasonable as the total duration of the trial is 60 months. The average number of events happened per arm is 115.