An investigation into unblinded sample size re-estimation methodology in clinical trials

The proposed research is a comparison of statistical methodologies for trial design purposes and interim analysis decisions only and is looking to address statistical as opposed to clinical questions.For the investigation the effect measure of interest, statistical analysis methods and planned adjustment of covariates will be the same as in the original study. Only the primary outcome is of interest, and no data analysis on secondary outcomes will be performed.Four key designs will be compared:    Promising zone (Mehta and Pocock, 2011[1]);   Weighted inverse normal combination tests (Jennison and Turnbull, 2015[2]);   Stepwise increase sample size rule (Liu and Hu, 2016[3]);   Fixed sample size designs (original design of all studies).Each key design will also look at the effect of incorporating stopping boundaries. Efficacy, futility, both efficacy and futility, or no stopping boundaries will be compared within each design. Boundaries will be based on conditional power values. Specific futility bounds to be investigated are at CP=0.1, and CP=0.2, and efficacy bounds at CP=0.9 and CP=0.95.Conditional power calculations will use six different future treatment effect assumptions:    The originally planned treatment effect (H1);   The observed current trend;   A combination of both originally planned and current trend;   Minimum clinically meaningful difference;   An 80% confidence limit of the observed trend;   A 90% confidence limit of the observed trend. Finally, interim analyses will be chosen at 25, 40, 50, 60, 75 and 90% through the originally planned trial. Two definitions of “percentage through the trial” have been established in the current literature; percentage of patients recruited, or percentage of patients with primary outcome data available. Both definitions will be used here, and compared to each other to investigate the impact of this on decision making at the interim analysis.Methodological designs:Conditional power, or the conditional probability of rejecting H0 at the final analysis given the first stage Wald statistic Z1=z1, is calculated by [1]:〖CP〗_δ (z_1,n_2 )=1- Φ{(z_α √n- z_1 √(n_1 ))/√(n-n_1 )-(z_1 √(n-n_1 ) )/√(n_1 )},where α denotes the Type I error, z_α= Φ^(-1) (1-α), δ is the assumed future treatment effect for the remainder of the trial, n is the originally planned final sample size, n1 is the first stage sample size (recruited prior to the interim analysis), and n2 is the second stage sample size, such that n=n1+n2.The promising zone is defined as the set℘={〖CP〗_δ (z_1,n_2^* ):b(z_1,n_2^* )≤z_α },whereb(z_1,n_2^*)=〖(n^*)〗^(-0.5) {√((n_2^*)/n_2 ) (z_α √n-z_1 √(n_1 ))+ z_1 √(n_1 )}If, at the interim analysis, the conditional power falls in the promising zone, the second stage sample size is increased to n_2^*, leading to a final sample size n^*, such that n^*=n_1+n_2^*. The new second stage sample size can be calculated asn_2^*=[n_1/(z_1^2 )][(z_α √n- z_1 √(n_1 ))/√(n-n_1 )+z_β ],where z_β=Φ^(-1) (1-β).For the restricted design, if n^* exceeds a prespecified maximum (n_max), n^* is replaced by this pre-specified maximum value.In the Jennison and Turnbull methodology framework, conditional power calculations are performed across all values of n^*, and n^* is chosen such that the objective function〖CP〗_δ (z_1,n^* (z_1 ))- γ(n^* (z_1)-n)is maximised, where n^* (z_1) is the sample size rule for choosing the sample size given Z1=z1.The stepwise design involves a step function, of the formn^*/n= ∑_(j=1)^J▒〖r_j I(z_1∈l_j)〗where rj≥1 is the pre-determined increased sample size ratio, or step value, chosen in advance by the investigator, when the interim test statistic z1 falls in the interval lj = (lj(L), lj(U)). Sample size increase:The original data only enrols patients up to the originally planned sample size. However, should the interim decision be to increase the number of patients, bootstrapping methods will be performed on the remainder of the patients (not analysed at the interim stage) to make up the additional required sample size.Key Outcomes:   Conditional power at the interim analysis   The decision at the interim analysis (Stop, continue as planned, or continue with increased sample size)   Sample size at the end of the trial compared to originally planned sample size   Number of pipeline patients and accrual ratesReferences:[1]   C. Mehta and S. Pocock, “Adaptive increase in sample size when interim results are promising: A practical guide with examples,” Stat. Med., vol. 30, no. 28, pp. 3267-3284, Dec. 2011.[2]   C. Jennison and B. W. Turnbull, “Adaptive sample size modification in clinical trials: Start small then ask for more?,” Stat. Med., 2015.[3]   Y. Liu and M. Hu, “Testing multiple primary endpoints in clinical trials with sample size adaptation,” Pharm. Stat., vol. 15, no. 1, pp. 37-45, 2016.