The Power And Sample Sizes And The Optimal Allocation Of Randomized Designs In Clinical Trials

Kinds of randomized designs are often used in clinical trials.In current,the popular problem in the study of randomized designs is the power and sample sizes.In literature,the sample sizes of randomized designs are usually obtained by ignoring the randomness of the allocation,that is we look it as fixed design.But in fact when we use the sample size calculated by the method that ignoring the randomness we can not achieve the target power with high probability.When we use a randomized design for a fixed sample size,the number of patients assigned to each treatment is a random variable,therefore the power of a randomized design with a fixed sample size is also a random variable.It is necessary to study the power and sample sizes of randomized designs. In this paper,we first give the random power function and discuss its properties for superiority two-arm clinical trials and non-inferiority three-arm clinical trials include a placebo.Through the Taylor expansion of the power function we can find that we lose the power compared to the fixed design because of the use of randomization,but the lose is usually very small.Second,based on the power function we derive the three sample sizes which defined according to the different requirement of the power for randomized designs.Through the calculation we find that the sample size n₁ is always greater than or equal to the sample size n₀,but the difference between them is usually very small.The sample size n₂ could be much larger than the sample size n₀-It means that when we want to achieve the target power with high probability we need much more sample sizes.We also find that response-adaptive randomization procedure can reduce sample sizes significantly in clinical trials.In addition,we discuss the optimal allocation of sample sizes,this optimization problem can be looked at from two perspectives:first,to minimize the total sample size for a given power,and second,to maximize the power for a fixed total sample size.Last,discussed the multi-arm clinical trials in power and sample sizes.For simplicity of calculation we approximating the quantiles of a non-central t-distribution by those of a central one and to get a rough solution we replace the t-quantile with those of a standard normal distribution,and we can find that the result is very good.