Hybrid classical-Bayesian approach to sample size determination for two-arm superiority clinical trials

A. Simulation procedure to obtain the average unconditional probability of correctly rejecting H ₀

We need to specify n ₂, r, α, θ _S , θ ^D , n ^D , θ 1 prior , θ 2 prior , m 1 prior and m 1 prior and a high number B. Then, AUPCR H 0 π D , π 1 , π 2 can be obtained via simulation by performing the following steps:

1. Simulate B values from π ₁(θ ₁), θ 1 i , i = 1 , … , B .

2. Simulate B values from π ₂(θ ₂), θ 2 i , i = 1 , … , B .

3. For i = 1, …, B, compute UPCR H 0 π D , k θ 1 i , θ 2 i .

4. Approximate AUPCR H 0 π D , π 1 , π 2 with the arithmetic mean of the B values obtained in step 3.

B. Effect of using prior distributions instead of point estimates for θ ₁ and θ ₂

In Section 3.2, we showed how to construct the average unconditional probability of correctly rejecting H ₀ by introducing two prior distributions for θ ₁ and θ ₂ in order to avoid the use of the point estimates θ ̂ 1 and θ ̂ 2 , that affect both UPCR H 0 π D and CPCR H 0 ( θ D ) . More specifically, we considered two beta prior distributions, π _j (θ _j ) = beta(θ _j ; α _j , β _j ), for j = 1, 2, and suggested to choose their hyperparameters by exploiting the expressions in (6). The effect of these prior densities on AUPCR H 0 π D , π 1 , π 2 depends on how they impact on the sample variance of the estimator Y n 2 , i.e. on σ Y n 2 2 ( θ 1 , θ 2 ) = k ( θ 1 , θ 2 ) 2 / n 2 .

When using the conditional and unconditional powers in (2) and (4), the sample variance is replaced by its plug-in estimate: all else being equal, the higher σ Y n 2 2 ( θ ̂ 1 , θ ̂ 2 ) , the lower the powers and, consequently, the larger the optimal sample sizes. Assuming that θ ₁ and θ ₂ are random variables that follow the beta distributions mentioned above, we can evaluate through simulation the characteristics of the corresponding distribution of the random quantity k(θ ₁, θ ₂), using the following steps.

1. Set B as a large number.

2. Simulate B values from π ₁(θ ₁), θ 1 i , i = 1 , … , B .

3. Simulate B values from π ₂(θ ₂), θ 2 i , i = 1 , … , B .

4. For i = 1, …, B, compute k θ 1 i , θ 2 i and display the corresponding density plot.

We can also approximate the probability that k(θ ₁, θ ₂) is lower (or higher) than k ( θ ̂ 1 , θ ̂ 2 ) with the proportion of the B values obtained in step 3 which are lower (or higher) than k ( θ ̂ 1 , θ ̂ 2 ) . These computations are useful to explain the behaviour of AUPCR H 0 π D , π 1 , π 2 .

Let us consider again the example in Section 4.1 with θ = log(OR). For the three couples of beta prior distributions represented in Figure 2, we perform the steps above to obtain the corresponding distributions of k ( θ 1 , θ 2 ) = 1 r θ 1 ( 1 − θ 1 ) + 1 θ 2 ( 1 − θ 2 ) , with r = 1, whose density plots are given in the left panel of Figure 6. The empirical probabilities that k(θ ₁, θ ₂) is lower than k ( θ ̂ 1 , θ ̂ 2 ) = k ( 0.75 , 0.6 ) = 9.5 are 0.488, 0.466 and 0.383, when m 1 prior = m 2 prior are equal to 12, 6 and 2, respectively. Thus, when introducing the beta prior densities, the probability that the variance of Y n 2 exceeds the fixed one based on point estimates is higher than 0.5. In other words, the two beta prior distributions convey jointly the information that, for each value of n ₂, the variance of the test statistic Y n 2 is most probably larger than its preliminarily estimated value. As a result, the curve of the average unconditional probability of correctly rejecting H ₀ results to be lower than that of the corresponding UPCR H 0 π D , as we can see in Figure 2.

Figure 6:

Left panel: Kernel density plots of k(θ ₁, θ ₂), with r = 1, when θ = log(OR) for different choices of the beta prior distributions of θ ₁ and θ ₂ (see example in Section 4.1). Right panel: Kernel density plots of k(θ ₁, θ ₂), with r = 2, when θ = log(RR) for different choices of the beta prior distributions of θ ₁ and θ ₂ (see example in Section 4.2).

We also consider the example provided in Section 4.2 with θ = log(RR). In this case k ( θ 1 , θ 2 ) = 1 − θ 1 r θ 1 + 1 − θ 2 θ 2 , with r = 2. In the right panel of Figure 6, we show the density plots of k(θ ₁, θ ₂) obtained through simulation, that correspond to the three couples of beta prior distributions represented Figure 4. The empirical probabilities that k(θ ₁, θ ₂) is lower than k ( θ ̂ 1 , θ ̂ 2 ) = k ( 0.004 , 0.015 ) = 190.2 are 0.872, 0.777 and 0.721, when m 1 prior = m 2 prior are equal to 30, 100 and 200, respectively. Therefore, with this choice of beta priors, we have quite low probability values that the variance of Y n 2 exceeds σ Y n 2 2 ( θ ̂ 1 , θ ̂ 2 ) . This leads to values of AUPCR H 0 π D , π 1 , π 2 larger than those of UPCR H 0 π D and the difference between the curves is more evident as the prior sample sizes of the beta prior densities decrease, as it is shown in Figure 4.