The design of the ViP trial is considered allowing for the allocation ratio to differ from 1:1. As with the design of ViP in Chapter 7, two approaches are taken to evaluate trial design; a fully Bayesian design using the Average Length Criterion (ALC) and Bayesian equivalents of Type I and Type II error rates.
As is shown from (8.7) the optimal allocation ratio for a trial depends both on the total sample size and the amount of information included through the prior dis- tributions, here denoted by the effective number of prior events. Table 8.3 shows the estimated optimal allocation ratios using equation (8.4) for varying total sample sizes and effective number of events in the prior distributions for the control arm.
Effective Prior Events
Sample Size 10 20 30 50
60 0.34 0.26 0.18 0.01 70 0.36 0.28 0.21 0.07 80 0.36 0.30 0.24 0.11 90 0.37 0.32 0.26 0.15 100 0.38 0.33 0.28 0.18 110 0.38 0.34 0.29 0.20 120 0.39 0.34 0.30 0.22 130 0.39 0.35 0.31 0.23 140 0.39 0.36 0.32 0.25 150 0.39 0.36 0.33 0.26
Table 8.3: Table to show estimated allocation ratios under differing total sample sizes and effective prior events.
Bayesian sample size calculation
The total sample size for the study is determined using the simulation approach detailed in Chapters 6 and 7. Data are simulated from the design priors for both ⁄ and — as detailed in Chapter 7. Only the number of patients allocated to each treatment arm differs based on table 8.3.
For each simulation, the length of the 90% credibility interval is estimated and the mean over all simulations is calculated to give the average length criterion (ALC).
Prior distributions for data analysis the baseline hazard function are defined based on Normal, Step and Trapezium priors as detailed in Chapter 7. Uninformative vague prior for the baseline hazard function are included as a reference. All priors on the log hazard ratio are vague and uninformative.
The results are displayed graphically in Figure 8.9. Here four separate figures show the ALC for each of the four different prior distributions that are used. In each figure, the estimated ALC across varying sample sizes are displayed for four different effective event scenarios. The intersection of the vertical and horizontal lines shows the design conditions obtained under equal allocation in Chapter 7. Here 98 events are required to obtain an ALC of 0.6.
Considering initially the non-informative reference priors, it is shown that there is an increase in the ALC for each sample size for the larger ‘effective prior events’ used to determine the trial allocation ratio. This is misleading however as effective prior
events are included to calculate the optimal allocation ratio but are not included in the analysis of the trial data. The deterioration of the design parameters here is the cost of an unequal allocation when all information from a trial is taken from the data.
Considering the informative priors, all reduce the ALC in comparison to the results obtained under equal allocation in Chapter 7. The Normal distribution has the most accentuated effect on the posterior ALC. The Trapezium priors have the least effect which is consistent with previous findings.
80 100 120 140
0.40.60.81.01.21.4
Reference Priors
Sample Size
Interval Length
EE − 10 EE − 20 EE − 30 EE − 50
80 100 120 140
0.40.50.60.70.8
Normal Priors
Sample Size
Interval Length
80 100 120 140
0.40.50.60.70.8
Step Priors
Sample Size
Interval Length
80 100 120 140
0.40.50.60.70.8
Trapezium Priors
Sample Size
Interval Length
Figure 8.9: Figure to show the ALC for different types of priors and different number of effective events.
The sample sizes required to obtain, on average, an ACL of 0.6 for each of set of priors and set of effective prior event are given in Table 8.4. These can be compared directly to the sample sizes given for equal allocation ratios given in Table 7.3. Com- parisons here show the normal priors have a greater effect over the locally flat priors
and also how the sample sizes with amended allocation ratio are consistently smaller than equivalent sample size calculations with equal allocation ratios.
Effective Sample Size
’ = 10 ’ = 20 ’ = 30 ’ = 50
Normal 89 78 74 67
Step 96 82 77 70
Trapezium 97 96 83 72
Table 8.4: Sample size estimates for the ViP trial under various differing priors on the baseline hazard function under the ALC.
Bayesian Type I and Type II error rates
Here design criteria are presented in terms of estimated Type I and Type II error rates.
As in Chapter 7, sample sizes are fixed at 120 to replicate the initial design of the ViP trial. Table 8.5 gives the Bayesian Type I and Type II error rates that are obtained.
The results obtained here can be directly compared to those obtained in Figure 7.4 for equal allocation ratios. There are two points of note to be made, firstly there is a reduction in the Type I error rate as the effective prior events increase. This can be attributed to the unequal allocation ratios that are obtained. As is consistent through- out, the optimum design parameters are obtained when normal prior distributions are used. Differences in the Type I error rates between distributions are small however.
The second point to note is that the effect of the Type II error rate. Here for the reference models, where trials are designed based on informative priors but vague uninformative prior distributions are included in the analysis, there is a increase in the Type II error rate for larger effective prior events which is a consequence of the unequal allocation ratio. This shows that if informative prior distributions are used to inform a trial design, not including the same prior distributions in the analysis of a trial can have adverse effects.
For the normal, step and trapezium priors there is a notable decrease in the Type II rate, to a greater extent than what is observed in Table 7.4 for equal allocation.