Parameters of the prior distribution

Additionally to the parameters in the baseline scenario with fixed treatment effects, we now need further input parameters:

• Differing from the baseline scenario we now have not only one treatment effect, but two: Delta1 and Delta2. The input parameter Delta1 is the one we got from some randomized controlled pilot trial that our team conducted earlier. Its value is given as the standardized difference in means (\(\Delta=\frac{\mu_{contro} - \mu_{exper}}{\sigma}\)) and its value was determined to be 0.625. However, we are not so sure about this result, as another research group conducted a similar study and got an treatment effect of 0.9, which will now be our value for Delta2. Of course, the choice of \(\Delta_1\) and \(\Delta_2\) need not be built on two clinical studies, but can also be derived from different sources for forming a prior belief, e.g. clinical experience.
• We now have to specify how sure we are about the two prior beliefs \(\Delta_1\) and \(\Delta_2\). This is done by the two additional parameters in1 and in2. We call these parameters the “amount of information”. They refer to the sample sizes of the studies on which we base our prior beliefs. If our pilot study was conducted with 300 participants, the value for in1 is set to be 300. Let’s assume that the study of the other research group was conducted with 600 participants, so the parameter value for in2 is 600. The higher amount of information, the lower the variance we attribute to that prior belief.
• Now, a weight parameter w has to be defined, that allows us to weigh the two treatment effects. If we want to trust our results more, than we can set a higher parameter value for w. (Note that w has to be between 0 and 1, a parameter value of 1 would put all the weight on our results and none on the results of the second study). If we think the results of the other group are more reliable we reduce the value for w, thus putting more weight on Delta2. Note that by exchanging the values of Delta1 and Delta2 (and the corresponding values for in1 and in2) and setting \(w_{new} = 1 - w_{old}\) our final results will not change. In our case, we want to put more trust on our results and thus set the parameter w to be 0.6. Setting the weight to \(1\) would effectively mean ignoring the second treatment effect, which is also possible.
• The prior distribution used in the package is the sum of two truncated normal distributions. Hence, we need truncation values a as the lower boundary for the truncation and b as the upper boundary. In our case we set a = 0.25 and b = 0.75.

The prior distribution for the standardized true difference in means is then given by \(\Delta ∼ w · \mathcal{N}^t_{[0.25,0.75]} (0.625, 4/300) + (1 − w) · \mathcal{N}^t_{[0.25,0.75]} (0.9, 4/600)\) where \(N^t_{[a,b]} (\mu, \sigma^2)\) denotes the truncated normal distribution with mean \(\mu\), variance \(\sigma^2\), truncated below at a and above at b. To see how different input values change the prior distribution we refer to the Shiny app.

 res <- optimal_normal(Delta1 = 0.625, Delta2 = 0.8, fixed = FALSE, # treatment effect
                       n2min = 20, n2max = 400, # sample size region
                       stepn2 = 4, # sample size step size
                       kappamin = 0.02, kappamax = 0.2, # threshold region
                       stepkappa = 0.02, # threshold step size
                       c2 = 0.675, c3 = 0.72, # maximal total trial costs
                       c02 = 15, c03 = 20, # maximal per-patient costs
                       b1 = 3000, b2 = 8000, b3 = 10000, # gains for patients
                       alpha = 0.025, # significance level
                       beta = 0.1, # 1 - power
                       w = 0.6, in1 = 300, in2 = 600, #weight and amount of information
                       a = 0.25, b = 0.75) #truncation values

Interpreting the output

After setting all these input parameters and running the function, let’s take a look at the output of the program.

res
#> Optimization result:
#>  Utility: 3147.32
#>  Sample size:
#>    phase II: 80, phase III: 188, total: 268
#>  Probability to go to phase III: 0.99
#>  Total cost:
#>    phase II: 69, phase III: 155, cost constraint: Inf
#>  Fixed cost:
#>    phase II: 15, phase III: 20
#>  Variable cost per patient:
#>    phase II: 0.675, phase III: 0.72
#>  Effect size categories (expected gains):
#>   small: 0 (3000), medium: 0.5 (8000), large: 0.8 (10000)
#>  Success probability: 0.85
#>  Success probability by effect size:
#>    small: 0.68, medium: 0.16, large: 0.01
#>  Significance level: 0.025
#>  Targeted power: 0.9
#>  Decision rule threshold: 0.06 [Kappa] 
#>  Parameters of the prior distribution: 
#>    Delta1: 0.625, Delta2: 0.9, in1: 300, in2: 600,
#>    a: 0.25, b: 0.75, w: 0.6
#>  Treatment effect offset between phase II and III: 0 [gamma]

The program returns a total of thirteen values and the input values. Once again, we will only focus at the most important ones:

• res$n2 is the optimal sample size for phase II and res$n3 the resulting sample size for phase III. We see that the optimal scenario requires 80 participants in phase II and 188 participants in phase III. The number of participants in both phases was reduced compared to the setting with a fixed treatment effect.
• res$Kappa is the optimal threshold value for the go/no-go decision rule. We see that we need a treatment effect of more than 0.06 in phase II in order to proceed to phase III, which is the same as in the baseline scenario.
• res$u is the expected utility of the program for the optimal sample size and threshold value. In our case it amounts to 3147.32, i.e. we have an expected utility of 314 732 000$. This is a 6.83% increase over the scenario without the prior distribution. The increase in the expected utility can be attributed to the additional weight which was put on the second treatment effect Delta2, which was more promising than our treatment effect.

Of course, the differences in the output values compared to the fixed setting heavily depend on the choice of the prior.