diff --git a/.DS_Store b/.DS_Store index 0664108..cc9ea07 100644 Binary files a/.DS_Store and b/.DS_Store differ diff --git a/01-SampleSize-Seamless.qmd b/01-SampleSize-Seamless.qmd new file mode 100644 index 0000000..edc9ab3 --- /dev/null +++ b/01-SampleSize-Seamless.qmd @@ -0,0 +1,24 @@ +# Seamless phase-II/III study + +```{r} +#| echo: false +#| warning: false + +library(ggplot2) +library(tidyverse) +library(splitstackshape) +library(extraDistr) +``` + +This vignette was motivated by a study of CTU Bern (1938). It is an example of a Bayesian seamless phase-II/III design. The primary endpoint is a binary endpoint, say, treatment discontinuation (TD). + + +## Methods + +## Author {-} + +André Moser, Senior Statistician\ +Iulia-Maia Muresan, Statistician\ +Department of Clinical Research\ +University of Bern\ +3012 Bern, Switzerland \ No newline at end of file diff --git a/_quarto.yml b/_quarto.yml index d074840..ef25a90 100644 --- a/_quarto.yml +++ b/_quarto.yml @@ -15,6 +15,7 @@ book: chapters: - 01-SampleSize-BinaryOutcome.qmd - 01-SampleSize-ContinuousOutcome.qmd + - 01-SampleSize-Seamless.qmd - 01-SampleSize_Bayesian_power_basics_examples.qmd - references.qmd diff --git a/docs/01-SampleSize-BinaryOutcome.html b/docs/01-SampleSize-BinaryOutcome.html index 1faf1a9..82e0dbe 100644 --- a/docs/01-SampleSize-BinaryOutcome.html +++ b/docs/01-SampleSize-BinaryOutcome.html @@ -208,11 +208,17 @@

1  2  Two-arm study with a continuous primary outcome + +
  • +
    + + 3  Seamless phase-II/III study +
  • - 3  External resources and references + 4  External resources and references
    • diff --git a/docs/01-SampleSize-ContinuousOutcome.html b/docs/01-SampleSize-ContinuousOutcome.html index 4cf06a4..60bed40 100644 --- a/docs/01-SampleSize-ContinuousOutcome.html +++ b/docs/01-SampleSize-ContinuousOutcome.html @@ -59,7 +59,7 @@ - + @@ -157,11 +157,17 @@

    2  2  Two-arm study with a continuous primary outcome + +
  • +
    + + 3  Seamless phase-II/III study +
  • - 3  External resources and references + 4  External resources and references
    • @@ -763,8 +769,8 @@

    Author

    - - 3  External resources and references + + 3  Seamless phase-II/III study
    diff --git a/docs/02-SampleSize-ContinuousOutcome.html b/docs/01-SampleSize-Seamless.html similarity index 51% rename from docs/02-SampleSize-ContinuousOutcome.html rename to docs/01-SampleSize-Seamless.html index 4dbd040..fdc1556 100644 --- a/docs/02-SampleSize-ContinuousOutcome.html +++ b/docs/01-SampleSize-Seamless.html @@ -7,7 +7,7 @@ -Bayesian Toolbox – 3  Two-arm study with a continuous primary outcome +Bayesian Toolbox – 3  Seamless phase-II/III study - - + + @@ -100,7 +66,7 @@ - + @@ -109,8 +75,8 @@
    -
    -

    3  Two-arm study with a continuous primary outcome

    +
    +

    3  Seamless phase-II/III study

    @@ -133,24 +99,45 @@

    3 
      -
    • +
    • + + Introduction +
      • +
    • +
      + + Sample size + + + +
      + +
    • @@ -163,22 +150,8 @@

      3  Table of contents

       @@ -193,131 +166,19 @@

      Table of contents

      -

      This vignette was motivated by a consulting of CTU Bern (C2956). It describes the sample size approach for a randomized Bayesian phase II trial with a continuous outcome that is corrected for the baseline value.

      -
      -

      3.1 Study description

      -
      -

      3.1.1 Primary objective

      -

      The study examines the impact of a treatment on upper limb motor activity in everyday life.

      -
      -
      -

      3.1.2 Primary outcome

      -

      The primary endpoint is bimanual hand function measured with the Assisting Hand Assessment (AHA) after treatment.

      -
      -
      -

      3.1.3 Study design

      -

      In this exploratory randomized Bayesian phase-II trial, we evaluate the probability that the investigational treatment is superior to the standard of care using Bayesian statistics. If this probability exceeds 90%, we deem the treatment as promising.

      +

      This vignette was motivated by a study of CTU Bern (1938). It is an example of a Bayesian seamless phase-II/III design. The primary endpoint is a binary endpoint, say, treatment discontinuation (TD).

      +
      +

      3.1 Methods

      -
      -
      -

      3.2 Methods and assumptions

      -

      We calculate the Bayesian probability of success (equivalent to the frequentist power) for the primary outcome using Monte-Carlo simulations. Success is defined as a difference in AHA greater than 0 in favor of the treatment. We simulate outcome and its respective baseline data from a multivariate normal distribution with a correlation of 0.5 and evaluate the treatment difference using Bayesian linear regression. We calculate the probability of success twice, first, using a non-informative prior (i.e. not integrating any prior information) and second, using an informative prior with an effect size of 1.0 and a precision of 1.0, which was based on a previous publication. This prior corresponds to approximately two patients in terms of weight.

      -
        -
      • Allocation ratio: 1:1
        • -
      • Effect measure: Difference in AHA (corrected for the baseline value)
        • -
      • Analysis approach: Bayesian linear regression using INLA
        • -
      • Expected effect size: 0.6
        • -
      • Expected correlation between baseline and outcome value: 0.5
        • -
      • Probability threshold to claim success: 0.9
        • -
      • Non-informative prior: Mean 0, precision 0.001
        • -
      • Informative prior: Mean 1.0, precision 1.0
        • -
      • Total sample size: 34
        • -
      • Number of simulations: 10,000
        • -
      -
      -
      -

      3.3 Calculation using R

      -
      -

      3.3.1 R-code

      -
      -
      -Show R code -
      #install.packages("C:/download/INLA_24.02.09.zip", repos = NULL, type = "bin") # manual download of latest version
-library(INLA)
-library(MASS)
-
-rm(list = ls())
-set.seed(123)
-
-# Set parameters
-n_simulations <- 10
-n_patients <- 17 # per arm
-effect_size <- 0.6
-correlation <- 0.5
-sd <- 1.0
-cor_matrix <- matrix(c(sd, correlation, correlation, sd), nrow = 2)
-
-# Define prior and threshold probability
-sd_p <- 1
-prec_p <- 1/sd_p^2
-mean_p <- 1.0
-prior_custom <- list(mean=list(treatment=mean_p), prec=list(treatment=prec_p))
-threshold_prob <- 0.9
-
-success_noprior <- numeric(n_simulations)
-success_prior <- numeric(n_simulations)
-
-# Simulate and fit the model multiple times
-for (i in 1:n_simulations) {
-  print(i)
-  # Generate simulated data
-  control_data <- mvrnorm(n = n_patients, mu = c(0, 0), Sigma = cor_matrix)
-  control_data <- data.frame(
-    baseline = control_data[, 1],
-    outcome = control_data[, 2],
-    treatment = 0
-  )
-  treatment_data <- mvrnorm(n = n_patients, mu = c(0, effect_size), Sigma = cor_matrix)
-  treatment_data <- data.frame(
-    baseline = treatment_data[, 1],
-    outcome = treatment_data[, 2],
-    treatment = 1
-  )
-  data <- rbind(control_data, treatment_data)
-  
-  # Model formula
-  formula <- outcome ~ treatment + baseline  # Model formula
-  
-  # Bayesian INLA model without prior (non-informative)
-  model_noprior <- inla(formula, data = data, family = "gaussian")
-  model_noprior$summary.fixed
-  1 - inla.pmarginal(0, model_noprior$marginals.fixed$treatment)
-  
-  # Model with informative prior
-  model_prior <- inla(formula, data = data, family = "gaussian", control.fixed=prior_custom)
-  model_prior$summary.fixed
-  1 - inla.pmarginal(0, model_prior$marginals.fixed$treatment)
-  
-  # Posterior of treatment coefficient
-  posterior_prob_noprior <- 1 - inla.pmarginal(0, model_noprior$marginals.fixed$treatment)
-  success_noprior[i] <- posterior_prob_noprior > threshold_prob
-  posterior_prob_prior <- 1 - inla.pmarginal(0, model_prior$marginals.fixed$treatment)
-  success_prior[i] <- posterior_prob_prior > threshold_prob
-  }
-  
-# Calculate probability of success
-prob_success_noprior <- mean(success_noprior)
-prob_success_power_prior <- mean(success_prior)
      -
      -
      -
      -
      -

      3.3.2 Result of simulation

      -

      The probability of success based on the non-informative and informative prior is 0.77 and 0.84, respectively.

      -
      -
      -
      -

      3.4 Analysis approach

      -

      The primary outcome will be assessed in both groups before and after the training. The difference in AHA between groups will be evaluated using a Bayesian linear regression model, adjusting for baseline values and stratification factors employed during randomization. An informative Gaussian prior, as described in the sample size section above, will be incorporated into the model. From the posterior distribution, we will calculate the mean difference along with a 95% credible interval as well as the probability that investigational treatment is superior to standard of care. In a sensitivity analysis, a non-informative prior will be employed.

      -
      -

      Author

      -

      Andreas Limacher, PhD
      +

      +

      Author

      +

      André Moser, Senior Statistician
      +Iulia-Maia Muresan, Statistician
      Department of Clinical Research
      University of Bern
      3012 Bern, Switzerland

      -
      @@ -743,13 +604,13 @@

      Author

      diff --git a/docs/01-SampleSize.html b/docs/01-SampleSize.html index 8fd9256..ef9f21b 100644 --- a/docs/01-SampleSize.html +++ b/docs/01-SampleSize.html @@ -123,11 +123,17 @@

      Sample size

      2  Two-arm study with a continuous primary outcome
      +
      • +
    • +
      + + 3  Seamless phase-II/III study +
    • - 3  External resources and references + 4  External resources and references
    @@ -153,7 +159,7 @@

    Sample size

    -

    This section covers specific the examples of the topic of ‘sample size’.

    +

    This section covers specific the examples related to the topic of ‘sample size’.

    diff --git a/docs/01-SampleSize_Bayesian_power_basics_examples.html b/docs/01-SampleSize_Bayesian_power_basics_examples.html index 623f901..28701e2 100644 --- a/docs/01-SampleSize_Bayesian_power_basics_examples.html +++ b/docs/01-SampleSize_Bayesian_power_basics_examples.html @@ -7,7 +7,7 @@ -Bayesian Toolbox – 3  External resources and references +Bayesian Toolbox – 4  External resources and references