library(BayesianMCPMod)
library(RBesT)
library(clinDR)
library(dplyr)
library(tibble)
library(reactable)

set.seed(7015)

#' Display Parameters Table
#'
#' This function generates a markdown table displaying the names and values of parameters
#' from a named list.
#'
#' @param named_list A named list where each name represents a parameter name and the list
#'   element represents the parameter value. Date values in the list are automatically
#'   converted to character strings for display purposes.
#'
#' @return Prints a markdown table with two columns: "Parameter Name" and "Parameter Values".
#'   The function does not return a value but displays the table directly to the output.
#'
#' @importFrom knitr kable
#' @examples
#' params <- list("Start Date" = as.Date("2020-01-01"),
#'                "End Date" = as.Date("2020-12-31"),
#'                "Threshold" = 10)
#' display_params_table(params)
#'
#' @export
display_params_table <- function(named_list) {
  display_table <- data.frame()
  value_names <- data.frame()
  for (i in 1:length(named_list)) {
    # dates will display as numeric by default, so convert to char first
    if (class(named_list[[i]]) == "Date") {
      named_list[[i]] = as.character(named_list[[i]])
    }
    if (!is.null(names(named_list[[i]]))) {
      value_names <- rbind(value_names, paste(names(named_list[[i]]), collapse = ', '))
    }
    values <- data.frame(I(list(named_list[[i]])))
    display_table <- rbind(display_table, values)
  }
  
  round_numeric <- function(x, digits = 3) {
    if (is.numeric(x)) {
      return(round(x, digits))
    } else {
      return(x)
    }
  }
  
  display_table[1] <- lapply(display_table[1], function(sublist) {
    lapply(sublist, round_numeric)
  })
  
  class(display_table[[1]]) <- "list"
  
  if (nrow(value_names) == 0) {
    knitr::kable(
      cbind(names(named_list), display_table),
      col.names = c("Name", "Value")
    )
  } else {
    knitr::kable(
      cbind(names(named_list), value_names, display_table),
      col.names = c("Name", "Value Labels", "Value")
    )
  }
}

Introduction

This vignette demonstrates the application of the BayesianMCPMod package for analyzing a phase 2 dose-finding trial using the Bayesian MCPMod approach.

A more general overview of the R package was provided with a poster presented during the PSI 2024 Conference.

This package makes use of the future framework for parallel processing, which can be set up as follows:

future::plan(future::multisession)

Calculation of a MAP Prior

In a first step, a meta analytic prior will be calculated using historical data from 4 trials with main endpoint Change from baseline in MADRS score after 8 weeks. Please note that only information from the control group will be integrated leading to an informative mixture prior for the control group, while for the active groups a non-informative prior will be specified.

data("metaData")
dataset     <- filter(as.data.frame(metaData), bname == "BRINTELLIX")
histcontrol <- filter(
  dataset,
  dose       == 0,
  primtime   == 8,
  indication == "MAJOR DEPRESSIVE DISORDER",
  protid     != 5)

hist_data   <- data.frame(
  trial = histcontrol$nctno,
  est   = histcontrol$rslt,
  se    = histcontrol$se,
  sd    = histcontrol$sd,
  n     = histcontrol$sampsize)

Here, we suggest a function to construct a list of prior distributions for the different dose groups. This function is adapted to the needs of this example. Other applications may need a different way to construct prior distributions.

getPriorList <- function (
  
  hist_data,
  dose_levels,
  dose_names    = NULL,
  robust_weight = 0.5
  
) {
  
  sd_tot <- with(hist_data, sum(sd * n) / sum(n))
  
  gmap <- RBesT::gMAP(
    formula    = cbind(est, se) ~ 1 | trial,
    weights    = hist_data$n,
    data       = hist_data,
    family     = gaussian,
    beta.prior = cbind(0, 100 * sd_tot),
    tau.dist   = "HalfNormal",
    tau.prior  = cbind(0, sd_tot / 4))
  
  prior_ctr <- RBesT::automixfit(gmap)
  
  if (!is.null(robust_weight)) {
    
    prior_ctr <- suppressMessages(RBesT::robustify(
      priormix = prior_ctr,
      weight   = robust_weight,
      sigma    = sd_tot))
    
  }
  
  prior_trt <- RBesT::mixnorm(
    comp1 = c(w = 1, m = summary(prior_ctr)[1], n = 1),
    sigma = sd_tot,
    param = "mn")
  
  prior_list <- c(list(prior_ctr),
                  rep(x     = list(prior_trt),
                      times = length(dose_levels[-1])))
  
  if (is.null(dose_names)) {
    
    dose_names <- c("Ctr", paste0("DG_", seq_along(dose_levels[-1])))
    
  }
  
  names(prior_list) <- dose_names
  
  return (prior_list)
  
}

With the dose levels to be investigated, the prior distribution can be constructed.

dose_levels <- c(0, 2.5, 5, 10)

prior_list  <- getPriorList(
  hist_data     = hist_data,
  dose_levels   = dose_levels,
  robust_weight = 0.3)

getESS(prior_list)
##  Ctr DG_1 DG_2 DG_3 
## 17.5  1.0  1.0  1.0

Dose-Response Model Shapes

Candidate models are specified using the {DoseFinding} package. Models can be parameterized using guesstimates or by directly providing distribution parameters. Note that the linear candidate model does not require parameterization.

Note: The LinLog model is rarely used and not currently supported by {BayesianMCPMod}.

In the code below, the models are “guesstimated” using the DoseFinding::guesst function. The d option usually takes a single value (a dose level), and the corresponding p for the maximum effect achieved at d.

# Guesstimate estimation
exp_guesst  <- DoseFinding::guesst(
  model = "exponential", 
  d = 5, p = 0.2, Maxd = max(dose_levels)
)
emax_guesst <- DoseFinding::guesst(
  model = "emax",
  d = 2.5, p = 0.9
)
sigEmax_guesst <- DoseFinding::guesst(
  model = "sigEmax",
  d = c(2.5, 5), p = c(0.5, 0.95)
)
logistic_guesst <- DoseFinding::guesst(
  model = "logistic",
  d = c(5, 10), p = c(0.1, 0.85)
)

In some cases, you need to provide more information. For instance, sigEmax requires a pair of d and p values, and exponential requires the specification of the maximum dose for the trial (Maxd).

See the help files for model specifications by typing ?DoseFinding::guesst in your console

Of course, you can also specify the models directly on the parameter scale (without using DoseFinding::guesst).

For example, you can get a betaMod model by specifying delta1 and delta2 parameters (scale is assumed to be 1.2 of the maximum dose), or a quadratic model with the delta2 parameter.

betaMod_params   <- c(delta1 = 1, delta2 = 1)
quadratic_params <- c(delta2 = -0.1)

Now, we can go ahead and create a Mods object, which will be used in the remainder of the vignette.

mods <- DoseFinding::Mods(
  linear      = NULL,
  # guesstimate scale
  exponential = exp_guesst,
  emax        = emax_guesst,
  sigEmax     = sigEmax_guesst,
  logistic    = logistic_guesst,
  # parameter scale
  betaMod     = betaMod_params,
  quadratic   = quadratic_params,
  # Options for all models
  doses       = dose_levels,
  maxEff      = -1,
  placEff     = -12.8
)

plot(mods)

The mods object we just created above contains the full model parameters, which can be helpful for understanding how the guesstimates are translated onto the parameter scale.

display_params_table(mods)
Name Value Labels Value
linear e0, delta -12.8, -0.1
exponential e0, e1, delta -12.800, -0.067, 3.607
emax e0, eMax, ed50 -12.800, -1.028, 0.278
sigEmax e0, eMax, ed50, h -12.800, -1.003, 2.500, 4.248
logistic e0, eMax, ed50, delta -12.797, -1.179, 7.794, 1.272
betaMod e0, eMax, delta1, delta2 -12.8, -1.0, 1.0, 1.0
quadratic e0, b1, b2 -12.80, -0.40, 0.04

And we can see the assumed treatment effects for the specified dose groups below:

knitr::kable(DoseFinding::getResp(mods, doses = dose_levels))
linear exponential emax sigEmax logistic betaMod quadratic
0 -12.80 -12.80000 -12.80000 -12.80000 -12.80000 -12.80000 -12.80
2.5 -13.05 -12.86667 -13.72500 -13.30138 -12.81551 -13.45972 -13.55
5 -13.30 -13.00000 -13.77368 -13.75263 -12.91538 -13.77222 -13.80
10 -13.80 -13.80000 -13.80000 -13.80000 -13.80000 -13.35556 -12.80

Trial Data

We will use the trial with ct.gov number NCT00735709 as our phase 2 trial data, available in the {clinDR} package [@nct00735709_2024a].

data("metaData")

trial_data <- dplyr::filter(
  dplyr::filter(tibble::tibble(metaData), bname == "BRINTELLIX"),
  primtime   == 8,
  indication == "MAJOR DEPRESSIVE DISORDER",
  protid     == 5
)

n_patients <- c(128, 124, 129, 122)

Posterior Calculation

In the first step of Bayesian MCPMod, the posterior is calculated by combining the prior information with the estimated results of the trial [@fleischer_2022].

posterior <- getPosterior(
  prior_list = prior_list,
  mu_hat     = trial_data$rslt,
  S_hat      = trial_data$se,
  calc_ess   = TRUE
)

knitr::kable(summary(posterior))
mean sd 2.5% 50.0% 97.5%
Ctr -11.21305 0.7040137 -12.55616 -11.22607 -9.801711
DG_1 -14.88094 0.7130817 -16.27856 -14.88094 -13.483329
DG_2 -15.07996 0.7101057 -16.47174 -15.07996 -13.688175
DG_3 -15.63639 0.7259755 -17.05928 -15.63639 -14.213506

Bayesian MCPMod Test Step

The testing step of Bayesian MCPMod is executed using a critical value on the probability scale and a pseudo-optimal contrast matrix.

The critical value is calculated using (re-estimated) contrasts for frequentist MCPMod to ensure error control when using weakly-informative priors.

A pseudo-optimal contrast matrix is generated based on the variability of the posterior distribution (see [@fleischer_2022] for more details).

crit_pval <- getCritProb(
  mods           = mods,
  dose_levels    = dose_levels,
  se_new_trial   = trial_data$se,
  alpha_crit_val = 0.05
)

contr_mat <- getContr(
  mods         = mods,
  dose_levels  = dose_levels,
  sd_posterior = summary(posterior)[, 2]
)

Please note that there are different ways to derive the contrasts. The following code shows the implementation of some of these ways but it is not executed and the contrast specification above is used.

# i) the frequentist contrast
contr_mat_prior <- getContr(
  mods           = mods,
  dose_levels    = dose_levels,
  dose_weights   = n_patients,
  prior_list     = prior_list)
# ii) re-estimated frequentist contrasts
contr_mat_prior <- getContr(
  mods           = mods,
  dose_levels    = dose_levels,
  se_new_trial   = trial_data$se)
# iii)  Bayesian approach using number of patients for new trial and prior distribution
contr_mat_prior <- getContr(
  mods           = mods,
  dose_levels    = dose_levels,
  dose_weights   = n_patients,
  prior_list     = prior_list)

The Bayesian MCP testing step is then executed:

BMCP_result <- performBayesianMCP(
  posterior_list = posterior,
  contr          = contr_mat, 
  crit_prob_adj  = crit_pval)

Summary information:

BMCP_result
## Bayesian Multiple Comparison Procedure
##   Significant:                   1 
##   Critical Probability:          0.9842778 
##   Maximum Posterior Probability: 0.9999999 
## Posterior Probabilities for Model Shapes
##                        lin       exp      emax      sigE       log     betaM      quad
##   Posterior Prob 0.9999616 0.9982012 0.9999999 0.9999987 0.9954502 0.9999938 0.9886949 
## Average Posterior ESS
##   Dose Level:     Ctr  DG_1  DG_2  DG_3 
##   Avg Post ESS: 192.2 186.6 188.2 180.0

The testing step is significant, indicating a non-flat dose-response shape. All models are significant, with the emax model indicating the greatest deviation from the null hypothesis.

Model Fitting and Visualization

In the model fitting step the posterior distribution is used as basis.

Both simplified and full fitting are performed.

For the simplified fit, the multivariate normal distribution of the control group is approximated and reduced by a one-dimensional normal distribution.

The actual fit (on this approximated posterior distribution) is then performed using generalized least squares criterion. In contrast, for the full fit, the non-linear optimization problem is addressed via the Nelder-Mead algorithm [@neldermead_2024a] implemented by the {nloptr} package.

The output of the fit includes information about the predicted effects for the included dose levels, the generalized AIC, and the corresponding weights.

For the considered case, the simplified and the full fit are very similar, so wepresent the full fit.

# If simple = TRUE, uses approx posterior
# Here we use complete posterior distribution
model_fits <- getModelFits(
  models      = mods,
  dose_levels = dose_levels,
  posterior   = posterior,
  simple      = FALSE)

Estimates for dose levels not included in the trial:

display_params_table(stats::predict(model_fits, doses = c(0, 2.5, 4, 5, 7, 10)))
Name Value
linear -12.447, -13.427, -14.015, -14.407, -15.190, -16.366
exponential -12.635, -13.394, -13.897, -14.253, -15.023, -16.331
emax -11.248, -14.805, -15.134, -15.257, -15.408, -15.528
sigEmax -11.247, -14.810, -15.092, -15.217, -15.394, -15.566
logistic -11.249, -14.836, -15.273, -15.352, -15.390, -15.395
betaMod -11.250, -14.786, -15.098, -15.246, -15.450, -15.556
quadratic -11.490, -14.108, -15.165, -15.655, -16.122, -15.536
avgFit -11.300, -14.683, -15.133, -15.314, -15.517, -15.531

Plots of fitted dose-response models and an AIC-based average model:

plot(model_fits)

To assess the uncertainty, one can additionally visualize credible bands (orange shaded areas, default levels are 50% and 95%).

These credible bands are calculated with a bootstrap method as follows:

plot(model_fits, cr_bands = TRUE)

The bootstrap-based quantiles can also be directly calculated via the getBootstrapQuantiles() function and a sample from the model fits can bootstrapped using getBootstrapSamples().

For this example, only 10 samples are bootstrapped for each model fit.

bootstrap_quantiles <- getBootstrapQuantiles(
  model_fits = model_fits,
  quantiles  = c(0.025, 0.5, 0.975),
  doses      = c(0, 2.5, 4, 5, 7, 10),
  n_samples  = 10)
reactable::reactable(
  data = bootstrap_quantiles |>
      tidyr::pivot_wider(names_from = q_probs, values_from = q_values),
  groupBy = "models",
  columns = list(
    doses   = colDef(aggregate = "count", format = list(aggregated = colFormat(suffix = " doses"))),
    "0.025"  = colDef(aggregate = "mean", format = list(aggregated = colFormat(prefix = "mean = ", digits = 2), cell = colFormat(digits = 4))),
    "0.5"   = colDef(aggregate = "mean", format = list(aggregated = colFormat(prefix = "mean = ", digits = 2), cell = colFormat(digits = 4))),
    "0.975" = colDef(aggregate = "mean", format = list(aggregated = colFormat(prefix = "mean = ", digits = 2), cell = colFormat(digits = 4)))
  )
)

Technical note: The median quantile of the bootstrap based procedure is not necessary similar to the main model fit, as they are derived via different procedures.

The main fit (black line) minimizes residuals for the posterior distribution, while the bootstrap median is the median fit of random sampling.

Assessment of the Minimally Efficacious Dose

The Minimally Efficacious Dose (MED) per model shape can be assessed with the function getMED().

getMED(delta       = 4,
       model_fits  = model_fits,
       dose_levels = seq(min(dose_levels), max(dose_levels), by = 0.01))
##             linear exponential emax sigEmax logistic betaMod quadratic avgFit
## med_reached      0           0 1.00    1.00     1.00    1.00      1.00   1.00
## med             NA          NA 4.92    5.29     3.83    5.04      4.63   4.92

For an optional Bayesian decision rule for the MED assessment and further details, please see ?getMED().

Additional Note

Testing, modeling, and MED assessment can also be combined via performBayesianMCPMod():

performBayesianMCPMod(
  posterior_list   = posterior,
  contr            = contr_mat,
  crit_prob_adj    = crit_pval,
  delta            = 4,
  simple           = FALSE)