setup
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.
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
19.4 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)
| 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))
| 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 (ClinicalTrials.gov 2024).
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 F 2022).
posterior <- getPosterior(
prior_list = prior_list,
mu_hat = trial_data$rslt,
S_hat = trial_data$se,
calc_ess = TRUE
)
knitr::kable(summary(posterior))
| Ctr |
-11.18718 |
0.7234646 |
-12.58113 |
-11.19002 |
-9.766636 |
| DG_1 |
-14.88089 |
0.7130817 |
-16.27850 |
-14.88089 |
-13.483275 |
| DG_2 |
-15.07990 |
0.7101057 |
-16.47168 |
-15.07990 |
-13.688121 |
| DG_3 |
-15.63634 |
0.7259755 |
-17.05922 |
-15.63634 |
-14.213450 |
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 F 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:
Bayesian Multiple Comparison Procedure
Summary:
Sign: 1
Critical Probability: 0.9842778
Maximum Posterior Probability: 0.9999999
Posterior Probabilities for Model Shapes:
Model Probability
linear 0.9999460
exponential 0.9977794
emax 0.9999999
sigEmax 0.9999983
logistic 0.9946123
betaMod 0.9999917
quadratic 0.9859196
Average Posterior ESS
Ctr DG_1 DG_2 DG_3
181.8 186.6 188.2 180.0
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 (Wikipedia 2024) 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 we present the full fit.
# If simple = TRUE, uses approx posterior
# Here we use complete posterior distribution
fit <- 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(fit, doses = c(0, 2.5, 4, 5, 7, 10)))
| linear |
-12.448, -13.427, -14.014, -14.406, -15.189, -16.364 |
| exponential |
-12.634, -13.393, -13.896, -14.253, -15.022, -16.331 |
| emax |
-11.245, -14.805, -15.134, -15.257, -15.408, -15.528 |
| sigEmax |
-11.245, -14.810, -15.092, -15.217, -15.394, -15.566 |
| logistic |
-11.245, -14.837, -15.273, -15.352, -15.389, -15.394 |
| betaMod |
-11.247, -14.786, -15.098, -15.246, -15.450, -15.556 |
| quadratic |
-11.488, -14.107, -15.165, -15.655, -16.122, -15.536 |
Plots of fitted dose-response models and an AIC-based average model:
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:
Samples from the posterior distribution are drawn and for every sample the simplified fitting step and a prediction is performed.
These predictions are then used to identify and visualize the specified quantiles.
plot(fit, cr_bands = TRUE)
The bootstrap based quantiles can also be directly calculated via the getBootstrapQuantiles() function.
For this example, only 6 quantiles are bootstrapped for each model fit.
bootstrap_quantiles <- getBootstrapQuantiles(
model_fits = fit,
quantiles = c(0.025, 0.5, 0.975),
doses = c(0, 2.5, 4, 5, 7, 10),
n_samples = 6
)
setup
reactable::reactable(
data = bootstrap_quantiles,
groupBy = "models",
columns = list(
doses = colDef(aggregate = "count", format = list(aggregated = colFormat(suffix = " doses"))),
"2.5%" = colDef(aggregate = "mean", format = list(aggregated = colFormat(prefix = "mean = ", digits = 2), cell = colFormat(digits = 4))),
"50%" = colDef(aggregate = "mean", format = list(aggregated = colFormat(prefix = "mean = ", digits = 2), cell = colFormat(digits = 4))),
"97.5%" = 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.
Additional note
Testing and modeling can also be combined via performBayesianMCPMod(), but this is not run here.
performBayesianMCPMod(
posterior_list = posterior,
contr = contr_mat,
crit_prob_adj = crit_pval,
simple = FALSE)
Fleischer F, Deng Q, Bossert S. 2022. “Bayesian MCPMod.” Pharmaceutical Statistics 21 (3): 654–70.
