Performs Bayesian multiple contrast test

This function performs a Bayesian multiple contrast test using normal mixture priors for the response on each dose, as proposed in Fleischer et al. (2022). For a general description of the multiple contrast test see MCTtest.

Usage

bMCTtest(
  dose,
  resp,
  data = NULL,
  models,
  S = NULL,
  type = c("normal", "general"),
  prior,
  alpha = 0.025,
  na.action = na.fail,
  mvtcontrol = mvtnorm.control(),
  contMat = NULL,
  critV = NULL
)

Arguments

dose, resp: Either vectors of equal length specifying dose and response values, or names of variables in the data frame specified in data.
data: Data frame containing the variables referenced in dose and resp if data is not specified it is assumed that dose and resp are variables referenced from data (and no vectors)
models: An object of class Mods, see Mods for details
S: The covariance matrix of resp when type = "general", see Description.
type: Determines whether inference is based on an ANCOVA model under a homoscedastic normality assumption (when type = "normal"), or estimates at the doses and their covariance matrix and degrees of freedom are specified directly in resp, S and df. See also fitMod and Pinheiro et al. (2014).
prior: List of length equal to the number of doses with the prior for each arm. Each element needs to be of class normMix (See RBesT package documentation). It is assumed that the i-th component of the prior list corresponds to the i-th largest dose. For example the first entry in the list is the prior for the placebo group, the second entry the prior for the second lowest dose and so on. Internally the priors across the different arms are combined (densities multiplied) assuming independence. The resulting multivariate normal mixture prior will have as many components as the product of the number of components of the individual mixture priors. The posterior mixture is part of the result object under "posterior".
alpha: Significance level for the frequentist multiple contrast test. If no critical values are supplied via critV this is used to derive critical values for Bayesian decision rule.
na.action: A function which indicates what should happen when the data contain NAs.
mvtcontrol: A list specifying additional control parameters for the qmvt and pmvt calls in the code, see also mvtnorm.control for details.
contMat: Contrast matrix to apply to the posterior dose-response estimates. The contrasts need to be in the columns of the matrix (i.e. the column sums need to be 0). If not specified optimal contrasts are calculated using optContr.
critV: Supply a critical value for the maximum posterior probability of the contrasts being greater than zero that needs to be surpassed to establish a non-flat dose-response. If this argument is NULL, this will be derived from critical values for frequentist MCP-Mod using the provided alpha.

Value

An object of class bMCTtest, a list containing the output.

Details

If type = "normal", an ANCOVA model based on a homoscedastic normality assumption is fitted and posteriors for dose-response and contrast vectors are obtained assuming a known variance.

For type = "general" it is assumed multivariate normally distributed estimates are specified in resp with covariance given by S, which define the likelihood. Posteriors for dose-response and contrast vectors are then obtained assuming a known covariance matrix S

The multiple contrast test decision is based on the maximum posterior probability of a contrast being greater than zero. Thresholds for the posterior probability can either be supplied or will be derived from frequentist critical values. In the latter case the Bayesian test will give approximately the same results as the frequentist multiple contrast test if uninformative priors are used.

For the default calculation of optimal contrasts the prior information is ignored (i.e. contrasts are calculated in the same way as in MCTtest). Fleischer et al. (2022) discuss using contrasts that take the prior effective sample sizes into account, which can be slightly more favourable for the Bayesian MCT test. Such alternative contrasts can be directly handed over via the contMat argument.

For analysis with covariate adjustment, covariate-adjusted resp and S can be supplied together with using type = "general". See `vignette("binary_data")` vignette "Design and analysis template MCP-Mod for binary data" for an example on how to obtain covariate adjusted estimates.

References

Fleischer, F., Bossert, S., Deng, Q., Loley, C. and Gierse, J. (2022). Bayesian MCP-Mod, Pharmaceutical Statistics, 21, 654–670

Author

Marius Thomas

Examples



if (require("RBesT")) {

###############################
## Normal outcome
###############################

data(biom)
## define shapes for which to calculate optimal contrasts
doses <- c(0, 0.05, 0.2, 0.6, 1)
modlist <- Mods(emax = 0.05, linear = NULL, logistic = c(0.5, 0.1),
                linInt = c(0, 1, 1, 1), doses = doses)
## specify an informative prior for placebo, weakly informative for other arms
plc_prior <- mixnorm(inf = c(0.8, 0.4, 0.1), rob = c(0.2, 0.4, 10))
vague_prior <- mixnorm(c(1, 0, 10))
## i-th component of the prior list corresponds to the i-th largest dose
## (e.g. 1st component -> placebo prior; last component prior for top dose)
prior <- list(plc_prior, vague_prior, vague_prior, vague_prior, vague_prior)

m1 <- bMCTtest(dose, resp, biom, models=modlist, prior = prior)
## now supply a critical value (= threshold for maxmimum posterior probability)
m2 <- bMCTtest(dose, resp, biom, models=modlist, prior = prior, critV = 0.99)

####################################
## Binary outcome with covariates
####################################
if (FALSE) { # \dontrun{
## generate data
logit <- function(p) log(p / (1 - p))
inv_logit <- function(y) 1 / (1 + exp(-y))
doses <- c(0, 0.5, 1.5, 2.5, 4)

## set seed and ensure reproducibility across R versions
set.seed(1, kind = "Mersenne-Twister", sample.kind = "Rejection", normal.kind = "Inversion")
group_size <- 100
dose_vector <- rep(doses, each = group_size)
N <- length(dose_vector)
## generate covariates
x1 <- rnorm(N, 0, 1)
x2 <- factor(sample(c("A", "B"), N, replace = TRUE, prob = c(0.6, 0.4)))
## assume approximately logit(10%) placebo and logit(35%) asymptotic response with ED50=0.5
prob <- inv_logit(emax(dose_vector, -2.2, 1.6, 0.5) + 0.3 * x1 + 0.3 * (x2 == "B"))
dat <- data.frame(y = rbinom(N, 1, prob),
                  dose = dose_vector, x1 = x1, x2 = x2)

## specify an informative prior for placebo (on logit scale), weakly informative for other arms
plc_prior <- mixnorm(inf = c(0.8, -2, 0.5), rob = c(0.2, -2, 10))
vague_prior <- mixnorm(c(1, 0, 10))
prior <- list(plc_prior, vague_prior, vague_prior, vague_prior, vague_prior)

## candidate models
mods <- Mods(emax = c(0.25, 1), sigEmax = rbind(c(1, 3), c(2.5, 4)), betaMod = c(1.1, 1.1),
             placEff = logit(0.1), maxEff = logit(0.35)-logit(0.1),
             doses = doses)

fit_cov <- glm(y~factor(dose) + 0 + x1 + x2, data = dat, family = binomial)

covariate_adjusted_estimates <- function(mu_hat, S_hat, formula_rhs,
                                         doses, other_covariates, n_sim) {
  ## predict every patient under *every* dose
  oc_rep <- as.data.frame(lapply(other_covariates, function(col) rep(col, times = length(doses))))
  d_rep <- rep(doses, each = nrow(other_covariates))
  pdat <- cbind(oc_rep, dose = d_rep)
  X <- model.matrix(formula_rhs, pdat)
  ## average on probability scale then backtransform to logit scale
  mu_star <- logit(tapply(inv_logit(X %*% mu_hat), pdat$dose, mean))
  ## estimate covariance matrix of mu_star
  pred <- replicate(n_sim, logit(tapply(inv_logit(X %*% drop(mvtnorm::rmvnorm(1, mu_hat, S_hat))),
                                        pdat$dose, mean)))
  return(list(mu_star = as.numeric(mu_star), S_star = cov(t(pred))))
}

ca <- covariate_adjusted_estimates(coef(fit_cov), vcov(fit_cov), ~factor(dose)+0+x1+x2,
                                   doses, dat[, c("x1", "x2")], 1000)
bMCTtest(doses, ca$mu_star, S = ca$S_star, type = "general", models = mods, prior = prior)
} # }
################################################
## example with contrasts handed over
################################################

data(biom)
## define shapes for which to calculate optimal contrasts
doses <- c(0, 0.05, 0.2, 0.6, 1)
modlist <- Mods(emax = 0.05, linear = NULL, sigEmax = c(0.5, 5),
                linInt = c(0, 1, 1, 1), doses = doses)

## specify an informative prior for placebo, weakly informative for other arms
plc_prior <- mixnorm(inf = c(0.8, 0.4, 0.1), rob = c(0.2, 0.4, 10), sigma = 0.7)
vague_prior <- mixnorm(c(1, 0, 10), sigma = 0.7)
prior <- list(plc_prior, vague_prior, vague_prior, vague_prior, vague_prior)

## use prior effective sample sizes to calculate optimal contrasts
prior_ess <- unlist(lapply(prior, ess))
n_grp <- as.numeric(table(biom$dose))
weights <- n_grp + prior_ess
cmat <- optContr(modlist, w = weights)

bMCTtest(dose, resp, biom, models=modlist, prior = prior, contMat = cmat)
}
#> Loading required package: RBesT
#> This is RBesT version 1.7.3 (released 2024-01-02, git-sha 511a0f1)
#> Using default prior reference scale 0.7
#> Using default prior reference scale 0.7
#> Using default prior reference scale 0.7
#> Using default prior reference scale 0.7
#> Using default prior reference scale 0.7
#> Bayesian MCP-Mod
#> 
#> Contrasts:
#>        emax linear sigEmax linInt
#> 0    -0.870 -0.681  -0.609 -0.762
#> 0.05  0.026 -0.191  -0.210 -0.263
#> 0.2   0.221 -0.060  -0.201  0.342
#> 0.6   0.302  0.291   0.400  0.342
#> 1     0.321  0.641   0.620  0.342
#> 
#> Posterior Mixture Weights:
#> Comp. 1 Comp. 2 
#>   0.995   0.005 
#> 
#> Bayesian t-statistics:
#>         Comp. 1 Comp. 2 posterior probability
#> linInt    4.167   3.372                     1
#> emax      4.111   2.996                     1
#> linear    3.709   3.198                     1
#> sigEmax   3.437   3.095                     1
#> 
#> Critical value (for maximum posterior probability): 0.987 (alpha = 0.025, one-sided)