bayescomp: Bayesian Compound DP-Optimal Designs

Description

Finds compound Bayesian DP-optimal designs that meet the dual goal of parameter estimation and increasing the probability of a particular outcome in a binary response model. A compound Bayesian DP-optimal design maximizes the product of the Bayesian efficiencies of a design $\xi$ with respect to D- and average P-optimality, weighted by a pre-defined mixing constant $0 \leq \alpha \leq 1$.

Usage

bayescomp(
  formula,
  predvars,
  parvars,
  family = binomial(),
  prior,
  alpha,
  prob,
  lx,
  ux,
  iter,
  k,
  fimfunc = NULL,
  ICA.control = list(),
  sens.control = list(),
  crt.bayes.control = list(),
  sens.bayes.control = list(),
  initial = NULL,
  npar = NULL,
  plot_3d = c("lattice", "rgl")
)

Value

An object of class 'sensminimax'. See sensminimax for structure details.

Arguments

formula: A linear or nonlinear model formula. A symbolic description of the model consists of predictors and the unknown model parameters. Will be coerced to a formula if necessary.
predvars: A vector of characters. Denotes the predictors in the formula.
parvars: A vector of characters. Denotes the unknown parameters in the formula.
family: A description of the response distribution and the link function to be used in the model. This can be a family function, a call to a family function or a character string naming the family. Every family function has a link argument allowing to specify the link function to be applied on the response variable. If not specified, default links are used. For details see family. By default, a linear gaussian model gaussian() is applied.
prior: An object of class cprior. User can also use one of the functions uniform, normal, skewnormal or student to create the prior. See 'Details' of bayes.
alpha: A value between 0 and 1. Compound or combined DP-criterion is the product of the efficiencies of a design with respect to D- and average P- optimality, weighted by alpha.
prob: Either formula or a function. When function, its argument are x and param, and they are the same as the arguments in fimfunc. prob as a function takes the design points and vector of parameters and returns the probability of success at each design points. See 'Examples'.
lx: Vector of lower bounds for the predictors. Should be in the same order as predvars.
ux: Vector of upper bounds for the predictors. Should be in the same order as predvars.
iter: Maximum number of iterations.
k: Number of design points. Must be at least equal to the number of model parameters to avoid singularity of the FIM.
fimfunc: A function. Returns the FIM as a matrix. Required when formula is missing. See 'Details' of minimax.
ICA.control: ICA control parameters. For details, see ICA.control.
sens.control: Control Parameters for Calculating the ELB. For details, see sens.control.
crt.bayes.control: A list. Control parameters to approximate the integral in the Bayesian criterion at a given design over the parameter space. For details, see crt.bayes.control.
sens.bayes.control: A list. Control parameters to verify the general equivalence theorem. For details, see sens.bayes.control.
initial: A matrix of the initial design points and weights that will be inserted into the initial solutions (countries) of the algorithm. Every row is a design, i.e. a concatenation of x and w. Will be coerced to a matrix if necessary. See 'Details' of minimax.
npar: Number of model parameters. Used when fimfunc is given instead of formula to specify the number of model parameters. If not specified correctly, the sensitivity (derivative) plot may be shifted below the y-axis. When NULL (default), it will be set to length(parvars) or prior$npar when missing(formula).
plot_3d: Which package should be used to plot the sensitivity (derivative) function for two-dimensional design space. Defaults to "lattice".

Details

Let $\Xi$ be the space of all approximate designs with $k$ design points (support points) at $x_1, x_2, ..., x_k$ from design space $\chi$ with corresponding weights $w_1,... ,w_k$. Let $M(\xi, \theta)$ be the Fisher information matrix (FIM) of a $k-$point design $\xi$, $\pi(\theta)$ is a user-given prior distribution for the vector of unknown parameters $\theta$ and $p(x_i, \theta)$ is the ith probability of success given by $x_i$ in a binary response model. A compound Bayesian DP-optimal design maximizes over $\Xi$ $$\int_{\theta \in \Theta} \frac{\alpha}{q}\log|M(\xi, \theta)| + (1- \alpha) \log \left( \sum_{i=1}^k w_ip(x_i, \theta) \right) \pi(\theta) d\theta.$$

To verify the equivalence theorem of the output design, use plot function or change the value of the checkfreq in the argument ICA.control.

To increase the speed of the algorithm, change the value of the tuning parameters tol and maxEval via the argument crt.bayes.control when its method component is equal to "cubature". In general, if the CPU time matters, the user should find an appropriate speed-accuracy trade-off for her/his own problem. See 'Examples' for more details.

References

McGree, J. M., Eccleston, J. A., and Duffull, S. B. (2008). Compound optimal design criteria for nonlinear models. Journal of Biopharmaceutical Statistics, 18(4), 646-661.

Examples

Run this code

##########################################################################
# DP-optimal design for a logitic model with two predictors: with formula
##########################################################################
p <- c(1, -2, 1, -1)
myprior <- uniform(p -1.5, p + 1.5)
myformula1 <- ~exp(b0+b1*x1+b2*x2+b3*x1*x2)/(1+exp(b0+b1*x1+b2*x2+b3*x1*x2))
res1 <- bayescomp(formula = myformula1,
                  predvars = c("x1", "x2"),
                  parvars = c("b0", "b1", "b2", "b3"),
                  family = binomial(),
                  lx = c(-1, -1), ux = c(1, 1),
                  prior = myprior, iter = 1, k = 7,
                  prob = ~1-1/(1+exp(b0 + b1 * x1 + b2 * x2 + b3 * x1 * x2)),
                  alpha = .5, ICA.control = list(rseed = 1366),
                  crt.bayes.control = list(cubature = list(tol = 1e-4, maxEval = 1000)))


 # \donttest{
  res1 <- update(res1, 1000)
  plot(res1, sens.bayes.control = list(cubature = list(tol = 1e-3, maxEval = 1000)))
  # or use quadrature method
  plot(res1, sens.bayes.control= list(method = "quadrature"))
# }

##########################################################################
# DP-optimal design for a logitic model with two predictors: with fimfunc
##########################################################################
# The function of the Fisher information matrix for this model is 'FIM_logistic_2pred'
# We should reparameterize it to match the standard of the argument 'fimfunc'
 # \donttest{
myfim <- function(x, w, param){
  npoint <- length(x)/2
  x1 <- x[1:npoint]
  x2 <- x[(npoint+1):(npoint*2)]
  FIM_logistic_2pred(x1 = x1,x2 = x2, w = w, param = param)
}

## The following function is equivalent to the function created
# by the formula: ~1-1/(1+exp(b0 + b1 * x1 + b2 * x2 + b3 * x1 * x2))
# It returns probability of success given x and param
# x = c(x1, x2) and param = c()

myprob <- function(x, param){
  npoint <- length(x)/2
  x1 <- x[1:npoint]
  x2 <- x[(npoint+1):(npoint*2)]
  b0 <- param[1]
  b1 <- param[2]
  b2 <- param[3]
  b3 <- param[4]
  out <- 1-1/(1+exp(b0 + b1 * x1 + b2 * x2 + b3 * x1 * x2))
  return(out)
}

res2 <- bayescomp(fimfunc = myfim,
                  lx = c(-1, -1), ux = c(1, 1),
                  prior = myprior, iter = 1000, k = 7,
                  prob = myprob, alpha = .5,
                  ICA.control = list(rseed = 1366))
  plot(res2, sens.bayes.control = list(cubature = list(maxEval = 1000, tol = 1e-4)))
  # quadrature with 6 nodes (default)
  plot(res2, sens.bayes.control= list(method = "quadrature"))
# }

Run the code above in your browser using DataLab