ICAOD (version 1.0.1)

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")
)

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".

Value

an object of class minimax that is a list including three sub-lists:

arg

A list of design and algorithm parameters.

evol

A list of length equal to the number of iterations that stores the information about the best design (design with the minimum criterion value) of each iteration as follows: evol[[iter]] contains:

iter		Iteration number.
x		Design points.
w		Design weights.
min_cost		Value of the criterion for the best imperialist (design).
mean_cost		Mean of the criterion values of all the imperialists.
sens		An object of class 'sensminimax'. See below.

empires

A list of all the empires of the last iteration.

alg

A list with following information:

nfeval		Number of function evaluations. It does not count the function evaluations from checking the general equivalence theorem.
nlocal		Number of successful local searches.
nrevol		Number of successful revolutions.
nimprove		Number of successful movements toward the imperialists in the assimilation step.
convergence		Stopped by 'maxiter' or 'equivalence'?

method

A type of optimal designs used.

design

Design points and weights at the final iteration.

out

A data frame of design points, weights, value of the criterion for the best imperialist (min_cost), and Mean of the criterion values of all the imperialistsat each iteration (mean_cost).

The list sens contains information about the design verification by the general equivalence theorem. See sensbayes for more Details. It is only given every ICA.control$checkfreq iterations and also the last iteration if ICA.control$checkfreq >= 0. Otherwise, NULL.

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.

See Also

sensbayescomp

Examples

# NOT RUN {
##########################################################################
# 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)))


# }
# NOT RUN {
  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"))
# }
# NOT RUN {
##########################################################################
# 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'
# }
# NOT RUN {
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"))
# }
# NOT RUN {

# }