ICAOD (version 1.0.1)

sensbayes: Verifying Optimality of Bayesian D-optimal Designs

Description

Plots the sensitivity (derivative) function and calculates the efficiency lower bound (ELB) for a given Bayesian design. Let \(\boldsymbol{x}\) belongs to \(\chi\) that denotes the design space. Based on the general equivalence theorem, a design \(\xi^*\) is optimal if and only if the value of the sensitivity (derivative) function is non-positive for all \(\boldsymbol{x}\) in \(\chi\) and zero when \(\boldsymbol{x}\) belongs to the support of \(\xi^*\) (be equal to the one of the design points).

For an approximate (continuous) design, when the design space is one or two-dimensional, the user can visually verify the optimality of the design by observing the sensitivity plot. Furthermore, the proximity of the design to the optimal design can be measured by the ELB without knowing the latter.

Usage

sensbayes(
  formula,
  predvars,
  parvars,
  family = gaussian(),
  x,
  w,
  lx,
  ux,
  fimfunc = NULL,
  prior = list(),
  sens.control = list(),
  sens.bayes.control = list(),
  crt.bayes.control = list(),
  plot_3d = c("lattice", "rgl"),
  plot_sens = TRUE,
  npar = NULL,
  calculate_criterion = TRUE,
  silent = FALSE,
  crtfunc = NULL,
  sensfunc = NULL
)

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.

x

A vector of candidate design (support) points. When is not set to NULL (default), the algorithm only finds the optimal weights for the candidate points in x. Should be set when the user has a finite number of candidate design points and the purpose is to find the optimal weight for each of them (when zero, they will be excluded from the design). For design points with more than one dimension, see 'Details' of sensminimax.

w

Vector of the corresponding design weights for x.

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.

fimfunc

A function. Returns the FIM as a matrix. Required when formula is missing. See 'Details' of minimax.

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.

sens.control

Control Parameters for Calculating the ELB. For details, see sens.control.

sens.bayes.control

A list. Control parameters to verify the general equivalence theorem. For details, see sens.bayes.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.

plot_3d

Which package should be used to plot the sensitivity (derivative) function for two-dimensional design space. Defaults to "lattice".

plot_sens

Plot the sensitivity (derivative) function? Defaults to TRUE.

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).

calculate_criterion

Calculate the optimality criterion? See 'Details' of sensminimax.

silent

Do not print anything? Defaults to FALSE.

crtfunc

(Optional) a function that specifies an arbitrary criterion. It must have especial arguments and output. See 'Details' of bayes.

sensfunc

(Optional) a function that specifies the sensitivity function for crtfunc. See 'Details' of bayes.

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 \(\xi\) and \(\pi(\theta)\) is a user-given prior distribution for the vector of unknown parameters \(\theta\). A design \(\xi^*\) is Bayesian D-optimal among all designs on \(\chi\) if and only if the following inequality holds for all \(\boldsymbol{x} \in \chi\) $$c(\boldsymbol{x}, \xi^*) = \int_{\theta \in Theta}tr M^{-1}(\xi^*, \theta)I(\boldsymbol{x}, \theta)-p \pi(\theta) d\theta\leq 0,$$ with equality at all support points of \(\xi^*\). Here, \(p\) is the number of model parameters. \(c(\boldsymbol{x},\xi^*)\) is called sensitivity or derivative function.

Depending on the complexity of the problem at hand, sometimes, the CPU time can be considerably reduced by choosing a set of less conservative values for the tuning parameters tol and maxEval in the function sens.bayes.control when sens.bayes.control$method = "cubature". Similarly, this applies when sens.bayes.control$method = "quadrature". 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.

Examples

# NOT RUN {
##################################################################
# Checking the Bayesian D-optimality of a design for the 2Pl model
##################################################################
skew2 <- skewnormal(xi = c(0, 1), Omega = matrix(c(1, -0.17, -0.17, .5), nrow = 2),
                    alpha = c(-1, 0), lower =  c(-3, .1), upper = c(3, 2))
# }
# NOT RUN {
  sensbayes(formula = ~1/(1 + exp(-b *(x - a))),
            predvars = "x", parvars = c("a", "b"),
            family = binomial(),
            x= c(-2.50914, -1.16780, -0.36904, 1.29227),
            w =c(0.35767, 0.11032, 0.15621, 0.37580),
            lx = -3, ux = 3,
            prior = skew2)
  # took 29 seconds on my system!
# }
# NOT RUN {
# It took very long.
# We re-adjust the tuning parameters in sens.bayes.control to be faster
# See how we drastically reduce the maxEval and increase the tolerance
# }
# NOT RUN {
  sensbayes(formula = ~1/(1 + exp(-b *(x - a))),
            predvars = "x", parvars = c("a", "b"),
            family = binomial(),
            x= c(-2.50914, -1.16780, -0.36904, 1.29227),
            w =c(0.35767, 0.11032, 0.15621, 0.37580),
            lx = -3, ux = 3,prior = skew2,
            sens.bayes.control = list(cubature = list(tol = 1e-4, maxEval = 300)))
  # took 5 Seconds on my system!
# }
# NOT RUN {


# Compare it with the following:
sensbayes(formula = ~1/(1 + exp(-b *(x - a))),
          predvars = "x", parvars = c("a", "b"),
          family = binomial(),
          x= c(-2.50914, -1.16780, -0.36904, 1.29227),
          w =c(0.35767, 0.11032, 0.15621, 0.37580),
          lx = -3, ux = 3,prior = skew2,
          sens.bayes.control = list(cubature = list(tol = 1e-4, maxEval = 200)))
# Look at the plot!
# took 3 seconds on my system


########################################################################################
# Checking the Bayesian D-optimality of a design for the 4-parameter sigmoid emax model
########################################################################################
lb <- c(4, 11, 100, 5)
ub <- c(9, 17, 140, 10)
# }
# NOT RUN {
  sensbayes(formula = ~ theta1 + (theta2 - theta1)*(x^theta4)/(x^theta4 + theta3^theta4),
            predvars = c("x"), parvars = c("theta1", "theta2", "theta3", "theta4"),
            x = c(0.78990, 95.66297, 118.42964,147.55809, 500),
            w = c(0.23426, 0.17071, 0.17684, 0.1827, 0.23549),
            lx = .001, ux = 500,  prior = uniform(lb, ub))
  # took 200 seconds on my system
# }
# NOT RUN {
# Re-adjust the tuning parameters to have it faster
# }
# NOT RUN {
  sensbayes(formula = ~ theta1 + (theta2 - theta1)*(x^theta4)/(x^theta4 + theta3^theta4),
            predvars = c("x"), parvars = c("theta1", "theta2", "theta3", "theta4"),
            x = c(0.78990, 95.66297, 118.42964,147.55809, 500),
            w = c(0.23426, 0.17071, 0.17684, 0.1827, 0.23549),
            lx = .001, ux = 500,  prior = uniform(lb, ub),
            sens.bayes.control = list(cubature = list(tol = 1e-3, maxEval = 300)))
  # took 4 seconds on my system. See how much it makes difference
# }
# NOT RUN {
# }
# NOT RUN {
  # Now we try it with quadrature. Default is 6 nodes
  sensbayes(formula = ~ theta1 + (theta2 - theta1)*(x^theta4)/(x^theta4 + theta3^theta4),
            predvars = c("x"), parvars = c("theta1", "theta2", "theta3", "theta4"),
            x = c(0.78990, 95.66297, 118.42964,147.55809, 500),
            w = c(0.23426, 0.17071, 0.17684, 0.1827, 0.23549),
            sens.bayes.control = list(method = "quadrature"),
            lx = .001, ux = 500,  prior = uniform(lb, ub))
  # 166.519 s

  # use less number of nodes  to see if we can reduce the CPU time
  sensbayes(formula = ~ theta1 + (theta2 - theta1)*(x^theta4)/(x^theta4 + theta3^theta4),
            predvars = c("x"), parvars = c("theta1", "theta2", "theta3", "theta4"),
            x = c(0.78990, 95.66297, 118.42964,147.55809, 500),
            w = c(0.23426, 0.17071, 0.17684, 0.1827, 0.23549),
            sens.bayes.control = list(method = "quadrature",
                                      quadrature = list(level = 3)),
            lx = .001, ux = 500,  prior = uniform(lb, ub))
  # we don't have an accurate plot

  # use less number of levels: use 4 nodes
  sensbayes(formula = ~ theta1 + (theta2 - theta1)*(x^theta4)/(x^theta4 + theta3^theta4),
            predvars = c("x"), parvars = c("theta1", "theta2", "theta3", "theta4"),
            x = c(0.78990, 95.66297, 118.42964,147.55809, 500),
            w = c(0.23426, 0.17071, 0.17684, 0.1827, 0.23549),
            sens.bayes.control = list(method = "quadrature",
                                      quadrature = list(level = 4)),
            lx = .001, ux = 500,  prior = uniform(lb, ub))

# }