ICAOD (version 1.0.1)

sensmultiple: Verifying Optimality of The Multiple Objective Designs for The 4-Parameter Hill Model

Description

This function uses general equivalence theorem to verify the optimality of a multiple objective optimal design found for the 4-Parameter Hill model and the 4-parameter logistic model. For more details, See Hyun and Wong (2015).

Usage

sensmultiple(
  dose,
  w,
  minDose,
  maxDose,
  inipars,
  lambda,
  delta,
  Hill_par = TRUE,
  sens.control = list(),
  calculate_criterion = TRUE,
  plot_sens = TRUE,
  tol = sqrt(.Machine$double.xmin),
  silent = FALSE
)

Arguments

dose

A vector of design points. It is either dose values or logarithm of dose values when Hill_par = TRUE.

w

A vector of design weights.

minDose

Minimum dose \(D\). For the 4-parameter logistic model, i.e. when Hill_par = FALSE, it is the minimum of \(log(D)\).

maxDose

Maximum dose \(D\). For the 4-parameter logistic model, i.e. when Hill_par = FALSE, it is the maximum of \(log(D)\).

inipars

A vector of initial estimates for the vector of parameters \((a, b, c, d)\). For the 4-parameter logistic model, i.e. when Hill_par = FALSE, it is a vector of initial estimates for \((\theta_1, \theta_2,\theta_3, \theta_4)\).

lambda

A vector of relative importance of each of the three criteria, i.e. \(\lambda = (\lambda_1, \lambda_2, \lambda_3)\). Here \(0 < \lambda_i < 1\) and s \(\sum \lambda_i = 1\).

delta

Predetermined meaningful value of the minimum effective dose MED. When \(\delta < 0 \), then \(\theta_2 > 0\) or when \(\delta > 0\), then \(\theta_2 < 0\).

Hill_par

Hill model parameterization? Defaults to TRUE.

sens.control

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

calculate_criterion

Calculate the criterion? Defaults to TRUE.

plot_sens

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

tol

Tolerance for finding the general inverse of the Fisher information matrix. Defaults to .Machine$double.xmin.

silent

Do not print anything? Defaults to FALSE.

Value

an object of class sensminimax that is a list with the following elements:

type

Argument type that is required for print methods.

optima

A matrix that stores all the local optima over the parameter space. The cost (criterion) values are stored in a column named Criterion_Value. The last column (Answering_Set) shows if the optimum belongs to the answering set (1) or not (0). See 'Details' of sens.minimax.control. Only applicable for minimax or standardized maximin designs.

mu

Probability measure on the answering set. Corresponds to the rows of optima for which the associated row in column Answering_Set is equal to 1. Only applicable for minimax or standardized maximin designs.

max_deriv

Global maximum of the sensitivity (derivative) function (\(\epsilon\) in 'Details').

ELB

D-efficiency lower bound. Can not be larger than 1. If negative, see 'Note' in sensminimax or sens.minimax.control.

merge_tol

Merging tolerance to create the answering set from the set of all local optima. See 'Details' in sens.minimax.control. Only applicable for minimax or standardized maximin designs.

crtval

Criterion value. Compare it with the column Crtiterion_Value in optima for minimax and standardized maximin designs.

time

Used CPU time (rough approximation).

Details

ELB is a measure of proximity of a design to the optimal design without knowing the latter. Given a design, let \(\epsilon\) be the global maximum of the sensitivity (derivative) function over \(x \in \chi\). ELB is given by $$ELB = p/(p + \epsilon),$$ where \(p\) is the number of model parameters. Obviously, calculating ELB requires finding \(\epsilon\) and another optimization problem to be solved. The tuning parameters of this optimization can be regulated via the argument sens.minimax.control. See, for more details, Masoudi et al. (2017).

References

Hyun, S. W., and Wong, W. K. (2015). Multiple-Objective Optimal Designs for Studying the Dose Response Function and Interesting Dose Levels. The international journal of biostatistics, 11(2), 253-271.

See Also

multiple

Examples

# NOT RUN {
#################################################################
# Verifying optimality of a design for the 4-parameter Hill model
#################################################################

## initial estiamtes for the parameters of the Hill model
a <- 0.008949  # ED50
b <- -1.79 # Hill constant
c <- 0.137 # lower limit
d <- 1.7 # upper limit
# D belongs to c(.001, 1000) ## dose in mg
## Hill parameters are c(a, b, c, d)
# dose, minDose and maxDose vector in mg scale

sensmultiple (dose = c(0.001, 0.009426562, 0.01973041, 999.9974),
              w =   c(0.4806477, 0.40815, 0.06114173, 0.05006055),
              minDose = .001, maxDose = 1000,
              Hill_par = TRUE,
              inipars = c(a, b, c, d),
              lambda = c(0.05, 0.05, .90),
              delta = -1)





# }