R: Verifying Optimality of The Multiple Objective Designs for...

sensmultiple {ICAOD}

R Documentation

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`.

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).

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).

Note

DO NOT use this function to verify c-optimal designs for estimating 'MED' or 'ED50' (verifying single objective optimal designs) because the results may be unstable. The reason is that for the c-optimal criterion the generalized inverse of the Fisher information matrix is not stable and depends on the tolerance value (tol).

Theoretically, ELB can not be larger than 1. But if so, it may have one of the following reasons:

max_deriv is not a GLOBAL maximum. Please increase the value of the parameter maxeval in sens.minimax.control to find the global maximum.
The sensitivity function is shifted below the y-axis because the number of model parameters has not been specified correctly (less value given). Please specify the correct number of model parameters via argument npar.

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.

Examples

#################################################################
# 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)

[Package ICAOD version 1.0.1 Index]