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

In ehsan66/ICAOD: Optimal Designs for Nonlinear Statistical Models by Imperialist Competitive Algorithm (ICA)

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 (θ_1, θ_2,θ_3, θ_4).
lambda	A vector of relative importance of each of the three criteria, i.e. λ = (λ_1, λ_2, λ_3). Here 0 < λ_i < 1 and s ∑ λ_i = 1.
delta	Predetermined meaningful value of the minimum effective dose MED. When δ < 0 , then θ_2 > 0 or when δ > 0, then θ_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 ε be the global maximum of the sensitivity (derivative) function over x belong to χ. ELB is given by

ELB = p/(p + ε),

where p is the number of model parameters. Obviously, calculating ELB requires finding ε 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 (ε 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.

See Also

multiple

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)

ehsan66/ICAOD documentation built on Oct. 16, 2020, 8:13 p.m.