ICAOD (version 0.9.2)

equivalence_ave: Checking the optimality of a design with respect to the optim-on-the-average criterion

Description

Let \(\Theta\) be the set of plausible parameter values and weighted by a probability measure \(\pi\), the measure having support in the parameter space \(\Theta\), and \(\Psi(\xi, \theta) =|M(\xi, \theta)|\). A design \(\xi^*\) is optimum-on-the-average with respect to prior \(\pi\) if the following inequality holds for all \(\boldsymbol{x} \in \chi\) $$c(\boldsymbol{x}, \pi, \xi^*) = \int_{\pi} tr M^{-1}(\xi^*, \theta)I(\boldsymbol{x}, \theta)\pi(\theta) d(\theta)-p \leq 0,$$ with equality at all support points of \(\xi^*\). Here, \(p\) is the number of model parameters.

Usage

equivalence_ave(fimfunc, x, w, lx, ux, prior, param,
  maxeval_equivalence = 6000, plot_sensitivity = TRUE, ...)

Arguments

fimfunc

Fisher information matrix. Can be the name of the Fisher information matrix from FIM family functions available in this package as a character string or a function that returns the information matrix. See "Details" of mica.

x

a vector of design points. If the model has \(n\) explanatory variables, let \(x_{ij}\) be the \(j\)th component of the $\(i\)th design point. The argument x is \(x = (x_{11}, x_{21},..., x_{k1},..., x_{1n}, x_{2n},... x_{kn})\). See "Examples" on how to set this argument when the design space does not have one dimension, e.g. is of two-dimension.

w

a vector of design weights

lx

lower bound of the design space \(\chi\)

ux

upper bound of the design space \(\chi\)

prior

a vector of the probability measure \(\pi\).

param

a matrix for set of parameters, i.e. support of \(\pi\). Every row is is a vector of values of a parameter. The number of its rows must be equal to the length of prior.

maxeval_equivalence

maximum number of evaluations (maxeval) that will be passed to optimization function directL to find the maximum of the sensitivity function required for calculating ELB. See "Details" of equivalence.

plot_sensitivity

logical, if TRUE, the sensitivity function will be plotted.

...

further argument to be passed to fimfunc

Value

an object of class 'equivalence' that is a list contains:

max_deriv

maximum of the sensitivity function

ELB

Efficiency lower bound. If it is negative, then the value of maxeval_equivalence should be increased to find the global maximum.

crtval

criterion value

See Also

equivalence and equivalence_multiple.

Examples

# NOT RUN {
equivalence_ave(fimfunc ="FIM_logistic",lx = -5, ux = 5, x = c(0.2603688, 1, 1.739631),
                      w = c(0.2750147, 0.4499705, 0.2750148),  prior = c(.25, .25, .25, .25),
                      param =  matrix(c(0.5, 1.5, 0.5, 1.5, 4.0, 4.0, 5.0, 5.0), 4, 2))
# }