senslocallycomp {ICAOD}

Verifying Optimality of The Locally DP-optimal Designs

Description

This function plot the sensitivity (derivative) function given an approximate (continuous) design and calculate the efficiency lower bound (ELB) for locally DP-optimal designs. Let \boldsymbol{x} belongs to \chi that denotes the design space. Based on the general equivalence theorem, generally, a design \xi^* is optimal if and only if the value of its sensitivity (derivative) function be non-positive for all \boldsymbol{x} in \chi and it only reaches zero when \boldsymbol{x} belong to the support of \xi^* (be equal to one of the design point). Therefore, the user can look at the sensitivity plot and the ELB to decide whether the design is optimal or close enough to the true optimal design.

Usage

senslocallycomp(
  formula,
  predvars,
  parvars,
  alpha,
  prob,
  family = gaussian(),
  x,
  w,
  lx,
  ux,
  inipars,
  fimfunc = NULL,
  sens.control = list(),
  calculate_criterion = TRUE,
  plot_3d = c("lattice", "rgl"),
  plot_sens = TRUE,
  npar = length(inipars),
  silent = FALSE
)

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.

alpha

A value between 0 and 1. Compound or combined DP-criterion is the product of the efficiencies of a design with respect to D- and average P- optimality, weighted by alpha.

prob

Either formula or a function. When function, its argument are x and param, and they are the same as the arguments in fimfunc. prob as a function takes the design points and vector of parameters and returns the probability of success at each design points. See 'Examples'.

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

Vector of the design (support) points. See 'Details' of sensminimax for models with more than one predictors.

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.

inipars

Vector of initial estimates for the unknown parameters. It must match parvars or argument param of the function provided in fimfunc.

fimfunc

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

sens.control

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

calculate_criterion

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

plot_3d

Which package should be used to plot the sensitivity (derivative) function for models with two predictors. Either "rgl" or "lattice" (default).

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 given, the sensitivity plot may be shifted below the y-axis. When NULL, it is set to length(inipars).

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).

References

McGree, J. M., Eccleston, J. A., and Duffull, S. B. (2008). Compound optimal design criteria for nonlinear models. Journal of Biopharmaceutical Statistics, 18(4), 646-661.

Examples

p <- c(1, -2, 1, -1)
prior4.4 <- uniform(p -1.5, p + 1.5)
formula4.4 <- ~exp(b0+b1*x1+b2*x2+b3*x1*x2)/(1+exp(b0+b1*x1+b2*x2+b3*x1*x2))
prob4.4 <- ~1-1/(1+exp(b0 + b1 * x1 + b2 * x2 + b3 * x1 * x2))
predvars4.4 <-  c("x1", "x2")
parvars4.4 <- c("b0", "b1", "b2", "b3")
lb <- c(-1, -1)
ub <- c(1, 1)

## That is the optimal design when alpha = .25, see ?locallycomp on how to find it
xopt <- c(-1, -0.389, 1, 0.802, -1, 1, -1, 1)
wopt <- c(0.198, 0.618, 0.084, 0.1)

# We want to verfiy the optimality of the optimal design by the general equivalence theorem.

senslocallycomp(formula = formula4.4, predvars = predvars4.4, parvars = parvars4.4,
                family = binomial(), prob = prob4.4, lx = lb, ux = ub,
                alpha = .25, inipars = p, x = xopt, w = wopt)

## Not run: 
# is this design also optimal when alpha = .3

senslocallycomp(formula = formula4.4, predvars = predvars4.4, parvars = parvars4.4,
                family = binomial(), prob = prob4.4, lx = lb, ux = ub,
                alpha = .3, inipars = p, x = xopt, w = wopt)

# when alpha = .3
senslocallycomp(formula = formula4.4, predvars = predvars4.4, parvars = parvars4.4,
                family = binomial(), prob = prob4.4, lx = lb, ux = ub,
                alpha = .5, inipars = p, x = xopt, w = wopt)
# when alpha = .8
senslocallycomp(formula = formula4.4, predvars = predvars4.4, parvars = parvars4.4,
                family = binomial(), prob = prob4.4, lx = lb, ux = ub,
                alpha = .8, inipars = p, x = xopt, w = wopt)


# when alpha = .9
senslocallycomp(formula = formula4.4, predvars = predvars4.4, parvars = parvars4.4,
                family = binomial(), prob = prob4.4, lx = lb, ux = ub,
                alpha = .9, inipars = p, x = xopt, w = wopt)

## As can be seen, the design looses efficiency as alpha increases.

## End(Not run)