R: Locally DP-Optimal Designs

locallycomp {ICAOD}

R Documentation

Locally DP-Optimal Designs

Description

Finds compound locally DP-optimal designs that meet the dual goal of parameter estimation and increasing the probability of a particular outcome in a binary response model. A compound locally DP-optimal design maximizes the product of the efficiencies of a design \xi with respect to D- and average P-optimality, weighted by a pre-defined mixing constant 0 \leq \alpha \leq 1.

Usage

locallycomp(
  formula,
  predvars,
  parvars,
  family = gaussian(),
  lx,
  ux,
  alpha,
  prob,
  iter,
  k,
  inipars,
  fimfunc = NULL,
  ICA.control = list(),
  sens.control = list(),
  initial = NULL,
  npar = length(inipars),
  plot_3d = c("lattice", "rgl")
)

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`.
`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.
`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`.
`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'.
`iter`	Maximum number of iterations.
`k`	Number of design points. When `alpha = 0`, then `k` can be less than the number of parameters.
`inipars`	Vector. Initial values for the unknown parameters. It will be passed to the information matrix and also probability function.
`fimfunc`	A function. Returns the FIM as a `matrix`. Required when `formula` is missing. See 'Details' of `minimax`.
`ICA.control`	ICA control parameters. For details, see `ICA.control`.
`sens.control`	Control Parameters for Calculating the ELB. For details, see `sens.control`.
`initial`	A matrix of the initial design points and weights that will be inserted into the initial solutions (countries) of the algorithm. Every row is a design, i.e. a concatenation of `x` and `w`. Will be coerced to a `matrix` if necessary. See 'Details' of `minimax`.
`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 will be set here to `length(inipars)`.
`plot_3d`	Which package should be used to plot the sensitivity (derivative) function for two-dimensional design space. Defaults to `"lattice"`.

Details

Let \Xi be the space of all approximate designs with k design points (support points) at x_1, x_2, ..., x_k from design space \chi with corresponding weights w_1,... ,w_k. Let M(\xi, \theta) be the Fisher information matrix (FIM) of a k-point design \xi, \theta_0 is a user-given vector of initial estimates for the unknown parameters \theta and p(x_i, \theta) is the ith probability of success given by x_i in a binary response model. A compound locally DP-optimal design maximizes over \Xi

\frac{\alpha}{q}\log|M(\xi, \theta_0)| + (1- \alpha) \log \left( \sum_{i=1}^k w_ip(x_i, \theta_0) \right).

Use plot function to verify the general equivalence theorem for the output design or change checkfreq in ICA.control.

One can adjust the tuning parameters in ICA.control to set a stopping rule based on the general equivalence theorem. See "Examples" in locally.

Value

an object of class minimax that is a list including three sub-lists:

arg

A list of design and algorithm parameters.

evol

A list of length equal to the number of iterations that stores the information about the best design (design with least criterion value) of each iteration. evol[[iter]] contains:

`iter`		Iteration number.
`x`		Design points.
`w`		Design weights.
`min_cost`		Value of the criterion for the best imperialist (design).
`mean_cost`		Mean of the criterion values of all the imperialists.
`sens`		An object of class `'sensminimax'`. See below.
`param`		Vector of parameters.

empires

A list of all the empires of the last iteration.

alg

A list with following information:

`nfeval`		Number of function evaluations. It does not count the function evaluations from checking the general equivalence theorem.
`nlocal`		Number of successful local searches.
`nrevol`		Number of successful revolutions.
`nimprove`		Number of successful movements toward the imperialists in the assimilation step.
`convergence`		Stopped by `'maxiter'` or `'equivalence'`?

method

A type of optimal designs used.

design

Design points and weights at the final iteration.

out

A data frame of design points, weights, value of the criterion for the best imperialist (min_cost), and Mean of the criterion values of all the imperialistsat each iteration (mean_cost).

The list sens contains information about the design verification by the general equivalence theorem. See sensminimax for more details. It is given every ICA.control$checkfreq iterations and also the last iteration if ICA.control$checkfreq >= 0. Otherwise, NULL.

param is a vector of parameters that is the global minimum of the minimax criterion or the global maximum of the standardized maximin criterion over the parameter space, given the current x, w.

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

## Here we produce the results of Table 2 in in McGree and Eccleston (2008)
# For D- and P-efficiency see, ?leff and ?peff

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)


# set checkfreq = Inf to ask for equivalence theorem at final step.
res.0 <- locallycomp(formula = formula4.4, predvars = predvars4.4, parvars = parvars4.4,
                     family = binomial(), prob = prob4.4, lx = lb, ux = ub,
                     alpha = 0, k = 1, inipars = p, iter = 10,
                     ICA.control = ICA.control(checkfreq = Inf))

## Not run: 
res.25 <- locallycomp(formula = formula4.4, predvars = predvars4.4, parvars = parvars4.4,
                      family = binomial(), prob = prob4.4, lx = lb, ux = ub,
                      alpha = .25, k = 4, inipars = p, iter = 350,
                      ICA.control = ICA.control(checkfreq = Inf))

res.5 <- locallycomp(formula = formula4.4, predvars = predvars4.4, parvars = parvars4.4,
                     family = binomial(), prob = prob4.4, lx = lb, ux = ub,
                     alpha = .5, k = 4, inipars = p, iter = 350,
                     ICA.control = ICA.control(checkfreq = Inf))
res.75 <- locallycomp(formula = formula4.4, predvars = predvars4.4, parvars = parvars4.4,
                      family = binomial(), prob = prob4.4, lx = lb, ux = ub,
                      alpha = .75, k = 4, inipars = p, iter = 350,
                      ICA.control = ICA.control(checkfreq = Inf))

res.1 <- locallycomp(formula = formula4.4, predvars = predvars4.4, parvars = parvars4.4,
                     family = binomial(), prob = prob4.4, lx = lb, ux = ub,
                     alpha = 1, k = 4, inipars = p, iter = 350,
                     ICA.control = ICA.control(checkfreq = Inf))

#### computing the D-efficiency
# locally D-optimal design is locally DP-optimal design when alpha = 1.

leff(formula = formula4.4, predvars = predvars4.4, parvars = parvars4.4, family = binomial(),
     x1 = res.0$evol[[10]]$x, w1 = res.0$evol[[10]]$w,
     inipars = p,
     x2 = res.1$evol[[350]]$x, w2 = res.1$evol[[350]]$w)

leff(formula = formula4.4, predvars = predvars4.4, parvars = parvars4.4, family = binomial(),
     x1 = res.25$evol[[350]]$x, w1 = res.25$evol[[350]]$w,
     inipars = p,
     x2 = res.1$evol[[350]]$x, w2 = res.1$evol[[350]]$w)

leff(formula = formula4.4, predvars = predvars4.4, parvars = parvars4.4, family = binomial(),
     x1 = res.5$evol[[350]]$x, w1 = res.5$evol[[350]]$w,
     inipars = p,
     x2 = res.1$evol[[350]]$x, w2 = res.1$evol[[350]]$w)


leff(formula = formula4.4, predvars = predvars4.4, parvars = parvars4.4, family = binomial(),
     x1 = res.75$evol[[350]]$x, w1 = res.75$evol[[350]]$w,
     inipars = p,
     x2 = res.1$evol[[350]]$x, w2 = res.1$evol[[350]]$w)



#### computing the P-efficiency
# locally p-optimal design is locally DP-optimal design when alpha = 0.

leff(formula = formula4.4, predvars = predvars4.4, parvars = parvars4.4, family = binomial(),
     x2 = res.0$evol[[10]]$x, w2 = res.0$evol[[10]]$w,
     prob = prob4.4,
     type = "PA",
     inipars = p,
     x1 = res.25$evol[[350]]$x, w1 = res.25$evol[[350]]$w)

leff(formula = formula4.4, predvars = predvars4.4, parvars = parvars4.4, family = binomial(),
     x2 = res.0$evol[[10]]$x, w2 = res.0$evol[[10]]$w,
     prob = prob4.4,
     inipars = p,
     type = "PA",
     x1 = res.5$evol[[350]]$x, w1 = res.5$evol[[350]]$w)

leff(formula = formula4.4, predvars = predvars4.4, parvars = parvars4.4, family = binomial(),
     x2 = res.0$evol[[10]]$x, w2 = res.0$evol[[10]]$w,
     prob = prob4.4,
     inipars = p,
     type = "PA",
     x1 = res.75$evol[[350]]$x, w1 = res.75$evol[[350]]$w)


leff(formula = formula4.4, predvars = predvars4.4, parvars = parvars4.4, family = binomial(),
     x2 = res.0$evol[[10]]$x, w2 = res.1$evol[[10]]$w,
     prob = prob4.4,
     type = "PA",
     inipars = p,
     x1 = res.1$evol[[350]]$x, w1 = res.1$evol[[350]]$w)



## End(Not run)

[Package ICAOD version 1.0.1 Index]