locallycomp: 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.

Vector of lower bounds for the predictors. Should be in the same order as predvars.

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.

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".

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.

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.

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

Run this code

# NOT RUN {
## 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)


# }

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