leff {ICAOD}

Calculates Relative Efficiency for Locally Optimal Designs

Description

Given a vector of initial estimates for the parameters, this function calculates the D-and PA- efficiency of a design \xi_1 with respect to a design \xi_2. Usually, \xi_2 is an optimal design.

Usage

leff(
  formula,
  predvars,
  parvars,
  family = gaussian(),
  inipars,
  type = c("D", "PA"),
  fimfunc = NULL,
  x2,
  w2,
  x1,
  w1,
  npar = length(inipars),
  prob = NULL
)

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.

inipars

Vector. Initial values for the unknown parameters. It will be passed to the information matrix and also probability function.

type

A character. "D" denotes the D-efficiency and "PA" denotes the average P-efficiency.

fimfunc

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

x2

Vector of design (support) points of the optimal design (\xi_2). Similar to x1.

w2

Vector of corresponding design weights for x2.

x1

Vector of design (support) points of \xi_1. See 'Details' of leff.

w1

Vector of corresponding design weights for x1.

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).

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'.

Details

For a known \theta_0, relative D-efficiency is

exp(\frac{log|M(\xi_1, \theta_0)| - log|M(\xi_2, \theta_0)|}{npar})

The relative P-efficiency is

\exp(\log(\sum_{i=1}^k w_{1i}p(x_{1i}, \theta_0) - \log(\sum_{i=1}^k w_{2i}p(x{2_i}, \theta_0))

where x_2 and w_2 are usually the support points and the corresponding weights of the optimal design, respectively.

The argument x1 is the vector of design points. For design points with more than one dimension (the models with more than one predictors), it is a concatenation of the design points, but dimension-wise. For example, let the model has three predictors (I, S, Z). Then, a two-point optimal design has the following points: \{\mbox{point1} = (I_1, S_1, Z_1), \mbox{point2} = (I_2, S_2, Z_2)\}. Then, the argument x1 is equal to x = c(I1, I2, S1, S2, Z1, Z2).

Value

A value between 0 and 1.

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")


# Locally D-optimal design is as follows:
## weight and point of D-optimal design
# Point1     Point2     Point3     Point4
# /1.00000 \ /-1.00000\ /0.06801 \ /1.00000 \
# \-1.00000/ \-1.00000/ \1.00000 / \1.00000 /
#   Weight1    Weight2    Weight3    Weight4
# 0.250      0.250      0.250      0.250

xopt_D <- c(1, -1, .0680, 1, -1, -1, 1, 1)
wopt_D <- rep(.25, 4)

# Let see if we use only three of the design points, what is the relative efficiency.
leff(formula = formula4.4, predvars = predvars4.4, parvars = parvars4.4, family = binomial(),
     x1 = c(1, -1, .0680,  -1, -1, 1), w1 = c(.33, .33, .33),
     inipars = p,
     x2 = xopt_D, w2 = wopt_D)
# Wow, it heavily drops!


# Locally P-optimal design has only one support point and is -1 and 1
xopt_P <- c(-1, 1)
wopt_P <- 1

# What is the relative P-efficiency of the D-optimal design with respect to P-optimal design?
leff(formula = formula4.4, predvars = predvars4.4, parvars = parvars4.4, family = binomial(),
     x1 = xopt_D, w1 = wopt_D,
     inipars = p,
     type = "PA",
     prob = prob4.4,
     x2 = xopt_P, w2 = wopt_P)
# .535