effbound {OptimalDesign}

Lower bound on efficiency

Description

Computes a lower bound on the efficiency of a design w in the class of all approximate designs of the same size as w.

Usage

effbound(Fx, w, crit="D", h=NULL, echo=TRUE)

Arguments

Fx

the n times m matrix of candidate regressors (as rows), where n is the number of candidate design points and m (where m>=2, m<=n) is the number of parameters.

w

a non-negative vector of length n representing the design.

crit

the criterion; possible values are "D", "A", "I", "C" and "c".

h

a non-zero vector of length m corresponding to the coefficients of the linear parameter combination of interest. If crit is not "C" nor "c" then h is ignored. If crit is "C" or "c" and h=NULL then h is assumed to be c(0,...,0,1).

echo

Print the call of the function?

Details

The lower bounds are based on the standard methods of convex analysis. See the reference paper at http://www.iam.fmph.uniba.sk/design/ for mathematical details.

Value

A lower bound on the D-, A-, I-, c-, or C-efficiency of w in the class of all approximate designs of the same size as w at the set of candidate regressors given by Fx.

Note

The design w should have a non-singular information matrix. Occasionally, the lower bound is very conservative. The exact value of the efficiency of w is the ratio of the criterion value of w and the criterion value of the optimal design.

Author(s)

Radoslav Harman, Lenka Filova

See Also

varfun, dirder

Examples

# A lower bound on the D-efficiencies of the uniform designs
# for the quadratic regression on a line grid

Fx <- Fx_cube(~x1 + I(x1^2), n.levels = 101)
effbound(Fx, rep(1/101, 101))

# The precise value of the D-efficiency
# requires computing the D-optimal design:

w.opt <- od_REX(Fx)$w.best
optcrit(Fx, rep(1/101, 101)) / optcrit(Fx, w.opt)

## Not run: 
# Let us do this for polynomial regressions of various degrees:

n <- 101; d.max <- 10; x <- seq(-1, 1, length = n)
effs <- matrix(0, ncol = 2, nrow = d.max)
Fx <- matrix(1, ncol = 1, nrow = n)
for(d in 1:d.max) {
  Fx <- cbind(Fx, x^d)
  effs[d, 1] <- effbound(Fx, rep(1/n, n))
  w.opt <- od_REX(Fx)$w.best
  effs[d, 2] <- optcrit(Fx, rep(1/n, n)) / optcrit(Fx, w.opt)
}
print(effs)

# We see that the lower bound becomes more and more conservative
# compared to the real efficiency which actually increases with d.


# Compute a D-optimal design for the main effects model
# on a random subsample of a 6D cube

n <- 1000000; m <- 6
Fx <- cbind(1, matrix(runif(n*m), ncol = m))
w <- od_REX(Fx, eff = 0.99)$w.best
Fx <- od_DEL(Fx, w)$Fx.keep
w <- od_REX(Fx)$w.best

# Now we will compute a lower bound on efficiency of such design
# on the entire (continuous) cube:
Fx <- rbind(Fx, Fx_cube(~x1 + x2 + x3 + x4 + x5 + x6, lower = rep(0, 6)))
w <- c(w, rep(0, 2^6))
effbound(Fx, w)

# The real D-efficiency of w on the entire cube is
optcrit(Fx, w)/od_REX(Fx)$Phi.best

## End(Not run)