# Lower bound on efficiency Computes a lower bound on the efficiency of a design `w` in the class of all approximate designs of the same size as `w`. ```r 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/](http://www.iam.fmph.uniba.sk/design/) for mathematical details. ## Returns 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`. ## Author(s) Radoslav Harman, Lenka Filova ## 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. ## See Also `varfun`, `dirder` ## Examples ```r # 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) ```

effbound function