# Evaluate the expectation of determinant the Fisher Information Matrix (FIM) using the Laplace approximation. Compute the expectation of the `det(FIM)` using the Laplace approximation to the expectation. Computations are made based on the model, parameters, distributions of parameter uncertainty, design and methods defined in the PopED database or as arguments to the function. ```r ed_laplace_ofv( model_switch, groupsize, ni, xtopto, xopto, aopto, bpopdescr, ddescr, covd, sigma, docc, poped.db, method = 1, return_gradient = FALSE, optxt = poped.db$settings$optsw[2], opta = poped.db$settings$optsw[4], x = c(), ... ) ``` ## Arguments - `model_switch`: A matrix that is the same size as xt, specifying which model each sample belongs to. - `groupsize`: A vector of the number of individuals in each group. - `ni`: A vector of the number of samples in each group. - `xtopto`: the sampling times - `xopto`: the discrete design variables - `aopto`: the continuous design variables - `bpopdescr`: Matrix defining the fixed effects, per row (row number = parameter_number) we should have: * column 1 the type of the distribution for E-family designs (0 = Fixed, 1 = Normal, 2 = Uniform, 3 = User Defined Distribution, 4 = lognormal and 5 = truncated normal) * column 2 defines the mean. * column 3 defines the variance of the distribution (or length of uniform distribution). - `ddescr`: Matrix defining the diagonals of the IIV (same logic as for the `bpopdescr`). - `covd`: Column major vector defining the covariances of the IIV variances. That is, from your full IIV matrix `covd <- IIV[lower.tri(IIV)]`. - `sigma`: Matrix defining the variances can covariances of the residual variability terms of the model. can also just supply the diagonal parameter values (variances) as a `c()`. - `docc`: Matrix defining the IOV, the IOV variances and the IOV distribution as for d and bpop. - `poped.db`: A PopED database. - `method`: If 0 then use an optimization routine translated from PopED code written in MATLAB to optimize the parameters in the Laplace approximation. If 1 then use `optim` to compute both k and the hessian of k (see Dodds et al, JPP, 2005 for more information). If 2 then use `fdHess` to compute the hessian. - `return_gradient`: Should the gradient be returned. - `optxt`: If sampling times are optimized - `opta`: If continuous design variables are optimized - `x`: The design parameters to compute the gradient on. - `...`: Arguments passed through from other functions, does not pass anything to another function. ## Returns The FIM and the hessian of the FIM. ## Details This computation follows the method outlined in Dodds et al, "Robust Population Pharmacokinetic Experiment Design" JPP, 2005, equation 16. Typically this function will not be run by the user. Instead use `evaluate.e.ofv.fim`. ## Examples ```r ## Warfarin example from software comparison in: ## Nyberg et al., "Methods and software tools for design evaluation ## for population pharmacokinetics-pharmacodynamics studies", ## Br. J. Clin. Pharm., 2014. ## Optimization using an additive + proportional reidual error to ## avoid sample times at very low concentrations (time 0 or very late samoples). library(PopED) ## find the parameters that are needed to define from the structural model ff.PK.1.comp.oral.sd.CL ## -- parameter definition function ## -- names match parameters in function ff sfg <- function(x,a,bpop,b,bocc){ parameters=c(CL=bpop[1]*exp(b[1]), V=bpop[2]*exp(b[2]), KA=bpop[3]*exp(b[3]), Favail=bpop[4], DOSE=a[1]) return(parameters) } ###################### # Normal distribution ###################### bpop_vals <- c(CL=0.15, V=8, KA=1.0, Favail=1) bpop_vals_ed_n <- cbind(ones(length(bpop_vals),1)*1, # normal distribution bpop_vals, ones(length(bpop_vals),1)*(bpop_vals*0.1)^2) # 10% of bpop value bpop_vals_ed_n["Favail",] <- c(0,1,0) bpop_vals_ed_n ## -- Define initial design and design space poped.db.n <- create.poped.database(ff_fun=ff.PK.1.comp.oral.sd.CL, fg_fun=sfg, fError_fun=feps.add.prop, bpop=bpop_vals_ed_n, notfixed_bpop=c(1,1,1,0), d=c(CL=0.07, V=0.02, KA=0.6), sigma=c(0.01,0.25), groupsize=32, xt=c( 0.5,1,2,6,24,36,72,120), minxt=0, maxxt=120, a=70, mina=0, maxa=100) ## ED evaluate using LaPlace approximation tic(); output <- evaluate.e.ofv.fim(poped.db.n,use_laplace=TRUE); toc() output$E_ofv ## Not run: ## ED value using MC integration (roughly) tic();e_ofv_mc_n <- evaluate.e.ofv.fim(poped.db.n,ED_samp_size=500,ofv_calc_type = 1);toc() e_ofv_mc_n$E_ofv ## Using ed_laplce_ofv directly ed_laplace_ofv(model_switch=poped.db.n$design$model_switch, groupsize=poped.db.n$design$groupsize, ni=poped.db.n$design$ni, xtopto=poped.db.n$design$xt, xopto=poped.db.n$design$x, aopto=poped.db.n$design$a, bpopdescr=poped.db.n$parameters$bpop, ddescr=poped.db.n$parameters$d, covd=poped.db.n$parameters$covd, sigma=poped.db.n$parameters$sigma, docc=poped.db.n$parameters$docc, poped.db.n) ###################### # Log-normal distribution ###################### # Adding 10% log-normal Uncertainty to fixed effects (not Favail) bpop_vals <- c(CL=0.15, V=8, KA=1.0, Favail=1) bpop_vals_ed_ln <- cbind(ones(length(bpop_vals),1)*4, # log-normal distribution bpop_vals, ones(length(bpop_vals),1)*(bpop_vals*0.1)^2) # 10% of bpop value bpop_vals_ed_ln["Favail",] <- c(0,1,0) bpop_vals_ed_ln ## -- Define initial design and design space poped.db.ln <- create.poped.database(ff_fun=ff.PK.1.comp.oral.sd.CL, fg_fun=sfg, fError_fun=feps.add.prop, bpop=bpop_vals_ed_ln, notfixed_bpop=c(1,1,1,0), d=c(CL=0.07, V=0.02, KA=0.6), sigma=c(0.01,0.25), groupsize=32, xt=c( 0.5,1,2,6,24,36,72,120), minxt=0, maxxt=120, a=70, mina=0, maxa=100) ## ED evaluate using LaPlace approximation tic() output <- evaluate.e.ofv.fim(poped.db.ln,use_laplace=TRUE) toc() output$E_ofv ## expected value (roughly) tic() e_ofv_mc_ln <- evaluate.e.ofv.fim(poped.db.ln,ED_samp_size=500,ofv_calc_type = 1)[["E_ofv"]] toc() e_ofv_mc_ln ## Using ed_laplce_ofv directly ed_laplace_ofv(model_switch=poped.db.ln$design$model_switch, groupsize=poped.db.ln$design$groupsize, ni=poped.db.ln$design$ni, xtopto=poped.db.ln$design$xt, xopto=poped.db.ln$design$x, aopto=poped.db.ln$design$a, bpopdescr=poped.db.ln$parameters$bpop, ddescr=poped.db.ln$parameters$d, covd=poped.db.ln$parameters$covd, sigma=poped.db.ln$parameters$sigma, docc=poped.db.ln$parameters$docc, poped.db.ln) ## End(Not run) ``` ## See Also Other FIM: `LinMatrixH()`, `LinMatrixLH()`, `LinMatrixL_occ()`, `calc_ofv_and_fim()`, `ed_mftot()`, `efficiency()`, `evaluate.e.ofv.fim()`, `evaluate.fim()`, `gradf_eps()`, `mf3()`, `mf7()`, `mftot()`, `ofv_criterion()`, `ofv_fim()` Other E-family: `calc_ofv_and_fim()`, `ed_mftot()`, `evaluate.e.ofv.fim()`

ed_laplace_ofv function