# optDesign: Function to calculate optimal designs In DoseFinding: Planning and Analyzing Dose Finding Experiments

## Description

Given a set of models (with full parameter values and model probabilities) the optDesign function calculates the optimal design for estimating the dose-response model parameters (D-optimal) or the design for estimating the target dose (TD-optimal design) (see Dette, Bretz, Pepelyshev and Pinheiro (2008)), or a mixture of these two criteria. The design can be plotted (together with the candidate models) using plot.design. calcCrit calculates the design criterion for a discrete set of design(s). rndDesign provides efficient rounding for the calculated continous design to a finite sample size.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18``` ```optDesign(models, probs, doses, designCrit = c("Dopt", "TD", "Dopt&TD", "userCrit"), Delta, standDopt = TRUE, weights, nold = rep(0, length(doses)), n, control=list(), optimizer = c("solnp", "Nelder-Mead", "nlminb", "exact"), lowbnd = rep(0, length(doses)), uppbnd = rep(1, length(doses)), userCrit, ...) ## S3 method for class 'DRdesign' plot(x, models, lwdDes = 10, colDes = rgb(0,0,0,0.3), ...) calcCrit(design, models, probs, doses, designCrit = c("Dopt", "TD", "Dopt&TD"), Delta, standDopt = TRUE, weights, nold = rep(0, length(doses)), n) rndDesign(design, n, eps = 0.0001) ```

## Arguments

 `models` An object of class c(Mods, fullMod), see the `Mods` function for details. When an TD optimal design should be calculated, the TD needs to exist for all models. If a D-optimal design should be calculated, you need at least as many doses as there are parameters in the specified models. `probs` Vector of model probabilities for the models specified in models, assumed in the same order as specified in models `doses` Optional argument. If this argument is missing the doses attribute in the c(Mods, fullMod) object specified in models is used. `designCrit` Determines which type of design to calculate. "TD&Dopt" uses both optimality criteria with equal weight. `Delta` Target effect needed for calculating "TD" and "TD&Dopt" type designs. `standDopt` Logical determining, whether the D-optimality criterion (specifically the log-determinant) should be standardized by the number of parameters in the model or not (only of interest if type = "Dopt" or type = "TD&Dopt"). This is of interest, when there is more than one model class in the candidate model set (traditionally standardization this is done in the optimal design literature). `weights` Vector of weights associated with the response at the doses. Needs to be of the same length as the doses. This can be used to calculate designs for heteroscedastic or for generalized linear model situations. `nold, n` When calculating an optimal design at an interim analysis, nold specifies the vector of sample sizes already allocated to the different doses, and n gives sample size for the next cohort. For optimizer = "exact" one always needs to specify the total sample size via n. `control` List containing control parameters passed down to numerical optimization algorithms (`optim`, `nlminb` or solnp function). For type = "exact" this should be a list with possible entries maxvls1 and maxvls2, determining the maximum number of designs allowed for passing to the criterion function (default maxvls2=1e5) and for creating the initial unrestricted matrix of designs (default maxvls1=1e6). In addition there can be an entry groupSize in case the patients are allocated a minimum group size is required. `optimizer` Algorithm used for calculating the optimal design. Options "Nelder-Mead" and "nlminb" use the `optim` and `nlminb` function and use a trigonometric transformation to turn the constrained optimization problem into an unconstrained one (see Atkinson, Donev and Tobias, 2007, pages 130,131). Option "solnp" uses the solnp function from the Rsolnp package, which implements an optimizer for non-linear optimization under general constraints. Option "exact" tries all given combinations of n patients to the given dose groups (subject to the bounds specified via lowbnd and uppbnd) and reports the best design. When patients are only allowed to be allocated in groups of a certain groupSize, this can be adjusted via the control argument. n/groupSize and length(doses) should be rather small for this approach to be feasible. When the number of doses is small (<8) usually "Nelder-Mead" and "nlminb" are best suited ("nlminb" is usually a bit faster but less stable than "Nelder-Mead"). For a larger number of doses "solnp" is the most reliable option (but also slowest) ("Nelder-Mead" and "nlminb" often fail). When the sample size is small "exact" provides the optimal solution rather quickly. `lowbnd, uppbnd` Vectors of the same length as dose vector specifying upper and lower limits for the allocation weights. This option is only available when using the "solnp" and "exact" optimizers. `userCrit` User defined design criterion, should be a function that given a vector of allocation weights and the doses returns the criterion function. When specified models does not need to be handed over. The first argument of userCrit should be the vector of design weights, while the second argument should be the doses argument (see example below). Additional arguments to userCrit can be passed via ... `...` For function optDesign these are additional arguments passed to userCrit. For function plot.design these are additional parameters passed to `plot.Mods`. `design` Argument for rndDesign and calcCrit functions: Numeric vector (or matrix) of allocation weights for the different doses. The rows of the matrices need to sum to 1. Alternatively also an object of class "DRdesign" can be used for rndDesign. Note that there should be at least as many design points available as there are parameters in the dose-response models selected in `models` (otherwise the code returns an NA). `eps` Argument for rndDesign function: Value under which elements of w will be regarded as 0. `x` Object of class DRdesign (for plot.design) `lwdDes, colDes` Line width and color of the lines plotted for the design (in plot.design)

## Details

Let M_m denote the Fisher information matrix under model m (up to proportionality). M_m is given by ∑ a_i w_i g_i^Tg_i, where a_i is the allocation weight to dose i, w_i the weight for dose i specified via weights and g_i the gradient vector of model m evaluated at dose i.

For designCrit = "Dopt" the code minimizes the design criterion

-sum_m p_m/k_m log(det(M_m))

where p_m is the probability for model m and k_m is the number of parameters for model m. When standDopt = FALSE the k_m are all assumed to be equal to one.

For designCrit = "TD" the code minimizes the design criterion

sum_m p_m log(v_m)

where p_m is the probability for model m and v_m is proportional to the asymptotic variance of the TD estimate and given by b_m'Minv_m b_m (see Dette et al. (2008), p. 1227 for details).

For designCrit = "Dopt&TD" the code minimizes the design criterion

sum_m p_m(-0.5log(det(M_m))/k_m+0.5log(v_m))

Again, for standDopt = FALSE the k_m are all assumed to be equal to one.

For details on the rndDesign function, see Pukelsheim (1993), Chapter 12.

## Note

In some cases (particularly when the number of doses is large, e.g. 7 or larger) it might be necessary to allow a larger number of iterations in the algorithm (via the argument control), particularly for the Nelder-Mead algorithm. Alternatively one can use the solnp optimizer that is usually the most reliable, but not fastest option.

Bjoern Bornkamp

## References

Atkinson, A.C., Donev, A.N. and Tobias, R.D. (2007). Optimum Experimental Designs, with SAS, Oxford University Press

Dette, H., Bretz, F., Pepelyshev, A. and Pinheiro, J. C. (2008). Optimal Designs for Dose Finding Studies, Journal of the American Statisical Association, 103, 1225–1237

Pukelsheim, F. (1993). Optimal Design of Experiments, Wiley

`Mods`, `drmodels`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84``` ```## calculate designs for Emax model doses <- c(0, 10, 100) emodel <- Mods(emax = 15, doses=doses, placEff = 0, maxEff = 1) optDesign(emodel, probs = 1) ## TD-optimal design optDesign(emodel, probs = 1, designCrit = "TD", Delta=0.5) ## 50-50 mixture of Dopt and TD optDesign(emodel, probs = 1, designCrit = "Dopt&TD", Delta=0.5) ## use dose levels different from the ones specified in emodel object des <- optDesign(emodel, probs = 1, doses = c(0, 5, 20, 100)) ## plot models overlaid by design plot(des, emodel) ## round des to a sample size of exactly 90 patients rndDesign(des, n=90) ## using the round function would lead to 91 patients ## illustrating different optimizers (see Note above for more comparison) optDesign(emodel, probs=1, optimizer="Nelder-Mead") optDesign(emodel, probs=1, optimizer="nlminb") ## optimizer solnp (the default) can deal with lower and upper bounds: optDesign(emodel, probs=1, designCrit = "TD", Delta=0.5, optimizer="solnp", lowbnd = rep(0.2,3)) ## exact design using enumeration of all possibilites optDesign(emodel, probs=1, optimizer="exact", n = 30) ## also allows to fix minimum groupSize optDesign(emodel, probs=1, designCrit = "TD", Delta=0.5, optimizer="exact", n = 30, control = list(groupSize=5)) ## optimal design at interim analysis ## assume there are already 10 patients on each dose and there are 30 ## left to randomize, this calculates the optimal increment design optDesign(emodel, 1, designCrit = "TD", Delta=0.5, nold = c(10, 10, 10), n=30) ## use a larger candidate model set doses <- c(0, 10, 25, 50, 100, 150) fmods <- Mods(linear = NULL, emax = 25, exponential = 85, linlog = NULL, logistic = c(50, 10.8811), doses = doses, addArgs=list(off=1), placEff=0, maxEff=0.4) probs <- rep(1/5, 5) # assume uniform prior desDopt <- optDesign(fmods, probs, optimizer = "nlminb") desTD <- optDesign(fmods, probs, designCrit = "TD", Delta = 0.2, optimizer = "nlminb") desMix <- optDesign(fmods, probs, designCrit = "Dopt&TD", Delta = 0.2) ## plot design and truth plot(desMix, fmods) ## illustrate calcCrit function ## calculate optimal design for beta model doses <- c(0, 0.49, 25.2, 108.07, 150) models <- Mods(betaMod = c(0.33, 2.31), doses=doses, addArgs=list(scal=200), placEff=0, maxEff=0.4) probs <- 1 deswgts <- optDesign(models, probs, designCrit = "Dopt", control=list(maxit=1000)) ## now compare this design to equal allocations on ## 0, 10, 25, 50, 100, 150 doses2 <- c(0, 10, 25, 50, 100, 150) design2 <- c(1/6, 1/6, 1/6, 1/6, 1/6, 1/6) crit2 <- calcCrit(design2, models, probs, doses2, designCrit = "Dopt") ## ratio of determinants (returned criterion value is on log scale) exp(deswgts\$crit-crit2) ## example for calculating an optimal design for logistic regression doses <- c(0, 0.35, 0.5, 0.65, 1) fMod <- Mods(linear = NULL, doses=doses, placEff=-5, maxEff = 10) ## now calculate weights to use in the covariance matrix mu <- as.numeric(getResp(fMod, doses=doses)) mu <- 1/(1+exp(-mu)) weights <- mu*(1-mu) des <- optDesign(fMod, 1, doses, weights = weights) ## one can also specify a user defined criterion function ## here D-optimality for cubic polynomial CubeCrit <- function(w, doses){ X <- cbind(1, doses, doses^2, doses^3) CVinv <- crossprod(X*w) -log(det(CVinv)) } optDesign(doses = c(0,0.05,0.2,0.6,1), designCrit = "userCrit", userCrit = CubeCrit, optimizer = "nlminb") ```