optDesign {DoseFinding}  R Documentation 
Function to calculate optimal designs
Description
Given a set of models (with full parameter values and model probabilities) the ‘optDesign’ function calculates the optimal design for estimating the doseresponse model parameters (Doptimal) or the design for estimating the target dose (TDoptimal 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
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", "NelderMead", "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

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 Doptimality criterion (specifically the logdeterminant) 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 ( 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
"NelderMead" and "nlminb" use the Option "solnp" uses the solnp function from the Rsolnp package, which implements an optimizer for nonlinear 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 ‘"NelderMead"’ and ‘"nlminb"’ are best suited (‘"nlminb"’ is usually a bit faster but less stable than ‘"NelderMead"’). For a larger number of doses ‘"solnp"’ is the most reliable option (but also slowest) (‘"NelderMead"’ 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’. 
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 doseresponse models selected in 
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 \sum 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'M_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(0.5\log(\det(M_m))/k_m+0.5\log(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 NelderMead algorithm. Alternatively one can use the solnp optimizer that is usually the most reliable, but not fastest option.
Author(s)
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
Pinheiro, J.C., Bornkamp, B. (2017) Designing Phase II DoseFinding Studies: Sample Size, Doses and Dose Allocation Weights, in O'Quigley, J., Iasonos, A. and Bornkamp, B. (eds) Handbook of methods for designing, monitoring, and analyzing dosefinding trials, CRC press
Pukelsheim, F. (1993). Optimal Design of Experiments, Wiley
See Also
Examples
## calculate designs for Emax model
doses < c(0, 10, 100)
emodel < Mods(emax = 15, doses=doses, placEff = 0, maxEff = 1)
optDesign(emodel, probs = 1)
## TDoptimal design
optDesign(emodel, probs = 1, designCrit = "TD", Delta=0.5)
## 5050 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="NelderMead")
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$critcrit2)
## 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*(1mu)
des < optDesign(fMod, 1, doses, weights = weights)
## one can also specify a user defined criterion function
## here Doptimality 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")