(This text relates to version 0.3.3) R package for approximate optimal experimental designs using generalised linear mixed models (GLMM) and combinatorial optimisation methods, built on the glmmrBase package. A discussion of the methods in this package can be found in Watson et al (2023).

The package is available on CRAN. A pre-compiled binary is also available with each release on this page. The package requires glmmrBase, it is recommended to build both glmmrBase and this package from source with the flags below, which can dramatically increase performance.

It is strongly recommended to build from source with the flags -fno-math-errno -O3 -g, this will cut the time to run many functions by as much as 90%. One way to do this is to set CPP_FLAGS in ~/.R/Makevars. Another alternative is to download the package source .tar.gz file and run from the command line

R CMD INSTALL --configure-args="CPPFLAGS=-fno-math-errno -O3 -g" glmmrOptim_0.3.3.tar.gz

For model specification see readme of glmmrBase. The glmmrOptim package adds the DesignSpace class. An instance of a DesignSpace object takes one or more Model objects, where each object specifies a particular design. Each object should be a complete enumeration of all possible observations. Observations can be nested within "experimental conditions", which are immutable collections of observations, for example, we may wish to choose $m$ of $n$ possible clusters of observations. Often the experimental conditions will comprise only a single observation. The aim is to identify an approximately c-optimal design with $m < n$ experimental conditions, or in the case of multiple designs a robust c-optimal design weighting over all designs.

The algorithm searches for a c-optimal design of size m from the design space using either local search, greedy search, or some combination of the two. The objective function is

$$ c^TM^{-1}c $$

where $M$ is the information matrix and $c$ is a vector. Typically $c$ will be a vector of zeros with a single 1 in the position of the parameter of interest. For example, if the columns of $X$ in the design are an intercept, the treatment indicator, and then time period indicators, the vector $c$ may be c(0,1,0,0,...), such that the objective function is the variance of that parameter. If there are multiple designs in the design space, the $c$ vectors do not have to be the same as the columns of X in each design might differ, in which case a list of vectors can be provided.

There are a variety of algorithms available:

• For design spaces with correlated experimental units, one can use either combinatorial algorithms: algo=1 local search, algo=2 greedy search, or algo=3; or an optimal mixed model weights algorithm, the "Girling" algorithm with algo="girling".
• For design spaces with uncorrelated experimental units by default the optimal experimental unit weights will be calculated using a second-order cone program. To instead use a combinatorial algorithm set use_combin=TRUE.

In some cases the optimal design will not be full rank with respect to the design matrix $X$ of the design space. This will result in a non-positive definite information matrix, and an error. The program will indicate which columns of $X$ are likely "empty" in the optimal design. The user can then optionally remove these columns in the algorithm using the rm_cols argument, which will delete the specified columns and linked observations before starting the algorithm.

The algorithm will also identify robust optimal designs if there are multiple designs in the design space. A weighted average of objective functions is used, where the weights are specified by the weights field in the design space with default $1/N$. The weights may represent the prior probability or plausibility of each design, for example. The objective function can be either a linear combination of variances, or a linear combination of log variances (robust_log=TRUE).

An example of model specification and optimal design search is below.

df <- nelder(formula(~ (cl(7) * t(6)) > ind(1)))
df$int <- 0
df[df$t >= df$cl,'int'] <- 1
des <- Model$new(formula = ~factor(t)  + int - 1 + (1|gr(cl)) + (1|gr(cl,t)),
                 covariance = c(0.04,0.01),
                 mean = rep(0,7),
                 var_par = sqrt(0.95),
                 data = df,
                 family=gaussian())
ds <- DesignSpace$new(des)
w1 <- ds$optimal(100,C = list(c(rep(0,6),1)),verbose = TRUE,algo="girling")

The design space supports any model specified in the glmmrBase package. Where there are non-linear functions of covariates in the fixed effects, a first-order approximation is used.