# Evaluating Algorithm Efficiency for Optimizing Experimental Designs with Correlated Data
## 1 Introduction  
Generating field experimental designs often requires the experimental units to be replicated, randomized and apply some form of blocking to reduce heterogeneity. These properties ensure that results of an experiment are unbiased, optimal, and allow to perform appropriate inferences to a larger population [1]. For plant breeding, field trials are an important component that help to evaluate and select the genotypes (or treatments) with superior performance to be used as future parents or commercial varieties [2]. Breeding trials are often characterized by testing a large number of genetic entries with limited replication. The effect of these entries is often estimated by fitting a linear mixed model (LMM) that considers genotypes as a random effects, and that incorporates genetic relationships (or correlations) by a variance-covariance matrix obtained based on pedigree information or molecular markers. Several proposed experimental designs exist for field trials, and these can be generated using widely available statistical software. However, the process of generating an optimal or near-optimal design, that maximizes the amount of information extracted with limited resources, is often ignored due to their intensive computational requirements, particularly for experimental designs with large number of treatments. Some authors have presented efficient procedures to construct experimental designs for breeding trials, including incomplete blocks, row-column and augmented designs (e.g., [PERSON] and [PERSON] (2011) and [PERSON] et al. (2013)). However, these are mostly restricted to the assumption of fixed treatments effects, and therefore ignore the information provided by the genetic relationships. At the same time, it has been shown that modelling field spatial correlations (e.g., by incorporating an autoregressive residual structure) results in more efficient designs than assuming that residuals are independent and identically distributed ([PERSON] and [PERSON], 2011; [PERSON] and [PERSON], 2013). Here, the framework of mixed models is advantageous over traditional linear models since they allow for specification of appropriate variance-covariance structures for both factors (e.g., genetic entries) and residuals (thorugh spatial correlation), providing greater flexibility and more efficient downstream statistical analyses. To generate experimental designs under the above framework an optimality criterion is used together with the implementation of an iterative search algorithm. \(A\)- and \(D\)-optimality information based criterion are the most widely used procedures in field experiments to generate optimal or near-optimal designs ([PERSON] and [PERSON], 2013; [PERSON] and [PERSON], 2013) and most recently were used by [PERSON] et al. (2013) and [PERSON] et al. (2014) to design experiments with correlated observations. These procedures are very useful in the process of selecting an optimal design ([PERSON] et al., 2014). \(A\)-optimality criterion seeks to minimize the average variance of random treatment effects and can be expressed as: \(A_{optim}=\text{argmin}\{trace[\mathbf{M}(\Omega)]\}\), where \(\mathbf{M}(\Omega)\) is the inverse of an information matrix of the treatment (or genetic) effects from a given design layout \(\Omega\). \(D\)-optimality was introduced by [PERSON] (1984) and minimizes the determinant of \(\mathbf{M}(\Omega)\) which can be interpreted as minimizing the generalized variance of the treatment effects ([PERSON] et al., 2014) by choosing designs which minimize the volume of the joint confidence ellipsoid ([PERSON], 1985) and is given by \(D_{optim}=\text{argmin}\{|\mathbf{M}(\Omega)|\}\) for \(|\mathbf{M}(\Omega)|\
eq 0\). Often, search algorithms involve interchanging the assigned treatments for a pair of experimental units and re-evaluating the efficiency of the new design to be compared against the previous one.