The blocksdesign package provides functionality for the construction
of nested or crossed block designs for general linear model treatment designs.
Block designs group experimental units into homogeneous blocks to provide maximum precision of estimation of treatment effects within blocks. The most basic type of block design is a complete randomized blocks design where each block contains one or more complete replicate sets of treatments. Complete randomized block designs estimate all treatment effects fully within individual blocks and are usually the best choice for small experiments. However, for large experiments, the variability within complete blocks can be large and then it may be beneficial to sub-divide each complete block into smaller more homogeneous incomplete blocks.
Block designs with a single level of nesting are widely used in practical research but sometimes for very large experiments a single set of nested blocks may still be too large to give good control of intra-block variability. In this situation, a second set of incomplete blocks can be nested within the first set to reduce the intra-block variability still further. This process of recursive nesting can be repeated as often as required until the bottom set of blocks is sufficiently small to give adequate control of intra-block variability.
Sometimes it can be advantageous to use a double blocking system in which one set of blocks, usually called row blocks, is crossed with a second set of blocks, usually called column blocks. Double blocking systems can be valuable for controlling block effects in two dimensions simultaneously.
The blocksdesign package provides functionality for the construction of general
multi-level block designs with nested or crossed blocks
for any feasible depth of nesting. The design algorithm proceeds recursively with each nested set of
blocks optimized conditionally within the levels of each preceding
set of blocks. The analysis of incomplete block designs is complex but the availability
of modern computers and modern software, for example the R mixed model
software package lme4 (Bates et. al. 2014), makes the analysis of any feasible
nested block designs with any depth of nesting practicable.
The blocksdesign package has two design functions:
i) blocks: This is a simple recursive function for nested block designs for unstructured treatment sets.
The function generates designs for treatments with arbitrary levels of replication and with arbitrary depth of nesting
where each successive set of blocks is optimized within the levels of each preceding set of blocks using conditional D-optimality.
Special block designs such as lattice designs or latin or Trojan square designs are constructed algebraically.
The outputs from the blocks function include a data frame showing the allocation of treatments to blocks for each plot of the design and a
table showing the achieved D- and A-efficiency factors for each set of nested blocks together with A-efficiency upper bounds, where available.
A plan showing the allocation of treatments to blocks in the bottom level of the design is also included in the output.
i) design: This is a general purpose function for unstructured or general qualitative or quantitative
factorial treatment sets. The function first finds a D-optimal or near D-optimal treatment design of the required size, possibly a simple
unstructured treatment set. The function then finds a D-optimal or near D-optimal block design for that treatment design based on a set of
defined block factors, if present. The blocks design algorithm builds the blocks design by sequentially adding
blocks factors where each block factor is optimized conditional on all previous block factors. Sequential optimization allows the
blocking factors to be fitted in order of importance with the largest and most important blocks fitted first and the smaller and less important
blocks fitted subsequently. If there are no defined bock factors, the algorithm assumes a completely randomised treatment design.
The outputs include a data frame of the block and treatment factors for each plot and a table showing the achieved D-efficiency
factors for each set of nested or crossed blocks. Fractional factorial efficiency factors based on
the generalized variance of the complete factorial design are also shown (see the design documentation for more details)
BATES D., MAECHLER M., BOLKER B., WALKER S. (2015). Fitting Linear Mixed-Effects Models Using lme4. Journal of Statistical Software, 67(1), 1-48. doi:10.18637/jss.v067.i01.