Overview
Partially replicated designs are commonly employed in early
generation field trials. This type of design is characterized by
replication of a portion of the entries, with the remaining entries only
appearing once in the experiment. Commonly, the part of treatments with
reps is due to an arbitrary decision by the research, also in some
cases, it is due to technical reasons. The replication ratio is
typically 1:4 (Cullis 2006), which means
that the portion of treatment repeated twice is p = 25%. However, the
design can be adapted to meet specific needs by adjusting the values of
\(p\) and the level of replication. For
example, standard varieties (checks) may be included with higher levels
of replication than test lines.
In FielDHub, users can set the number of entries that
will have reps, as well as the number of entries that will only appear
once. You can also choose to run the same experiment over multiple
locations.
Optimization
Each partially replicated (p-rep) design location undergoes an
optimization process that involves the following procedure:
Given a matrix \(X\) of integers
(p-rep design within location), we want to ensure that the distance
between any two occurrences of the same treatment is at least a distance
\(d\). More specifically, we want to
modify \(X\) to ensure that no
treatments appear twice within a distance less than \(d\) in the resulting matrix.
The goal of the optimization process is to find a modified matrix
that satisfies this constraint while maximizing some measure of
deviation from the original matrix \(X\). In this case, the measure of deviation
is the pairwise Euclidean distance between occurrences of the same
treatment. The process is done by the function swap_pairs()
that uses a heuristic algorithm that starts with a distance of \(d = 3\) between pairs of occurrences of the
same treatment, and increases this distance by \(1\) and repeats the process until either
the algorithm no longer converges or the maximum number of iterations is
reached.
The algorithm works by first identifying all pairs of occurrences of
the same treatment that are closer than \(d\). For each such pair, the function
selects a random occurrence of a different integer that is at least
\(d\) away, and swaps the two
occurrences. This process is repeated until no further swaps can be made
that increase the pairwise Euclidean distances between occurrences of
the same treatment.
Toy Example
Consider a p-rep design where ten treatments are replicated twice and
forty only once. The matrix (field layout) for this experiment has 6
rows and 10 columns.
\(X =\)
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] 21 40 17 25 26 3 11 31 36 6
[2,] 5 5 33 8 48 29 43 23 1 45
[3,] 41 27 38 39 7 28 14 22 24 4
[4,] 4 47 18 7 2 35 6 20 12 46
[5,] 3 15 9 34 49 50 2 10 42 8
[6,] 32 16 19 9 10 13 37 1 44 30
In this initial p-rep design, we notice that the two instances of
treatment 5 are positioned next to each other.
Additionally, treatments 7 and 9 are
also situated in adjacent cells. These suboptimal allocations could lead
to issues or inaccurate results when analyzing the data from this
experiment due to the short distance between replicated treatments and
the likely spatial correlation between them.
The following table shows the pairwise distances for the replicated
treatments
geno Pos1 Pos2 DIST rA cA rB cB
5 5 2 8 1.000000 2 1 2 2
7 7 22 27 1.414214 4 4 3 5
9 9 17 24 1.414214 5 3 6 4
2 2 28 41 2.236068 4 5 5 7
10 10 30 47 3.162278 6 5 5 8
1 1 48 50 4.123106 6 8 2 9
6 6 40 55 4.242641 4 7 1 10
3 3 5 31 6.403124 5 1 1 6
8 8 20 59 6.708204 2 4 5 10
4 4 4 57 9.055385 4 1 3 10
Swap pairs
We can improve the efficiency of the design by swapping the
treatments that are close and next to each other by using the function
swap_pairs() from FielDHub R package.
library(FielDHub)
B <- swap_pairs(X, starting_dist = 3)
The new matrix or the optimized p-rep design is,
print(B$optim_design)
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] 8 35 6 2 33 44 3 4 37 6
[2,] 43 30 25 5 39 29 19 11 36 45
[3,] 40 13 38 10 20 28 15 41 10 17
[4,] 1 27 18 31 32 22 24 21 12 5
[5,] 23 47 3 34 49 50 16 46 14 48
[6,] 7 26 2 42 9 1 8 7 4 9
The distances for each pairwise of treatments are,
print(B$pairwise_distance)
geno Pos1 Pos2 DIST rA cA rB cB
9 9 30 60 5.000000 6 5 6 10
10 10 21 51 5.000000 3 4 3 9
2 2 18 19 5.099020 6 3 1 4
4 4 43 54 5.099020 1 8 6 9
1 1 4 36 5.385165 4 1 6 6
3 3 17 37 5.656854 5 3 1 7
5 5 20 58 6.324555 2 4 4 10
6 6 13 55 7.000000 1 3 1 10
7 7 6 48 7.000000 6 1 6 8
8 8 1 42 7.810250 1 1 6 7
As we can see, the minimum distance that the algorithm reached is 5.
This means no treatments appear twice within a distance less than 5 in
the resulting prep design. It is a considerable improvement from the
first version of the p-rep design
The FielDHub function
partially_replicated() does internally all the optimization
process and uses the function swap_pairs() to maximize the
distance between replicated treatments.
Acknowledge
We would like to acknowledge Mr. Jean-Marc Montpetit for contributing
code and ideas for the swap_pairs() and
pairs_distance() functions. His contributions have had a
significant impact on improving the partially replicated (p-rep) design
in the R package FielDHub. We thank him for his valuable
contributions.
Use case
Consider a plant breeding field trial with 300 plots containing 75
entries appearing two times each, and 150 entries only appearing once.
This field trial is arranged in a field of 15 rows by 20 columns. In
this case, the breeder decided to replicate the genotypes that do not
share significant generic information with each other (75), as well as
leave with just one copy the genotypes that are siblings (150).
1. Using the FielDHub Shiny App
Once the app is running, click the tab Partially Replicated
Design and select Single and Multi-Location
p-rep from the dropdown.
Then, follow the following steps where we will show how to generate a
partially replicated design.
Outputs
After you run a single diagonal arrangement in FielDHub and set the
dimensions of the field, there are several ways to display the
information contained in the field book. The first tab, Get
Random, shows the option to change the dimensions of the field
and re-randomize, as well as a reference guide for experiment
design.
Randomized Field
The Randomized Field tab displays a graphical
representation of the randomization of the entries in a field of the
specified dimensions. The checks are the green colored cells, with the
The display includes numbered labels for the rows and columns. You can
copy the field as a table or save it directly as an Excel file with the
Copy and Excel buttons at the top.
Plot Number Field
On the Plot Number Field tab, there is a table
display of the field with the plots numbered according to the Plot Order
Layout specified, either serpentine or cartesian. You
can see the corresponding entries for each plot number in the field
book. Like the Randomized Field tab, you can copy the table or save it
as an Excel file with the Copy and Excel buttons.
Field Book
The Field Book displays all the information on the
experimental design in a table format. It contains the specific plot
number and the row and column address of each entry, as well as the
corresponding treatment on that plot. This table is searchable, and we
can filter the data in relevant columns.
2. Using the FielDHub function:
partially_replicated().
You can run the same design with the function
partially_replicated() in the FielDHub
package.
First, you need to load the FielDHub package typing,
Then, you can enter the information describing the above design like
this:
prep <- partially_replicated(
nrows = 15,
ncols = 20,
repGens = c(75,150),
repUnits = c(2,1),
planter = "serpentine",
plotNumber = 101,
l = 1,
exptName = "Expt1",
locationNames = "PALMIRA",
seed = 1245,
)
Print prep object
To print a summary of the information that is in the object
prep, we can use the generic function
print().
Partially Replicated Design
Replications within location:
LOCATION Replicated Unreplicated
1 PALMIRA 75 150
Information on the design parameters:
List of 7
$ rows : num 15
$ columns : num 20
$ min_distance : num 7
$ incidence_in_rows: num 3
$ locations : num 1
$ planter : chr "serpentine"
$ seed : num 1245
10 First observations of the data frame with the partially_replicated field book:
ID EXPT LOCATION YEAR PLOT ROW COLUMN CHECKS ENTRY TREATMENT
1 1 Expt1 PALMIRA 2024 101 1 1 44 44 G44
2 2 Expt1 PALMIRA 2024 102 1 2 0 102 G102
3 3 Expt1 PALMIRA 2024 103 1 3 71 71 G71
4 4 Expt1 PALMIRA 2024 104 1 4 0 107 G107
5 5 Expt1 PALMIRA 2024 105 1 5 8 8 G8
6 6 Expt1 PALMIRA 2024 106 1 6 13 13 G13
7 7 Expt1 PALMIRA 2024 107 1 7 0 170 G170
8 8 Expt1 PALMIRA 2024 108 1 8 67 67 G67
9 9 Expt1 PALMIRA 2024 109 1 9 0 123 G123
10 10 Expt1 PALMIRA 2024 110 1 10 0 105 G105
Access to prep output
The partially_replicated() function returns a list
consisting of all the information displayed in the output tabs in the
FielDHub app: design information, plot layout, plot numbering, entries
list, and field book. These are Accessible by the $
operator, i.e. prep$layoutRandom or
prep$fieldBook.
prep$fieldBook is a list containing information about
every plot in the field, with information about the location of the plot
and the treatment in each plot. As seen in the output below, the field
book has columns for ID, EXPT,
LOCATION, YEAR, PLOT,
ROW, COLUMN, CHECKS,
ENTRY, and TREATMENT.
Let us see the first 10 rows of the field book for this
experiment.
field_book <- prep$fieldBook
head(field_book, 10)
ID EXPT LOCATION YEAR PLOT ROW COLUMN CHECKS ENTRY TREATMENT
1 1 Expt1 PALMIRA 2024 101 1 1 44 44 G44
2 2 Expt1 PALMIRA 2024 102 1 2 0 102 G102
3 3 Expt1 PALMIRA 2024 103 1 3 71 71 G71
4 4 Expt1 PALMIRA 2024 104 1 4 0 107 G107
5 5 Expt1 PALMIRA 2024 105 1 5 8 8 G8
6 6 Expt1 PALMIRA 2024 106 1 6 13 13 G13
7 7 Expt1 PALMIRA 2024 107 1 7 0 170 G170
8 8 Expt1 PALMIRA 2024 108 1 8 67 67 G67
9 9 Expt1 PALMIRA 2024 109 1 9 0 123 G123
10 10 Expt1 PALMIRA 2024 110 1 10 0 105 G105
Plot field layout
For plotting the layout in function of the coordinates
ROW and COLUMN in the field book object we can
use the generic function plot() as follow,
In the figure above, green plots contain replicated entries, and gray
plots contain entries that only appear once.
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.