Overview
Partially replicated (p-rep) 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, the optimized multi-location p-rep design
employs the principles of incomplete block designs (IBD) to determine
the distribution of replicated and non-replicated treatments across
multiple locations.
Optimization
Across Location
The function multi_location_prep() uses the incomplete
blocks design approach (Edmondson 2020) to
optimize the allocation of replicated and un-replicated treatments
across the environments.
Within Location
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
40 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 multi_location_prep()
does internally all the optimization process and uses the function
swap_pairs() to maximize the distance between replicated
treatments.
Use case (Multi-Location P-rep Design)
Suppose there is a plant breeding field trial with 150 entries to be
tested across five environments, where up to seven replications of each
entry are allowed. Additionally, the project includes three checks; each
replicated six times. We can generate an optimized multi-location
partially replicated design using these parameters. This strategy
guarantees that all treatments are present in all environments but in
different amounts of replications.
We can generate this design using the FielDHub Shiny app and the
FielDHub multi_location_prep() standalone
function in R.
1. Using the FielDHub Shiny App
Once the app is running, click the tab Partially Replicated
Design and select Optimized Multi-Location
p-rep from the dropdown.
Then, follow the following steps where we will show how to generate
an optimized partially replicated design.
Outputs
After you run a Optimized Multi-Location P-rep Design 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 the genotype allocation matrix
generated for the optimized p-rep design, which displays the
replications of each genotype over each location, much like the matrix
generated in sparse allocation.
Randomized Field
The Randomized Field tab displays a graphical
representation of the randomization of the entries in a field of the
specified dimensions. The replicated entries are the green colored
cells, with the which cells appearing only once in the location. 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/genotype on that plot. This table is searchable,
and we can filter the data in relevant columns.
2. Using the FielDHub function:
multi_location_prep().
You can run the same design with the function
multi_location_prep() 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:
optim_multi_prep <- multi_location_prep(
lines = 150,
l = 5,
copies_per_entry = 7,
checks = 3,
rep_checks = c(6,6,6),
plotNumber = c(1,1001,2001,3001,4001),
locationNames = c("LOC1", "LOC2", "LOC3", "LOC4", "LOC5"),
seed = 2456
)
Print optim_multi_prep object
To print a summary of the information that is in the object
optim_multi_prep, we can use the generic function
print().
The multi_location_prep() function returns all the same
objects as in partially_replicated() and in addition
list_locs, allocation, and
size_locations. The object list_locs is a list
of data frames. Each data frame has three columns; ENTRY,
NAME and REPS with the information to
randomize to each environment. The object allocation is the
binary allocation matrix of genotypes to locations, and
size_locations is a data frame with a column for each
location and a row indicating the size of the location (number of field
plots).
For example, we can display the allocation object. Let
us print the first ten genotypes allocation.
print(head(optim_multi_prep$allocation, 10))
LOC1 LOC2 LOC3 LOC4 LOC5
1 1 2 2 1 1
2 2 1 1 1 2
3 1 2 1 1 2
4 1 1 2 1 2
5 1 1 2 2 1
6 2 1 1 2 1
7 2 1 2 1 1
8 1 2 2 1 1
9 1 1 2 1 2
10 1 2 1 2 1
Let us add two new columns to the allocation table. We can add the
number of copies by genotype; it should be 7 for all of them. We can
also add the average allocation by genotype. Each treatment will appear
1.4 times in average.
|
|
LOC1
|
LOC2
|
LOC3
|
LOC4
|
LOC5
|
Copies
|
Avg
|
|
Gen-1
|
1
|
2
|
2
|
1
|
1
|
7
|
1.4
|
|
Gen-2
|
2
|
1
|
1
|
1
|
2
|
7
|
1.4
|
|
Gen-3
|
1
|
2
|
1
|
1
|
2
|
7
|
1.4
|
|
Gen-4
|
1
|
1
|
2
|
1
|
2
|
7
|
1.4
|
|
Gen-5
|
1
|
1
|
2
|
2
|
1
|
7
|
1.4
|
|
Gen-6
|
2
|
1
|
1
|
2
|
1
|
7
|
1.4
|
|
Gen-7
|
2
|
1
|
2
|
1
|
1
|
7
|
1.4
|
|
Gen-8
|
1
|
2
|
2
|
1
|
1
|
7
|
1.4
|
|
Gen-9
|
1
|
1
|
2
|
1
|
2
|
7
|
1.4
|
|
Gen-10
|
1
|
2
|
1
|
2
|
1
|
7
|
1.4
|
We can manipulate the optim_multi_prep object as any
other list in R. We can first display the design parameters for the
randomizations with the following code:
which outputs:
Multi-Location Partially Replicated Design
Replications within location:
LOCATION Replicated Unreplicated
1 LOC1 63 90
2 LOC2 63 90
3 LOC3 63 90
4 LOC4 63 90
5 LOC5 63 90
Information on the design parameters:
List of 7
$ rows : num [1:5] 19 19 19 19 19
$ columns : num [1:5] 12 12 12 12 12
$ min_distance : num [1:5] 3 3 3 3 3
$ incidence_in_rows: num [1:5] 4 2 3 5 2
$ locations : num 5
$ planter : chr "serpentine"
$ seed : num 2456
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 PrepExpt LOC1 2024 1 1 1 77 77 G-77
2 2 PrepExpt LOC1 2024 2 1 2 0 117 G-117
3 3 PrepExpt LOC1 2024 3 1 3 132 132 G-132
4 4 PrepExpt LOC1 2024 4 1 4 58 58 G-58
5 5 PrepExpt LOC1 2024 5 1 5 0 130 G-130
6 6 PrepExpt LOC1 2024 6 1 6 0 41 G-41
7 7 PrepExpt LOC1 2024 7 1 7 120 120 G-120
8 8 PrepExpt LOC1 2024 8 1 8 0 48 G-48
9 9 PrepExpt LOC1 2024 9 1 9 66 66 G-66
10 10 PrepExpt LOC1 2024 10 1 10 125 125 G-125
Access to optim_multi_prep output
All objects are accessible by the $ operator,
i.e. optim_multi_prep$layoutRandom[[1]] for
LOC1, optim_multi_prep$fieldBook for the
fieldBook with all locations.
optim_multi_prep$fieldBook is a data frame containing
information about every plot in the field, with information about the
location of the plot and the treatment in each plot. As seen in the
output below, the field book has columns for ID,
EXPT, LOCATION, YEAR,
PLOT, ROW, COLUMN,
CHECKS, ENTRY, and TREATMENT.
Let us see the first 10 rows of the field book for this
experiment.
field_book <- optim_multi_prep$fieldBook
head(field_book, 10)
ID EXPT LOCATION YEAR PLOT ROW COLUMN CHECKS ENTRY TREATMENT
1 1 PrepExpt LOC1 2024 1 1 1 77 77 G-77
2 2 PrepExpt LOC1 2024 2 1 2 0 117 G-117
3 3 PrepExpt LOC1 2024 3 1 3 132 132 G-132
4 4 PrepExpt LOC1 2024 4 1 4 58 58 G-58
5 5 PrepExpt LOC1 2024 5 1 5 0 130 G-130
6 6 PrepExpt LOC1 2024 6 1 6 0 41 G-41
7 7 PrepExpt LOC1 2024 7 1 7 120 120 G-120
8 8 PrepExpt LOC1 2024 8 1 8 0 48 G-48
9 9 PrepExpt LOC1 2024 9 1 9 66 66 G-66
10 10 PrepExpt LOC1 2024 10 1 10 125 125 G-125
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 follows. This plots only
the first location, but this is indexable by location using the dollar
sign operator as well.
Field Layout for Location 1
plot(optim_multi_prep, l = 1)
In the figure above, green plots contain replicated entries, and gray
plots contain entries that only appear once.
Field Layout for Location 5
Also, for example the location five:
plot(optim_multi_prep, l = 5)
