# Partially Replicated (p-rep) Design

This vignette shows how to generate a partially replicated (p-rep) design using both the FielDHub Shiny App and the scripting function partially_replicated() from the FielDHub R package.

## 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).

## Running the Shiny App

To launch the app you need to run either

FielDHub::run_app()

or

library(FielDHub)
run_app()

## 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.

## Inputs

1. Import entries’ list? Choose whether to import a list with entry numbers and names for genotypes or treatments.
• If the selection is No, that means the app is going to generate synthetic data for entries and names of the treatment/genotypes based on the user inputs.

• If the selection is Yes, the entries list must fulfill a specific format and must be a .csv file. The file must have the columns ENTRY, NAME, and REPS. The ENTRY column must have a unique entry integer number for each treatment/genotype. The column NAME must have a unique name that identifies each treatment/genotype. The REPS column must have an integer number for the replications of the groups. Both ENTRY and NAME must be unique, duplicates are not allowed. In the following table, we show an example of the entries list format. This example has an entry list with four treatments/genotypes that will appear twice and 8 that appear just once.

ENTRY NAME REPS
1 GenotypeA 2
2 GenotypeB 2
3 GenotypeC 2
4 GenotypeD 2
5 GenotypeE 1
6 GenotypeF 1
7 GenotypeG 1
8 GenotypeH 1
9 GenotypeI 1
10 GenotypeJ 1
11 GenotypeK 1
12 GenotypeL 1
1. Enter the number of entries per replicate group in the # of Entries per Rep Group box as a comma separated list. In our example we will have 2 groups with 85 and 130 entries. So, we enter 75, 150 in the box for our sample experiment.

2. Enter the number of replications per group in the # of Rep per Group box. In our example we will have 2 and 1 replications for the 2 groups, so we enter 2, 1 in this box.

3. Enter the number of locations in Input # of Locations. We will run this experiment over a single location, so set Input # of Locations to 1.

4. Select serpentine or cartesian in the Plot Order Layout. For this example we will use the default serpentine layout.

5. To ensure that randomizations are consistent across sessions, we can set a random seed in the box labeled random seed. In this example, we will set it to 1245.

6. Enter the starting plot number in the Starting Plot Number box. If the experiment has multiple locations, you must enter a comma separated list of numbers the length of the number of locations for the input to be valid.

7. Once we have entered the information for our experiment on the left side panel, click the Run! button to run the design.

8. You will then be prompted to select the dimensions of the field from the list of options in the drop down in the middle of the screen with the box labeled Select dimensions of field. In our case, we will select 15 x 20.

9. Click the Randomize! button to randomize the experiment with the set field dimensions and to see the output plots. If you change the dimensions again, you must re-randomize.

If you change any of the inputs on the left side panel after running an experiment initially, you have to click the Run and Randomize buttons again, to re-run with the new inputs.

## 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.

### Data Input

On the second tab, Data Input, you can see all the entries in the randomization in a list, as well as a table of the checks with the number of times they appear in the field. In the list of entries, the reps for each check is included as well.

### 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,

library(FielDHub)

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,
)

#### Details on the inputs entered in optimized_arrangement() above

The description for the inputs that we used to generate the design,

• nrows = 15 is the number of rows in the field.
• ncols = 20 is the number of columns in the field.
• repGens = c(75,150) are the values for the groups to replicate
• repUnits = c(2,1) are the values for representing respective replicates of each group.
• planter = "serpentine" is the layout order.
• plotNumber = 101 is the starting plot number for the experiment.
• l = 1 is the number of locations.
• exptName = "Expt1" is an optional name for experiment.
• locationNames = "PALMIRA" is the optional name for the locations.
• seed = 1245 is the random seed to replicate identical randomizations.

### 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,

plot(prep)

In the figure above, green plots contain replicated entries, and gray plots contain entries that only appear once.

# References

Cullis, et al., B. R. 2006. On the Design of Early Generation Variety Trials with Correlated Data.” Journal of Agricultural, Biological, and Environmental Statistics 11 (4): 381–93.