Author: Lucy SinkeDavy CatsBas Heijmans

# Omixer: multivariate and reproducible randomization to proactively counter batch effects in omics studies

# Introduction

Batch effects can have a major impact on the results of omics studies (Leek et al. 2010). Randomization is the first, and arguably most influential, step in handling them. However, its implementation suffers from a few key issues:

- A single,ATTEMPT314 random draw can inadvertently result in high correlation between technical covariates and biological factors. Particularly in studies with large numbers of batches and outcomes of interest, minimizing these correlations is crucial.
- Long, randomized sample lists are unintuitive and translate poorly into any wet lab that is not fully automated. This can result in errors and sample mixups.
- The randomization process is inherently unclear in many publications, rarely described despite the varying efficacy of methods.
- Randomized layouts are not always reproducible, resulting in inconsistent results.

To combat these problems, we developed Omixer - an R package for multivariate randomization and reproducible generation of intuitive sample layouts.

## Dependencies

This document has the following dependencies.

    library(Omixer)
    library(tibble)
    library(forcats)
    library(stringr)
    library(dplyr)
    library(ggplot2)
    library(magick)

## Workflow

Omixer randomizes input sample lists multiple times (default: 1,000) and then combines these randomized lists with plate layouts, which can be selected from commonly used setups or custom-made. It can handle paired samples, keeping these adjacent but shuffling their order, and allows explicit masking of specific wells if, for example, plates are shared between different studies.

After performing robust tests of correlation between technical covariates and selected randomization factors, a layout is chosen using these criteria:

- No test provided sufficient evidence to suggest correlation between the variables (all p-values over 0.05).
- From the remaining layouts, return one where the absolute sum of correlations is minimized.

The optimal randomized list can then be processed by `omixerSheet`, returning intuitive sample layouts for the wet lab.

# Creating Layouts

In order to establish correlations between technical covariates and biological factors, Omixer needs to know the plate layout that your samples will be randomized to. There are several options for automatically creating some common layouts. Alternatively, a data frame can be input to the `layout` option alongside specified `techVars`. Possibilities are discussed in more detail below.

## Automated Layouts

Several options can be used to automatically generate common layouts:

- `wells` specifies the number of wells on a plate, which can be 96, 48, or 24.
- `plateNum` determines how many copies of the plate your samples will need.
- `div` is optional, and subdivides the plate into batches. This can be used to specify chips within a plate, for example.
- `positional` allows positions within `div` to also be treated as batches. This is useful for 450K experiments where positional batch effects have been identified (Jiao et al. 2018).

## Subdivisions

By default, `div` is set to “none”, but it can be set to “col”, “row”, “col-block”, or “row-block”.

- `col` treats each column in the plate as a batch
- `row` treats each row in the plate as a batch
- `col-block` will separate the plate into batches that are 2 columns wide
- `row-block` separates the plate into 2 row wide batches

So, for `wells=48, div="col"`, each column of a 48-well plate will be treated as a batch (different colours in the image below).

If you instead specify `div="row"`, the rows will be treated as batches.

Similarly, you can set `div="col-block"` or `div="row-block"` for batches that are 2 columns or rows wide, respectively. The image below shows how a 48 well plate would be subdivided with the `div="col-block"` option.

Combining the above will allow you to create a large number of layouts commonly used in omics experiments.

## Masking

If your experiment requires certain wells to be left empty, then these can be specified with the `mask` option. By default, no wells are masked, but you can specify masking, with `1` representing a masked well and `0` signifying that a sample should be randomized to this position.

Wells are numbered along each row sequentially. In the images above, row A includes wells 1 through 8, row B is wells 9 to 16, and so on until well 48 at F8.

## Custom Layouts

If none of the automated layouts represent your setup you can input your own plate layout as a data frame. The only requirement is that the number of unmasked wells is equal to the number of samples in your experiment, and that you input the names of technical covariate columns to the `techVars` option.

For example, if we wanted to create a 96-well plate to send for 450K DNA methylation profiling, we might submit the following `layout` and `techVars`.

    layout <- tibble(plate=rep(1, 96), well=1:96,
        row=factor(rep(1:8, each=12), labels=toupper(letters[1:8])),
        column=rep(1:12, 8), chip=as.integer(ceiling(column/2)),
        chipPos=ifelse(column %% 2 == 0, as.numeric(row)+8, row))

    techVars <- c("chip", "chipPos")

    layout
    #> # A tibble: 96 × 6
    #>    plate  well row   column  chip chipPos
    #>    <dbl> <int> <fct>  <int> <int>   <dbl>
    #>  1     1     1 A          1     1       1
    #>  2     1     2 A          2     1       9
    #>  3     1     3 A          3     2       1
    #>  4     1     4 A          4     2       9
    #>  5     1     5 A          5     3       1
    #>  6     1     6 A          6     3       9
    #>  7     1     7 A          7     4       1
    #>  8     1     8 A          8     4       9
    #>  9     1     9 A          9     5       1
    #> 10     1    10 A         10     5       9
    #> # ℹ 86 more rows

# Simple example

We create toy data, representing 48 samples to be sent off for RNA sequencing. All samples will be on a single 48-well flowcell, with each of the 8-sample rows being pipetting onto a lane, resulting in 6 lanes. This setup can be represented using provided Omixer layouts, as is described below.

First, we build the sample list that will be provided to Omixer, with information on the age, sex, and smoking status of individuals alongside sample isolation dates. We want to optimize distribution of these randomization variables across lanes on the flowcell to minimize batch effects.

    sampleList <- tibble(sampleId=str_pad(1:48, 4, pad="0"),
        sex=as_factor(sample(c("m", "f"), 48, replace=TRUE)),
        age=round(rnorm(48, mean=30, sd=8), 0),
        smoke=as_factor(sample(c("yes", "ex", "never"), 48,
            replace=TRUE)),
        date=sample(seq(as.Date('2008/01/01'), as.Date('2016/01/01'),
            by="day"), 48))

    sampleList
    #> # A tibble: 48 × 5
    #>    sampleId sex     age smoke date
    #>    <chr>    <fct> <dbl> <fct> <date>
    #>  1 0001     m        25 ex    2013-02-27
    #>  2 0002     m        17 ex    2010-05-25
    #>  3 0003     m        37 ex    2015-01-25
    #>  4 0004     f        31 never 2014-02-02
    #>  5 0005     m        21 ex    2009-07-07
    #>  6 0006     f        40 never 2015-11-14
    #>  7 0007     f        33 ex    2013-11-01
    #>  8 0008     f        28 never 2012-03-26
    #>  9 0009     m        37 yes   2015-05-04
    #> 10 0010     m        37 never 2011-08-18
    #> # ℹ 38 more rows

## Randomization Variables

Using the `randVars` option, we inform Omixer which columns in our data represent randomization variables. You can specify any number of variables, but with increasing numbers it will become more difficult to optimize their distribution across batches.

    randVars <- c("sex", "age", "smoke", "date")

## Running Omixer

To perform multivariate randomization use the `omixerRand` function. For our example, we have one 96-well flowcell `wells=96, plateNum=1` and want to optimize sample distribution across lanes `div="row"`.

Following randomization, `omixerRand` will display a visualization of correlations between randomization and technical variables. If the returning correlations are higher than you would like, you can increase the `iterNum` or decrease the number of randomization variables.

    omixerLayout <- omixerRand(sampleList, sampleId="sampleId",
        block="block", iterNum=100, wells=48, div="row",
        plateNum=1, randVars=randVars)
    #> Random seed saved to working directory
    #> Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
    #> ℹ Please use `linewidth` instead.
    #> ℹ The deprecated feature was likely used in the Omixer package.
    #>   Please report the issue at <https://github.com/molepi/Omixer/issues>.
    #> This warning is displayed once per session.
    #> Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
    #> generated.

Following `omixerRand`, an optimal randomized sample list is returned. This can be used as is or processed by `omixerSheet` to create lab-friendly sample sheets, which will be shown below.

    head(omixerLayout[1:11])
    #>   sampleId sex age smoke       date plate well row column mask chip
    #> 1     0009   m  37   yes 2015-05-04     1    1   A      1    0    1
    #> 2     0028   f  30    ex 2014-02-25     1    2   B      1    0    2
    #> 3     0001   m  25    ex 2013-02-27     1    3   C      1    0    3
    #> 4     0042   f  32   yes 2011-08-05     1    4   D      1    0    4
    #> 5     0017   f  24    ex 2009-11-02     1    5   E      1    0    5
    #> 6     0026   m  29 never 2015-01-13     1    6   F      1    0    6

## Regenerating Layouts

Since multivariate randomization can take some time with large datasets and many randomization variables, we provide the `omixerSpecific` function to reproduce previously generated layouts. After running `omixerRand`, the seed of the optimal layout is saved to the working directory.

After setting the global variable `.Random.seed`, you can run `omixerSpecific` to regenerate the optimal layout.

    load("randomSeed.Rdata")
    .GlobalEnv$.Random.seed <- randomSeed

    omixerLayout <- omixerSpecific(sampleList, sampleId="sampleId",
        block="block", wells=96, div="row",
        plateNum=1, randVars=randVars)

## Sample Sheets

Once the multivariate randomization is complete, the resulting data frame can be input into `omixerSheet` to produce lab-friendly sample layouts. These will be saved in your working directory as a PDF document.

It is possible to colour code these sheets by a specific factor using the `group` option, and this is demonstrated in the extended example.

    omixerSheet(omixerLayout)
    #> Warning: The `size` argument of `element_rect()` is deprecated as of ggplot2 3.4.0.
    #> ℹ Please use the `linewidth` argument instead.
    #> ℹ The deprecated feature was likely used in the Omixer package.
    #>   Please report the issue at <https://github.com/molepi/Omixer/issues>.
    #> This warning is displayed once per session.
    #> Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
    #> generated.

# Extended example

To demonstrate the full functionality of Omixer, we present an extended example.

Here, our toy data represents 616 samples ready to be sent off for EPIC DNA methylation profiling. These samples will be randomized to 7 96-well plates where each of the 8 columns are transferred to a 12-sample EPIC chip. The last chip on each plate needs to be kept empty for control samples, and we will communicate this to Omixer using the `mask` option.

Our samples are taken from 4 different tissues of 77 individuals, and we are interested in how DNA methylation changes over 2 timepoints. Given our primary research question, we would like to keep the timepoints adjacent on the array but randomize their order. We can do this in Omixer with the `block` option, as demonstrated below.

## Creating Toy Data

As well as a sample ID, we need to tell Omixer which variables specify paired sample blocks using a blocking variable, which we name `block`.

    sampleList<- tibble(sampleId=str_pad(1:616, 4, pad="0"),
        block=rep(1:308, each=2),
        time=rep(0:1, 308),
        tissue=as_factor(rep(c("blood", "fat", "muscle", "saliva"),
            each=2, 77)),
        sex=as_factor(rep(sample(c("male", "female"), 77, replace=TRUE),
            each=8)),
        age=round(rep(rnorm(77, mean=60, sd=10), each=8), 0),
        bmi=round(rep(rnorm(77, mean=25, sd=2), each=8) , 1),
        date=rep(sample(seq(as.Date('2015/01/01'), as.Date('2020/01/01'),
            by="day"), 77), each=8))

    sampleList
    #> # A tibble: 616 × 8
    #>    sampleId block  time tissue sex      age   bmi date
    #>    <chr>    <int> <int> <fct>  <fct>  <dbl> <dbl> <date>
    #>  1 0001         1     0 blood  male      61  26.9 2016-02-28
    #>  2 0002         1     1 blood  male      61  26.9 2016-02-28
    #>  3 0003         2     0 fat    male      61  26.9 2016-02-28
    #>  4 0004         2     1 fat    male      61  26.9 2016-02-28
    #>  5 0005         3     0 muscle male      61  26.9 2016-02-28
    #>  6 0006         3     1 muscle male      61  26.9 2016-02-28
    #>  7 0007         4     0 saliva male      61  26.9 2016-02-28
    #>  8 0008         4     1 saliva male      61  26.9 2016-02-28
    #>  9 0009         5     0 blood  female    42  24.9 2018-07-16
    #> 10 0010         5     1 blood  female    42  24.9 2018-07-16
    #> # ℹ 606 more rows
    save(sampleList, file="sampleList.Rdata")

## Setting up Variables

We set up our randomization variables to optimize distribution of our biological factors across chips and plates. Randomization variables in our example are tissue, sex, age, body mass index (BMI), and isolation date.

    randVars <- c("tissue", "sex", "age", "bmi", "date")

Since the last chip on each plate needs to be reserved, we specify a mask so that Omixer knows not to assign samples to these wells.

In the mask, a `0` indicates that a sample will be assigned to that well, and a `1` indicates that it should be left empty.

    mask <- rep(c(rep(0, 88), rep(1, 8)), 7)

## Running Omixer

Now we are ready to perform multivariate randomization with the `omixerRand` function. We specify 7 96-well plates `wells=96, plateNum=7` subdivided into 8-sample EPIC chips `div="col"`.

    omixerLayout <- omixerRand(sampleList, sampleId="sampleId",
        block="block", iterNum=100, wells=96, div="col", plateNum=7,
        randVars=randVars, mask=mask)
    #> Random seed saved to working directory

## Simple Randomization

Looking at the above correlations, you may wonder how Omixer compares to simple randomization. Briefly, we will investigate this.

Simple randomization can be replicated using `omixerRand` with a `iterNum=1`. Here, only one randomized layout will be created. If this is not optimal, a warning will print but the layout will still be returned.

    simpleLayout <- omixerRand(sampleList, sampleId="sampleId",
        block="block",iterNum=1, wells=96, div="col", plateNum=7,
        randVars=randVars, mask=mask)
    #> Warning: There was 1 warning in `filter()`.
    #> ℹ In argument: `absSum == min(absSum)`.
    #> Caused by warning in `min()`:
    #> ! no non-missing arguments to min; returning Inf
    #> Warning in omixerRand(sampleList, sampleId = "sampleId", block = "block", : All
    #> randomized layouts contained unwanted correlations.
    #> Warning in omixerRand(sampleList, sampleId = "sampleId", block = "block", :
    #> Returning best possible layout.
    #> Random seed saved to working directory

Here, we see strong evidence of a correlation between:

- Age and chip (τ = -0.082, p = 0.005)
- Age and plate (τ = -0.073, p = 0.011)
- Date and plate (τ = 0.082, p = 0.005)

These patterns threaten the validity of our future inferences, as the effects of biological factors are entangled with technical variations.

In comparison, there is insufficient evidence to suggest correlation between any biological factor and technical covariate in the Omixer produced layout, and the largest correlation coefficient returned is 0.048, which is considerably lower than many seen in the simple randomized layout.

## Regenerating layouts

As in the simple example, any Omixer layouts can be regenerated using the saved random environment in the `omixerSpecific` function.

After setting the global variable `.Random.seed`, you can run `omixerSpecific` to regenerate the optimal layout.

    load("randomSeed.Rdata")
    .GlobalEnv$.Random.seed <- randomSeed

    omixerLayout <- omixerSpecific(sampleList, sampleId="sampleId",
        block="block", wells=96, div="col", plateNum=7,
        randVars=randVars, mask=mask)

## Sample Sheets

The bridge between dry and wet labs can be precarious. Technicians are often faced with long, monotonous lists of samples, which they need to pipette accurately to minimize sample mixups. This is especially prevalent in more complicated setups as in this extended example.

The `omixerSheet` function smooths this transition by creating lab-friendly PDFs of sample layouts.

You can colour code wells by another variable. In our example, we want to highlight wells of each tissue, since samples from one tissue are likely to be stored together `group="tissue"`. The first page of the resulting PDF is shown below.

    omixerSheet(omixerLayout, group="tissue")

# References

Jiao, Chuan, Chunling Zhang, Rujia Dai, et al. 2018. “Positional effects revealed in Illumina methylation array and the impact on analysis.” Epigenomics 10 (5): 643–59. https://doi.org/10.2217/epi-2017-0105.
Leek, Jeffrey T., Robert B. Scharpf, Héctor Corrada Bravo, et al. 2010. Tackling the widespread and critical impact of batch effects in high-throughput data. 11 (10): 733–39. https://doi.org/10.1038/nrg2825.

# Session Info

    sessionInfo()
    #> R version 4.6.0 (2026-04-24)
    #> Platform: x86_64-pc-linux-gnu
    #> Running under: Ubuntu 24.04.4 LTS
    #>
    #> Matrix products: default
    #> BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3
    #> LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so;  LAPACK version 3.12.0
    #>
    #> locale:
    #>  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C
    #>  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8
    #>  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8
    #>  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C
    #>  [9] LC_ADDRESS=C               LC_TELEPHONE=C
    #> [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
    #>
    #> time zone: Etc/UTC
    #> tzcode source: system (glibc)
    #>
    #> attached base packages:
    #> [1] stats     graphics  grDevices utils     datasets  methods   base
    #>
    #> other attached packages:
    #> [1] magick_2.9.1     ggplot2_4.0.3    dplyr_1.2.1      stringr_1.6.0
    #> [5] forcats_1.0.1    tibble_3.3.1     Omixer_1.22.0    BiocStyle_2.40.0
    #>
    #> loaded via a namespace (and not attached):
    #>  [1] gtable_0.3.6        jsonlite_2.0.0      compiler_4.6.0
    #>  [4] BiocManager_1.30.27 Rcpp_1.1.1-1.1      tidyselect_1.2.1
    #>  [7] gridExtra_2.3       jquerylib_0.1.4     scales_1.4.0
    #> [10] yaml_2.3.12         fastmap_1.2.0       readr_2.2.0
    #> [13] R6_2.6.1            labeling_0.4.3      generics_0.1.4
    #> [16] knitr_1.51          maketools_1.3.2     tzdb_0.5.0
    #> [19] RColorBrewer_1.1-3  bslib_0.10.0        pillar_1.11.1
    #> [22] rlang_1.2.0         utf8_1.2.6          stringi_1.8.7
    #> [25] cachem_1.1.0        xfun_0.57           S7_0.2.2
    #> [28] sass_0.4.10         sys_3.4.3           cli_3.6.6
    #> [31] withr_3.0.2         magrittr_2.0.5      digest_0.6.39
    #> [34] grid_4.6.0          hms_1.1.4           lifecycle_1.0.5
    #> [37] vctrs_0.7.3         evaluate_1.0.5      glue_1.8.1
    #> [40] farver_2.1.2        buildtools_1.0.0    rmarkdown_2.31
    #> [43] tools_4.6.0         pkgconfig_2.0.3     htmltools_0.5.9