Author: Julian Taylor
Date: 2024-08-25

# A Quick Introduction to wgaim QTL Analysis

#### Julian Taylor

#### 2024-08-25

Welcome to version 2.0.x of the wgaim R package! This package is an implementation of the whole genome average interval mapping (WGAIM) QTL analysis algorithm discussed in Verbyla, Cullis & Thompson (2007) and Verbyla, Taylor & Verbyla (2012). Although slightly out of date, the definitive reference for this software is Taylor & Verbyla (2011) with full reference given as

Taylor, J. and Verbyla, A (2011) R package wgaim: QTL Analysis in Bi-Parental Populations using Linear Mixed Models, Journal of Statistical Software, 40(7).

Note: The QTL analysis functions in wgaim explicitly use and build upon the functionality provided by the linear mixed modelling ASReml-R package (currently version 4). This is a commercial package available from VSNi at https://vsni.co.uk/software/asreml-r/ with pricing dependent on the institution. Users will require a fully licensed version of ASReml-R to use the QTL analysis functionality of the wgaim package and to run the code in this vignette. Users should consult the ASReml-R documentation for thorough details on the model syntax and extensive peripheral features of the package.

This introductory vignette presents the workflow of a wgaim QTL analysis. More in depth analyses can be found in an upcoming sister vignette, “A deeper look at the wgaim functionality.” The analysis workflow can be summarised simply with three steps:

1. With phenotypic data, build a base linear mixed model using the functionality of ASReml-R.
2. Construct a genotypic linkage map, store it as a qtl cross object and convert it to a wgaim interval object.
3. Use the base model from 1. and the interval genotype object from 2. to conduct a wgaim QTL analysis.

Package restrictions: The current version of wgaim provides functionality for QTL analysis of Double Haploid, Backcross, Advanced Recombinant Inbred and F2 populations.

## Package data

The wgaim package contains several pre-packaged phenotypic data sets with matching genetic linkage maps ready for QTL analysis.

    data(package = "wgaim")

The data has also been placed in a second location to provide the ability to read in manually.

    wgpath <- system.file("extdata", package = "wgaim")
    list.files(wgpath)

    ## [1] "genoCxR.csv"  "genoRxK.csv"  "genoSxT.csv"  "phenoCxR.csv" "phenoRxK.csv" "phenoSxT.csv"

## Example: RAC875 x Kukri phenotypic and genotypic data

### Phenotypic data and base model

This example consists of phenotypic and genotypic data sets involving a Doubled Haploid (DH) population derived from the crossing of wheat varieties RAC875 and Kukri (Bonneau et al., 2013). The main goal of the experiment was to find causal links between grain yield related traits and genetic markers associated with the population.

    data(phenoRxK, package = "wgaim")
    head(phenoRxK)

    ##     Genotype Type Row Range Rep    yld  tgw  lrow lrange
    ## 1    DH_R003   DH   1     1   1 2.2384 33.4 -12.5   -9.5
    ## 40   DH_R055   DH   2     1   1 1.1576 31.6 -11.5   -9.5
    ## 41   DH_R056   DH   3     1   1 1.6424 48.3 -10.5   -9.5
    ## 80   DH_R111   DH   4     1   1 2.3991 31.6  -9.5   -9.5
    ## 81   DH_R112   DH   5     1   1 1.9744 33.4  -8.5   -9.5
    ## 120  DH_R170   DH   6     1   1 1.2741 26.3  -7.5   -9.5

The RAC875 x Kukri phenotypic data relates to a field trial consisting of 520 plots. Two replicates of 256 DH lines (`Genotype`) from the RAC875 x Kukri population were allocated to a 20 `Row` by 26 `Range` layout using a randomized complete block design with 2 Blocks (`Rep`). The additional plots remaining in each block were filled with one of each of the parents and controls (ATIL, SOKOLL, WEEBILL). A `Type` factor is included to distinguish the set of DH lines from each of the parents and controls. `lrow` and `lrange` are numerically encoded and zero centred row and range covariates. A number of yield related trait measurements were collected including grain yield (t/ha) (`yld`) and thousand grain weight (`tgw`).

The analysis in this vignette concentrates on grain yield (`yld`). Before using the QTL analysis functions in wgaim, an appropriate initial base ASReml-R linear mixed model needs to be built and fitted.

    rkyld.asi <- asreml::asreml(yld ~ Type, random = ~ Genotype + Rep, residual =
         ~ ar1(Range):ar1(Row), data = phenoRxK)

    ## Online License checked out Sun Aug 25 16:54:36 2024
    ## Model fitted using the gamma parameterization.
    ## ASReml 4.1.0 Sun Aug 25 16:54:36 2024
    ##           LogLik        Sigma2     DF     wall    cpu
    ##  1       128.285      0.202517    514 16:54:36    0.0
    ##  2       178.124      0.126231    514 16:54:36    0.0
    ##  3       211.555      0.086862    514 16:54:36    0.0
    ##  4       221.240      0.074148    514 16:54:36    0.0
    ##  5       222.515      0.071463    514 16:54:36    0.0
    ##  6       222.595      0.072380    514 16:54:36    0.0
    ##  7       222.606      0.072730    514 16:54:36    0.0
    ##  8       222.607      0.072856    514 16:54:36    0.0

The focus of this model is the accurate calculation of the genetic variance of the DH progeny using `Genotype`. This accuracy is dramatically enhanced through the addition of terms used to account for extraneous variation arising from the experimental design (random term `Rep`) as well as potential correlation of the observations due to the similarity of neighbouring field trial plots (separable residual correlation structure `ar1(Row):ar1(Range)`)(Gilmour, 2007; Verbyla et al., 2007). Additionally, the inclusion of a `Type` factor as a fixed effect ensures the random `Genotype` factor only contains non-zero effects for the DH progeny.

A summary of the models variance parameter estimates shows a moderate correlation exists in the Range direction with a small correlation existing across the Rows.

    summary(rkyld.asi)$varcomp

    ##                       component   std.error    z.ratio bound %ch
    ## Rep                 0.001733554 0.003934291  0.4406269     P 0.7
    ## Genotype            0.167952916 0.017092886  9.8258958     P 0.0
    ## Range:Row!R         0.072856232 0.007130535 10.2174994     P 0.0
    ## Range:Row!Range!cor 0.240159289 0.068828138  3.4892603     U 0.3
    ## Range:Row!Row!cor   0.506495872 0.048864188 10.3653798     U 0.1

ASReml-R provides functionality for diagnostically checking the linear mixed model residuals. The variogram of the residuals indicates there is potential trends in the row and range directions of the experimental layout.

    plot(asreml::varioGram.asreml(rkyld.asi))

plot of chunk variogram

A faceted plot of the residuals confirm these trends.

    phenoRxKd <- cbind.data.frame(phenoRxK, Residuals = resid(rkyld.asi))
    ggplot(phenoRxKd, aes(y = Residuals, x = as.numeric(Range))) + facet_wrap(~ Row) +
        geom_hline(yintercept = 0, linetype = 2) + geom_point(shape = 16, colour = "blue") +
        xlab("Range") + theme_bw()

plot of chunk residuals

To account for these trends, terms `lrow` and `Range` are added to the fixed and random components of the `asreml` model and the model is refitted. An `lrange` fixed term would also be a suitable alternative to the random `Range` term.

    rkyld.asf <- asreml::asreml(yld ~ Type + lrow, random = ~ Genotype + Range, residual =
         ~ ar1(Range):ar1(Row), data = phenoRxK)

    ## Model fitted using the gamma parameterization.
    ## ASReml 4.1.0 Sun Aug 25 16:54:40 2024
    ##           LogLik        Sigma2     DF     wall    cpu
    ##  1       130.554      0.189858    513 16:54:40    0.0
    ##  2       182.276      0.113102    513 16:54:40    0.0
    ##  3       214.329      0.074662    513 16:54:40    0.0
    ##  4       223.385      0.060682    513 16:54:40    0.0
    ##  5       224.625      0.055965    513 16:54:40    0.0
    ##  6       224.654      0.055857    513 16:54:40    0.0
    ##  7       224.658      0.055957    513 16:54:40    0.0
    ##  8       224.658      0.056000    513 16:54:40    0.0

Users can diagnostically re-check this model to see model assumptions are more appropriately satisfied.

### Genetic linkage map

The wgaim package uses Karl Bromans qtl package (Broman et al., 2003) “cross” class objects to store and manipulate genetic data. The RAC875 x Kukri cross object can be accessed using

    data(genoRxK, package = "wgaim")

However, in this vignette we will read in the external CSV file using the qtl package function `read.cross()`. This function is a highly flexible importation function that handles many types of genetic marker data. It is advised to read the help file for this function thoroughly to understand the arguments you require to import your genetic data successfully. Noting the external genetic genoRxK data is in rotated CSV format, the importing occurs using `

    genoRxK <- read.cross(format = "csvr", file="genoRxK.csv", genotypes=c("AA","BB"),
         dir = wgpath, na.strings = c("-", "NA"))

    ##  --Read the following data:
    ##   368  individuals
    ##   500  markers
    ##   1  phenotypes
    ##  --Cross type: bc

The importation message indicates there are 500 markers. These are a combination of SSR and Diversity Array Technology (DArT) markers. The returned cross object is given a class `"bc"` (abbrev. for back-cross) by default. This can be changed to a `"dh"` class to match the population type, however, for this QTL analysis workflow the two classes are numerically equivalent.

    summary(genoRxK)

    ##     Backcross
    ##
    ##     No. individuals:    368
    ##
    ##     No. phenotypes:     1
    ##     Percent phenotyped: 100
    ##
    ##     No. chromosomes:    27
    ##         Autosomes:      1A 1B 1D 2A 2B 2D1 2D2 3A 3B 3D 4A 4B1 4B2 4D 5A1 5A2 5B 5D1 5D2 6A 6B
    ##                         6D 7A1 7A2 7B 7D1 7D2
    ##
    ##     Total markers:      500
    ##     No. markers:        44 37 26 22 23 3 13 24 57 21 22 16 3 6 7 5 19 2 3 21 32 8 13 34 21 11 7
    ##     Percent genotyped:  89.5
    ##     Genotypes (%):      AA:51.1  AB:48.9

    names(genoRxK$pheno)

    ## [1] "Genotype"

A quick summary of the object reveals the genotype data is a pre-constructed linkage map with 27 linkage groups and ~ 10% missing values. Additionally, note the object contains its own `pheno` element with a column named by `Genotype`. The contents of this column MUST match (at least in part) to the contents of the `Genotype` column in the phenotype data `phenoRxK` used in the fitting of the base model.

As the `genoRxK` cross object is a finalized linkage map, it is ready for conversion to an interval object for use in wgaim. This is achieved using the `cross2int()` function available in wgaim.

    genoRxKi <- cross2int(genoRxK, consensus.mark = TRUE, impute = "MartinezCurnow",
                          id = "Genotype")

By default, this function sequentially performs two very important tasks.

1. With `consensus.mark = TRUE` it will collapse each set of co-located markers to form unique consensus markers. As a consequence, each marker in the reduced linkage map will have a unique position.
2. Missing values are imputed using flanking marker information.

The returned `genoRxKi` object contains updated linkage group elements.

    names(genoRxKi$geno[[1]])

    ## [1] "data"          "map"           "imputed.data"  "dist"          "theta"
    ## [6] "interval.data"

The elements are:

- `data`: numerically encoded set of unique ordered markers (with missing values).
- `map`: genetic distances for the ordered set of unique markers
- `imputed.data`: numerically encoded set of unique ordered markers with missing values imputed
- `dist`: genetic distances between markers
- `theta`: recombination fractions between markers
- `interval.data`: numerically encoded set of unique ordered interval markers calculated using the derivations in Verbyla et al. (2007)

`genoRxKi` is also given an additional `"interval"` class and is now ready for marker or interval QTL analysis with the main wgaim analysis function.

Note, this vignette does not discuss the complex task of linkage map construction and diagnosis. For efficient construction of a linkage map ready for use with the functions in wgaim, we can highly recommend the combination of the qtl and ASMap R packages (Broman et al., 2003; Taylor & Butler, 2017). ASMap uses the very efficient and robust MSTmap algorithm discussed in Wu et al. (2008) to cluster and order markers. It also contains functionality for flexible pre/post construction map diagnostics as well as methods for incorporating additional markers in established linkage maps.

### QTL analysis

We now have a baseline phenotypic asreml model for grain yield and a matching linkage map containing a unique set of imputed markers. QTL analysis can then be conducted using the `wgaim` function. Before proceeding with the QTL analysis, and for the purpose of presentation in this vignette, it is prudent to discuss some of the relevant arguments that will be used in the `wgaim` call.

    rkyld.qtl <- wgaim(rkyld.asf, genoRxKi, merge.by = "Genotype", fix.lines = TRUE,
                        gen.type = "interval", method = "random", selection = "interval",
                        trace = "rxk.txt", na.action = asreml::na.method(x = "include"))

The first two arguments are the baseline phenotypic asreml model (`rkyld.asf`) and the matching genotypic data (`genoRxKi`). The phenotypic data is not required as it is internally recalled through from the baseline model. The other relevant arguments in this call are:

- `merge.by`: named column in the phenotype and genotype data for matching
- `fix.lines`: whether lines in the phenotype data not in the genotype data are fixed in the baseline and subsequent QTL models
- `gen.type`: whether an `"interval"` or a `"marker"` analysis is conducted
- `method`: whether selected putative QTL are additively fitted in the `"fixed"` or `"random"` components of the linear mixed model
- `selection`: whether `"interval"` or `"chromosome"` outlier statistics are inspected first

As `asreml` outputs optimisation numerics for each of the models, the `trace = "rxk.txt"` argument ensures this output is piped to a file for later inspection if required. As `gen.type = "interval"` has been set the wgaim algorithm will use the `"interval.data"` components of the linkage groups to form the complete set of genetic data for analysis. In this analysis, `fix.lines = TRUE` has been set and this will place a factor in the fixed model to fix the lines that do not exist in the genetic map. This new factor will now be partially confounded with the `Type` factor in the base model and a slew of messages will appear indicating some terms have zero degrees of freedom. Although harmless, these messages can be avoided by removing `Type` from the model and letting `fix.lines = TRUE`in the `wgaim` call handle the constraints.

    rkyld.asf <- asreml::update.asreml(rkyld.asf, fixed. = . ~ . - Type)

    ## Model fitted using the gamma parameterization.
    ## ASReml 4.1.0 Sun Aug 25 16:54:41 2024
    ##           LogLik        Sigma2     DF     wall    cpu
    ##  1       213.621     0.0588101    516 16:54:41    0.0
    ##  2       213.817     0.0581169    516 16:54:41    0.0
    ##  3       213.964     0.0573401    516 16:54:41    0.0
    ##  4       214.009     0.0568377    516 16:54:41    0.0
    ##  5       214.012     0.0569412    516 16:54:41    0.0
    ##  6       214.012     0.0569805    516 16:54:41    0.0

    rkyld.qtl <- wgaim(rkyld.asf, genoRxKi, merge.by = "Genotype", fix.lines = TRUE,
                        gen.type = "interval", method = "random", selection = "interval",
                        trace = "rxk.txt", na.action = asreml::na.method(x = "include"))

    ## Found QTL on chromosome 3B interval 30

    ## Found QTL on chromosome 4D interval 2

    ## Found QTL on chromosome 3B interval 10

    ## Found QTL on chromosome 3D interval 3

    ## Found QTL on chromosome 2B interval 21

    ## Found QTL on chromosome 1D interval 12

    ## Found QTL on chromosome 3B interval 2

    ## Found QTL on chromosome 2B interval 4

    ## Found QTL on chromosome 2A interval 8

    ## Found QTL on chromosome 4B1 interval 8

The iterative wgaim QTL analysis algorithm finds 10 putative QTL. Relevant diagnostic and summary information about the QTL are stored in the QTL element of the returned object ie `rkyld.qtl$QTL`. The returned object is also given a `"wgaim"` class.

    names(rkyld.qtl$QTL)

    ## [1] "selection"  "method"     "type"       "diag"       "iterations" "breakout"   "qtl"
    ## [8] "effects"    "veffects"

    names(rkyld.qtl$QTL$diag)

    ## [1] "oint"         "blups"        "lik"          "coef.list"    "vcoef.list"   "lik.mat"
    ## [7] "state"        "genetic.term" "rel.scale"

    class(rkyld.qtl)

    ## [1] "wgaim"  "asreml"

Method functions `summary` and `print` are available to conveniently summarize the significant QTL selected.

    summary(rkyld.qtl, genoRxKi)

    ##    Chromosome Left Marker dist(cM) Right Marker dist(cM)    Size   Prob % Var     LOD
    ## 1          1D    wPt-1799   128.29     wPt-1263   166.85 -0.0783 0.0015   2.7  1.9030
    ## 2          2A barc0220(C)    87.47      cfa2263    87.76  0.0629 0.0003   2.8  2.5099
    ## 3          2B    wPt-9644    25.24     wPt-5672    29.97 -0.0895 0.0000   5.3  3.6105
    ## 4          2B    wPt-3378   135.93     wPt-7360   136.11 -0.0790 0.0000   4.4  3.4345
    ## 5          3B    wPt-7984     6.65     barc0075        7  0.0648 0.0002   3.0  2.6647
    ## 6          3B     wmc0043    68.14  wPt-6973(C)    79.41  0.0859 0.0000   4.5  3.4353
    ## 7          3B    wPt-8021   244.67     gwm0114b   256.42 -0.1838 0.0000  19.8 14.4915
    ## 8          3D     cfd0064    53.42      cfd0034    61.54  0.1030 0.0000   6.7  5.6227
    ## 9         4B1    barc0114    48.32     wPt-0391    53.55 -0.0623 0.0009   2.7  2.1311
    ## 10         4D     wmc0457     6.56     barc0288     7.32  0.1012 0.0000   6.9  6.4987

At each iteration of the wgaim algorithm, the set of marker outlier statistics and scaled marker Best Linear Unbiased Predictions (BLUPs) are returned for diagnostic assessment. These can be viewed using the `outStat` function.

    outStat(rkyld.qtl, genoRxKi, iter = 1:2, statistic = "outlier")

plot of chunk out

    outStat(rkyld.qtl, genoRxKi, iter = 1:2, statistic = "blups")

plot of chunk out

There is also a simplistic linkage map plotting function that provides flexibility for overlaying the significant QTL obtained in `rkyld.qtl`.

    linkMap(rkyld.qtl, genoRxKi)

plot of chunk link

## References

Bonneau, J., Taylor, J.D., Parent, B., Reynolds, M., Feuillet, C., Langridge, P., et al. (2013) Multi-environment analysis and fine mapping of a yield related QTL on chromosome 3B of wheat. Theoretical and Applied Genetics, 126, 747–761.
Broman, K.W., Wu, H., Sen, S. & Churchill, G.A. (2003) R/qtl: QTL mapping in experimental crosses. Bioinformatics, 19, 889–890.
Gilmour, A.R. (2007) Mixed Model Regression Mapping for QTL Detection in Experimental Crosses. Computational Statistics and Data Analysis, 51, 3749–3764.
Taylor, J.D. & Butler, D. (2017) package : Efficient genetic linkage map construction and diagnosis. Journal of Statistical Software, 79.
Taylor, J. & Verbyla, A. (2011) R package wgaim: QTL analysis in bi-parental populations using linear mixed models. Journal of Statistical Software, 40.
Verbyla, A.P., Cullis, B.R. & Thompson, R. (2007) The analysis of QTL by simultaneous use of the of the full linkage map. Theoretical and Applied Genetics, 116, 95–111.
Verbyla, A.P., Taylor, J.D. & Verbyla, K.L. (2012) RWGAIM: An efficient high dimensional random whole genome average (QTL) interval mapping approach. Genetics Research, 94, 291–306.
Wu, Y., Bhat, P.R., Close, T.J. & Lonardi, S. (2008) Efficient and accurate construction of genetic linkage maps from the minimum spanning tree of a graph. PLoS genetics, 4, e1000212.