Modeling field trials using statgenSTA

Bart-Jan van Rossum

2020-01-09

1 The statgenSTA Package

The statgenSTA (Single Site Analysis) package is developed as an easy-to-use package for analyzing data of plant breeding experiments with many options for plotting and reporting the results of the analyses.

This vignette describes in detail how to prepare data for analysis, perform analyses using different modeling engines and extract the results from the models.

2 Data preparation

The first step when modeling field trial data with the statgenSTA package is creating an object of class TD (Trial Data). This object is used throughout the statgenSTA package as input for analyses.

2.1 Creating a TD object

A TD object can be created from a data.frame with the function createTD. This function does a number of things:

After creating a TD object, data for new trials can be added to it using addTD. This function works in exactly the same way as createTD except that it adds data to an existing TD object instead of creating a new one.
Dropping one or more trials from a TD object can be done using the function dropTD.

2.2 Example

Field data from an experiment with wheat in Chile described in detail by Lado et al. (2013) will be used as an example throughout this vignette. The experiment was performed in two locations in Chile, Santa Rosa and Cauquenes, with two different drought regimes in 2011 and 2012 for the Santa Rosa and one trial in 2012 for Cauquenes. For 384 genotypes four traits were measured in 2011, but in 2012 only grain yield (GY) was measured and the examples will focus on this trait.

For the example, first a TD object is created for the first location only. The data for the second location is then added later on. In practice all this could be done in one go.

The TD object just created is a list with four items, one for each trial (combination of location, drought regime and year) in the original data.frame. The column “trt” in the original data is renamed to “genotype” and converted to a factor. The columns “rep” and “bl” are renamed and converted likewise. The columns “row” and “col” are renamed to “rowCoord” and “colCoord” respectively. Simultaneously two columns “rowId” and “colId” are created containing the same information converted to a factor. This seemingly duplicate information is needed for spatial analysis. It is possible to define different columns as “rowId” and “colId” than the ones used as “rowCoord” and “colCoord”. The information about which columns have been renamed when creating a TD object is stored as an attribute of each individual data.frame in the object.

2.3 Meta data

The meta data will be a data.frame with four rows, one for each trial in wheatTD. The data.frame has the following columns:

Content Usage
trLocation Location Default names for plots and reports
trDate Date
trDesign Design Model specification (see 3.1)
trLat Latitude Position in a map plot (see 2.6.2)
trLong Longitude Position in a map plot (see 2.6.2)
trPlWidth Plot width Ratio of width/height in a layout plot (see 2.6.1)
trPlLength Plot length Ratio of width/height in a layout plot (see 2.6.1)

After extracting the meta data, it can be modified and then added back to the original TD object.

2.4 Add extra data to a TD object

To add the data for the final trial to the TD object the function addTD can be used. Since now only one new trial is added, it makes sense to immediately add the meta data for this trial as well, using the appropriate parameters in addTD.

2.5 Summarizing a TD object

The summary function can be used to get an idea of the content of the data in the TD object. Multiple traits can be summarized at once but for clarity here a summary is only made for GY.

Using the default options, nine summary statistics are printed, but many more are available. These can be accessed using the parameter what in the summary function. For a full list of available statistics, use help(summary.TD). It is also possible to output all statistics using what = "all".

It is possible to summarize the data in a TD object for different groups. This can be done using the parameter groupBy. It will display three main summary statistics per group. Again, more statistics can be displayed using the parameter what.

2.6 Plotting a TD object

Several plots can be made to further investigate the contents of a TD object.

2.6.1 Layout plot

The default plot creates plots for the layout of all trials in the TD object. This can be restricted to selected trials using the trials parameter. The width and length of the plot are derived from “trPlWidth” and “trPlLength” in the meta data if these are available.

This plot can be extended by highlighting interesting genotypes in the layout.

It is also possible to color the subBlocks within the field.

Highlighting genotypes and coloring subBlocks cannot be done simultaneously. If both options are specified, only highlighting is done.

Finally, it is possible to add the names of the genotypes to the layout.

2.6.2 Map plot

A second type of plot displays the trial locations on a map. This plot is made based on trLat and trLong in the meta data. If latitude or longitude is not available for a certain location, then this location is not plotted. If the locations are very close together, the resulting map can become quite small. The parameters minLatRange and minLongRange can be used to extend the minimum range of latitude and longitude respectively to address this issue.

2.6.3 Box plot

Boxplots can be made to get an idea of the contents of the data in the TD object. By default a box is plotted per trial in the data for the specified traits. Boxplots for multiple traits can be made at once.

The trials in the boxplot can be grouped using the parameter groupBy. Colors can be applied to groups within trials using the parameter colorBy. The boxes for the (groups of) trials can be ordered using orderBy. Boxes can be ordered by an external vector or by the group mean.

2.6.4 Correlation plot

The last plot that can be made is a plot of the correlations between the fields for a specified trait. The order of the plotted fields is determined by clustering them and plotting closely related field close to each other.

3 Modeling

After creating a TD object, a model can be fitted on the trial data. This is done using the function fitTD. The exact model fitted depends on the design of the trial (see 3.1). The design can be given as a parameter in the function or included in the meta data of the TD object as described in 2.3. In the former case, the same model will be fitted for all trials, in the latter, different models can be fitted for different trials. If both are available the function parameter will always be leading.

The output of fitTD is an object of class STA (Single Site Analysis), a list of fitted models with one item for each trial the model was fitted for.

3.1 Model types

fitTD uses three different engines for fitting the models, namely SpATS (Rodríguez-Álvarez et al. 2017), lme4 (Bates et al. 2015) and asreml (Butler et al. 2017). For models with row column or resolvable row column design, SpATS is the default engine, for the other models lme4. This can always be overruled by specifying the function parameter engine.

Models can be fitted for five different trial designs. These are listed in the following table with their respective model specifications.

design code model fitted
incomplete block design ibd trait = genotype + subBlock + \(\epsilon\)
resovable incomplete block design res.ibd trait = genotype + repId + repId:subBlock + \(\epsilon\)
randomized complete block design rcbd trait = genotype + repId + \(\epsilon\)
row column design rowcol trait = genotype + rowId + colId + \(\epsilon\)
resolvable row column design res.rowcol trait = genotype + repId + repId:rowId + repId:colId + \(\epsilon\)

In the models above, fixed effects are indicated in italics whereas random effects are indicated in bold. genotype can be fitted as fixed or as random effect depending on the value of the parameter what. Extra fixed effects may be fitted using the parameter covariates.
If SpATS is used as modeling engine, an extra spatial term is always included in the model (see 3.2). A spatial term is also included when the modeling engine is asreml and the function parameter spatial is set to TRUE.

Using the TD object wheatTD from the previous section, a model for the trial SR_FI_11 and trait GY can now be fitted on the data. The trial was set up using a resolvable row column design. This is specified in fitTD using the parameter design. Since engine is not supplied as a parameter, SpATS is used for fitting the model.

Note that by not supplying the what argument to the function, two models are fitted, one with genotype as a fixed effect and one with genotype as a random effect. The results of both these models are stored in the STA object modWheatSp. This is very useful for extracting different results from the model later on. A trade-off is that fitting two models takes more time than fitting only one so when fitting models on large datasets it is sensible to explicitly define what if only a subset of the results is needed as output.

3.2 Spatial models

When using SpATS as a modeling engine for fitting a model, an extra spatial component is always included in the model. This spatial component is composed using the PSANOVA function in the SpATS package which uses 2-dimensional smoothing with P-splines as described in Lee, Durbán, and Eilers (2013) and in Rodríguez-Álvarez et al. (2017). See help(PSANOVA, SpATS) for a detailed description. The parameters nseg and nest.div of PSANOVA can be modified using the control parameter in fitTD.
Fitting the model in the previous section specifying the number of segments for both rows and columns as 20 works as follows

Alternatively, spatial models can be fitted using asreml as modeling engine and setting the parameter spatial = TRUE. In this case seven models are fitted and the best model, based on a goodness-of-fit criterion, either AIC or BIC, is chosen. For a full specification of the models fitted see help(fitTD). The criterion to be used can be specified using the control parameter in fitTD.
Fitting a model similar to the one above using asreml with BIC as goodness-of-fit criterion works as follows

The fitted models and the best model are stored in the output together with a summary table with details on the fitted models.

As the overview shows, the best model, the model with the lowest BIC, is AR1(x)AR1 with units in the random part of the model.

3.3 Model summary

Since genotype has been fitted both as fixed and as random factor in modWheatSp it is possible to calculate both the Best Linear Unbiased Estimators (BLUEs) and the Best Linear Unbiased Predictors (BLUPs). Therefore both are printed in the summary of the model together with their respective standard errors.

3.4 Model plots

Two types of plots can be made for fitted models.

3.4.1 Base plots

The first is a series of four plots, a histogram of the residuals, normal quantiles of the residuals, a scatter plot of residuals against fitted values and a scatter plot of absolute value of residuals against fitted values. These plots can be made by calling plot on the STA object. Plots can be made for multiple trials and multiple traits simultaneously, either for the model with genotype as fixed effect or for the model with genotype as random effect. By default plots are made for all trials and all traits, but this can be restricted using the parameters trials and traits. If only one model is fitted the results of the fitted model will be plotted. In case both models were fitted, as a default the results will be plotted for the model with genotype fixed. This can be changed using the parameter what.

3.4.2 Spatial plots

The second type of plot consists of five plots, spatial plots of the raw data, fitted values, residuals and either BLUEs or BLUPs, and a histogram of the BLUEs or BLUPs. If SpATS was used for modeling an extra plot with the fitted spatial trend is included. The spatial trend can be displayed on the original scale (spaTrend = "raw") or as percentega (spaTrend = "percentage"). In the latter case the spatial trend is scaled (i.e., divided by the average of the observed response variable of interest across the field) and displayed as percentage. The call for creating these plots differs from the base plots only by an extra parameter plotType = "spatial". Note that spatial plots can only be made if spatial information, i.e. rowCoord and colCoord, is available in the TD object.

3.5 Outlier detection

After fitting a model, it is possible to run an outlier detection on the results. This outlier detection che cks the residuals in the fitted model and compares them to a limit. Observations with a residual higher than this limit are marked as outliers. The default limit is calculated based on the number of observations in the data. Setting a custom limit is also possible.

The output of the outlier detection function is a data.frame with the outliers and of vector of indices of the outliers in the original data. This vector can be used for removing outliers from the data.

It is possible to run outlier detection on a model with genotype fitted as fixed, but when doing so it is only possible to detect outliers in genotypes that are replicated in the data. For unreplicated genotypes the residual will always be 0 and so these will never be marked as outliers.

To get an idea of the magnitude of the outliers, the parameter commonFactors can be used to see the values of observations with similar characteristics. In the example below, for all outliers also the other observations for the same genotype are shown.

3.6 Model reports

For STA objects there is a report function available in the statgenSTA package. This function creates a pdf report summarizing the main results of the fitted model. Also the tex file and figures used for generating the pdf report are saved. By editing the tex file it is possible to modify the report to ones needs, creating high flexibility.

When no outfile is specified, reports will be created with a default names, e.g. “modelReport_trial_trait_fixed_timestamp.pdf”, in the current working directory. The parameter outfile can be used to change the name and location of the report. The value of this parameter should be a valid location and name for a pdf file, i.e. including the postfix “.pdf”. Non-existing directories are created by the report function. When an outfile is supplied, trial, trait and modeltype (fixed or random) are concatenated to it in the name of the report.

The reports contains general information on the model fitted, a summary of the results, the plots described in the previous section, a list of best (highest BLUEs or BLUPs) genotypes and a scatter plot of all genotypes and their BLUEs or BLUPs. For some traits a low value might mean a genotype is performing well. To correctly show the best genotypes in the report in this case, set the parameter descending to TRUE in the report function.

Reporting for an STA object can be done for multiple trials, traits and modeltypes simultaneously. For each combination of trial, trait and modeltype, a separate .pdf report will be created. Parameters traits, trials and what can be used for specifying the models for which the reports should be made.

4 Extracting model results

After fitting a model, various results can be extracted or calculated from the fitted model using the function extract. This can be anything from a single result for one trait and one trial to a list of different results for all models in an STA object. The results that can be extracted depend on the type of model fitted and sometimes on the modeling engine as well. For example, BLUEs can only extracted if genotype was fitted as fixed effect. On the other hand, BLUPs and heritabilities can only be calculated and extracted if genotype was fitted as random effect.

All results that can be extracted are shown in the table below. In the first column is the result. This is also the value to be used for the parameter what in extract needed to extract the corresponding result. The second column shows what model needs to be fitted in order to be able to extract the result. Here F stands for genotype as fixed effect and R for genotype as random effect. The last column gives a short description of the result that will be extracted and, where needed, also states for which modeling engines it can be extracted.

result model description
BLUEs F Best Linear Unbiased Estimators
seBLUEs F standard errors of the BLUEs
ue F unit errors - only for lme4 and asreml
varCompF F variance components for the model with genotype as fixed component
fitted F fitted values for the model with genotype as fixed component
residF F residuals for the model with genotype as fixed component
stdResF F standardized residuals for the model with genotype as fixed component
wald F results of the wald test - only for lme4 and asreml
CV F Coefficient of Variation - only for lme4 and asreml
rDfF F residual degrees of freedom for the model with genotype as fixed component
sed F standard error of difference - only for asreml
lsd F least significant difference - only for asreml
BLUPs R Best Linear Unbiased Predictors
seBLUPs R standard errors of the BLUPs
heritability R broad sense heritability
varCompR R variance components for the model with genotype as random component
varGen R genetic variance component
varErr R residual variance component - only for lme4 and asreml
varSpat R spatial variance components - only for SpATS
rMeans R fitted values for the model with genotype as random component
ranEf R random genetic effects
residR R residuals for the model with genotype as random component
stdResR R standardized residuals for the model with genotype as random component
rDfR R residual degrees of freedom for the model with genotype as random component
effDim R effective dimensions - only for SpATS
ratEffDim R ratios of the effective dimensions - only for SpATS

Using what = "all" in the function call, extracts all results possible for the fitted model. This is also the default.

Below are some examples of extracting results from a fitted model. Recall that modWheatSp contains two fitted models, one with genotype as fixed effect and one with genotype as random effect.

## Extract BLUEs
BLUEsWheat <- extract(STA = modWheatSp, what = "BLUEs")
## Extract BLUEs and BLUPs
predWheat <- extract(STA = modWheatSp, what = c("BLUEs", "BLUPs"))

Both BLUEsWheat and predWheat are lists with one item, the trial used for modeling. In general, when extracting results, there will be an item in the list for every trial for which results were extracted. These items are a list themselves with an item for every statistic that has been extracted, so one item for BLUEsWheat, a data.frame containing the BLUEs, and two for predWheat, the data.frames containing BLUEs and BLUPs respectively.

4.1 Adding extra variables

The data.frame BLUEs in either of the lists consists of only two columns, genotype and GY. If the model would have been fitted for multiple traits all these traits would become columns in the data.frame. It might be useful to add extra columns from the data used to fit the model to the output. This can be achieved using the paramater keep in extract. To include the trial in the output, useful for easily combining several data.frames with BLUEs and using them for a GxE analysis, use the following command

Not every column from the original TD object can be included in the extracted data in this way. Only columns that, for each genotype, have only one value per trial can be included. For example, the column repId containing replicates, that has several different values for a single genotype within each trial, cannot be included. When trying to do so it will be ignored with a warning.
It is however possible to include repId when extracting fitted values, since for each observation in the original data a fitted value is computed.

4.2 Prepare data for GxE analysis

To use the BLUEs or BLUPs from the fitted model in a GxE analysis they have to be converted again into a TD object. For this the function STAtoTD can be used. It creates a TD object from a fitted model outputting one or more of the following: BLUEs, standard errors of BLUEs, BLUPs and standard errors of BLUPs. Optionally, a column wt with weights (calculated as \((1 / seBLUEs)^2\)) can be added as well. In the same way as described in the previous section extra columns can be added to the output using the parameter keep.

Performing GxE analysis on the TD object can be done using the statgenGxE package.

References

Bates, Douglas, Martin Mächler, Ben Bolker, and Steve Walker. 2015. “Fitting Linear Mixed-Effects Models Using Lme4.” Journal of Statistical Software 67 (1). https://doi.org/10.18637/jss.v067.i01.

Butler, D. G., B. R. Cullis, A. R. Gilmour, B. G. Gogel, and R. Thompson. 2017. “ASReml-R Reference Manual Version 4.” https://www.vsni.co.uk/.

Lado, Bettina, Ivan Matus, Alejandra Rodríguez, Luis Inostroza, Jesse Poland, François Belzile, Alejandro del Pozo, Martín Quincke, Marina Castro, and Jarislav von Zitzewitz. 2013. “Increased Genomic Prediction Accuracy in Wheat Breeding Through Spatial Adjustment of Field Trial Data.” G3: Genes\(\vert\)Genomes\(\vert\)Genetics 3 (12): 2105–14. https://doi.org/10.1534/g3.113.007807.

Lee, Dae-Jin, María Durbán, and Paul Eilers. 2013. “Efficient Two-Dimensional Smoothing with P-Spline Anova Mixed Models and Nested Bases.” Computational Statistics & Data Analysis 61 (May): 22–37. https://doi.org/10.1016/j.csda.2012.11.013.

Rodríguez-Álvarez, María, Martin P. Boer, Fred van Eeuwijk, and Paul H. C. Eilers. 2017. “Correcting for Spatial Heterogeneity in Plant Breeding Experiments with P-Splines.” Spatial Statistics 23 (October): 52–71. https://doi.org/10.1016/j.spasta.2017.10.003.