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**|**BMC Proc**|**v.4(Suppl 1); 2010**|**PMC2857850

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BMC Proc. 2010; 4(Suppl 1): S8.

Published online 2010 March 31. doi: 10.1186/1753-6561-4-S1-S8

PMCID: PMC2857850

Torben Schulz-Streeck: ed.miehnehoh-inu@kceerts-zluhcs.nebrot; Hans-Peter Piepho: ed.miehnehoh-inu@ohpeip

Proceedings of the 13th European workshop on QTL mapping and marker assisted selection

Marco Bink, John Bastiaansen, Mario Calus and Chris Maliepaard

Publication of this supplement was supported by EADGENE (European Animal Disease Genomics Network of Excellence) and the LEB Foundation

http://www.biomedcentral.com/content/pdf/1753-6561-4-S1-info.pdf13th European workshop on QTL mapping and marker assisted selection,

20–21 April 2009

Wageningen, The Netherlands

Copyright ©2010 Piepho and Schulz-Streeck; licensee BioMed Central Ltd.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This article has been cited by other articles in PMC.

The success of genome-wide selection (GS) approaches will depend crucially on the availability of efficient and easy-to-use computational tools. Therefore, approaches that can be implemented using mixed models hold particular promise and deserve detailed study. A particular class of mixed models suitable for GS is given by geostatistical mixed models, when genetic distance is treated analogously to spatial distance in geostatistics.

We consider various spatial mixed models for use in GS. The analyses presented for the QTL-MAS 2009 dataset pay particular attention to the modelling of residual errors as well as of polygenetic effects.

It is shown that geostatistical models are viable alternatives to ridge regression, one of the common approaches to GS. Correlations between genome-wide estimated breeding values and true breeding values were between 0.879 and 0.889. In the example considered, we did not find a large effect of the residual error variance modelling, largely because error variances were very small. A variance components model reflecting the pedigree of the crosses did not provide an improved fit.

We conclude that geostatistical models deserve further study as a tool to GS that is easily implemented in a mixed model package.

Genome-wide selection (GS) is a marker-based method that predicts breeding values on the basis of a large number of molecular markers, which typically cover the entire genome [1]. The idea is to estimate the effects of all genes or chromosomal segments simultaneously and to integrate these estimates in order to predict the total breeding value.

One basic approach for GS is ridge regression (RR) [1]. An interesting alternative to RR is to use spatial models [2] to model genetic correlation among relatives [3].

This study compares RR models and spatial models for estimating genome-wide breeding values for the common dataset provided by the 13^{th} QTL-MAS workshop. The focus is on methods that can be easily implemented using a standard mixed model package with facilities for spatial covariance structures.

The dataset was simulated as part of the 13^{th} QTL-MAS workshop (see [4] for details). Phenotypes of 1000 of 2025 individuals were recorded at five different times (0, 132, 265, 397 and 530), so there is a series of five repeated measurements for each phenotyped individual. Breeding values for the non-phenotyped individuals were to be predicted for time=600, which constitutes an extrapolation.

Careful inspection of the data revealed that a logistic model

would give a reasonable fit to the data, where* y*_{it} is the trait value of the *i*-th individual at time* t *(*t* = 0, 132, 256, 397, 530 or 600) and *α _{i}*,

The marker covariate* z _{ik}* for the

The approach to GS closely follows [2]. Our basic model was

where* µ* is an intercept,* h _{i}* is the total genotypic effect of the

[2]. There were two options regarding the model for var(*e _{i}*), the variance of

We considered different models for the variance of* g*' = (

for some matrix **Γ** that is a function of* Z* and
is a variance component. The models that were used are identical to those used in [2]. Under the mixed model for RR the matrix

where* d _{ii'}* is the Euclidean distance of genotypes

Genotypic covariance models of the form **Γ** = {*f*(*d*_{ii'})}, where* d* is the Euclidean distance computed from marker data and θ is a parameter.

We also considered an extended model , where **Ω** represents the covariance due to simple random effects, i.e., , where , and are the variance components for random effects of father and mother of crosses and of the crosses themselves, respectively, and** V_{f} , V_{m }** and

For each fitted model we obtained BLUPs of *µ* + *h _{i}* corresponding to genome-wide estimated breeding values (GEBV). For non-phenotyped individuals in the case of models with independent polygenic effects and for the extended model when

After the 13^{th} QTL-MAS workshop the organizers reported the true breeding values (TBV). The TBV of the non-phenotyped individuals were compared to the GEBV by the Pearson correlation.

The nonlinear regression was done by the NLMIXED procedure of the SAS System, while all mixed models were fitted by the REML method using the MIXED procedure of SAS.

First the models were fitted without fixing the residual variance (Table (Table2).2). The RR and spatial methods give better fits than the model with independent genotypic effects. The AIC values of the RR and the spatial models are relatively close. The spatial linear, power, exponential and spherical models have a smaller residual variance than the RR/quadratic spatial model and Gaussian spatial model. The latter models show a higher correlation of GEBV with TBV (Table (Table2).2). Overall, the RR/quadratic model has the best AIC value, but this model shows a relatively low correlation of GEBV and *y _{i}*

Model fits of different genetic covariance models and Pearson correlation between GEBV and fitted value and between GEBV and true breeding value (TBV).

The fits from the extended models that include **Ω** are shown in Table Table3.3. The AIC values are a little bit higher than in the models without considering the effects of the parents, the independent model being an exception. The ranking of genotypes remains unaltered. Only when do we find a non-zero variance for mother effects in **Ω** . Throughout, there is a non-zero estimate for the variance of father effects (), while the variance for cross effects () is zero. The correlations of GEBV with TBV are almost the same as those when **Ω** was omitted (Table (Table33).

The results of the models with a fixed residual variance at the squared standard errors of predictions of *y _{i}*

There are only minor differences of the AIC values between RR and spatial models, like in [2]. Thus, some of the spatial models are viable alternatives to RR. Among the spatial models, the Gaussian model gave almost the same fit as RR. This can be explained by a Taylor expansion argument. The correlation function under the Gaussian model is exp(– *d*^{2} /* θ ^{2}*). When

In this study we have used AIC defined as AIC = -2 log(*likelihood*) + 2 * *number of parameters* as printed by mixed model packages such as the MIXED procedure of SAS. The models with the lowest AIC values showed the highest correlation between the GEBV and TBV. But the correlation between the model ranks produced by AIC and by the correlation between GEBV and TBV is not perfect. For smoothing methods modifications of the AIC have been proposed (e.g. corrected AIC (AIC_{C}: [9]) and different other criteria (e.g. generalized cross-validation (GCV: [10]). The main difference to the AIC is that the complexity of the fitted model is calculated as the trace of the so-called smoother matrix tr(* S_{λ}*) described in [11], which relates to the effective degrees of freedom of the fit. It is important to realize that GS may be regarded as a smoothing exercise that replaces observed data (adjusted genotype means) by smoothed fitted values. Thus, model selection criteria developed for smoothing can be a useful extension for selecting a preferable model in GS.

The comparison between GEBV and phenotypes is not a good indictor for accuracy of breeding values, when all individuals are involved in the prediction and no independent validation set is left (Tables (Tables2,2, ,33 and and4).4). Cross-validation is one option to avoid this problem. The leave-one-out cross-validation procedure is equivalent to the cross-validation criterion, which is related to other selection criteria (AIC, AIC_{C} and GCV) [12]. Therefore one idea is to replace the cross-validation procedure by model selection criteria, which would entail a considerable saving of computing time.

In the present study the data were simulated without polygenic effects. Nevertheless, it is prudent to cater for the case that the total genotypic variance can not be fully explained by the markers alone. We think that this unexplained part should be modelled by a polygenic effect.

We modelled the polygenic effect as independent. Alternatively, one can assume that the polygenic effect is correlated due to the pedigree. We modelled the pedigree of the crosses by variance components, however, reflecting the pedigree did not provide an improved fit in the present case.

We also think it is important to separate the polygenic effect from error in order to avoid overfitting [2]. In the present study, however, fixing the residual variance did not have much of an effect because essentially we only had a within-individual error variance estimate. This ignored between-individual error variance, which is therefore expected to be confounded with the variance component for polygenic effects (). In plant breeding trials, where replication is available, one can separate polygenic effects from non-genetic between-individual errors. We expect that such separation will be crucial to the success of GS approaches in plant breeding.

Our study has shown that geostatistical models are viable alternatives to RR that deserve further study as a tool to GS. With respect to our analyses for the QTL-MAS 2009 dataset, however, we prefer the RR/quadratic model without fixed residual variance for predicting GEBV.

The authors declare no competing interests.

TSS participated in the design of the study, performed all analyses and drafted the paper. HPP conceived the study, and participated in its design, and helped in the final editing of the paper.

Two anonymous reviewers are thanked for constructive comments.

This article has been published as part of BMC Proceedings Volume 4 Supplement 1, 2009: Proceedings of 13th European workshop on QTL mapping and marker assisted selection.

The full contents of the supplement are available online at http://www.biomedcentral.com/1753-6561/4?issue=S1.

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