Volume 6 Supplement 2
Proceedings of the 15th European workshop on QTL mapping and marker assisted selection (QTLMAS)
Linear models for breeding values prediction in haplotypeassisted selection  an analysis of QTLMAS Workshop 2011 Data
 Anna Mucha†^{1}Email author and
 Heliodor Wierzbicki†^{1}
DOI: 10.1186/175365616S2S11
© Mucha and Wierzbicki; licensee BioMed Central Ltd. 2012
Published: 21 May 2012
Abstract
Background
The aim of this study was to estimate haplotype effects and then to predict breeding values using linear models. The haplotype based analysis enables avoidance of loosing information due to linkage disequilibrium between single markers. There are also less explanatory variables in the linear model which makes the estimation more reliable.
Methods
Different methods and criteria for marker and haplotype selection were considered. First, markers with MAF lower than 5% where excluded from the data set. Then, SNPs in complete linkage disequilibrium where selected. Next step was to construct haplotypes and to estimate their frequencies basing on selected SNPs. The haplotypes with a frequency lower than 1% were not considered in further analysis. Chosen haplotypes were used as the explanatory variables in the linear models for breeding values prediction. Linear models with fixed and random haplotype effects as well as animal model were tested.
Results
The number of markers was limited to 1206, 1189, 1249, 1288 and 1167 for chromosome 1, 2, 3, 4 and 5, respectively due to MAF criterion. In total 409 subsets of SNPs with r^{2}=1 were found. 1476 haplotypes with different lengths were inferred. The frequencies of 817 haplotypes were higher than 1%  184 for the first chromosome, 172 for the second, 131 for the third, 146 for the forth and 184 haplotypes for the fifth chromosome. The haplotype effects estimated using random models were comparable and more precise in prediction for individuals with unknown phenotypes. A few haplotypes with large effects were found when their effects were defined as fixed in the linear model . The correlations of the predicted breeding values with true breeding values were not that high. This could be brought about by selection criteria imposed on the genotype data which led to substantial reduction of number of markers.
Conclusions
Although not many markers were considered in the study, the results obtained show that the implemented approach can be considered as quite promising. The haplotype approach let to avoid high dimensional models as compared with single SNPs models.
Background
Single Nucleotide Polymorphisms (SNPs) are the most widely used genetic markers for breeding value prediction [1]. Nonetheless, each SNP has relatively low content of genetic information. The haplotype approach gives a possibility to accumulate genetic information in haplotype blocks and to keep the Linkage Disequilibrium (LD) information in the statistical model [2]. Thus, the haplotypeassisted selection can be a very powerful tool in animal breeding [3].
Methods
The QTL MAS 2011 simulated dataset was analysed to predict breeding values of individuals with known (2000 observations) and unknown (1000 observations) phenotypes. Genotype data were selected according to three criteria. Markers with Minor Allele Frequency (MAF) lower than 5% were excluded from the dataset. Then, LD between markers was measured using r^{2}. SNPs in complete LD with at least one other SNP were picked out for further analysis. Basing on subsets of closely linked markers (MAF>5%, r^{2}=1), haplotypes were constructed. Bayesian algorithm implemented in PHASE was used for haplotypes construction and for their frequencies estimation [4]. Haplotypes with population frequency lower than 1% were omitted in further analysis [5]. Inferred haplotype effects were estimated using statistical models for breeding values prediction. Four statistical models were considered. Fixed model (FM) handled haplotypes effects as fixed. The fitted model was the following: y = 1_{ n }μ_{1}+Xg_{1}+e_{1}, where y is a vector of phenotypes, 1_{ n } is a vector of ones, n is number of known phenotypes, μ_{1} is an overall mean, X is a design matrix of haplotype effects, g_{1} is a vector of fixed haplotype effects, e_{1} is a vector of random residual effects and ${e}_{1}~N\left(0,{\sigma}_{e1}^{2}\right)$. Two random models (RM1 and RM2) treated haplotype effects as random. RM1 was the following: y=1_{ n }μ_{2}+Xg_{2}+e_{2}, where y,1_{ n }, n, μ_{2}, X are defined analogically as above, g_{2} is a vector of random haplotype effects and ${g}_{2}~N\left(0,\frac{{\sigma}_{g2}^{2}}{\#haplotypes}\right)$, e_{2} is a vector of random residual effects and ${e}_{2}~N\left(0,{\sigma}_{e2}^{2}\right)$. RM2 was the following: y=1_{ n }μ_{3}+Xg_{3}+e_{3}, where y,1_{ n },n, μ_{3}, X are defined analogically as above, g_{3} is a vector of random haplotype effects and ${g}_{3}~N\left(0,{\sigma}_{g3}^{2}\frac{haplotypelength}{\#alleles}\right)$, e_{3} is a vector of random residual effects and ${e}_{3}~N\left(0,{\sigma}_{e3}^{2}\right)$. In RM1 the homogeneous variance whatever haplotype length, and in RM2 the heterogeneous variance depending of the haplotype length was assumed. Animal model (AM) was also fitted to the data to predict breeding values and to compare results obtained with previous models. AM was defined as follows: y=1_{ n }μ+Zg+e, where y,1_{ n }, n,μ are defined as in previous models, Z is a design matrix of random additive polygenic effects, g is a vector of random additive polygenic effects and $g~N\left(0,A{\sigma}_{g}^{2}\right)$, A is the numerator relationship matrix,e is a vector of random residual effects and $e~N\left(0,{\sigma}_{e}^{2}\right)$. The breeding values for individual j estimated using FM, RM1 and RM2 were defined as a sum of haplotype effects of the individual. The results of considered models were compared using the Pearson's correlation coefficients. All computations were performed using Rpackage.
Results
MAF and LD reduction
MAF and LD reduction
chromosome:  1  2  3  4  5 

all markers  1998  1998  1998  1998  1998 
markers with MAF>0.05  1206  1189  1249  1288  1167 
markers with MAF>0.05 and r^{2}=1  211  201  150  166  216 
Subsets of SNPs after MAF and LD reduction
chromosome  subset of SNPs  

all  2SNP  3SNP  4SNP  5SNP  6SNP  7SNP  8SNP  
1  92  75  12  3  1      1 
2  87  67  16  3      1   
3  65  52  8  3  2       
4  73  57  12  4         
5  92  71  14  4  2  1     
TOTAL  409  322  62  17  5  1  1  1 
Reduction by haplotype frequencies
Number of haplotypes according to chromosome, haplotype length and frequency
Haplotype length  subset  chromosome  TOTAL  

1  2  3  4  5  
all  all  328  309  240  262  337  1476 
freq>1%  184  172  131  146  184  817  
2  all  232  215  173  187  235  1042 
freq>1%  150  134  104  114  142  644  
3  all  56  70  35  49  44  254 
freq>1%  24  30  17  24  28  123  
4  all  18  14  16  26  24  98 
freq>1%  6  6  6  8  8  34  
5  all  8    16    22  46 
freq>1%  2    4    4  10  
6  all          12  12 
freq>1%          2  2  
7  all    10        10 
freq>1%    2        2  
8  all  14          14 
freq>1%  2          2 
Breeding values prediction
Correlations between true and predicted phenotypes
MODEL  true  FM  RM1  RM2  AM 

true  1  0.4873 (0.4385, 0.5331)  0.7043 (0.6716, 0.7342)  0.7052 (0.6726, 0.7351)  0.6081 (0.5675, 0.6458) 
FM  0.7145 (0.6924, 0.7353)  1  0.5396 (0.4942, 0.5821)  0.5443 (0.4991, 0.5865)  0.4306 (0.3787, 0.4797) 
RM1  0.4872 (0.4531, 0.5200)  0.6819 (0.6577, 0.7047)  1  0.9972 (0.9968, 0.9975)  0.7631 (0.7360, 0.7879 
RM2  0.4911 (0.4571, 0.5236)  0.6873 (0.6635, 0.7097)  0.9974 (0.9972, 0.9976)  1  0.7616 (0.7342, 0.7864) 
AM  0.7315 (0.7105, 0.7513)  0.7174 (0.6955, 0.7381)  0.7921 (0.7751, 0.8078)  0.7939 (0.7771, 0.8096)  1 
Discussion
The MAF and LD reduction results were comparable and there were not substantial differences between chromosomes. The haplotypes consisted of 2 alleles were predominant. The longest haplotype length was 8 alleles. The longer haplotype, the lower was its frequency and the less haplotypes fulfilled the threshold of 1%. A few haplotypes with large effects were found using the fixed model. The negligible differences between results obtained using RM1 and RM2 were probably caused by small disparities between haplotype lengths (from 2 to 8 alleles). Regardless of heterogeneous (RM2) or homogeneous (RM1) variance assumption, the breeding values prediction results were comparable. FM and AM gave better results for the individuals with known phenotypes, whereas RM1 and RM2 were more precise in prediction for individuals with unknown phenotypes. The correlations of the predicted breeding values with true breeding values were not high and ranged from 0.4872 to 0.7315. This could be brought about by selection criteria imposed on the genotype data which led to substantial reduction of number of markers.
Conclusions
Although not many markers were considered in the study (outcome of complete LD as a marker selection criterion), the results obtained show that the implemented approach can be considered as quite promising. The random models (RM1 and RM2) gave highly comparable results, more precise for individuals with unknown phenotypes. The haplotype approach let to avoid high dimensional models as compared with single SNPs models.
Notes
List of abbreviations used
 SNP:

Single Nucleotide Polymorphisms
 LD:

Linkage Disequilibrium
 MAF:

Minor Allele Frequency
 FM:

Fixed model
 RM:

Random models
 AM:

Animal model.
Declarations
Acknowledgements
The QTLMAS Workshop 2011 organizers are acknowledged for simulating the dataset and providing true phenotype values.
This article has been published as part of BMC Proceedings Volume 6 Supplement 2, 2012: Proceedings of the 15th European workshop on QTL mapping and marker assisted selection (QTLMAS). The full contents of the supplement are available online at http://www.biomedcentral.com/bmcproc/supplements/6/S2.
Authors’ Affiliations
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