Impact of imputation methods on the amount of genetic variation captured by a single-nucleotide polymorphism panel in soybeans
© Xavier et al. 2016
Received: 8 October 2015
Accepted: 19 January 2016
Published: 2 February 2016
Success in genome-wide association studies and marker-assisted selection depends on good phenotypic and genotypic data. The more complete this data is, the more powerful will be the results of analysis. Nevertheless, there are next-generation technologies that seek to provide genotypic information in spite of great proportions of missing data. The procedures these technologies use to impute genetic data, therefore, greatly affect downstream analyses. This study aims to (1) compare the genetic variance in a single-nucleotide polymorphism panel of soybean with missing data imputed using various methods, (2) evaluate the imputation accuracy and post-imputation quality associated with these methods, and (3) evaluate the impact of imputation method on heritability and the accuracy of genome-wide prediction of soybean traits. The imputation methods we evaluated were as follows: multivariate mixed model, hidden Markov model, logical algorithm, k-nearest neighbor, single value decomposition, and random forest. We used raw genotypes from the SoyNAM project and the following phenotypes: plant height, days to maturity, grain yield, and seed protein composition.
We propose an imputation method based on multivariate mixed models using pedigree information. Our methods comparison indicate that heritability of traits can be affected by the imputation method. Genotypes with missing values imputed with methods that make use of genealogic information can favor genetic analysis of highly polygenic traits, but not genome-wide prediction accuracy. The genotypic matrix captured the highest amount of genetic variance when missing loci were imputed by the method proposed in this paper.
We concluded that hidden Markov models and random forest imputation are more suitable to studies that aim analyses of highly heritable traits while pedigree-based methods can be used to best analyze traits with low heritability. Despite the notable contribution to heritability, advantages in genomic prediction were not observed by changing the imputation method. We identified significant differences across imputation methods in a dataset missing 20 % of the genotypic values. It means that genotypic data from genotyping technologies that provide a high proportion of missing values, such as GBS, should be handled carefully because the imputation method will impact downstream analysis.
Marker-assisted selection (MAS) is a powerful tool for accelerating genetic improvement of plants and animals  through introgression of quantitative trait loci (QTL) and selection using genomic-enhanced breeding values (GEBV) [18, 27]. Meuwissen et al.  and Xu  first incorporated genome-wide molecular markers into the estimation of breeding values, and these so-called genomic selection (GS) methods have improved over time to handle a large number of loci [24, 59] with increased accuracy .
For genetic improvement and genetic studies in the post-genomic era, new genotyping platforms such as genotyping by sequencing (GBS)  and single-nucleotide polymorphism (SNP) arrays  provide a large number of molecular markers, but often with reduced genotyping accuracy [16, 52] and missing genotypes due to other factors. The implementation of such technology in crop breeding programs [18, 22, 48] helps maximize genetic gain through genomic selection [27, 56, 57] and also benefits genome wide association studies (GWAS) [17, 37, 52]. When there is a high proportion of missing data, consistent imputation may be challenging [22, 42, 53]. Accurate imputation of missing values and correction of genotyping errors is essential for eliminating gaps in genome coverage, integrating data across different arrays, and allowing robust genome-wide association mapping and prediction [21, 30, 49].
Researchers have proposed a variety of procedures for adjusting genotypic data to impute missing values and correct SNP miscalls. Most imputation algorithms, such as Hidden Markov Models , linear models [8, 65] and pedigree-based haplotyping , were designed for ordered markers which are known because there is a reference genome available. But some methods of imputation do not require information on marker order or phase [17, 44] such as k-Nearest Neighbors , Single Value Decomposition , or Random Forest . These, therefore, are well-suited for de novo genotyping [3, 43].
Incorporating genealogical information boosts the accuracy of genotypic imputation, phasing, and the power of association analysis [16, 23]. Therefore, here we describe a new method of imputation based on the pedigree relationship matrix  and covariance among markers. The proposed method treats genotypes as response variables within a multivariate empirical Bayes model traditionally used in plant and animal breeding . The expectation of narrow-sense heritability across a set of traits can be translated into a machine learning approach of pattern recognition to measure the amount of additive genetic information captured by the genomic information. Thus, we then compare the accuracy of each imputation method based on the amount of genetic variation that can be accounted for by a SNP panel .
This study employed genotypes and phenotypes from the SoyNAM project (soynam.org), downloaded on September 10th, 2014. SoyNAM is a soybean (Glycine max, Merr.) maturity group III nested association panel composed of 40 bi-parental crosses sharing a common parent and inbred for five generations. The panel comprises 5,596 recombinant inbred lines, phenotyped in 18 combinations of year and location, for four agronomic traits: grain yield, plant height, days to maturity, and seed protein composition. The panel was genotyped with a designed 5 k SNP chip. After removing non-segregating markers, low quality SNPs based on marker heritability (0.99)  and minor allele frequency (0.05) , we selected a set of 4,246 SNPs for this study that had 1 % of its genotypic data missing. To compare imputation performance under conditions with more missing data, we generated a second dataset by randomly deleting 20 % of genotypic data across all individuals using the prodNA function from the R package missForest . If differences in imputation methods are detectable for a strict amount of 20 % missing values, these results will apply to datasets with a larger proportion of missing values.
We compared imputation approaches using a variety of methods: a multivariate mixed model (MMM); Hidden Markov Models (HMM) implemented in three software packages commonly employed in genetic studies ; a logical algorithm; and three non-parametric methods k-Nearest Neighbor (kNN), Single Value Decomposition (SVD), and Random Forest (RF). Raw and imputed datasets are available upon request.
Multivariate mixed model
We adapted the multivariate mixed model method for imputation from Gengler et al.  and Yang et al. , based on the concept that marker inheritance will proceed in a Mendelian manner and therefore, follow the pedigree . The method bases the numerator relationship matrix on the expectation of Mendelian allele inheritance with shared identity by descent (IBD). This contrasts with expectation-maximization imputation algorithms that rely on observed kinship [47, 65] and imputation through coalescent analysis  that attempts to recreate the pedigree.
where A is the additive numerator relationship matrix and I is an identity matrix. MMM uses restricted maximum likelihood (REML) to estimate genetic covariances and then replaces missing values for the central position by the multivariate empirical Bayes estimate , also referred to as the best linear unbiased predictor (BLUP) . The model is then incremented to the next position and repeated.
This study used the software BLUPF90 [32, 33] to compute the covariance components. However, any existing software that allows multivariate mixed models incorporating pedigree information can implement the model, such as Wombat, ASReml, or SAS. Efficient algorithms to compute mixed models are described by Zhou and Stephens , Legarra and Misztal  and VanRaden .
Hidden Markov models
The HMM method is commonly employed in genetics and genomics for stochastic modeling of Markov processes (such as the computation of haplotypes). Assuming ordered markers, the HMM estimates the most likely path of states (i.e. genotype) based on the transition probability of marker Mt to change state given the previous marker Mt − 1. In genetic terms, the four possible states for a diploid organism with alleles M1 and M2 for locus M, are: M1M1, M1M2, M2M1 and M2M2.
This study evaluated three HMM software programs: fastPHASE , Beagle , and MaCH . Beagle, fastPHASE, and MaCH implement HMM with iterative updating via Expectation-Maximization (Baum–Welch algorithm). Missing values of each chromosome were imputed separately and pedigree information was not provided. HMM is the most common method of imputation and is shown to boost power and resolution of genome-wide association studies [21, 30].
This method is implemented in the program findhap.f90 . The program is computationally efficient, suited for large datasets, and becomes increasingly accurate as the proportion of pedigree data increases . Thus it is advantageous for populations that pursue genotyped pedigrees.
Findhap.f90 first generates a list of possible haplotypes, then adds a genotype into the haplotype list and searches for a matching haplotype, then finds the second haplotype by subtracting the first haplotype from the genotype. It then compares each genotype to the haplotype list and imputes unknown alleles from the haplotypes. The first two iterations use population-wide genotypic data, and subsequent iterations locate matching haplotypes from pedigree data .
kNN is a non-parametric method commonly used for prediction and classification. kNN is a memory-based learning algorithm based on voting . It relies on filling missing data points with the weighted mean of the k most similar genotypes based on Euclidean distance (root sum-of-squares of differences) between standardized observations. Rutkoski et al.  evaluated kNN for genotypic imputation and suggested it was a promising method. We used the package knnGarden  with a setting of k = 10 to perform our computation.
Single value decomposition
SVD relies on orthogonal expression of the genotypic matrix  from the decomposition M = UDV T. SVD uses the most significant eigenvectors (columns of U) to predict missing values. We performed SVD imputations by chromosome using the bi-cross-validation algorithm described in Owen & Perry  and implemented in package bcv .
Random Forest  is a non-parametric method of prediction, classification, and imputation of mixed data types . It establishes a combination of decision tree predictors, where the trees are bootstrapped random independent vectors that constitute training forests . Imputation studies of GBS data in wheat breeding have reported promising results for RF [42–44]. We performed imputations employing random forest by chromosome to generate more informative trees and reduce computational burden, using the package missForest .
Comparison of methods
given the phenotypic variance σ Y 2 = σ A 2 + σ E 2 + σ C 2 + σ ε 2 ; where σ A 2 is the additive genetic variance, σ E 2 is the variance due to environment, σ C 2 is the variance due to microenvironment (i.e. controls), and σ ε 2 is the residual variance. In genetic terms, ICC represents narrow-sense heritability, which is defined as the amount of genetic variation that alleles can transmit to the following generation, and it quantifies the population response to selection . The rational for using narrow-sense heritability estimator as a measure of how well genetic variability is captured by the genotypic information comes from its practical application of pattern recognition in machine learning [15, 34]. Genomic relation matrices generated from a SNP panel imputed through distinct methods would yield different values of variance components. Consequently, kernels that provide enhanced genetic variance and higher heritability reflect better estimators of the genetic term .
Imputation accuracy and prediction accuracy
This study evaluated imputation accuracy by determining the proportion of data imputed that was identical to the standard dataset . For each method, we compared the imputed genotypic matrix with 20 % of the loci missing to the genotypic matrix corrected using the same method with 1 % of loci missing. Thus, the level of accuracy provides insight into both imputation method and modifications of the dataset.
To measure prediction accuracy in the context of genomic selection, we used the correlation of observed and predicted values in a five-fold cross validation divided by the square root of the heritability as described by Lehermeier et al. . We used the following whole-genome regression methods [7, 14] to generate the predicted values: BayesA, BayesB, BayesCπ, Bayesian LASSO, and Bayesian Ridge Regression, as implemented by Pérez and de los Campos  with default settings of hyperpriors. The statistical testing of imputation method on prediction accuracy followed the same ANOVA model previously described with an additional term to accommodate the whole-genome regression method.
Effect of imputation method on heritability
We found a significant association (p-value < 0.01) between heritability values (yijkl) and all terms in the statistical model: imputation method (τi), percentage of missing values (βj) and agronomic trait (γk). The fitted model provided a coefficient of determination R2 = 0.95 and a coefficient of variation CV = 6.72. The group means procedure (Fig. 1) showed that most imputation methods were similar. MMM had provided the highest across-trait heritability, although not significantly better than the other methods except for the kNN and SVD methods, which were inferior to the others, with SVD being the worst.
Post-imputation quality parameters
We observed a higher number of markers with full linkage disequilibrium (i.e. repeated markers) after imputation with the HMM implementations. MaCH and Beagle (Fig. 2b) also increased the number of repeated markers as the percentage of missing data increased. Non-parametric methods, SVD and kNN, did not provide repeated markers. Random forest and both pedigree-based methods, MMM and findhap, showed fewer repeated markers as the missing data increased.
The third measure of accuracy of imputation and data correction is the difference between the observed and expected proportion of heterozygosity in the data. The expected proportion of heterozygosity in this dataset is 6.25 % of loci given that the parents were homozygous and the plants were genotyped in the F5 generation (i.e. after 5 generations of selfing), in which the expected inbreeding coefficient is 93.75 %. The results show that the proportion of heterozygous loci tends to increase as the proportion of missing data increases (Fig. 2c). We observed this in all but two methods, Beagle and MaCH. SVD and kNN were more likely to increase the number of heterozygous loci with more missing data, especially SVD for which proportions of heterozygosity reached 24 %, meaning that heterozygous loci were assigned to most missing loci.
Imputation methods and genetic variation captured
The prediction accuracy across traits ranged from 0.606 (MMM) to 0.628 (RF), but did not differ significantly (p-value = 0.537). However, it appears that the genotypic data imputed with RF may have a slightly better performance. Whole-genome regression methods did not provide statistically significant differences in prediction accuracy (data not shown).
The capture of genetic variation
Differences between imputation methods were clearly detectable on a SNP panel containing 20 % missing values, raising questions about the quality and reliability of genotypic data obtained by imputing datasets with an 80 % or greater proportion of missing values, a common scenario for GBS.
Our results indicate that genotypes imputed with methods that rely on pedigree information, such as MMM, better capture genetic variance in complex low heritability traits, while imputations performed using HMM may favor traits with high heritability. The choice of imputation method will affect downstream analysis. Thus, if the genetic architecture of the traits to be analyzed is known, it is possible to achieve more accurate results by selecting the most suitable imputation method.
The genetic architecture of the trait should also influence the choice of genomic prediction method , although this study found no statistically significant differences, likely due to the similar nature of evaluated models also reported by Howard et al. . Chen et al.  and Poland et al.  reported interactions between imputation method and prediction accuracy for different traits. For example, genotypic imputation through SVD showed inferior capture of genetic variance, but it did not affect prediction accuracy.
MMM: strengths and weaknesses
In this study we presented a mixed model method of imputation using pedigree information that displayed interesting properties regarding the capture of genetic variance and may have advantages for downstream analysis of complex traits. Some reported advantages of MMM include imputation of un-genotyped individuals  or with a large percentage of missing data  without losing robustness; and the identity link function of MMM allows imputation regardless of the allelic coding .
On the other hand, genotypic information imputed by MMM was not advantageous for analyzing highly heritable traits and it did not improve prediction accuracy. However, NAM populations have a very simple pedigree structure and more complex pedigree structures may lead to better results . Another drawback was the method’s computational burden because the mixed model computes each molecular marker, a limitation that can be addressed through parallel computing.
Much as in HMM, the change-of-state of a haplotype in MMM is controlled by the covariance, with flanking markers considering linkage disequilibrium (LD), and the population structure allied to the pedigree information simultaneously, while forward-backward HMM algorithms work one direction at time using just a subset of random samples . Imputation methods that incorporate pedigree do not rely on samples from a reference dataset that is assumed to comprise all populations . Using the genomic relationship instead of pedigree, Yang et al.  have reported the superiority of a similar MMM method over HMM emphasizing better incorporation of information on LD and IBD.
Choice of the imputation method
Summary of properties of imputation methods under evaluation
Better fit for low heritability traits
High imputation accuracy
Enlarges LD blocks
Accommodates pedigree information
Suitable for unordered markers
For GBS and other technologies where a high percentage of missing values is expected, our results support the use of MaCH, fastPHASE, and RF, methods that have shown insensitivity to the percentage of missing loci. In a similar study, ) observed inferior performance of RF over HMM in wheat that could be attributed to the whole-genome imputation at once as opposed to one chromosome at a time, resulting into less informative decision trees and inconsistent imputation. The imputation of each chromosome separately is also a common practice with HMM .
Quality of the imputation disturbs the genetic variance captured by the genotypic data. We were able to show that the imputation of genotypic data in the case where proportion of missing values as low as 20 % can affect the quality of the genotypic representation by the SNP panel. This results must be seen as a word of care for technologies based on low-coverage genotyping, such as GBS, where the amount of missing information commonly achieves values of 80 %. Yet, for the scenario in study it was not possible to identify significant impact of imputation method on genomic prediction. Thus, based on imputation accuracy and genetic variance captured by the SNP panel, the imputation method choice is hidden Markov models and random forest for general analysis. Pedigree-based methods of imputation were recognized to enhance the heritability of grain yield, the lowest heritable trait in this study.
Availability of supporting data
This study employed genotypes and phenotypes from the SoyNAM project (soynam.org). Data is available through the R package SoyNAM  and USDA-ARS soybean database SoyBase (soybase.org).
We thank the SoyNAM collaborators for their contributions to the experiment: Dr. Bill Beavis for experimental design, Qijan Song and Perry Cregan for genotyping, and Jim Specht and Brian Diers for creating the germplasm resource. A.X. was supported by funds from Dow AgroSciences, and the SoyNAM experiment was supported by funds from the North Central Soybean Research Program, the United Soybean Board, and Dow AgroSciences.
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