Genome wide association studies in presence of misclassified binary responses
© Smith et al.; licensee BioMed Central Ltd. 2013
Received: 6 May 2013
Accepted: 17 December 2013
Published: 26 December 2013
Misclassification has been shown to have a high prevalence in binary responses in both livestock and human populations. Leaving these errors uncorrected before analyses will have a negative impact on the overall goal of genome-wide association studies (GWAS) including reducing predictive power. A liability threshold model that contemplates misclassification was developed to assess the effects of mis-diagnostic errors on GWAS. Four simulated scenarios of case–control datasets were generated. Each dataset consisted of 2000 individuals and was analyzed with varying odds ratios of the influential SNPs and misclassification rates of 5% and 10%.
Analyses of binary responses subject to misclassification resulted in underestimation of influential SNPs and failed to estimate the true magnitude and direction of the effects. Once the misclassification algorithm was applied there was a 12% to 29% increase in accuracy, and a substantial reduction in bias. The proposed method was able to capture the majority of the most significant SNPs that were not identified in the analysis of the misclassified data. In fact, in one of the simulation scenarios, 33% of the influential SNPs were not identified using the misclassified data, compared with the analysis using the data without misclassification. However, using the proposed method, only 13% were not identified. Furthermore, the proposed method was able to identify with high probability a large portion of the truly misclassified observations.
The proposed model provides a statistical tool to correct or at least attenuate the negative effects of misclassified binary responses in GWAS. Across different levels of misclassification probability as well as odds ratios of significant SNPs, the model proved to be robust. In fact, SNP effects, and misclassification probability were accurately estimated and the truly misclassified observations were identified with high probabilities compared to non-misclassified responses. This study was limited to situations where the misclassification probability was assumed to be the same in cases and controls which is not always the case based on real human disease data. Thus, it is of interest to evaluate the performance of the proposed model in that situation which is the current focus of our research.
KeywordsMisclassification Genome wide association Discrete responses
Misclassification of dependent variables is a major issue in many areas of science that can arise when indirect markers are used to classify subjects or continuous traits are treated as categorical . Binary responses are typically subjective measurements which can lead to error in assigning individuals to relevant groups in case–control studies. Many quantitative traits have precise guidelines for measurements but in qualitative diagnosis different individuals will understand conditions in their own way . Some disorders require structured evaluations but these can be time consuming and very costly and not readily available for all patients . This sometimes requires clinicians to use heuristics rather than following strict diagnostic criteria , leading to diagnoses based on personal opinions and experience. It was found that physicians will disagree with one another one third of the time as well as with themselves (on later review) one fifth of the time. This lack of consistency leads to large variation and error [5, 6].
Researchers indicated that there is a common assumption under most approaches that disorders can be distinguished without error which is seldom the case . For instance, a longitudinal study was carried out over 10 years where 15% of subjects initially diagnosed with bipolar disorder were re-diagnosed with schizophrenia, whereas 4% were reclassified in the opposite direction . Reports have shown an error rate of more than 5-10% for some discrete responses [9, 10]. In some instances, these rates have proven to be significantly higher. The frequency of medical misdiagnosis and clinical errors has reached error rates as high as 47% as documented in several autopsy studies . Error rates in clinical practices have shown to be higher than perceptual specialties , but still these areas have demonstrated high rates as well. In radiology areas, failure to detect abnormalities when they were present (false negative) ranged between 25-30%, and when the cases were normal but incorrectly diagnosed as diseased (false positive) ranged between 1.5-2% . Some stated that these errors are not due to failure of not showing on film but due to perceptual errors . These findings are similar to recent published studies [3, 6, 15, 16].
Unfortunately, finding these errors in clinical data is not trivial. Even in the best case scenario when well-founded suspicion exists about a sample, re-testing is often not possible and the best that could be done is to remove the sample leading to power reduction. Recently, several research groups [17–19] have proposed using single nucleotide polymorphisms (SNPs) to evaluate the association between discrete responses and genomic variations. Genome-wide association studies (GWAS) provide researchers with the opportunity of discovering genomic variations affecting important traits such as diseases in humans, and production and fitness responses in livestock and plant species. Several authors have indicated that the precision and validity of GWAS relies heavily on the accuracy of the SNP genotype data as well as the certainty of the response variable [20–25]. Thus, analyzing misclassified discrete data without correcting or accounting for these errors may cause algorithms to select polymorphisms with little or no predictive ability. This could lead to varying and even contradictory conclusions. In fact, it was reported that only 6 out of 600 gene-disease associations reported in the literature were significant in more than 75% of the studies published . In majority of cases, heterogeneity, population stratification, and potential misclassification in the discrete dependent variables were at the top of the list of potential reasons for these inconsistent results [22, 27–30].
In supervised learning, if individuals are wrongly assigned to subclasses, false positive and erroneous effects will result if these phenotypes are used when trying to identify which markers or genes can distinguish between disease subclasses. Researchers carried out a study of misclassification using gene expression data with application to human breast cancer . They looked at the influence of misclassification on gene selection. It was found that even when only one sample is misclassified, 20% of the most significant genes were not identified. Further results showed that with misclassification rates between 3-13%, there could be unfavorable effect on detecting the most significant genes for disease classification. Furthermore, if some genes are identified as significant while misclassification is present, this will lead to the inability to replicate the results due to the fact it is only relevant to the specific data.
To overcome these issues it would be advantageous to develop a statistical model that is able to account for misclassification in discrete responses. There have been several approaches proposed on how to handle misclassification. Researchers have suggested Bayesian methods [32–34], some described a latent Markov model for longitudinal binary data , others proposed marginal analysis methods , and some considered two-state Markov models with misclassified responses [37, 38].
In 2001, a Bayesian approach was proposed for dealing with misclassified binary data . This procedure, with the use of Gibbs sampling, “made the analysis of binary data subject to misclassification tractable”. It was concluded that failure to account for errors in responses results in adverse effects related to the parameters of interest including genetic variance. The analysis was applied to simulated cow fertility data and was later implemented with the use of real data which resulted in similar findings [10, 31]. One study found considering a potential for misdiagnosis in the data could increase prediction power by 25% . To extend their ideas we simulated a typical case–control study to measure and understand the effects of misclassification on GWAS using a threshold model and misclassification algorithm. Three analyses were conducted: (M1) the true data was analyzed with a standard threshold model; (M2) the noisy (5% and 10% miscoding) data analyzed with standard threshold model ignoring miscoding; (M3) the noisy data analyzed with threshold model with probability of being miscoded (π) included.
Detecting discrete phenotype errors
With q i = p i (1 - π) + (1 - p i )π
where p i (β) indicates that p i is a function of the vector of parameters β.
Notice that when α i = 0(no switching), the formula in (2) reduces to r i = y i
where βmin, βmax, a and b are known hyper-parameters. In our case a and b were set heuristically to 1 and 4, respectively, in order to convey limited prior information. From our previous experience, these values for the hyper-parameters have little effects on the posterior inferences and the results were similar to those obtained using a flat prior for π. Obviously, the effect of these hyper-parameters depends on the magnitude of n (number of observations). Thus, a special attention has to be placed on specifying these parameters when using small samples and a sensitivity analysis is recommended. For the SNP effects, βmin and βmax were set to -100 and 100 respectively conveying, thus, a very vague bounded prior. With real data, it is often the case that the number of SNPs is much larger than the number of observations. In such scenario, an informative prior is needed to make the model identifiable and often a normal prior is assumed.
where T is an arbitrary threshold value.
where μ is the overall mean, x ij is the genotype for SNP j for individual i, β j is the effect of SNP j (j = 1,1000) and e i is the residual term. To make the model in (4) identifiable, two restrictions are needed. It was assumed that the T = 0 and var(e i ) = 1.
where α-i is vector α without αi.
Hence, π is distributed as Beta(a + ∑ α i , b + n - ∑ α i ) with ∑ α i is the total number of misclassified (switched) observations.
where n = 2000 is the number of data points.
where with x j is a column vector of genotypes for SNP j, X is an nxp matrix of SNP genotypes with the jth row and column set to zero and β-jis the vector β excluding the jth position.
This is a truncated normal (TN) distribution to the left if y i = 1 and to the right if y i = 0 (Sorensen et al., 1995) where and l-i is the vector l excluding the ith position.
In all simulation scenarios, the Gibbs sampler was run for a unique chain of 50,000 iterations of which the first 10,000 iterations were discarded as burn-in period. The convergence of the chain was assessed heuristically based on the inspection of the trace plot of the sampling process.
PLINK software  was used to simulate a case–control type data sets using the SNP simulation routine. Four simulation scenarios were generated to determine the effects of misclassification of binary status on GWAS. In each scenario, a dataset of 2000 individuals consisting of 1000 cases and 1000 controls was simulated. All individuals were genotyped for 1000 SNPs with minor allele frequencies generated from a uniform distribution between 0.05 and 0.49. SNPs were coded following an additive model (AA = 0, Aa = 1, and aa = 2). Of the 1000 SNPs, 850 SNPs were assumed non-influential and the remaining 150 SNPs were assumed to be associated with the disease status. To mimic realistic scenarios, a series of bins were specified for the 150 influential SNPs to build a spectrum of odds ratios (OR) for disease susceptibility. Two different series of odds ratios were considered. The first group was generated with “moderate” ratios where 25 of the 150 disease associated SNPs were assumed to have an odds ratio of 1:4, 35 with OR of 1:2, and 90 with OR of 1:1.8. The second group was generated using the same distribution except the ratios increased to a more extreme range; 25 with OR 1:10, 35 with OR of 1:4, and 90 with OR of 1:2. Once these parameters were established, PLINK generated a quantitative phenotype based on the disease variants and a random component or error term. Then a median split of that trait was performed thereafter each individual was assigned a binary status. When the “true” binary data were generated as described above, randomly 5 or 10% of the true binary records were miscoded, meaning binary records from cases were switched to controls and vice versa.
Based on the OR distribution (moderate and extreme) and the level of misclassification (5 or 10%), four data sets were generated: 5% misclassification rate and moderate OR (D1); 5% misclassification and extreme OR (D2); 10% misclassification rate and moderate OR (D3); and 10% misclassification rate and extreme OR (D4). For each dataset, 10 replicates were generated.
To further test our proposed method, a more diverse and representative data was simulated using the basic simulation procedure previously indicated. For this second simulation, a dataset consisting of 1800 individuals (1200 controls and 600 cases) was genotyped for 40,000 linked SNPs assuming an additive model. Five hundred SNPs were assumed to be influential with OR set equal to 1:4 (80 SNPs), 1:2 (120 SNPs), and 1:1.8 (300 SNPs). Only the 5% misclassification rate scenario was considered.
Results and discussion
Summary of the posterior distribution of the misclassification probability ( π ) for the four simulation scenarios (averaged over 10 replicates)
Correlation between true 1 and estimated SNP effects under four simulation scenarios using noisy data (M2) and the proposed approach (M3)
Using the data set simulated under a more realistic scenario (imbalance between cases and controls, larger SNP panel) the results were similar in trend and magnitude to those observed using the first four simulations. In fact, the posterior mean of the misclassification probability was 0.04 and the true value (0.05) was well within the HPD95% interval. Furthermore, the correlations between SNP effect estimates using M2 and M3 were 0.54 and 0.70, respectively. This 30% increase in accuracy using M3 indicates a substantial improvement of the model when our proposed method is used. This is of special practical importance as the superiority of the method was maintained with a dataset similar to what is currently used in GWAS.
It is clear that across all simulation scenarios our proposed method (M3) showed superior performance. Accounting for misclassification in the model increases the predictive power by eliminating or at least by attenuating the negative effects caused by these errors, allowing for better estimates of the true SNP effects. This is essential in GWA studies for correctly estimating the proportion of variation in cause of disease associated with SNPs. Complex diseases which are under the control of several genes and genetic mechanisms are moderately to highly heritable [42–44].
Number of the top 10% (15 SNPs) most influential SNPs that were correctly identified for all simulation scenarios using the noisy data (M2) and the proposed approach (M3)
Percent of misclassified individuals correctly identified based on two cutoff probabilities across the four simulation scenarios
Misclassification of discrete responses has been shown to occur often in datasets and has proven to be difficult and often expensive to resolve before analyses are run. Ignoring misclassified observations increases the uncertainty of significant associations that may be found leading to inaccurate estimates of the effects of relevant genetic variants. The method proposed in this study was capable of identifying miscoded observations, and in fact these individuals were distinguished from the correctly coded set and were detected at higher probabilities over all four simulation scenarios. This is essential as it shows the capability of our algorithm to maintain its superior performance across different levels of misclassification as well as different odds ratios of the influential SNPs.
More notably, our method was able to estimate SNP effects with higher accuracy compared to estimation using the “noisy” data. Running analyses on data that do not account for potential misclassification of binary responses, such as M2 in this study, will lead to non-replicative results as well as causing an inaccurate estimation of the effect of polymorphisms which can be correlated to the disease of interest. This severely reduces the power of the study. For instance, it was determined that conducting a study on 5000 cases and 5000 controls with 20% of the samples being misdiagnosed has the power equivalent to only 64% of the actual sample size . Implementing our proposed method provides the ability to produce more reliable estimates of SNP effects increasing predictive power and reducing any bias that may have been caused by misclassification. Our results suggested that the proposed method is effective for implementation of association studies for binary responses subject to misclassification.
Single nucleotide polymorphism
Genome-wide association studies
High posterior density 95% interval.
The first author was supported financially by the graduate school and the department of Animal and Dairy science at the University of Georgia.
- Fabris C, Smirne C, Toniutto P, Colletta C, Rapetti R, Minisini R, Falleti E, Leutner M, Pirisi M: Usefulness of six non-proprietary indirect markers of liver fibrosis in patients with chronic hepatitis C. Clin Chem. 2008, 46 (2): 253-259.Google Scholar
- Barendse W: The effect of measurement error of phenotypes on genome wide association studies. BMC Genomics. 2011, 12: 232-243. 10.1186/1471-2164-12-232.PubMed CentralView ArticlePubMedGoogle Scholar
- Theodore RS, Basco MR, Biggan JR: Diagnostic disagreements in bipolar disorder: the role of substance abuse comorbidities. Depression Research and Treatment. 2012, 2012: 6-Article ID 435486, doi:10.1155/2012/435486Google Scholar
- Meyer F, Meyer TD: The misdiagnosis of bipolar disorder as a psychotic disorder: some of its causes and their influence on therapy. J Affect Disord. 2009, 112: 105-115.Google Scholar
- Garland LH: Studies on the accuracy of diagnostic procedures. Am J Roentgenol. 1959, 82: 25-38.Google Scholar
- Berlin L: Accuracy of diagnostic procedures: has it improved over the past five decades?. Am J Roentgenol. 2007, 188: 1173-1178. 10.2214/AJR.06.1270.View ArticleGoogle Scholar
- Wray N, Lee SH, Kendler KS: Impact of diagnostic misclassification on estimation of genetic correlations using genome-wide genotypes. Eur J Hum Genet. 2012, 20: 668-674. 10.1038/ejhg.2011.257.PubMed CentralView ArticlePubMedGoogle Scholar
- Bromet EJ, Kotov R, Fochtmann LJ, Carlson GA, Tanenberg-Karant M, Ruggero C, Chang SW: Diagnostic shifts during the decade following first admission for psychosis. Am J Psychiat. 2011, 168: 1186-1194.PubMed CentralView ArticlePubMedGoogle Scholar
- West M, Blanchette C, Dressman H, Huang E, Ishida S, Spang R, Zuzan H, Olson JA, Marks JR, Nevins JR: Predicting the clinical status of human breast cancer by using gene expression profiles. Proc Natl Acad Sci U S A. 2001, 98: 11462-11467. 10.1073/pnas.201162998.PubMed CentralView ArticlePubMedGoogle Scholar
- Robbins K, Joseph S, Zhang W, Rekaya R, Bertrand JK: Classification of incipient Alzheimer patients using gene expression data: dealing with potential misdiagnosis. Online J. Bioniformatics. 2006, 7: 22-31.Google Scholar
- Anderson RE, Hill RB, Key CR: The sensitivity and specificity of clinical diagnostics during five decades: toward an understanding of necessary fallibility. JAMA. 1989, 261: 1610-1617. 10.1001/jama.1989.03420110086029.View ArticlePubMedGoogle Scholar
- Berner ES, Graber ML: Overconfidence as a cause of diagnostic error in medicine. Am J Med. 2008, 121: S2-S23.View ArticlePubMedGoogle Scholar
- Renfrew DL, Franken EA, Berbaum KS, Weigelt FH, Abu-Yousef MM: Error in radiology: classification and lessons in 182 cases presented at a problem case conference. Radiology. 1992, 183: 145-150.View ArticlePubMedGoogle Scholar
- Shively CM: Quality in management radiology. Imaging Economics. 2003, 11: 6-Google Scholar
- Landro L: Hospitals move to cut dangerous lab errors. Wall Street Journal. in press
- Plebani M: Errors in clinical laboratories or errors in laboratory medicine?. Clin Chem Lab Med. 2006, 44: 750-759.PubMedGoogle Scholar
- Hirschhorn JN, Daly MJ: Genome-wide association studies for common diseases and complex traits. Nat Rev Genet. 2005, 6: 98-108.View ArticleGoogle Scholar
- Manolio TA, Brooks LD, Collins FS: A HapMap harvest of insights into the genetics of common disease. J Clin Invest. 2008, 118: 1590-1605. 10.1172/JCI34772.PubMed CentralView ArticlePubMedGoogle Scholar
- McCarthy MI, Abecasis GR, Cardon LR, Goldstein DB, Little J, Ioannidis JPA, Hirschhorn JN: Genome-wide association studies for complex traits: consensus, uncertainty and challenges. Nat Rev Genet. 2008, 9: 356-369. 10.1038/nrg2344.View ArticlePubMedGoogle Scholar
- Thomas A: GMCheck: Bayesian error checking for pedigree genotypes and phenotypes. Bioinformatics. 2005, 21: 3187-3188. 10.1093/bioinformatics/bti485.View ArticlePubMedGoogle Scholar
- Kennedy J, Mandoiu I, Pasaniuc B: Genotype error detection using hidden markov models of haplotype diversity. J Comp Bio. 2008, 15: 1155-1171. 10.1089/cmb.2007.0133.View ArticleGoogle Scholar
- Avery CL, Monda KL, North KE: Genetic association studies and the effect of misclassification and selection bias in putative confounders. BMC Proc. 2009, 3: S48-10.1186/1753-6561-3-s7-s48.PubMed CentralView ArticlePubMedGoogle Scholar
- Wilcox MA, Paterson AD: Phenotype definition and development—contributions from Group 7. Genet Epidemiol. 2009, 33 (Suppl 1): S40-S44.PubMed CentralView ArticlePubMedGoogle Scholar
- Huang X, Feng Q, Qian Q, Zhao Q, Wang L, Wang A, Guan J, Fan D, Weng Q, Huang T, Dong G, Sang T, Han B: High-throughput genotyping by whole genome resequencing. Genome Res. 2009, 19: 1068-1076. 10.1101/gr.089516.108.PubMed CentralView ArticlePubMedGoogle Scholar
- Hossain S, Le ND, Brooks-Wilson AR, Spinelli JJ: Impact of genotype misclassification on genetic association estimates and the Bayesian adjustment. Am J Epidemiol. 2009, 170: 994-1004. 10.1093/aje/kwp243.View ArticlePubMedGoogle Scholar
- Hirschhorn JN, Lohmueller K, Byrne E, Hirschhorn K: A comprehensive review of genetic association studies. Genet Med. 2002, 2: 45-61.View ArticleGoogle Scholar
- Skafidas E, Testa R, Zantomio D, Chana G, Everall IP, Pantelis C: Predicting the diagnosis of autism spectrum disorder using gene pathway analysis. Mol Psychiatry. 2012, doi:10.1038/mp.2012.126Google Scholar
- Li A, Meyre D: Challenges in reproducibility of genetic association studies: lessons learned from the obesity field. Int J Obes (Lond). 2012, doi:10.1038/ijo.2012.82Google Scholar
- Galvan A, Ioannidis JPA, Dragani TA: Beyond genome-wide association studies: genetic heterogeneity and individual predisposition to cancer. Trends Genet. 2010, 26: 132-141. 10.1016/j.tig.2009.12.008.PubMed CentralView ArticlePubMedGoogle Scholar
- Wu C, DeWan A, Hoh J, Wang Z: A comparison of association methods correcting for population stratification in case–control studies. Annals of human genetics. 2011, 418-427. doi:10.1111/j.1469-1809.2010.00639Google Scholar
- Zhang W, Rekaya R, Bertrand JK: A method for predicting disease subtypes in presence of misclassification among training samples using gene expression: application to human breast cancer. Bioinformatics. 2006, 22: 317-325. 10.1093/bioinformatics/bti738.View ArticlePubMedGoogle Scholar
- Paulino CD, Soares P, Neuhaus J: Binomial regression with misclassification. Biometrics. 2003, 59: 670-675. 10.1111/1541-0420.00077.View ArticlePubMedGoogle Scholar
- Paulino CD, Silva G, Achcar JA: Bayesian analysis of correlated misclassified binary data. Comp Statist Data Anal. 2005, 49: 1120-1131. 10.1016/j.csda.2004.07.004.View ArticleGoogle Scholar
- Rekaya R, Weigel KA, Gianola D: Threshold model for misclassified binary responses with applications to animal breeding. Biometrics. 2001, 57: 1123-1129. 10.1111/j.0006-341X.2001.01123.x.View ArticlePubMedGoogle Scholar
- Cook RJ, Ng ETM, MEADE, MO: Estimation of operating characteristics for dependent diagnostic tests based on latent Markov models. Biometrics. 2000, 56: 1109-1117. 10.1111/j.0006-341X.2000.01109.x.View ArticlePubMedGoogle Scholar
- Chen Z, Yi GY, Wu C: Marginal methods for correlated binary data with misclassified responses. Biometrika. 2011, 98: 647-662. 10.1093/biomet/asr035.View ArticleGoogle Scholar
- Rosychuck RJ, Thompson ME: A semi-Markov model for binary longitudinal responses subject to misclassification. Can J Statist. 2001, 29: 395-404. 10.2307/3316036.View ArticleGoogle Scholar
- Rosychuck RJ, Thompson ME: Bias correction of two-state latent Markov process parameter estimates under misclassification. Statist Med. 2003, 22: 2035-2055. 10.1002/sim.1473.View ArticleGoogle Scholar
- Sorensen DA, Andersen S, Gianola D, Korsgaard I: Bayesian inference in threshold using Gibbs sampling. Genet Sel Evol. 1995, 27: 229-249. 10.1186/1297-9686-27-3-229.PubMed CentralView ArticleGoogle Scholar
- Sapp RL, Spangler ML, Rekaya R, Bertrand JK: a simulation study for analysis of uncertain binary responses: application to first insemination success in beef cattle. Genet Sel Evol. 2005, 37: 615-634. 10.1186/1297-9686-37-7-615.PubMed CentralView ArticlePubMedGoogle Scholar
- Purcell S, Neale B, Todd-Brown K, Thomas L, Ferreira MAR, Bender D, Maller J, Sklar P, de Bakker PIW, Daly MJ, Sham PC: PLINK: a toolset for whole-genome association and population-based linkage analysis. Am J Hum Genet. 2007, 81: 559-575. 10.1086/519795.PubMed CentralView ArticlePubMedGoogle Scholar
- Hardy J, Singleton A: Genome wide association studies and human disease. N Engl J Med. 2009, 360: 1759-1768. 10.1056/NEJMra0808700.PubMed CentralView ArticlePubMedGoogle Scholar
- Wray NR, Goddard ME: Multi-locus models of genetic risk of disease. Genome Med. 2010, 2: 10-10.1186/gm131.PubMed CentralView ArticlePubMedGoogle Scholar
- Cambien F: Heritability, weak effects, and rare variants in genome wide association studies. Clin Chem. 2011, 57: 1263-1266. 10.1373/clinchem.2010.155655.View ArticlePubMedGoogle Scholar
- Spencer C, Hechter E, Vukcevic D, Donnelly P: Quantifying the underestimation of relative risks from genome-wide association studies. PLoS Genet. 2011, 7: e1001337-10.1371/journal.pgen.1001337.PubMed CentralView ArticlePubMedGoogle Scholar
- Stringer S, Wray NR, Kahn RS, Derks EM: Underestimated effect sizes in GWAS: fundamental limitations of single SNP analysis for dichotomous phenotypes. PLoS ONE. 2011, 6: e27964-10.1371/journal.pone.0027964.PubMed CentralView ArticlePubMedGoogle Scholar
- Feng JY, Zhang J, Zhang WJ, Wang SB, Han SF, Zhang YM: An efficient hierarchical generalized linear mixed model for mapping QTL of ordinal traits in crop cultivars. PLoS ONE. 2013, 8: e59541-10.1371/journal.pone.0059541.PubMed CentralView ArticlePubMedGoogle Scholar
- Yi N, Liu N, Zhi D, Li J: Hierarchical generalized model for multiple groups of rare and common variants: jointly estimating group and individual-variant effects. PLoS Genet. 2011, 7: e1002382-10.1371/journal.pgen.1002382.PubMed CentralView ArticlePubMedGoogle Scholar
- Huang A, Xu S, Cai X: Empirical Bayesian LASSO-logistic regression for multiple binary trait locus mapping. BMC Genet. 2013, 14: 5-10.1186/1471-2156-14-5.PubMed CentralView ArticlePubMedGoogle Scholar
This article is published under license to 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.