- Open Access
PSMIX: an R package for population structure inference via maximum likelihood method
© Wu et al; licensee BioMed Central Ltd. 2006
Received: 23 January 2006
Accepted: 22 June 2006
Published: 22 June 2006
Inference of population stratification and individual admixture from genetic markers is an integrative part of a study in diverse situations, such as association mapping and evolutionary studies. Bayesian methods have been proposed for population stratification and admixture inference using multilocus genotypes and widely used in practice. However, these Bayesian methods demand intensive computation resources and may run into convergence problem in Markov Chain Monte Carlo based posterior samplings.
We have developed PSMIX, an R package based on maximum likelihood method using expectation-maximization algorithm, for inference of population stratification and individual admixture.
Compared with software based on Bayesian methods (e.g., STRUCTURE), PSMIX has similar accuracy, but more efficient computations.
PSMIX and its supplemental documents are freely available at http://bioinformatics.med.yale.edu/PSMIX.
Information about population structure, namely population stratification and admixture, is useful in a variety of situations, such as association studies of genes underlying complex traits, subspecies classification, genetic barrier detection, and evolutionary study [1–10]. For example, it is very important to identify genetic ancestry and admixture in admixture mapping [7, 8]. The presence of population stratification or admixture may pose a practical nuisance as well. In association studies, case-control design is often used to identify genetic variants underlying complex traits by comparing allele frequencies between unrelated individuals that are affected and those unaffected. However, the presence of population stratification or admixture in the sample can lead to spurious associations between a candidate marker and a phenotype [5, 10, 11]. In forensic studies, the identification of reference groups is central but becomes difficult when there exists population stratification [12, 13]. In the estimation of the magnitude of inbreeding, it is useful to distinguish between the causes for the excess homozygosity which might be consanguineous mating or population substructure, or an artifact due to factors like null alleles . In all these situations, identifying population stratification or admixture has been an important component.
Commonly used software for population structure inference
Bayesian, MCMC is used when the number of populations ≥ 9
Latent class analysis, EM
Clustering analysis, EM
Population structure inference
Process geo-referenced individual multilocus genetic data for population structure inference
Population structure inference. Use geographical sampling design of the individuals
Mainly for analysis of datasets that consist of trait measurements and genotype data on a sample of individuals from an admixed or stratified population
Population structure inference
Population structure inference
Population structure inference
HWE and LE between loci
HWE, LE between loci, and spatial distribution of sub-populations
HWE and LE between loci
Ancestry state is the same at all loci within a compound locus on any gamete. Mating is not assortative for admixture in the population from which the parental gametes were drawn
HWE and LE between loci. The underlying population genetic model is appropriate for out-crossing diploid organisms.
HWE and LE between loci
HWE and LE between loci
Parameters for running MCMC, parameters for ancestry model and allele frequency model, and the number of populations
In addition to genetic and spatial data, the user must provide parameters for the maximum number of populations, the way geographical information is handled and the allele frequency model
When MCMC is used, need parameters for running MCMC
Parameters for running MCMC, allele frequencies (number of population is specified here), and mating model. Disease information (outcome variable is suggested even if focus on population structure). Parameters for tests and output
Parameters for running MCMC, maximum number of populations, prior parameter for allelic diversity, and prior parameter for number of populations
Number of populations, admixture option, data format options, model options, output format options, and convergence criterion
Number of populations and convergence criterion
One file for estimates and some files for plots. Main parameter estimates are inferred ancestry of individuals and estimated allele frequencies in each population
Main parameter estimates are the number of populations, population membership of each individual, maps giving the population memberships of each geographical pixel of a given size to locate genetic discontinuities between populations
Main parameter estimates are the number of populations and population membership of each individual
Individual/gamete level admixture variables. Ancestry-specific allele or haplotype frequencies. Results for association analysis and model parameters
The output file contains a list of the parameter settings followed by the sequence of observations of the Markov chain. A companion program PartitionView is provided to obtain useful information from the PARTITION output file.
Main outputs are estimates of allele frequencies, posterior class probabilities, and class-specific allele frequencies
Main parameter estimates are inferred ancestry of individuals
Easy to use. Once number of populations is given, the estimates are accurate
Easy to use. Flexible to extend. Can work with or without spatial information. Can estimate number of populations
Easy to use. Provide good estimate for number of populations. When geographical sampling information is applicable, can improve the statistical power to detect clusters in the data
In addition to population structure inference, can perform association analysis on structured populations. Can deal with tightly linked loci using haplotypes
Easy to use. Can estimate number of populations and calculate a Bayes factor in support of a single source population against the alternative of more than one source population.
Easy to use. Computationally efficient. Flexible to extend
Computationally intensive. Can detect number of populations but does not work well.#
Does not handle admixture. Computationally intensive, especially when "Falush" is used as allele frequency model, or number of populations needs to be estimated.
Very memory intensive. When MCMC is used, becomes relatively computationally intensive. Only provides membership partition, does not handle admixture
Difficult to use. Computationally intensive. Does not estimate number of populations *
Computationally intensive, especially when number of populations needs to be estimated
Parameter configuration is difficult to use, works OK for discrete populations but not for admixed populations. Does not estimate number of populations
Does not estimate number of populations
Windows/Linux/Mac (R package)
Windows, Unix/Linux. R statistical package is required
Windows (DOS), Unix/Linux
Windows/Linux/Mac (R package)
Pritchard et al. (2000), Falush et al. (2003)
Guillot et al. (2005)
Corander et al. (2003, 2004)
McKeigue et al. (2000), Hoggart et al. (2003, 2004)
Dawson and Belkhir (2001)
Purcell and Sham (2004)
Tang et al. (2005), Liu et al. (2005)
We have developed an efficient R package, named PSMIX (Population Structure inference via MIXture model), for population stratification and individual admixture inference. Since R can be slow when computation is intensive, we implemented the expectation-maximization (EM) algorithm  using C programming language. PSMIX is mainly based on the methods proposed in Tang et al.  and Liu et al. . Three models (described in section 2.2, 2.3, and 2.4, respectively) are discussed in full detail in . The second one is equivalent to the model proposed in Tang et al. . The first model is a special case of the second one. In Tang et al. the method itself has been fully assessed by simulation studies .
We used two real datasets from Rosenberg et al.  and one simulated dataset from Tang et al.  to demonstrate the functionality of PSMIX. One real dataset contains two American populations, Pima and Surui with 25 and 21 individuals, respectively; the other contains two European populations, Sardinian and Russian with 28 and 25 individuals, respectively. The simulated data set contains 50 individuals from each of the two ancestral populations, and 200 individuals from the admixed population. The true individual admixture values of the admixed individuals are also available.
To evaluate the efficiency of PSMIX, we randomly selected 100 markers from the Pima-Surui dataset with no missing values and tried the four models available in STRUCTURE2.0. Burnin length and number of MCMC replications after burnin were both set to be 10,000 in the analyses. The time needed for each run of STRUCTURE2.0 increased almost linearly with the increase of number of clusters. On our PC with Pentium III 500 MHZ CPU and 384 MB SDRAM, when K = 2, about two and a half minutes were needed for each run of STRUCTURE2.0. For all PSMIX runs, we set the stopping criterion to be that the parameter difference <10-6 between consecutive iterations, or 10,000 steps, whichever was reached first. For the same Pima-Surui data with 100 markers, each run of PSMIX needed about 6 seconds.
We have implemented a likelihood based method of population structure inference into an efficient R package, PSMIX. PSMIX can be used in population genetics and disease gene mapping, wherever population stratification or individual admixture is needed to be estimated from genetic markers. Compared with other available similar programs, PSMIX has several advantages. First, it is computationally efficient and provides similar accuracy under realistic situations (Tang, et al.  and Liu et al., Technical Report ). And thus the confidence intervals of the estimates can be constructed via resampling methods, e.g., the bootstrap method . Second, as shown in Tang et al. , it performs a little better (compared with STRUCTURE) under some conditions involving a small number of ancestors and markers. We note that L-POP is also computationally efficient. However, it is not clear if L-POP can perform better under such conditions. Third, it is very flexible. It is likelihood based and can be easily incorporated into study designs, such as marker choice . The program is implemented as a public R package and can be easily extended and incorporated into other packages. This is an advantage of PSMIX over STRUCTURE and L-POP, which has only executable programs.
We would like to note that the examples used in this work are mainly for the purpose of demonstrating the R package, not for the purpose of the assessing the underlying method. Please refer to Tang et al.  for a detailed assessment of the methodology.
In our simulation and application to real data, PSMIX and STRUCTURE gave very similar results. This is not surprising because estimating parameters via maximum likelihood and maximum a posterior with flat prior is formally strictly similar, where PSMIX belongs to the former and STRUCTURE belongs to the latter.
Many studies have been performed to assess the ability of STRUCTURE in assigning individuals to their populations of origin using either real data or simulated data [3, 44–47]. However, very limited studies have been performed to assess the ability of STRUCTURE in detecting the number of populations. Recently, Evanno et al.  performed a systematic study on this issue using simulations. They simulated amplified fragment length polymorphism (AFLP) and microsatellite genetic data under three population structure models: the island model, a contact zone, and a hierarchical island model . Their major finding is that the "log probability of data", an ad hoc criterion suggested by Pritchard et al.  for detecting the number of populations, does not provide a correct estimation of the number of populations most of the time . However, they found that another ad hoc statistic, which is based on the rate of change in the log probability of the data between successive numbers of populations, can accurately detect the uppermost hierarchical level of structure . They also found some other factors that can affect the detection of the number of populations . These findings are important and useful in that with the increasing usage of STRUCTURE, they provide guidance on how to use STRUCTURE to detect the number of populations. However, PSMIX does not directly detect the number of populations in this version. Due to its computation efficiency, model selection methods such as Akaike information criterion (AIC) [28, 48], Bayesian information criterion (BIC) , and even more general, penalized likelihood based methods [50, 51] can be used for this purpose. The findings of Evanno et al.  may be incorporated into PSMIX as well. This is one of our future works.
EM approach and Bayesian MCMC approach have their own advantages and disadvantages. They both can trap in local modes, although theoretically speaking, Bayesian MCMC approach can converge to the true value eventually, maybe after an unrealistic long time. However, the Bayesian MCMC approach, in addition, has the label switching problem. Two authors (Stephens and Donnelly) of the paper where the method of STRUCTURE was proposed  mentioned in other papers [52, 53] on methods to deal with this problem. Although this issue is believed to be well addressed by STRUCTURE, it does make the Bayesian MCMC approach more complicated. However, this topic is beyond the scope of this work. From the users' point of view, they only see the computation efficiency and stability of the methods.
We think that it may be necessary to explicitly explain some details about the models mentioned in this work. First, the orientation of Tang et al.  is different from that of STRUCTURE, L-POP, and Liu et al. . The goal of the former was to estimate individual admixture for the admixed individuals. The original focus of the latter was to "identify discrete clusters roughly corresponding to subpopulations" . STRUCTURE, L-POP, and Liu et al.  use methods for clustering, although they "can also be applied to an admixture model" . So initially, Tang et al.  faced a population (the "admixed group" in their paper) that is currently in Hardy-Weinberg equilibrium, but was created as the result of admixture at some point in the past. However, as emphasized in Tang et al. , the problem may not be identifiable without the inclusion of pseudo-ancestors who are proxies of the true pure ancestry . Here the nonidentifiablity issue is related to the problem, and by no means pertains to the method. In other words, the nonidentifiablity issue exists and has nothing to do with the statistical methods to be used, if pseudo-ancestors are not included. Therefore, the actual data Tang et al.  dealt with consist of "I0 individuals from the admixed group, as well as IK subjects from each of the K ancestral populations" , that is, a stratified "pooled" population. So the actual data all these methods deal with are the same in the sense that the data consist of stratified populations within which Hardy-Weinberg equilibrium holds. One major difference is that Tang et al.  only focus on the individual admixture of the people in the admixed population (their original population). Facing the same data, the method in Tang et al.  is for clustering as well, in spirit. They included pseudo-ancestors and used clustering method in order to estimate individual admixture. In other words, all the aforementioned methods are for population stratification, and can be applied to estimate individual admixture. Thus the comparisons made in this work are appropriate. We also want to emphasize here the importance of inclusion of ancestral populations or their surrogates when individual admixture is needed; otherwise the problem may not be identifiable no matter what method to use.
In summary, we have implemented a new, likelihood based method for inference of population stratification and individual admixture which is available as a public R package. Although the package has several advantages over its peers, we strongly suggest that the users use different software in their analysis. If the results from these software are consistent; this may provide more support for the results; if the results are not consistent, further investigation is needed. A potential limitation is the assumption of independence among markers behind PSMIX, which will be addressed in future versions of PSMIX.
Availability and requirements
Project name: PSMIX
Project home page: http://bioinformatics.med.yale.edu/PSMIX
Operating system(s): MS Windows, Linux, Mac
Programming language: C, R
Other requirements: R 2.0 or higher
Any restrictions to use by non-academics: none
We thank the two anonymous reviewers for their insightful and constructive comments which have greatly improved the presentation of our work. We thank Dr. Hua Tang for sending us her presentation slides. This work was supported in part by NIH grants GM59507 and GM57672.
- Patterson N, Hattangadi N, Lane B, Lohmueller KE, Hafler DA, Oksenberg JR, Hauser SL, Smith MW, O'Brien SJ, Altshuler D, Daly M, Reich D: Methods for high-density admixture mapping of disease genes. Am J Hum Genet 2004,74(5):979–1000. 10.1086/420871PubMed CentralView ArticlePubMedGoogle Scholar
- Pritchard JK, Stephens M, Donnelly P: Inference of population structure using multilocus genotype data. Genetics 2000,155(2):945–959.PubMed CentralPubMedGoogle Scholar
- Rosenberg NA, Burke T, Elo K, Feldman MW, Freidlin PJ, Groenen MA, Hillel J, Maki-Tanila A, Tixier-Boichard M, Vignal A, Wimmersh K, Weigend S: Empirical evaluation of genetic clustering methods using multilocus genotypes from 20 chicken breeds. Genetics 2001,159(2):699–713.PubMed CentralPubMedGoogle Scholar
- Rosenberg NA, Pritchard JK, Weber JL, Cann HM, Kidd KK, Zhivotovsky LA, Feldman MW: Genetic structure of human populations. Science 2002,298(5602):2381–2385. 10.1126/science.1078311View ArticlePubMedGoogle Scholar
- Cardon LR, Palmer LJ: Population stratification and spurious allelic association. Lancet 2003,361(9357):598–604. 10.1016/S0140-6736(03)12520-2View ArticlePubMedGoogle Scholar
- Freedman ML, Reich D, Penney KL, McDonald GJ, Mignault AA, Patterson N, Gabriel SB, Topol EJ, Smoller JW, Pato CN, Pato MT, Petryshen TL, Kolonel LN, Lander ES, Sklar P, Henderson B, Hirschhorn JN, Altshuler D: Assessing the impact of population stratification on genetic association studies. Nat Genet 2004,36(4):388–393. 10.1038/ng1333View ArticlePubMedGoogle Scholar
- Montana G, Pritchard JK: Statistical tests for admixture mapping with case-control and cases-only data. Am J Hum Genet 2004,75(5):771–789. 10.1086/425281PubMed CentralView ArticlePubMedGoogle Scholar
- Reich D, Patterson N: Will admixture mapping work to find disease genes? Philos Trans R Soc Lond B Biol Sci 2005,360(1460):1605–1607. 10.1098/rstb.2005.1691PubMed CentralView ArticlePubMedGoogle Scholar
- Marchini J, Cardon LR, Phillips MS, Donnelly P: The effects of human population structure on large genetic association studies. Nat Genet 2004,36(5):512–517. 10.1038/ng1337View ArticlePubMedGoogle Scholar
- Chen HS, Zhu X, Zhao H, Zhang S: Qualitative semi-parametric test for genetic associations in case-control designs under structured populations. Ann Hum Genet 2003,67(Pt 3):250–264. 10.1046/j.1469-1809.2003.00036.xView ArticlePubMedGoogle Scholar
- Risch NJ: Searching for genetic determinants in the new millennium. Nature 2000,405(6788):847–856. 10.1038/35015718View ArticlePubMedGoogle Scholar
- National research council: The Evaluation of Forensic DNA Evidence. 1996.Google Scholar
- Kim JJ, Verdu P, Pakstis AJ, Speed WC, Kidd JR, Kidd KK: Use of autosomal loci for clustering individuals and populations of East Asian origin. Hum Genet 2005,117(6):511–519. 10.1007/s00439-005-1334-8View ArticlePubMedGoogle Scholar
- Overall AD, Nichols RA: A method for distinguishing consanguinity and population substructure using multilocus genotype data. Mol Biol Evol 2001,18(11):2048–2056.View ArticlePubMedGoogle Scholar
- Evanno G, Regnaut S, Goudet J: Detecting the number of clusters of individuals using the software STRUCTURE: a simulation study. Mol Ecol 2005,14(8):2611–2620. 10.1111/j.1365-294X.2005.02553.xView ArticlePubMedGoogle Scholar
- Petit E, Balloux F, Goudet J: Sex-biased dispersal in a migratory bat: a characterization using sex-specific demographic parameters. Evolution Int J Org Evolution 2001,55(3):635–640.View ArticleGoogle Scholar
- Corander J, Waldmann P, Marttinen P, Sillanpaa MJ: BAPS 2: enhanced possibilities for the analysis of genetic population structure. Bioinformatics 2004,20(15):2363–2369. 10.1093/bioinformatics/bth250View ArticlePubMedGoogle Scholar
- Corander J, Waldmann P, Sillanpaa MJ: Bayesian analysis of genetic differentiation between populations. Genetics 2003,163(1):367–374.PubMed CentralPubMedGoogle Scholar
- Dawson KJ, Belkhir K: A Bayesian approach to the identification of panmictic populations and the assignment of individuals. Genet Res 2001,78(1):59–77. 10.1017/S001667230100502XView ArticlePubMedGoogle Scholar
- Excoffier L, Estoup A, Cornuet JM: Bayesian analysis of an admixture model with mutations and arbitrarily linked markers. Genetics 2005,169(3):1727–1738. 10.1534/genetics.104.036236PubMed CentralView ArticlePubMedGoogle Scholar
- Falush D, Stephens M, Pritchard JK: Inference of population structure using multilocus genotype data:linked loci and correlated allele frequencies. Genetics 2003,164(4):1567–1587.PubMed CentralPubMedGoogle Scholar
- Fu R, Dey DK, Holsinger KE: Bayesian models for the analysis of genetic structure when populations are correlated. Bioinformatics 2005,21(8):1516–1529. 10.1093/bioinformatics/bti178View ArticlePubMedGoogle Scholar
- Guillot G, Estoup A, Mortier F, Cosson JF: A spatial statistical model for landscape genetics. Genetics 2005,170(3):1261–1280. 10.1534/genetics.104.033803PubMed CentralView ArticlePubMedGoogle Scholar
- Guillot G, Mortier F, Estoup A: Geneland: A computer package for landscape genetics. Molecular Ecology Notes 2005,5(3):708–711.View ArticleGoogle Scholar
- Hoggart CJ, Parra EJ, Shriver MD, Bonilla C, Kittles RA, Clayton DG, McKeigue PM: Control of confounding of genetic associations in stratified populations. Am J Hum Genet 2003,72(6):1492–1504. 10.1086/375613PubMed CentralView ArticlePubMedGoogle Scholar
- Holsinger KE, Wallace LE: Bayesian approaches for the analysis of population genetic structure: an example from Platanthera leucophaea (Orchidaceae). Mol Ecol 2004,13(4):887–894. 10.1111/j.1365-294X.2004.02052.xView ArticlePubMedGoogle Scholar
- Liu N, Wu B, Zhao H: Inference of population structure using mixture model. Technical report 2005. [http://bioinformatics.med.yale.edu/psmix]Google Scholar
- Purcell S, Sham P: Properties of structured association approaches to detecting population stratification. Hum Hered 2004,58(2):93–107. 10.1159/000083030View ArticlePubMedGoogle Scholar
- Satten GA, Flanders WD, Yang Q: Accounting for unmeasured population substructure in case-control studies of genetic association using a novel latent-class model. Am J Hum Genet 2001,68(2):466–477. 10.1086/318195PubMed CentralView ArticlePubMedGoogle Scholar
- Tang H, Peng J, Wang P, Risch NJ: Estimation of individual admixture: analytical and study design considerations. Genet Epidemiol 2005,28(4):289–301. [http://www.fhcrc.org/science/labs/tang] 10.1002/gepi.20064View ArticlePubMedGoogle Scholar
- Wang J: Maximum-likelihood estimation of admixture proportions from genetic data. Genetics 2003,164(2):747–765.PubMed CentralPubMedGoogle Scholar
- Manel S, Gaggiotti OE, Waples RS: Assignment methods: matching biological questions with appropriate techniques. TRENDS in Ecology and Evolution 2005,20(3):136–142. 10.1016/j.tree.2004.12.004View ArticlePubMedGoogle Scholar
- Coulon A, Guillot G, Cosson J-F, Angibault JMA, Aulagnier S, Cargnelutti B, Galan M, Hewison AJM: Genetics structure is influenced by lansdcape features. Empirical evidence from a roe deer population. Molecular Ecology Notes, in press.Google Scholar
- Banks MA, Eichert W: WHICHRUN (version 3.2): a computer program for population assignment of individuals based on multilocus genotype data. J Hered 2000,91(1):87–89. 10.1093/jhered/91.1.87View ArticlePubMedGoogle Scholar
- Cornuet JM, Piry S, Luikart G, Estoup A, Solignac M: New methods employing multilocus genotypes to select or exclude populations as origins of individuals. Genetics 1999,153(4):1989–2000.PubMed CentralPubMedGoogle Scholar
- Rannala B, Mountain JL: Detecting immigration by using multilocus genotypes. Proc Natl Acad Sci USA 1997,94(17):9197–9201. 10.1073/pnas.94.17.9197PubMed CentralView ArticlePubMedGoogle Scholar
- McKeigue PM, Carpenter JR, Parra EJ, Shriver MD: Estimation of admixture and detection of linkage in admixed populations by a Bayesian approach: application to African-American populations. Ann Hum Genet 2000,64(Pt 2):171–186. 10.1046/j.1469-1809.2000.6420171.xView ArticlePubMedGoogle Scholar
- Hoggart CJ, Shriver MD, Kittles RA, Clayton DG, McKeigue PM: Design and analysis of admixture mapping studies. AmJ Hum Genet 2004,74(5):965–978. 10.1086/420855View ArticleGoogle Scholar
- Li SL, Yamamoto T, Yoshimoto T, Uchihi R, Mizutani M, Kurimoto Y, Tokunaga K, Jin F, Katsumata Y, Saitou N: Phylogenetic relationship of the populations within and around Japan using 105 short tandem repeat polymorphic loci. Hum Genet 2006,118(6):695–707. 10.1007/s00439-005-0106-9View ArticlePubMedGoogle Scholar
- Kuroda Y, Kaga A, Tomooka N, Vaughan DA: Population genetic structure of Japanese wild soybean (Glycine soja) based on microsatellite variation. Mol Ecol 2006,15(4):959–974.View ArticlePubMedGoogle Scholar
- Manel S, Bellemain E, Swenson JE, Francois O: Assumed and inferred spatial structure of populations: the Scandinavian brown bears revisited. Mol Ecol 2004,13(5):1327–1331. 10.1111/j.1365-294X.2004.02074.xView ArticlePubMedGoogle Scholar
- Pearse DE, Arndt AD, Valenzuela N, Miller BA, Cantarelli V, Sites JW Jr: Estimating population structure under nonequilibrium conditions in a conservation context: continent-wide population genetics of the giant Amazon river turtle, Podocnemis expansa (Chelonia; Podocnemididae). Mol Ecol 2006,15(4):985–1006.View ArticlePubMedGoogle Scholar
- Dempster AP, Laird NM, Rubin DB: Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society Series B 1977, 34: 1–38.Google Scholar
- Yang BZ, Zhao H, Kranzler HR, Gelernter J: Practical population group assignment with selected informative markers: characteristics and properties of Bayesian clustering via STRUCTURE. Genet Epidemiol 2005,28(4):302–312. 10.1002/gepi.20070View ArticlePubMedGoogle Scholar
- Pritchard JK, Donnelly P: Case-control studies ofassociation in structured or admixed populations. Theor Popul Biol 2001,60(3):227–237. 10.1006/tpbi.2001.1543View ArticlePubMedGoogle Scholar
- Turakulov R, Easteal S: Number of SNPS loci needed to detect population structure. Hum Hered 2003,55(1):37–45. 10.1159/000071808View ArticlePubMedGoogle Scholar
- Manel S, Berthier P, Luikart G: Detecting Wildlife Poaching: Identifying the Origin of Individuals with Bayesian Assignment Tests and Multilocus Genotypes. Conservation Biology 2002,16(3):650–659. 10.1046/j.1523-1739.2002.00576.xView ArticleGoogle Scholar
- Akaike H: A new look at the statistical identification model. IEEE Trans Automatic Control 1974, 19: 716–723. 10.1109/TAC.1974.1100705View ArticleGoogle Scholar
- Zhu X, Zhang S, Zhao H, Cooper RS: Association mapping, using a mixture model for complex traits. Genet Epidemiol 2002,23(2):181–196. 10.1002/gepi.210View ArticlePubMedGoogle Scholar
- Chen H, Chen J, Kalbfleisch JD: A modied likelihood ratio test for homogeneity in finite mixture models. Journal of Royal Statistical Society B 2001, 63: 19–29. 10.1111/1467-9868.00273View ArticleGoogle Scholar
- Chen H, Chen J, Kalbfleisch JD: Testing for a finite mixture model with two components. Journal of Royal Statistical Society B 2004, 66: 95–115. 10.1111/j.1467-9868.2004.00434.xView ArticleGoogle Scholar
- Stephens M: Dealing with label-switching in mixture models. Journal of Royal Statistical Society B 2000, 62: 795–809. 10.1111/1467-9868.00265View ArticleGoogle Scholar
- Stephens M, Smith NJ, Donnelly P: A new statistical method for haplotype reconstruction from population data. Am J Hum Genet 2001,68(4):978–989. 10.1086/319501PubMed 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.