- Software
- Open Access

# PSMIX: an R package for population structure inference via maximum likelihood method

- Baolin Wu
^{1}, - Nianjun Liu
^{2}and - Hongyu Zhao
^{3, 4}Email author

**7**:317

https://doi.org/10.1186/1471-2105-7-317

© Wu et al; licensee BioMed Central Ltd. 2006

**Received:**23 January 2006**Accepted:**22 June 2006**Published:**22 June 2006

## Abstract

### Background

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.

### Results

We have developed PSMIX, an R package based on maximum likelihood method using expectation-maximization algorithm, for inference of population stratification and individual admixture.

### Conclusion

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.

## Keywords

- Markov Chain Monte Carlo
- Latent Class Analysis
- Population Stratification
- Approximate Bayesian Computation
- Admix Population

## Background

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 [14]. In all these situations, identifying population stratification or admixture has been an important component.

Commonly used software for population structure inference

Software | STRUCTURE | GENELAND | BAPS/BAPS2 | ADMIXMAP | PARTITION | L-POP | PSMIX |
---|---|---|---|---|---|---|---|

Method | Bayesian MCMC | Bayesian MCMC | Bayesian, MCMC is used when the number of populations ≥ 9 | Bayesian MCMC | Bayesian MCMC | Latent class analysis, EM | Clustering analysis, EM |

Features | 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 |

Assumptions | 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 |

Input parameters | 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 |

Output | 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 |

Advantages | 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. | Computationally efficient | Easy to use. Computationally efficient. Flexible to extend |

Limitations | 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 |

Platforms | Windows, Unix/Linux | Windows/Linux/Mac (R package) | Windows | Windows, Unix/Linux. R statistical package is required | Windows | Windows (DOS), Unix/Linux | Windows/Linux/Mac (R package) |

References | 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) |

URL | http://www.inapg.inra.fr/ens_rech/mathinfo/personnel/guillot/Geneland.html | http://www.genetix.univ-montp2.fr/partition/partition.htm#analyse_+exe |

## Implementation

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 [43] using C programming language. PSMIX is mainly based on the methods proposed in Tang et al. [30] and Liu et al. [27]. Three models (described in section 2.2, 2.3, and 2.4, respectively) are discussed in full detail in [27]. The second one is equivalent to the model proposed in Tang et al. [30]. 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 [30].

## Results

We used two real datasets from Rosenberg et al. [4] and one simulated dataset from Tang et al. [30] 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.

## Discussion

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. [30] and Liu et al., Technical Report [27]). And thus the confidence intervals of the estimates can be constructed via resampling methods, e.g., the bootstrap method [30]. Second, as shown in Tang et al. [30], 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 [30]. 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. [30] 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. [15] 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 [15]. Their major finding is that the "log probability of data", an ad hoc criterion suggested by Pritchard et al. [2] for detecting the number of populations, does not provide a correct estimation of the number of populations most of the time [15]. 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 [15]. They also found some other factors that can affect the detection of the number of populations [15]. 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) [49], and even more general, penalized likelihood based methods [50, 51] can be used for this purpose. The findings of Evanno et al. [15] 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 [2] 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. [30] is different from that of STRUCTURE, L-POP, and Liu et al. [27]. 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" [30]. STRUCTURE, L-POP, and Liu et al. [27] use methods for clustering, although they "can also be applied to an admixture model" [30]. So initially, Tang et al. [30] 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. [30], the problem may not be identifiable without the inclusion of pseudo-ancestors who are proxies of the true pure ancestry [25]. 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. [30] dealt with consist of "I^{0} individuals from the admixed group, as well as I^{K} subjects from each of the K ancestral populations" [30], 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. [30] 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. [30] 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.

## Conclusion

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

**License:** GPL

**Any restrictions to use by non-academics:** none

## Declarations

### Acknowledgements

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.

## Authors’ Affiliations

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## Copyright

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