A note on estimating the posterior density of a quantitative trait locus from a Markov chain Monte Carlo sample

2002 ◽  
Vol 22 (4) ◽  
pp. 369-376 ◽  
Author(s):  
Fabian J. Hoti ◽  
Mikko J. Sillanpää ◽  
Lasse Holmström
Keyword(s):  
Quantitative Trait Locus ◽  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Quantitative Trait ◽  
Posterior Density ◽  
Monte Carlo Sample ◽  
Trait Locus

scholarly journals Markov chain Monte Carlo for mapping a quantitative trait locus in outbred populations

2000 ◽  
Vol 75 (2) ◽  
pp. 231-241 ◽  
Author(s):  
M. C. A. M. BINK ◽  
L. L. G. JANSS ◽  
R. L. QUAAS
Keyword(s):  
Quantitative Trait Locus ◽  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Quantitative Trait ◽  
Covariance Structure ◽  
Simulated Data ◽  
Simulated Tempering ◽  
Variance Ratios ◽  
Trait Locus

A Bayesian approach is presented for mapping a quantitative trait locus (QTL) using the ‘Fernando and Grossman’ multivariate Normal approximation to QTL inheritance. For this model, a Bayesian implementation that includes QTL position is problematic because standard Markov chain Monte Carlo (MCMC) algorithms do not mix, i.e. the QTL position gets stuck in one marker interval. This is because of the dependence of the covariance structure for the QTL effects on the adjacent markers and may be typical of the ‘Fernando and Grossman’ model. A relatively new MCMC technique, simulated tempering, allows mixing and so makes possible inferences about QTL position based on marginal posterior probabilities. The model was implemented for estimating variance ratios and QTL position using a continuous grid of allowed positions and was applied to simulated data of a standard granddaughter design. The results showed a smooth mixing of QTL position after implementation of the simulated tempering sampler. In this implementation, map distance between QTL and its flanking markers was artificially stretched to reduce the dependence of markers and covariance. The method generalizes easily to more complicated applications and can ultimately contribute to QTL mapping in complex, heterogeneous, human, animal or plant populations.

scholarly journals Mapping-Linked Quantitative Trait Loci Using Bayesian Analysis and Markov Chain Monte Carlo Algorithms

Genetics ◽  
1997 ◽  
Vol 146 (2) ◽  
pp. 735-743 ◽  
Author(s):  
Pekka Uimari ◽  
Ina Hoeschele
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Quantitative Trait Loci ◽  
Quantitative Trait ◽  
Allele Frequencies ◽  
Mcmc Methods ◽  
Substitution Effects ◽  
Indicator Variable ◽  
Trait Loci

A Bayesian method for mapping linked quantitative trait loci (QTL) using multiple linked genetic markers is presented. Parameter estimation and hypothesis testing was implemented via Markov chain Monte Carlo (MCMC) algorithms. Parameters included were allele frequencies and substitution effects for two biallelic QTL, map positions of the QTL and markers, allele frequencies of the markers, and polygenic and residual variances. Missing data were polygenic effects and multi-locus marker-QTL genotypes. Three different MCMC schemes for testing the presence of a single or two linked QTL on the chromosome were compared. The first approach includes a model indicator variable representing two unlinked QTL affecting the trait, one linked and one unlinked QTL, or both QTL linked with the markers. The second approach incorporates an indicator variable for each QTL into the model for phenotype, allowing or not allowing for a substitution effect of a QTL on phenotype, and the third approach is based on model determination by reversible jump MCMC. Methods were evaluated empirically by analyzing simulated granddaughter designs. All methods identified correctly a second, linked QTL and did not reject the one-QTL model when there was only a single QTL and no additional or an unlinked QTL.

Statistical Approach on Grading the Student Achievement via Normal Mixture Modeling

2012 ◽  
Author(s):  
Zairul Nor Deana Md. Desa ◽  
Ismail Mohamad ◽  
Zarina Mohd. Khalid ◽  
Hanafiah Md. Zin
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Standard Deviation ◽  
Bayesian Methods ◽  
Mixture Distribution ◽  
Coefficient Of Determination ◽  
Normal Mixture ◽  
Posterior Density ◽  
Letter Grades

Kajian dijalankan untuk membanding keputusan yang didapati daripada tiga kaedah penggredan terhadap pencapaian pelajar. Kaedah konvensional yang popular adalah kaedah Skala Tegak. Pendekatan statistik yang menggunakan kaedah Sisihan Piawai dan kaedah Bayesian bersyarat dipertimbangkan untuk memberi gred. Dalam model Bayesian, dianggapkan bahawa data adalah mengikut taburan Normal Tergabung di mana setiap gred adalah dipisahkan secara berasingan oleh parameter; min dan kadar bandingan dari taburan Normal Tergabung. Masalah yang timbul adalah sukar untuk menganggarkan ketumpatan posterior bagi parameter tersebut secara analitik. Satu penyelesaiannya adalah dengan menggunakan pendekatan Markov Chain Monte Carlo iaitu melalui algoritma pensampelan Gibbs. Kaedah Skala Tegak, kaedah Sisihan Piawai dan kaedah Bayesian bersyarat diaplikasikan untuk markah mentah peperiksaan bagi dua kumpulan pelajar. Pencapaian ketiga–tiga kaedah dibandingkan melalui nilai Kehilangan Kelas Neutral, Kehilangan Kelas Tidak Tegas dan Pekali Penentuan. Didapati keputusan dari kaedah Bayesian bersyarat menunjukkan penggredan yang lebih baik berbanding kaedah Skala Tegak dan kaedah Sisihan Piawai. Kata kunci: Kaedah penggredan, pengukuran pendidikan, Skala Tegak, kaedah Sisihan Piawai, Normal Tergabung, Markov Chain Monte Carlo, pensampelan Gibbs The purpose of this study is to compare results obtained from three methods of assigning letter grades to students’ achievement. The conventional and the most popular method to assign grades is the Straight Scale method (SS). Statistical approaches which used the Standard Deviation (GC) and conditional Bayesian methods are considered to assign the grades. In the conditional Bayesian model, we assume the data to follow the Normal Mixture distribution where the grades are distinctively separated by the parameters: means and proportions of the Normal Mixture distribution. The problem lies in estimating the posterior density of the parameters which is analytically intractable. A solution to this problem is using the Markov Chain Monte Carlo approach namely Gibbs sampler algorithm. The Straight Scale, Standard Deviation and Conditional Bayesian methods are applied to the examination raw scores of two sets of students. The performances of these methods are measured using the Neutral Class Loss, Lenient Class Loss and Coefficient of Determination. The results showed that Conditional Bayesian outperformed the Conventional Methods of assigning grades. Key words: Grading methods, educational measurement, Straight Scale, Standard Deviation method, Normal Mixture, Markov Chain Monte Carlo, Gibbs sampling

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