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A complete classification of epistatic two-locus models.

Hallgrímsdóttir IB, Yuster DS - BMC Genet. (2008)

Bottom Line: Our approach is geometric and we show that there are 387 distinct types of two-locus models, which can be reduced to 69 when symmetry between loci and alleles is accounted for.The circuits provide information on epistasis beyond that contained in the additive x additive, additive x dominance, and dominance x dominance interaction terms.We discuss the connection between our classification and standard epistatic models and demonstrate its utility by analyzing a previously published dataset.

View Article: PubMed Central - HTML - PubMed

Affiliation: Department of Statistics, University of Oxford, 1 South Parks Road, OX1 3TG, UK. ingileif@stats.ox.ac.uk

ABSTRACT

Background: The study of epistasis is of great importance in statistical genetics in fields such as linkage and association analysis and QTL mapping. In an effort to classify the types of epistasis in the case of two biallelic loci Li and Reich listed and described all models in the simplest case of 0/1 penetrance values. However, they left open the problem of finding a classification of two-locus models with continuous penetrance values.

Results: We provide a complete classification of biallelic two-locus models. In addition to solving the classification problem for dichotomous trait disease models, our results apply to any instance where real numbers are assigned to genotypes, and provide a complete framework for studying epistasis in QTL data. Our approach is geometric and we show that there are 387 distinct types of two-locus models, which can be reduced to 69 when symmetry between loci and alleles is accounted for. The model types are defined by 86 circuits, which are linear combinations of genotype values, each of which measures a fundamental unit of interaction.

Conclusion: The circuits provide information on epistasis beyond that contained in the additive x additive, additive x dominance, and dominance x dominance interaction terms. We discuss the connection between our classification and standard epistatic models and demonstrate its utility by analyzing a previously published dataset.

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Power to detect association. Power to detect association. The plots show the maximum value of the likelihood ratio test statistic observed for randomly generated data from each of the 387 shapes.
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Figure 7: Power to detect association. Power to detect association. The plots show the maximum value of the likelihood ratio test statistic observed for randomly generated data from each of the 387 shapes.

Mentions: We generated 50, 000 random vectors of length 9. For the QTL analysis we fixed the population mean of the phenotype, fixed the allele frequencies of A and B, and then normalized each random vector to give the desired population mean. For each vector we generated 10 datasets, each with sample size 300, and fit both the full model and an additive model (note that for all of the models the 6 interaction coordinates are non-zero so the tests all have the same degrees of freedom). We used the average likelihood ratio statistic as an indicator of the power to detect interaction for that particular model. For each random model we then recorded which of the 387 model shapes it fell into and for each shape looked at maximum of the likelihood ratio statistic. In the first panel of Figure 7 we show the maximum for each shape. These maxima are highly variable between shapes, indicating that some types of interactions are easier to detect than others. We also observed that there is a strong association between large values of the likelihood ratio test statistic and the number of polygons a shape divides the square into.


A complete classification of epistatic two-locus models.

Hallgrímsdóttir IB, Yuster DS - BMC Genet. (2008)

Power to detect association. Power to detect association. The plots show the maximum value of the likelihood ratio test statistic observed for randomly generated data from each of the 387 shapes.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC2289835&req=5

Figure 7: Power to detect association. Power to detect association. The plots show the maximum value of the likelihood ratio test statistic observed for randomly generated data from each of the 387 shapes.
Mentions: We generated 50, 000 random vectors of length 9. For the QTL analysis we fixed the population mean of the phenotype, fixed the allele frequencies of A and B, and then normalized each random vector to give the desired population mean. For each vector we generated 10 datasets, each with sample size 300, and fit both the full model and an additive model (note that for all of the models the 6 interaction coordinates are non-zero so the tests all have the same degrees of freedom). We used the average likelihood ratio statistic as an indicator of the power to detect interaction for that particular model. For each random model we then recorded which of the 387 model shapes it fell into and for each shape looked at maximum of the likelihood ratio statistic. In the first panel of Figure 7 we show the maximum for each shape. These maxima are highly variable between shapes, indicating that some types of interactions are easier to detect than others. We also observed that there is a strong association between large values of the likelihood ratio test statistic and the number of polygons a shape divides the square into.

Bottom Line: Our approach is geometric and we show that there are 387 distinct types of two-locus models, which can be reduced to 69 when symmetry between loci and alleles is accounted for.The circuits provide information on epistasis beyond that contained in the additive x additive, additive x dominance, and dominance x dominance interaction terms.We discuss the connection between our classification and standard epistatic models and demonstrate its utility by analyzing a previously published dataset.

View Article: PubMed Central - HTML - PubMed

Affiliation: Department of Statistics, University of Oxford, 1 South Parks Road, OX1 3TG, UK. ingileif@stats.ox.ac.uk

ABSTRACT

Background: The study of epistasis is of great importance in statistical genetics in fields such as linkage and association analysis and QTL mapping. In an effort to classify the types of epistasis in the case of two biallelic loci Li and Reich listed and described all models in the simplest case of 0/1 penetrance values. However, they left open the problem of finding a classification of two-locus models with continuous penetrance values.

Results: We provide a complete classification of biallelic two-locus models. In addition to solving the classification problem for dichotomous trait disease models, our results apply to any instance where real numbers are assigned to genotypes, and provide a complete framework for studying epistasis in QTL data. Our approach is geometric and we show that there are 387 distinct types of two-locus models, which can be reduced to 69 when symmetry between loci and alleles is accounted for. The model types are defined by 86 circuits, which are linear combinations of genotype values, each of which measures a fundamental unit of interaction.

Conclusion: The circuits provide information on epistasis beyond that contained in the additive x additive, additive x dominance, and dominance x dominance interaction terms. We discuss the connection between our classification and standard epistatic models and demonstrate its utility by analyzing a previously published dataset.

Show MeSH
Related in: MedlinePlus