# A Bayesian approach to detect QTL affecting a simulated binary and quantitative trait

- Aniek C Bouwman
^{1}Email author, - Luc LG Janss
^{2}and - Henri CM Heuven
^{3}

**5(Suppl 3)**:S4

https://doi.org/10.1186/1753-6561-5-S3-S4

© Bouwman et al; licensee BioMed Central Ltd. 2011

**Published: **27 May 2011

## Abstract

### Background

We analyzed simulated data from the 14^{th} QTL-MAS workshop using a Bayesian approach implemented in the program iBay. The data contained individuals genotypes for 10,031 SNPs and phenotyped for a quantitative and a binary trait.

### Results

For the quantitative trait we mapped 8 out of 30 additive QTL, 1 out of 3 imprinted QTL and both epistatic pairs of QTL successfully. For the binary trait we mapped 11 out of 22 additive QTL successfully. Four out of 22 pleiotropic QTL were detected as such.

### Conclusions

The Bayesian variable selection method showed to be a successful method for genome-wide association. This method was reasonably fast using dense marker maps.

## Background

Discovering the genetic architecture of traits is not a trivial task, but it is important for our understanding of complex phenotypes. Dense marker maps make it possible to perform genome-wide association (GWA) studies to detect QTL. Bayesian variable selection methods [1] are powerful in association studies, because they can simultaneous take polygenic and all SNP effects into account. This is implemented in packages such as ‘Genomic Selection’ [2] and ‘iBay’ [3]. Meuwissen and Goddard [4] describe how this method could be extended to multi-trait models.

In this paper we analyzed simulated data using a Bayesian approach implemented in the program iBay. The QTL-MAS workshop gives the opportunity to test this method on data with a QTL structure that is unknown beforehand. Although it is hypothesized that the quantitative and binary trait in the dataset are to some degree affected by the same QTL we used an univariate approach because the multivariate version of iBay is still in progress.

## Methods

### Data

The pedigree contained 5 generations, all generations were genotyped but only the first 4 generations (2,326 individuals) were phenotyped for a quantitative and a binary trait. The genome consisted of 5 chromosomes and was genotyped for 10,031 SNPs. A full description of the dataset can be found at the 14^{th} QTL-MAS workshop website [5].

### ASReml analysis

First both traits were analyzed in ASReml [6]. An animal model was applied to estimate the heritability of both traits. A bivariate animal model was applied to estimate the genetic correlation between both traits. In this bivariate analysis the binary trait was analyzed in a linear model. Univariate analysis of the binary trait showed that a linear model gives similar estimates as a threshold model (results not shown).

### QTL analysis

where **y** is the quantitative phenotype or the underlying liabilities of the binary phenotype for each individual. Terms
fit marker association effects where
is a vector with allele substitution effects, with
*~ N*(0, **I**); **X**_{
k
} is the incidence matrix relating allele substitution effects to observed marker genotypes and
is a scaling factor that shrinks allele effects and models the variance explained by the marker. The scaling factors are conditionally estimated as simple normally distributed regressions, and can be interpreted as a standard deviation. **Z**_{
u
}, **Z**_{
s
} and **Z**_{
d
} are known incidence matrices relating observations to random genetic effects **u**, with
, sire **s**, with
, and dam **d**, with
, respectively. **A** is the numerator relationship matrix, for the sire-dam model the progeny was not included in the relationship matrix. The error vector is
, with identity matrix **I**.

*σ*

_{ k }, is done in a dualistic manner by applying a mixture distribution on scaling factors that heavily shrink the effects for most of the markers, effectively removing most of the markers from the model. Only a small part of marker effects are less severely shrunken, identifying markers with important associations. This prior mixture distribution is a mixture of a normal and a truncated normal distribution:

where the first distribution is referred to as the ‘null’ distribution that models the majority of markers with no effect using π_{0} = 0.95 and setting
to a small value. Here
was set to 0.015 for the quantitative trait and to 0.005 for the binary trait (‘null’ markers explain ~2% of phenotypic variation,
. The second distribution models markers with important effects. For this second distribution a truncated normal is used so that the signs of estimated allele effects will be identifiable, and the parameter
is estimated from the data, using a flat prior. In this case π_{0}/π_{1} was set at 0.95/0.05.

For the mixture prior, the model estimates a ‘mixture indicator’ which indicates for each marker whether it was estimated to belong to the first distribution or the second distribution. The first distribution is indicated by 0 and the second one with 1, so that, after averaging in the MCMC, a value ranging from 0 to 1 which is a posterior probability for each marker to have a large effect (i.e. the probability to belong to the second distribution) and can be used for model selection [1].

### Applied MCMC techniques

All samplers were single site Gibbs samplers. The particular parameterization with scaling factors was chosen so that scaling factors *σ*_{
k
} can be sampled as ‘regressions’ from normal distributions (N(0,1)) and with normal prior distributions.

Multiple MCMC chains of 50,000 cycles with a burn-in period of 1,000 cycles were run until the estimated effective number of samples was >100 for all parameters. The estimated effective number of samples was used as convergence diagnostic based on comparison of within and between chain variances.

### Identification of associated markers

As indicated above, the posterior probability for a marker to come from the second mixture distribution can be used for model selection. We used two approaches to determine a cut-off on these posterior probabilities for the selection of significant associations, denoting the estimated posterior probability by
and the prior probabilities used in the model by *π*_{
0
} and *π*_{
1
}.

Using guidelines by Kass and Raftery [7] to judge Bayes Factors, a value above 3.2 is ‘substantial’, a value above 10 is ‘strong’, and a value above 100 is ‘decisive’.

### Post-marker analysis

Using a simultaneous fit of all markers as in the Bayesian variable selection method can cause the signal of a QTL to be spread over multiple markers. In that case individual marker have a moderate posterior probability, but the group of markers has a high joint posterior probability. The primary joint Gibbs samples for the mixture indicators were used, which take account of the switches for adjacent markers being on or off, to derive the joint probability for having a signal in a window. Different grouping-windows with size of 1 up to and including 11 SNP in a window were tried on the output. First, a probability for the presence of a QTL at all is given. Secondly, if there is a QTL present in the window, the probability of multiple QTL in the window is given. If the mixture indicators show that more than one SNP within a window has a high probability of being in the model, this is counted to determine the probability of multiple QTL.

## Results

### ASReml

Phenotypic and genetic parameters for the quantitative (Q) and binary (B) trait.

Trait | Phenotypic variance | Genetic parameters | |
---|---|---|---|

Q | 104.35 | 0.54 | |

B | 0.21 | 0.66 | 0.23 |

### iBay

Loci associated with the quantitative (Q) and binary (B) trait, their parameter-wise Bayes Factor (pwBF) and posterior probability (Prob(2ndMix))

Trait | Locus | Chr | Position | pwBF | Prob(2ndMix) |
---|---|---|---|---|---|

Q | 5488 | 3 | 71,610,807 | 551.0 | 0.97 |

3623 | 2 | 78,604,040 | 32.8 | 0.63 | |

4485 | 3 | 22,443,619 | 20.3 | 0.52 | |

4480 | 3 | 22,030,629 | 18.7 | 0.50 | |

6703 | 4 | 27,663,560 | 16.8 | 0.47 | |

2719 | 2 | 32,741,451 | 15.3 | 0.45 | |

3405 | 2 | 66,759,090 | 15.3 | 0.45 | |

3905 | 2 | 92,573,498 | 15.3 | 0.45 | |

954 | 1 | 50,009,335 | 11.6 | 0.38 | |

952 | 1 | 49,965,266 | 9.6 | 0.34 | |

3948 | 2 | 94,982,901 | 6.9 | 0.27 | |

3402 | 2 | 66,632,577 | 6.7 | 0.26 | |

947 | 1 | 49,825,082 | 6.0 | 0.24 | |

2465 | 2 | 20,369,230 | 4.9 | 0.21 | |

4477 | 3 | 21,919,975 | 4.8 | 0.20 | |

2658 | 2 | 29,667,353 | 4.7 | 0.20 | |

4411 | 3 | 18,509,382 | 4.5 | 0.19 | |

2810 | 2 | 37,448,320 | 4.3 | 0.18 | |

4559 | 3 | 26,890,769 | 4.2 | 0.18 | |

959 | 1 | 50,316,379 | 4.2 | 0.18 | |

3381 | 2 | 65,270,284 | 4.0 | 0.17 | |

1215 | 1 | 63,017,238 | 3.9 | 0.17 | |

939 | 1 | 49,185,089 | 3.8 | 0.17 | |

3498 | 2 | 71,583,451 | 3.7 | 0.16 | |

2827 | 2 | 37,933,865 | 3.4 | 0.15 | |

B | 4480 | 3 | 22,030,629 | 1881.0 | 1.00 |

145 | 1 | 7,149,725 | 133.0 | 0.88 | |

1215 | 1 | 63,017,238 | 55.5 | 0.75 | |

3961 | 2 | 95,493,425 | 13.2 | 0.41 | |

3948 | 2 | 94,982,901 | 10.0 | 0.34 | |

6217 | 4 | 5,977,635 | 9.8 | 0.34 | |

8030 | 4 | 97,774,814 | 7.8 | 0.29 | |

2033 | 2 | 2,213,453 | 7.0 | 0.27 | |

3405 | 2 | 66,759,090 | 6.3 | 0.25 | |

4511 | 3 | 23,981,734 | 6.0 | 0.24 | |

1913 | 1 | 97,688,161 | 5.5 | 0.23 | |

3421 | 2 | 67,468,328 | 5.2 | 0.22 | |

5616 | 3 | 78,155,543 | 5.1 | 0.21 | |

6127 | 4 | 1,456,752 | 4.6 | 0.19 | |

7887 | 4 | 90,517,506 | 4.2 | 0.18 | |

1631 | 1 | 82,409,839 | 3.6 | 0.16 | |

1102 | 1 | 57,850,647 | 3.4 | 0.15 | |

1383 | 1 | 70,982,584 | 3.4 | 0.15 |

Comparison simulated and detected QTL for the quantitative (Q) and binary (B) trait.

Simulated | Detected | ||||||||
---|---|---|---|---|---|---|---|---|---|

Q | B | var | Q | B | |||||

QTL | QTL | Chr | Position | SNP | QTL | SNP | Position | SNP | Position |

1 | * | 1 | 7,536,081 | R152 | 1.84 | 145 | 7,149,725 | ||

2 | * | 1 | 50,389,545 | R960 | 1.13 | 959 | 50,316,379 | ||

3 | * | 1 | 58,038,782 | R1106 | 1.09 | 1102 | 57,850,647 | ||

4 | * | 1 | 63,386,317 | L1226 | 1.19 | 1215 | 63,017,238 | 1215 | 63,017,238 |

5 | * | 2 | 2,289,495 | R2036 | 0.97 | 2033 | 2,213,453 | ||

6 | * | 2 | 30,511,220 | L2675 | 0.48 | 2658 | 29,667,353 | ||

8 | * | 2 | 67,248,417 | L3414 | 0.87 | 3405 | 66,759,090 | 3421 | 67,468,328 |

11 | * | 2 | 94,680,408 | L3946 | 0.40 | 3948 | 94,982,901 | 3948 | 94,982,901 |

12 | * | 2 | 95,449,160 | R3959 | 1.13 | 3961 | 95,493,425 | ||

14 | * | 3 | 22,415,527 | L4483 | 4.50 | 4485 | 22,443,619 | 4480 | 22,030,629 |

17 | 3 | 71,610,807 | 5488 | 4.49 | 5488 | 71,610,807 | |||

18 | * | 3 | 78,153,081 | R5616 | 0.29 | 5616 | 78,155,543 | ||

22 | * | 4 | 6,296,223 | R6224 | 0.57 | 6217 | 5,977,635 | ||

24 | 4 | 26,749,857 | R6684 | 0.14 | 6703 | 27,663,560 | |||

30 | * | 4 | 97,651,414 | R8024 | 0.72 | 8030 | 97,774,814 | ||

31 | Epi | 1 | 49,185,089 | 939 | 7.01 | 939 | 79,185,089 | ||

32 | Epi | 1 | 50,316,379 | 959 | 959 | 50,316,379 | |||

33 | Epi | 2 | 32,617,381 | 2715 | 4.18 | 2719 | 32,741,451 | ||

34 | Epi | 2 | 33,139,075 | 2727 | 2719 | 32,741,451 | |||

36 | Imp | 2 | 78,604,040 | 3623 | 2.20 | 3623 | 78,604,040 |

Post-marker analysis of the quantitative (Q) and binary (B) trait

Trait | Region Size | Pr(≥1) | Pr(>1) | Marker start | Marker end |
---|---|---|---|---|---|

Q | 5 | 1.00 | 0.25 | 946 | 950 |

1 | 1.00 | 0.00 | 5488 | 5488 | |

6 | 0.96 | 0.11 | 4479 | 4484 | |

10 | 0.78 | 0.22 | 951 | 960 | |

10 | 0.78 | 0.15 | 3901 | 3910 | |

5 | 0.76 | 0.25 | 4485 | 4489 | |

9 | 0.60 | 0.00 | 6696 | 6704 | |

B | 1 | 1.00 | 0.00 | 4480 | 4480 |

3 | 1.00 | 0.19 | 4482 | 4484 | |

9 | 0.88 | 0.05 | 137 | 145 | |

10 | 0.86 | 0.08 | 1207 | 1216 |

### Pleiotropy

Pleiotropic SNPs and their parameter-wise Bayes Factors (pwBF) for the quantitative (Q) and binary (B) trait

SNP | Chr | Position | Q pwBF | B pwBF |
---|---|---|---|---|

4480 | 3 | 22,030,629 | 18.7 | 1881.0 |

3405 | 2 | 66,759,090 | 15.3 | 6.3 |

3948 | 2 | 94,982,901 | 6.9 | 10.0 |

1215 | 1 | 63,017,238 | 3.9 | 55.5 |

## Discussion

The technique used by iBay are a Bayesian hierarchical regression model similar to Bayesian Lasso, by introduction of a variance parameter per marker, and a model using a mixture model following the version of the Bayesian variable selection method by George and McCullogh [1]. The SNP variance originates from a mixture of two distributions, one for the SNP with an effect on the phenotype and the other for SNPs without an effect on the phenotype. The method is similar to BayesB [8]. However, BayesB uses an informative prior which is estimated from the data, in contrast iBay uses a fixed prior.

For the quantitative trait we ran 6 MCMC chains of 50,000 cycles with a burn-in period of 1,000 cycles. One chain took approximately 2.5 hour on a dual core Intel 2.33 GHz processor, so in total it took 15 hours. For the binary trait only 4 MCMC chains were needed, which took 10 hours.

A univariate QTL analysis was performed on the simulated data. However, a multivariate QTL analysis would increase the power and the precision of the pleiotropic QTL position [9, 10]. Multivariate analysis is especially beneficial when one of the traits has a low heritability [10]. The simulated data contained two traits with relatively high heritabilities, therefore, the univariate analysis was able to detect the main QTL for either trait. A multivariate analysis might be able to detect the pleiotropic QTL with small effects as well.

## Conclusions

The Bayesian variable selection method showed to be a successful method for GWA. This method was reasonably fast using dense marker maps. The univariate Bayesian analysis was able to detect the main QTL, however, a multivariate approach might be able to detect more pleiotropic QTL and to a more precise position.

## Declarations

### Acknowledgements

This article has been published as part of *BMC Proceedings* Volume 5 Supplement 3, 2011: Proceedings of the 14th QTL-MAS Workshop. The full contents of the supplement are available online at http://www.biomedcentral.com/1753-6561/5?issue=S3.

## Authors’ Affiliations

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## Copyright

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