# A Bayesian Algorithm for Functional Mapping of Dynamic Complex Traits

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## Abstract

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## 1. Introduction

## 2. Bayesian Functional Mapping

#### 2.1. Linear Model

#### 2.2. Modeling the Mean-Covariance Structures

#### 2.3. Likelihood

#### 2.4. Parameter Estimation and Algorithm

#### Estimation theory

#### Algorithm implementation

#### Estimation issues

#### 2.5. Structuring the Covariance Matrix

#### Autoregressive model

#### Structured Antedependence Model

#### 2.6. Bayes Factor

**y**, ${B}_{{l}_{1}{l}_{2}}=\frac{P\left(\mathbf{y}\right|\kappa ={l}_{1})}{P\left(\mathbf{y}\right|\kappa ={l}_{2})}$, is known as the Bayes factor for comparing $\kappa ={l}_{1}$ with $\kappa ={l}_{2}$. The Bayes factor does not depend on the prior distribution of κ. Recently, the Bayesian analysis of mixtures with an unknown number of components has received great attention [13,35,46]. In practice, a Bayes factor larger than 100 can often be regarded as an evidence for the preference of ${l}_{1}$ QTLs over ${l}_{2}$ QTLs.

## 3. A Worked Example

#### 3.1. Mapping Population

#### 3.2. Results

**Table 1.**Bayesian estimates of QTL locations and genotype-specific growth curves for the QTLs detected on mouse chromosome 6, 7 and 10. Numbers in parentheses are the $95\%$ equal-tail confidence intervals.

Parameter | $QQ$ | $Qq$ | $qq$ | |||

Chromosome 6 | ||||||

Location, cM from the first marker 82.68 (67.77, 92.96) | ||||||

α | 36.09 | (35.20,37.04) | 34.94 | (34.36,35.52) | 33.12 | (32.36,33.93) |

β | 11.93 | (11.44,12.45) | 11.58 | (11.16,12.03) | 11.07 | (10.65,11.51) |

γ | 0.65 | (0.64,0.66) | 0.65 | (0.64,0.66) | 0.65 | (0.64,0.67) |

Chromosome 7 | ||||||

Location, cM from the first marker 46.84 (38.80,56.02) | ||||||

α | 36.55 | (35.50, 37.73) | 35.61 | (34.56,36.50) | 33.38 | (32.54,34.33) |

β | 11.83 | (11.43,12.34) | 11.27 | (10.90,11.73) | 11.25 | (10.76, 11.70) |

γ | 0.65 | (0.63,0.66) | 0.64 | (0.63,0.65) | 0.65 | (0.63,0.66) |

Chromosome 10 | ||||||

Location, cM from the first marker 77.78 (68.75,80.96) | ||||||

α | 35.41 | (34.33, 36.52) | 34.71 | (33.67,35.70) | 33.59 | (32.61,34.42) |

β | 11.67 | (11.23,12.19) | 11.47 | (11.23,11.81) | 11.01 | (10.61, 11.44) |

γ | 0.65 | (0.64,0.66) | 0.64 | (0.63,0.66) | 0.65 | (0.63,0.66) |

**Figure 1.**A profile of Estimated marginal posterior distribution of the QTL location by assuming that exactly one QTL is located on one of the chromosome respectively.

## 4. Monte Carlo Simulation

**Figure 2.**Fitted growth curves for the three QTL genotypes ($qq$, red; $Qq$, blue; $QQ$, black) assuming a single QTL is located on mouse chromosome 6, 7 and 10.

**Figure 3.**Dynamic changes of the additive and dominant effect due to the QTL located on mouse chromosome 6,7,and 10 respectively.

**Figure 4.**Estimated marginal posterior distributions of the QTL location over a simulated linkage group with different matric-structuring approaches, unstructured (top), SAD(1)-structured (middle) and AR(1)-structured (bottom).

## 5. Discussion

**Figure 5.**The profile of the log-likelihood ratio (LR) test statistics between the full (there is a QTL) and reduced (there is no QTL) models across a simulated linkage group. The covariance matrix was structured by the SAD(1) model.

**Table 2.**Bayesian estimates of QTL locations and genotype-specific growth curves for an assumed QTL from the simulated data set for an F${}_{2}$ population of 450 individuals based on different covariance-structuring approaches. Numbers in parentheses are the $95\%$ equal-tail confidence intervals.

Parameter | $QQ$ | $Qq$ | $qq$ | |||

Unstructured approach | ||||||

Location | 32.74 (28.02, 38.16) | |||||

α | 36.67 | (35.81, 37.46) | 36.03 | (35.47, 36.63) | 33.67 | (32.67, 34.31) |

β | 11.83 | (11.95, 12.64) | 11.22 | (10.97, 11.50) | 11.30 | (10.94, 11.60) |

γ | 0.66 | (0.64, 0.67) | 0.64 | (0.63, 0.65) | 0.64 | (0.63, 0.66) |

SAD(1)-structured approach | ||||||

Location | 34.63 (33.01, 36.30) | |||||

α | 36.58 | (35.85, 37.36) | 35.88 | (35.37, 36.39) | 33.81 | (33.09, 34.55) |

β | 12.04 | (11.65, 12.45) | 11.27 | (11.01, 11.54) | 11.46 | (11.09, 11.83) |

γ | 0.66 | (0.65, 0.67) | 0.64 | (0.63, 0.65) | 0.65 | (0.64, 0.65) |

AR(1)-structured approach | ||||||

Location | 33.54 (25.54, 41.39) | |||||

α | 36.60 | (35.59, 37.66) | 35.57 | (34.88, 36.35) | 33.61 | (32.65, 34.54) |

β | 12.04 | (11.61, 12.48) | 11.23 | (10.99, 11.56) | 11.42 | (11.05, 11.83) |

γ | 0.65 | (0.64, 0.66) | 0.63 | (0.62, 0.65) | 0.66 | (0.64, 0.67) |

## Appendix A: Estimating (λ,Q,Θ,Σ)

**Parameter Estimation and Algorithm**section is constructed as follows:

**Step 1**. Initialize the iteration at an arbitrary point $({\lambda}^{0},{\mathbf{Q}}^{0},{\Theta}^{0},{\Sigma}^{0})$, which has a positive posterior density;

**Step 2**. Modify four blocks of the unknowns parameters and move to a new state from the previous step $({\lambda}^{k-1},{\mathbf{Q}}^{k-1},{\Theta}^{k-1},{\Sigma}^{k-1})$ through a successive generation of new values ${\lambda}^{k},{\mathbf{Q}}^{k},{\Theta}^{k}$, and ${\Sigma}^{k}$. More specifically, given the values of the unknowns $(\lambda ,\mathbf{Q},\Theta ,\Sigma )$ from the current state, we proceed as follows:

## Appendix B. Estimating $({\sigma}^{2},\rho )$

## Appendix C. Estimating $({\nu}^{2},\varphi )$

## Appendix D. Alternative Approach for Modeling Σ

- (1)
- Given the current positive-definitive matrix ${\Sigma}_{k}$, we set ${\mathbf{W}}_{k}=\mathrm{log}{\Sigma}_{k}$, in the sense that ${\mathbf{W}}_{k}={\sum}_{i=0}^{\infty}\frac{{\left({\Sigma}_{k}\right)}^{i}}{i!}$;
- (2)
- Randomly generate a symmetric p by p matrix T, with elements ${t}_{ij}={z}_{ij}/{\left({\sum}_{l\u2a7dm}{z}_{lm}^{2}\right)}^{1/2},$ where ${z}_{ij}\backsim \phantom{\rule{0.166667em}{0ex}}i.i.d.N(0,1),\phantom{\rule{0.277778em}{0ex}}for\phantom{\rule{0.277778em}{0ex}}i\u2a7dj$;
- (3)
- Set ${\mathbf{W}}^{*}={\mathbf{W}}_{k}+\nu T$ where ν is generated from $N(0,1)$;
- (4)
- Update ${\mathbf{W}}_{k}$ with an acceptance probability $\mathrm{min}(1,{\pi}^{*}\left({\mathbf{W}}^{*}\right|\mathbf{y},\Theta ,\mathbf{Q},\lambda )/{\pi}^{*}\left({\Sigma}_{k}\right|\mathbf{y},\Theta ,\mathbf{Q},\lambda ))$.

## Acknowledgements

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Liu, T.; Wu, R.
A Bayesian Algorithm for Functional Mapping of Dynamic Complex Traits. *Algorithms* **2009**, *2*, 667-691.
https://doi.org/10.3390/a2020667

**AMA Style**

Liu T, Wu R.
A Bayesian Algorithm for Functional Mapping of Dynamic Complex Traits. *Algorithms*. 2009; 2(2):667-691.
https://doi.org/10.3390/a2020667

**Chicago/Turabian Style**

Liu, Tian, and Rongling Wu.
2009. "A Bayesian Algorithm for Functional Mapping of Dynamic Complex Traits" *Algorithms* 2, no. 2: 667-691.
https://doi.org/10.3390/a2020667