# An Efficient Bayesian Method for Estimating the Degree of the Skewness of X Chromosome Inactivation Based on the Mixture of General Pedigrees and Unrelated Females

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Notations

#### 2.2. Building Bayesian Models

#### 2.3. Eigenvalue Decomposition and Cholesky Decomposition for Accelerating Computation Speed

**,**respectively, are the transformed ${\mathit{X}}_{1}$, ${\mathit{X}}_{2}$ and $\mathit{Z}$ based on $\mathit{\phi}={\mathit{Q}}^{T}\mathit{\Sigma}\mathit{Q}$ by the EVD; $\mathit{C}$ is a lower triangular matrix satisfying $\mathit{\phi}=\mathit{C}{\mathit{C}}^{T}$ by Cholesky decomposition; and $\mathit{h}$ follows $MVN(\mathbf{0},{\mathit{I}}_{{n}_{f}\times {n}_{f}})$ and satisfies ${\sigma}_{g}\mathit{C}\mathit{h}~MVN(\mathbf{0},{\sigma}_{g}^{2}\mathit{\phi})$. The details refer to Supplementary Appendices SA and SB. From Table 1, we find that using the EVD and Cholesky decomposition in the posterior sampling process can greatly reduce running time (the details can be seen in Section 3).

#### 2.4. HMC Algorithm and Priors

#### 2.5. Situations When Considering General Pedigrees and Unrelated Females, Respectively

#### 2.6. Situation When There Are Missing Genotypes for Some Individuals from General Pedigrees

#### 2.7. Simulation Settings

## 3. Results

#### 3.1. Simulation Results under the Situations of Homoscedasticity and Allele Frequencies in Females and Males Being the Same

#### 3.2. Simulation Results When Allele Frequencies in Females and Males Being Different

#### 3.3. Simulation Results under Heteroscedasticity

#### 3.4. Application to MCTFR Data

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Pedigree structure used for the simulation studies. The squares are males, the circles are females and the rhombus could be any gender. The numbers are used to encode the family members. (

**a**) Nuclear family; (

**b**) three-generation family; and (

**c**) four-generation family.

**Figure 2.**Scatter plots of six point estimates of $\gamma $ against true values of $\gamma $ with ${N}_{p}=150$, ${n}_{If}=650$, ${p}_{f}={p}_{m}=0.3$, ${\sigma}_{g}^{2}=1/3$, $\left({\sigma}_{e0}^{2},{\sigma}_{e1}^{2},{\sigma}_{e2}^{2}\right)=(1,1,1)$, and $MR=\{0,0.4\}$ for quantitative trait. The upper side and the right side of each subplot are the distribution of the true value of $\gamma $ and that of the point estimate of $\gamma $, respectively. (

**a**) ${\widehat{\gamma}}_{BNM}$ with $MR=0$; (

**b**) ${\widehat{\gamma}}_{BNM}$ with $MR=0.4$; (

**c**) ${\widehat{\gamma}}_{BUM}$ with $MR=0$; (

**d**) ${\widehat{\gamma}}_{BUM}$ with $MR=0.4$; (

**e**) ${\widehat{\gamma}}_{BNP}$ with $MR=0$; (

**f**) ${\widehat{\gamma}}_{BNP}$ with $MR=0.4$; (

**g**) ${\widehat{\gamma}}_{BUP}$ with $MR=0$; (

**h**) ${\widehat{\gamma}}_{BUP}$ with $MR=0.4$; (

**i**) ${\widehat{\gamma}}_{BN}$ with $MR=0$; (

**j**) ${\widehat{\gamma}}_{BN}$ with $MR=0.4$; (

**k**) ${\widehat{\gamma}}_{BU}$ with $MR=0$; and (

**l**) ${\widehat{\gamma}}_{BU}$ with $MR=0.4$.

**Figure 3.**Scatter plots of six point estimates of $\gamma $ against true values of $\gamma $ with ${N}_{p}=150$, ${n}_{If}=650$, ${p}_{f}={p}_{m}=0.3$, ${\sigma}_{g}^{2}=1/3$, $\left({\sigma}_{e0}^{2},{\sigma}_{e1}^{2},{\sigma}_{e2}^{2}\right)=(1,1,1)$, and $MR=\{0,0.4\}$ for qualitative trait. The upper side and the right side of each subplot are the distribution of the true value of $\gamma $ and that of the point estimate of $\gamma $, respectively. (

**a**) ${\widehat{\gamma}}_{BNM}$ with $MR=0$; (

**b**) ${\widehat{\gamma}}_{BNM}$ with $MR=0.4$; (

**c**) ${\widehat{\gamma}}_{BUM}$ with $MR=0$; (

**d**) ${\widehat{\gamma}}_{BUM}$ with $MR=0.4$; (

**e**) ${\widehat{\gamma}}_{BNP}$ with $MR=0$; (

**f**) ${\widehat{\gamma}}_{BNP}$ with $MR=0.4$; (

**g**) ${\widehat{\gamma}}_{BUP}$ with $MR=0$; (

**h**) ${\widehat{\gamma}}_{BUP}$ with $MR=0.4$; (

**i**) ${\widehat{\gamma}}_{BN}$ with $MR=0$; (

**j**) ${\widehat{\gamma}}_{BN}$ with $MR=0.4$; (

**k**) ${\widehat{\gamma}}_{BU}$ with $MR=0$; and (

**l**) ${\widehat{\gamma}}_{BU}$ with $MR=0.4$.

**Figure 4.**Scatter plots of widths of HPDIs based on six methods against true values of $\gamma $ with ${N}_{p}=150$, ${n}_{If}=650$, ${p}_{f}={p}_{m}=0.3$, ${\sigma}_{g}^{2}=1/3$, $\left({\sigma}_{e0}^{2},{\sigma}_{e1}^{2},{\sigma}_{e2}^{2}\right)=(1,1,1)$, and $MR=\{0,0.4\}$ for quantitative trait. The upper side and the right side of each subplot are the distribution of the true value of $\gamma $ and that of the width of the HPDI of $\gamma $, respectively. (

**a**) BNM with $MR=0$; (

**b**) BNM with $MR=0.4$; (

**c**) BUM with $MR=0$; (

**d**) BUM with $MR=0.4$; (

**e**) BNP with $MR=0$; (

**f**) BNP with $MR=0.4$; (

**g**) BUP with $MR=0$; (

**h**) BUP with $MR=0.4$; (

**i**) BN with $MR=0$; (

**j**) BN with $MR=0.4$; (

**k**) BU with $MR=0$; and (

**l**) BU with $MR=0.4$.

**Figure 5.**Scatter plots of widths of HPDIs based on six methods against true values of $\gamma $ with ${N}_{p}=150$, ${n}_{If}=650$, ${p}_{f}={p}_{m}=0.3$, ${\sigma}_{g}^{2}=1/3$, $\left({\sigma}_{e0}^{2},{\sigma}_{e1}^{2},{\sigma}_{e2}^{2}\right)=(1,1,1)$, and $MR=\{0,0.4\}$ for qualitative trait. The upper side and the right side of each subplot are the distribution of the true value of $\gamma $ and that of the width of the HPDI of $\gamma $, respectively. (

**a**) BNM with $MR=0$; (

**b**) BNM with $MR=0.4$; (

**c**) BUM with $MR=0$; (

**d**) BUM with $MR=0.4$; (

**e**) BNP with $MR=0$; (

**f**) BNP with $MR=0.4$; (

**g**) BUP with $MR=0$; (

**h**) BUP with $MR=0.4$; (

**i**) BN with $MR=0$; (

**j**) BN with $MR=0.4$; (

**k**) BU with $MR=0$; and (

**l**) BU with $MR=0.4$.

**Table 1.**Mean running time of the BNP method with a posterior sampling process based on EVD or Cholesky decomposition and an original posterior sampling process for general pedigrees.

${\mathit{N}}_{\mathit{p}}$ | Quantitative Trait | Qualitative Trait | ||
---|---|---|---|---|

EVD ^{a} | Original ^{b} | Cholesky ^{a} | Original ^{b} | |

150 | 10 s | 1.8 h | 1.3 min | 7.2 h |

600 | 40 s | 7 days | 47 min | 30 days |

^{a}The mean running time is based on 500 replicates;

^{b}the mean running time is based on 10 replicates.

**Table 2.**Mean squared errors (MSEs) of point estimates ${\widehat{\gamma}}_{BNM}$, ${\widehat{\gamma}}_{BUM}$, ${\widehat{\gamma}}_{BNP}$, ${\widehat{\gamma}}_{BUP}$, ${\widehat{\gamma}}_{BN}$, and ${\widehat{\gamma}}_{BU}$ under ${p}_{f}={p}_{m}$ and $\left({\sigma}_{e0}^{2},{\sigma}_{e1}^{2},{\sigma}_{e2}^{2}\right)=(1,1,1)$ among 500 replicates for mixed data, only general pedigrees, and only unrelated females, respectively.

Trait | $({\mathit{N}}_{\mathit{p}},{\mathit{n}}_{\mathit{I}\mathit{f}})$ | ${\mathit{p}}_{\mathit{f}}$ | ${\mathit{\sigma}}_{\mathit{g}}^{2}$ | $\mathit{M}\mathit{R}$ | Mixed Data | Pedigrees | Unrelated Females | |||
---|---|---|---|---|---|---|---|---|---|---|

${\widehat{\mathit{\gamma}}}_{\mathit{B}\mathit{N}\mathit{M}}$ | ${\widehat{\mathit{\gamma}}}_{\mathit{B}\mathit{U}\mathit{M}}$ | ${\widehat{\mathit{\gamma}}}_{\mathit{B}\mathit{N}\mathit{P}}$ | ${\widehat{\mathit{\gamma}}}_{\mathit{B}\mathit{U}\mathit{P}}$ | ${\widehat{\mathit{\gamma}}}_{\mathit{B}\mathit{N}}$ | ${\widehat{\mathit{\gamma}}}_{\mathit{B}\mathit{U}}$ | |||||

Quantitative | (150, 650) | 0.3 | 1/3 | 0 | 0.0643 | 0.0707 | 0.1167 | 0.1342 | 0.1123 | 0.1258 |

0.3 | 1/3 | 0.4 | 0.0943 | 0.1032 | 0.1564 | 0.1757 | 0.1532 | 0.1704 | ||

0.3 | 1 | 0 | 0.0889 | 0.0966 | 0.1528 | 0.1646 | 0.1391 | 0.1556 | ||

0.3 | 1 | 0.4 | 0.1323 | 0.1473 | 0.2086 | 0.2409 | 0.2017 | 0.2354 | ||

0.1 | 1/3 | 0 | 0.1850 | 0.1968 | 0.2959 | 0.3247 | 0.2284 | 0.2499 | ||

0.1 | 1/3 | 0.4 | 0.2455 | 0.2670 | 0.3781 | 0.4325 | 0.3304 | 0.3706 | ||

0.1 | 1 | 0 | 0.2010 | 0.2192 | 0.3399 | 0.3792 | 0.3260 | 0.3595 | ||

0.1 | 1 | 0.4 | 0.2754 | 0.3064 | 0.4224 | 0.4900 | 0.4046 | 0.4709 | ||

(600, 2600) | 0.3 | 1/3 | 0 | 0.0229 | 0.0236 | 0.0407 | 0.0421 | 0.0377 | 0.0383 | |

0.3 | 1/3 | 0.4 | 0.0359 | 0.0377 | 0.0570 | 0.0606 | 0.0558 | 0.0596 | ||

0.3 | 1 | 0 | 0.0256 | 0.0260 | 0.0543 | 0.0576 | 0.0540 | 0.0571 | ||

0.3 | 1 | 0.4 | 0.0416 | 0.0445 | 0.0805 | 0.0879 | 0.0764 | 0.0813 | ||

0.1 | 1/3 | 0 | 0.0786 | 0.0846 | 0.1210 | 0.1300 | 0.1169 | 0.1209 | ||

0.1 | 1/3 | 0.4 | 0.1147 | 0.1205 | 0.1689 | 0.1750 | 0.1593 | 0.1684 | ||

0.1 | 1 | 0 | 0.0962 | 0.1039 | 0.1650 | 0.1773 | 0.1512 | 0.1660 | ||

0.1 | 1 | 0.4 | 0.1353 | 0.1415 | 0.2080 | 0.2252 | 0.1910 | 0.2056 | ||

Qualitative | (150, 650) | 0.3 | 1/3 | 0 | 0.0862 | 0.0926 | 0.1551 | 0.1774 | 0.1499 | 0.1621 |

0.3 | 1/3 | 0.4 | 0.1243 | 0.1298 | 0.2181 | 0.2593 | 0.1922 | 0.2197 | ||

0.3 | 1 | 0 | 0.1175 | 0.1249 | 0.2112 | 0.2381 | 0.1923 | 0.2138 | ||

0.3 | 1 | 0.4 | 0.1588 | 0.1863 | 0.2768 | 0.3405 | 0.2459 | 0.2793 | ||

0.1 | 1/3 | 0 | 0.2654 | 0.2815 | 0.3911 | 0.4343 | 0.3822 | 0.4107 | ||

0.1 | 1/3 | 0.4 | 0.4051 | 0.4250 | 0.5403 | 0.6122 | 0.5383 | 0.6103 | ||

0.1 | 1 | 0 | 0.2910 | 0.3316 | 0.4342 | 0.5159 | 0.4322 | 0.4921 | ||

0.1 | 1 | 0.4 | 0.4430 | 0.5020 | 0.6024 | 0.6967 | 0.5978 | 0.6838 | ||

(600, 2600) | 0.3 | 1/3 | 0 | 0.0344 | 0.0347 | 0.0551 | 0.0599 | 0.0540 | 0.0562 | |

0.3 | 1/3 | 0.4 | 0.0564 | 0.0581 | 0.0847 | 0.0921 | 0.0811 | 0.0885 | ||

0.3 | 1 | 0 | 0.0441 | 0.0459 | 0.0816 | 0.0872 | 0.0667 | 0.0700 | ||

0.3 | 1 | 0.4 | 0.0727 | 0.0743 | 0.1154 | 0.1261 | 0.1045 | 0.1091 | ||

0.1 | 1/3 | 0 | 0.1092 | 0.1214 | 0.2405 | 0.2546 | 0.1549 | 0.1615 | ||

0.1 | 1/3 | 0.4 | 0.1577 | 0.1665 | 0.3585 | 0.3722 | 0.2225 | 0.2273 | ||

0.1 | 1 | 0 | 0.1429 | 0.1496 | 0.2427 | 0.2570 | 0.1718 | 0.1832 | ||

0.1 | 1 | 0.4 | 0.1783 | 0.2017 | 0.3614 | 0.3857 | 0.2472 | 0.2614 |

**Table 3.**Coverage probabilities (CPs, in %) of the BNM, BUM, BNP, BUP, BN, and BU methods under ${p}_{f}={p}_{m}$ and $\left({\sigma}_{e0}^{2},{\sigma}_{e1}^{2},{\sigma}_{e2}^{2}\right)=(1,1,1)$ among 500 replicates for mixed data, only general pedigrees, and only unrelated females, respectively

^{a}.

Trait | $({\mathit{N}}_{\mathit{p}},{\mathit{n}}_{\mathit{I}\mathit{f}})$ | ${\mathit{p}}_{\mathit{f}}$ | ${\mathit{\sigma}}_{\mathit{g}}^{2}$ | $\mathit{M}\mathit{R}$ | Mixed Data | Pedigrees | Unrelated Females | |||
---|---|---|---|---|---|---|---|---|---|---|

BNM | BUM | BNP | BUP | BN | BU | |||||

Quantitative | (150, 650) | 0.3 | 1/3 | 0 | 94.6 | 96.2 | 95.4 | 96.2 | 95.2 | 94.8 |

0.3 | 1/3 | 0.4 | 95.0 | 95.6 | 95.2 | 95.6 | 95.4 | 96.0 | ||

0.3 | 1 | 0 | 95.0 | 95.0 | 93.8 | 94.6 | 95.2 | 95.2 | ||

0.3 | 1 | 0.4 | 94.4 | 95.2 | 94.2 | 95.0 | 94.0 | 94.4 | ||

0.1 | 1/3 | 0 | 93.8 | 93.6 | 95.8 | 94.4 | 95.0 | 95.2 | ||

0.1 | 1/3 | 0.4 | 93.4 | 95.4 | 95.0 | 94.2 | 94.4 | 95.4 | ||

0.1 | 1 | 0 | 96.2 | 94.6 | 94.0 | 95.0 | 94.2 | 94.8 | ||

0.1 | 1 | 0.4 | 94.4 | 95.8 | 94.2 | 95.4 | 93.8 | 94.2 | ||

(600, 2600) | 0.3 | 1/3 | 0 | 94.2 | 94.2 | 94.0 | 94.4 | 94.8 | 95.0 | |

0.3 | 1/3 | 0.4 | 94.2 | 94.4 | 95.6 | 95.4 | 95.6 | 95.4 | ||

0.3 | 1 | 0 | 95.0 | 96.4 | 95.6 | 95.8 | 95.2 | 95.2 | ||

0.3 | 1 | 0.4 | 95.8 | 95.8 | 94.6 | 94.6 | 95.4 | 94.8 | ||

0.1 | 1/3 | 0 | 94.2 | 94.2 | 95.0 | 94.6 | 94.4 | 95.4 | ||

0.1 | 1/3 | 0.4 | 94.0 | 95.0 | 94.6 | 95.2 | 95.2 | 96.2 | ||

0.1 | 1 | 0 | 95.6 | 95.6 | 95.6 | 96.0 | 93.6 | 95.2 | ||

0.1 | 1 | 0.4 | 95.0 | 95.4 | 94.8 | 95.6 | 95.6 | 94.6 | ||

Qualitative | (150, 650) | 0.3 | 1/3 | 0 | 95.8 | 94.8 | 95.0 | 94.8 | 94.2 | 95.2 |

0.3 | 1/3 | 0.4 | 94.8 | 95.0 | 94.2 | 95.0 | 94.6 | 95.2 | ||

0.3 | 1 | 0 | 95.2 | 95.6 | 95.4 | 94.8 | 95.0 | 95.6 | ||

0.3 | 1 | 0.4 | 94.8 | 96.4 | 94.2 | 95.4 | 94.8 | 93.8 | ||

0.1 | 1/3 | 0 | 95.4 | 94.8 | 95.2 | 94.8 | 95.4 | 94.6 | ||

0.1 | 1/3 | 0.4 | 95.2 | 95.0 | 94.8 | 95.2 | 94.8 | 95.6 | ||

0.1 | 1 | 0 | 93.6 | 94.6 | 95.0 | 95.8 | 95.4 | 94.8 | ||

0.1 | 1 | 0.4 | 94.8 | 95.2 | 94.6 | 94.8 | 95.0 | 95.4 | ||

(600, 2600) | 0.3 | 1/3 | 0 | 95.4 | 94.8 | 95.0 | 94.8 | 96.2 | 95.6 | |

0.3 | 1/3 | 0.4 | 93.8 | 95.2 | 94.2 | 95.4 | 95.0 | 94.2 | ||

0.3 | 1 | 0 | 94.2 | 94.2 | 95.0 | 94.8 | 94.2 | 95.4 | ||

0.3 | 1 | 0.4 | 95.6 | 95.2 | 95.0 | 95.6 | 95.8 | 94.4 | ||

0.1 | 1/3 | 0 | 93.6 | 94.6 | 96.0 | 94.4 | 94.6 | 95.8 | ||

0.1 | 1/3 | 0.4 | 95.0 | 95.0 | 94.4 | 95.6 | 94.0 | 95.0 | ||

0.1 | 1 | 0 | 94.2 | 93.8 | 95.2 | 93.4 | 95.2 | 95.0 | ||

0.1 | 1 | 0.4 | 95.6 | 95.2 | 94.4 | 95.4 | 95.0 | 95.8 |

^{a}The empirical CP should be between 93.05% and 96.95% ($0.95\pm 2\times \sqrt{\frac{0.95\times 0.05}{500}}$) with 95% probability.

**Table 4.**${W}_{median}$s of the BNM, BUM, BNP, BUP, BN, and BU methods under ${p}_{f}={p}_{m}$ and $\left({\sigma}_{e0}^{2},{\sigma}_{e1}^{2},{\sigma}_{e2}^{2}\right)=(1,1,1)$ among 500 replicates for mixed data, only general pedigrees, and unrelated females, respectively.

Trait | $({\mathit{N}}_{\mathit{p}},{\mathit{n}}_{\mathit{I}\mathit{f}})$ | ${\mathit{p}}_{\mathit{f}}$ | ${\mathit{\sigma}}_{\mathit{g}}^{2}$ | $\mathit{M}\mathit{R}$ | Mixed Data | Pedigrees | Unrelated Females | |||
---|---|---|---|---|---|---|---|---|---|---|

BNM | BUM | BNP | BUP | BN | BU | |||||

Quantitative | (150, 650) | 0.3 | 1/3 | 0 | 0.9770 | 0.9815 | 1.2152 | 1.2336 | 1.2103 | 1.2249 |

0.3 | 1/3 | 0.4 | 1.1601 | 1.1801 | 1.4467 | 1.4781 | 1.3937 | 1.4373 | ||

0.3 | 1 | 0 | 1.0627 | 1.0636 | 1.3667 | 1.3966 | 1.3405 | 1.3653 | ||

0.3 | 1 | 0.4 | 1.2572 | 1.2627 | 1.5525 | 1.6017 | 1.5260 | 1.5821 | ||

0.1 | 1/3 | 0 | 1.4258 | 1.4452 | 1.5863 | 1.6305 | 1.5729 | 1.6297 | ||

0.1 | 1/3 | 0.4 | 1.5502 | 1.5935 | 1.6720 | 1.7201 | 1.6584 | 1.7103 | ||

0.1 | 1 | 0 | 1.5453 | 1.5940 | 1.6713 | 1.7166 | 1.6637 | 1.7112 | ||

0.1 | 1 | 0.4 | 1.6493 | 1.6999 | 1.7109 | 1.7633 | 1.7106 | 1.7629 | ||

(600, 2600) | 0.3 | 1/3 | 0 | 0.5350 | 0.5378 | 0.7332 | 0.7324 | 0.7292 | 0.7278 | |

0.3 | 1/3 | 0.4 | 0.6658 | 0.6642 | 0.8894 | 0.8840 | 0.8639 | 0.8650 | ||

0.3 | 1 | 0 | 0.6216 | 0.6272 | 0.8754 | 0.8861 | 0.8388 | 0.8433 | ||

0.3 | 1 | 0.4 | 0.7755 | 0.7868 | 1.0389 | 1.0515 | 1.0265 | 1.0282 | ||

0.1 | 1/3 | 0 | 1.0073 | 1.0217 | 1.2340 | 1.2505 | 1.2201 | 1.2450 | ||

0.1 | 1/3 | 0.4 | 1.1494 | 1.1790 | 1.3633 | 1.3885 | 1.3538 | 1.3814 | ||

0.1 | 1 | 0 | 1.1208 | 1.1298 | 1.3468 | 1.3654 | 1.3392 | 1.3562 | ||

0.1 | 1 | 0.4 | 1.3067 | 1.3313 | 1.4664 | 1.5054 | 1.4634 | 1.5038 | ||

Qualitative | (150, 650) | 0.3 | 1/3 | 0 | 1.0719 | 1.0821 | 1.4111 | 1.4353 | 1.3991 | 1.4332 |

0.3 | 1/3 | 0.4 | 1.2857 | 1.3103 | 1.6186 | 1.6573 | 1.5767 | 1.6258 | ||

0.3 | 1 | 0 | 1.2224 | 1.2394 | 1.5703 | 1.6178 | 1.5559 | 1.6176 | ||

0.3 | 1 | 0.4 | 1.4206 | 1.4531 | 1.6704 | 1.7288 | 1.6644 | 1.7193 | ||

0.1 | 1/3 | 0 | 1.5167 | 1.5485 | 1.6562 | 1.7093 | 1.6551 | 1.6972 | ||

0.1 | 1/3 | 0.4 | 1.5914 | 1.6449 | 1.7038 | 1.7543 | 1.6725 | 1.7236 | ||

0.1 | 1 | 0 | 1.6121 | 1.6757 | 1.7135 | 1.7649 | 1.6907 | 1.7450 | ||

0.1 | 1 | 0.4 | 1.6751 | 1.7284 | 1.7203 | 1.7690 | 1.7160 | 1.7642 | ||

(600, 2600) | 0.3 | 1/3 | 0 | 0.6888 | 0.6920 | 0.8648 | 0.8684 | 0.8369 | 0.8448 | |

0.3 | 1/3 | 0.4 | 0.8207 | 0.8274 | 1.0530 | 1.0555 | 1.0117 | 1.0057 | ||

0.3 | 1 | 0 | 0.7781 | 0.7777 | 1.0235 | 1.0199 | 0.9527 | 0.9559 | ||

0.3 | 1 | 0.4 | 0.9714 | 0.9739 | 1.1988 | 1.2089 | 1.1522 | 1.1660 | ||

0.1 | 1/3 | 0 | 1.1414 | 1.1550 | 1.3164 | 1.3447 | 1.3027 | 1.3380 | ||

0.1 | 1/3 | 0.4 | 1.3028 | 1.3184 | 1.4463 | 1.4831 | 1.4182 | 1.4421 | ||

0.1 | 1 | 0 | 1.2614 | 1.2848 | 1.4063 | 1.4336 | 1.4022 | 1.4294 | ||

0.1 | 1 | 0.4 | 1.4044 | 1.4328 | 1.5227 | 1.5681 | 1.5185 | 1.5555 |

**Table 5.**${W}_{iqr}$s of the BNM, BUM, BNP, BUP, BN, and BU methods under ${p}_{f}={p}_{m}$ and $\left({\sigma}_{e0}^{2},{\sigma}_{e1}^{2},{\sigma}_{e2}^{2}\right)=(1,1,1)$ among 500 replicates for mixed data, only general pedigrees, and only unrelated females, respectively.

Trait | $({\mathit{N}}_{\mathit{p}},{\mathit{n}}_{\mathit{I}\mathit{f}})$ | ${\mathit{p}}_{\mathit{f}}$ | ${\mathit{\sigma}}_{\mathit{g}}^{2}$ | $\mathit{M}\mathit{R}$ | Mixed Data | Pedigrees | Unrelated Females | |||
---|---|---|---|---|---|---|---|---|---|---|

BNM | BUM | BNP | BUP | BN | BU | |||||

Quantitative | (150, 650) | 0.3 | 1/3 | 0 | 0.3333 | 0.3700 | 0.4620 | 0.5268 | 0.4530 | 0.5122 |

0.3 | 1/3 | 0.4 | 0.4106 | 0.4680 | 0.4760 | 0.5466 | 0.4435 | 0.4987 | ||

0.3 | 1 | 0 | 0.4218 | 0.4755 | 0.4739 | 0.5418 | 0.4291 | 0.5261 | ||

0.3 | 1 | 0.4 | 0.4351 | 0.5176 | 0.4162 | 0.4673 | 0.4069 | 0.4314 | ||

0.1 | 1/3 | 0 | 0.4058 | 0.4620 | 0.3309 | 0.3603 | 0.3193 | 0.3392 | ||

0.1 | 1/3 | 0.4 | 0.3389 | 0.3812 | 0.2603 | 0.2690 | 0.2267 | 0.2243 | ||

0.1 | 1 | 0 | 0.3345 | 0.3817 | 0.2612 | 0.2806 | 0.2586 | 0.2585 | ||

0.1 | 1 | 0.4 | 0.2467 | 0.2585 | 0.1618 | 0.1658 | 0.1590 | 0.1488 | ||

(600, 2600) | 0.3 | 1/3 | 0 | 0.1590 | 0.1629 | 0.2391 | 0.2655 | 0.2272 | 0.2582 | |

0.3 | 1/3 | 0.4 | 0.2184 | 0.2344 | 0.3295 | 0.3803 | 0.2823 | 0.3190 | ||

0.3 | 1 | 0 | 0.1946 | 0.2087 | 0.3241 | 0.3795 | 0.2968 | 0.3297 | ||

0.3 | 1 | 0.4 | 0.2733 | 0.3080 | 0.3892 | 0.4434 | 0.3643 | 0.4125 | ||

0.1 | 1/3 | 0 | 0.3785 | 0.4380 | 0.3878 | 0.4345 | 0.3790 | 0.4319 | ||

0.1 | 1/3 | 0.4 | 0.4386 | 0.4744 | 0.3998 | 0.4649 | 0.3880 | 0.4338 | ||

0.1 | 1 | 0 | 0.3744 | 0.4297 | 0.4101 | 0.4489 | 0.3670 | 0.4449 | ||

0.1 | 1 | 0.4 | 0.3653 | 0.3961 | 0.3640 | 0.4160 | 0.3524 | 0.4152 | ||

Qualitative | (150, 650) | 0.3 | 1/3 | 0 | 0.4290 | 0.4922 | 0.4841 | 0.5399 | 0.4561 | 0.5148 |

0.3 | 1/3 | 0.4 | 0.4542 | 0.5314 | 0.3988 | 0.4512 | 0.3902 | 0.4319 | ||

0.3 | 1 | 0 | 0.4539 | 0.5204 | 0.4009 | 0.4466 | 0.3970 | 0.4309 | ||

0.3 | 1 | 0.4 | 0.4424 | 0.5062 | 0.2980 | 0.3132 | 0.2893 | 0.3005 | ||

0.1 | 1/3 | 0 | 0.3811 | 0.4325 | 0.2866 | 0.3086 | 0.2694 | 0.2752 | ||

0.1 | 1/3 | 0.4 | 0.3118 | 0.3636 | 0.2822 | 0.3342 | 0.2245 | 0.2540 | ||

0.1 | 1 | 0 | 0.3195 | 0.3387 | 0.2239 | 0.2082 | 0.1856 | 0.1867 | ||

0.1 | 1 | 0.4 | 0.2468 | 0.2913 | 0.1973 | 0.2209 | 0.1932 | 0.2027 | ||

(600, 2600) | 0.3 | 1/3 | 0 | 0.1886 | 0.2109 | 0.2929 | 0.3232 | 0.2715 | 0.3036 | |

0.3 | 1/3 | 0.4 | 0.2580 | 0.2884 | 0.3947 | 0.4478 | 0.3536 | 0.4152 | ||

0.3 | 1 | 0 | 0.2680 | 0.2937 | 0.3801 | 0.4009 | 0.3617 | 0.3981 | ||

0.3 | 1 | 0.4 | 0.3576 | 0.4061 | 0.4404 | 0.5107 | 0.4060 | 0.4630 | ||

0.1 | 1/3 | 0 | 0.3826 | 0.4368 | 0.3421 | 0.4035 | 0.3350 | 0.3903 | ||

0.1 | 1/3 | 0.4 | 0.3333 | 0.3700 | 0.4620 | 0.5268 | 0.4530 | 0.5122 | ||

0.1 | 1 | 0 | 0.4106 | 0.4680 | 0.4760 | 0.5466 | 0.4435 | 0.4987 | ||

0.1 | 1 | 0.4 | 0.4218 | 0.4755 | 0.4739 | 0.5418 | 0.4291 | 0.5261 |

SNP | Position | Alleles | MAF ^{a} | Trait | p-Value | Gene |
---|---|---|---|---|---|---|

rs10522027 | 34630163 | G > A | 0.141 | DEP | ${3.64\times 10}^{-7}$ | TMEM47 |

rs12860832 | 151643064 | G > A | 0.263 | DEP | ${2.00\times 10}^{-6}$ | PASD1 |

rs12849233 | 151645704 | C > A | 0.329 | DEP | ${1.26\times 10}^{-6}$ | PASD1 |

^{a}MAF represents the minor allele frequency.

**Table 7.**Application of the six methods to SNPs detected in association analysis for the MCTFR data.

SNP | Point Estimate | 95% HPDI | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\widehat{\mathit{\gamma}}}_{\mathit{B}\mathit{N}\mathit{M}}$ | ${\widehat{\mathit{\gamma}}}_{\mathit{B}\mathit{U}\mathit{M}}$ | ${\widehat{\mathit{\gamma}}}_{\mathit{B}\mathit{N}\mathit{P}}$ | ${\widehat{\mathit{\gamma}}}_{\mathit{B}\mathit{U}\mathit{P}}$ | ${\widehat{\mathit{\gamma}}}_{\mathit{B}\mathit{N}}$ | ${\widehat{\mathit{\gamma}}}_{\mathit{B}\mathit{U}}$ | BNM | BUM | BNP | BUP | BN | BU | |

rs10522027 | 0.6922 | 0.6895 | 0.6394 | 0.6494 | 0.7238 | 0.7429 | (0.2451, 1.3518) | (0.2316, 1.4420) | (0.0156, 1.5816) | (0.0063, 1.5567) | (0.1791, 1.6615) | (0.1870, 1.6384) |

rs12860832 | 0.8371 | 0.8288 | 0.9422 | 0.9448 | 0.7281 | 0.7200 | (0.3266, 1.4935) | (0.3942, 1.5788) | (0.1878, 1.6258) | (0.2077, 1.6698) | (0.0945, 1.6294) | (0.1214, 1.6503) |

rs12849233 | 0.7633 | 0.7426 | 0.8843 | 0.8736 | 0.6906 | 0.6968 | (0.2236, 1.2934) | (0.2133, 1.3054) | (0.1054, 1.5392) | (0.1361, 1.5964) | (0.0211, 1.5229) | (0.0764, 1.5490) |

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## Share and Cite

**MDPI and ACS Style**

Kong, Y.-F.; Li, S.-Z.; Wang, K.-W.; Zhu, B.; Yuan, Y.-X.; Li, M.-K.; Zhou, J.-Y. An Efficient Bayesian Method for Estimating the Degree of the Skewness of X Chromosome Inactivation Based on the Mixture of General Pedigrees and Unrelated Females. *Biomolecules* **2023**, *13*, 543.
https://doi.org/10.3390/biom13030543

**AMA Style**

Kong Y-F, Li S-Z, Wang K-W, Zhu B, Yuan Y-X, Li M-K, Zhou J-Y. An Efficient Bayesian Method for Estimating the Degree of the Skewness of X Chromosome Inactivation Based on the Mixture of General Pedigrees and Unrelated Females. *Biomolecules*. 2023; 13(3):543.
https://doi.org/10.3390/biom13030543

**Chicago/Turabian Style**

Kong, Yi-Fan, Shi-Zhu Li, Kai-Wen Wang, Bin Zhu, Yu-Xin Yuan, Meng-Kai Li, and Ji-Yuan Zhou. 2023. "An Efficient Bayesian Method for Estimating the Degree of the Skewness of X Chromosome Inactivation Based on the Mixture of General Pedigrees and Unrelated Females" *Biomolecules* 13, no. 3: 543.
https://doi.org/10.3390/biom13030543