Abstract
To assess the effect of chemotherapy on mitochondrial genome mutations in cancer survivors and their offspring, a study sequenced the full mitochondrial genome and determined the mitochondrial DNA heteroplasmic (mtDNA) mutation rate. To build a model for counts of heteroplasmic mutations in mothers and their offspring, bivariate Poisson regression was used to examine the relationship between mutation count and clinical information while accounting for the paired correlation. However, if the sequencing depth is not adequate, a limited fraction of the mtDNA will be available for variant calling. The classical bivariate Poisson regression model treats the offset term as equal within pairs; thus, it cannot be applied directly. In this research, we propose an extended bivariate Poisson regression model that has a more general offset term to adjust the length of the accessible genome for each observation. We evaluate the performance of the proposed method with comprehensive simulations, and the results show that the regression model provides unbiased parameter estimations. The use of the model is also demonstrated using the paired mtDNA dataset.
Acknowledgment
The authors thank the editor, the associate deitor and referees for their valuable comments nad suggestions. The work was partly supported by a Ministry of Science and Technology grant, MOST 105-2118-M-006-007-MY2 and the Mathematics Division of the National Center for Theoretical Sciences in Taiwan.
Appendix I : Recursive relation
We refine the notation BPO(Yi1 = yi1, Yi2 = yi2) equal to P(yi1, yi2). Because the offset ti0, ti1, ti2 vary, the recursive relationships proposed by Kawamura (1985) need to be re-driven. In this appendix, we would like to prove
for every integers yi1, yi2 = 1, 2, …, n. First, we prove this relationship
Let’s consider two cases: the first case is yi1 ≤ yi2 and second case is yi2 ≤ yi1. Considering yi1 ≤ yi2,
To calculate yi1P(yi1, yi2), the above formula becomes
Therefore, yi1P(yi1, yi2) can be rearrange as
Similarly, for the case of yi2 ≤ yi1, we use the same method to get the result
Therefore, the second relation can be derived in the same way.
Appendix II: Solve E-step by recursive relationship
We know Xi1 and Xi0 are independent, and Yi1 = Xi0 + Xi1. Then the conditional expectation
As long as we derive the formual of
Using the recursive relationship, the formula can be rewritten as
Comparing (6) and (7), we know
Similarly,
Then, we can get the results that
References
Andrews, R. M., I. Kubacka, P. F. Chinnery, R. N. Lightowlers, D. M. Turnbull and N. Howell (1999): “Reanalysis and revision of the Cambridge reference sequence for human mitochondrial DNA,” Nat. Genet., 23, 147.10.1038/13779Search in Google Scholar PubMed
Bermúdez, L. and D. Karlis (2012): “A finite mixture of bivariate Poisson regression models with an application to insurance ratemaking,” Comput. Stat. Data Anal., 56, 3988–3999.10.1016/j.csda.2012.05.016Search in Google Scholar
Famoye, F. (2010): “On the bivariate negative binomial regression model,” J. Appl. Stat., 37, 969–981.10.1080/02664760902984618Search in Google Scholar
Guo, Y., Q. Cai, D. Samuels, F. Ye, J. Long, C. Li, J. Winther, E. J. Tawn, M. Stovall, P. Lähteenmäki, N. Malila, S. Levy, C. Shaffer, Y. Shyr, X. Shu, and J. Boice (2012): “The use of next generation sequencing technology to study the effect of radiation therapy on mitochondrial DNA mutation,” Mutat. Res., 744, 154–160.10.1016/j.mrgentox.2012.02.006Search in Google Scholar PubMed PubMed Central
Johnson, N., S. Kotz, and N. Balakrishnan (1997): Discrete multivariate distributions, Wiley, New York.Search in Google Scholar
Jung, R. and R. Winkelmann (1993): “Two aspects of labor mobility: a bivariate Poisson regression approach,” Empir. Econ., 18, 543–556.10.1007/BF01176203Search in Google Scholar
Karlis, D. (2003): “An em algorithm for multivariate Poisson distribution and related models,” J. Appl. Stat., 30, 63–77.10.1080/0266476022000018510Search in Google Scholar
Karlis, D. and L. Meligkotsidou (2005): “Multivariate Poisson regression with covariance structure,” Stat. Comput., 15, 255–265.10.1007/s11222-005-4069-4Search in Google Scholar
Karlis, D. and I. Ntzoufras (2005): “Bivariate Poisson and diagonal inflated bivariate Poisson regression models in r,” J. Stat. Softw., 14, 1–36.10.18637/jss.v014.i10Search in Google Scholar
Karlis, D. and L. Ntzoufras (2003): “Analysis of sports data by using bivariate Poisson models,” Statistician, 52, 381–393.10.1111/1467-9884.00366Search in Google Scholar
Kawamura, K. (1985): “A note on the recurrent relations for the bivariate Poisson distribution,” Kodai Math. J., 8, 70–78.10.2996/kmj/1138036998Search in Google Scholar
Kocherlakota, S. and K. Kocherlakota (2001): “Regression in the bivariate Poisson distribution,” Commun. Stat. Theory Methods, 30, 815–825.10.1081/STA-100002259Search in Google Scholar
McLachlan, G. and T. Krishnan (1997): The EM algorithm and extensions, Wiley, New York.Search in Google Scholar
Pakendorf, B. and M. Stoneking (2005): “Mitochondrial DNA and human evolution,” Annu. Rev. Genomics Hum. Genet., 6, 165–183.10.1146/annurev.genom.6.080604.162249Search in Google Scholar PubMed
Robinson, M. and A. Oshlack (2010): “A scaling normalization method for differential pression analysis of RNA-seq data,” Genome Biol., 11, R25.10.1186/gb-2010-11-3-r25Search in Google Scholar PubMed PubMed Central
Srivastava, S. and L. Chen (2010): “A two-parameter generalized Poisson model to improve the analysis of RNA-seq data,” Nucleic Acids Res., 38, e170.10.1093/nar/gkq670Search in Google Scholar PubMed PubMed Central
Verma, M. and D. Kumar (2007): “Application of mitochondrial genome information in cancer epidemiology,” Clin. Chim. Acta, 383, 41–50.10.1016/j.cca.2007.04.018Search in Google Scholar PubMed
Wang, L., Z. Feng, X. Wang, X. Wang, and X. Zhang (2010): “Degseq: an R package for identifying differentially expressed genes from RNA-seq data,” Bioinformatics, 26, 136–138.10.1093/bioinformatics/btp612Search in Google Scholar PubMed
Supplemental Material:
The online version of this article (DOI: 10.1515/sagmb-2016-0040) offers supplementary material, available to authorized users.
©2017 Walter de Gruyter GmbH, Berlin/Boston
