The Annals of Statistics

Multiple regression approach to mapping of quantitative trait loci (QTL) based on sib-pair data: a theoretical analysis

Sunwei Guo and Momiao Xiong

Full-text: Open access


The interval mapping method has been shown to be a powerful tool for mapping QTL. However, it is still a challenge to perform a simultaneous analysis of several linked QTLs, and to isolate multiple linked QTLs. To circumvent these problems, multiple regression analysis has been suggested for experimental species. In this paper, the multiple regression approach is extended to human sib-pair data through multiple regression of the squared difference in trait values between two sibs on the proportions of alleles shared identical by descent by sib pairs at marker loci.We conduct an asymptotic analysis of the partial regression coeffcients, which provide a basis for the estimation of the additive genetic variance and of locations of the QTLs. We demonstrate how the magnitude of the regression coefficients can be used to separate multiple linked QTLs. Further, we shall show that the multiple regression model using sib pairs is identifiable, and our proposed procedure for locating QTLs is robust in the sense that it can detect the number of QTLs and their locations in the presence of several linked (QTLs) in an interval, unlike a simple regression model which may find a “ghost” QTL with no effect on the trait in the interval with several linked QTLs. Moreover, we give procedures for computing the threshold values for prespecified significance levels and for computing the power for detecting (QTLs). Finally, we investigate the consistency of the estimator for QTL locations. Using the concept of epi-convergence and variation analysis theory, we shall prove the consistency of the estimator of map location in the framework of the multiple regression approach. Since the true IBD status is not always known, the multiple regression of the squared sib difference on the estimated IBD sharing is also considered.

Article information

Ann. Statist., Volume 28, Number 5 (2000), 1245-1278.

First available in Project Euclid: 12 March 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F03: Hypothesis testing
Secondary: 92D30: Epidemiology 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40] 15A18: Eigenvalues, singular values, and eigenvectors

Mapping QTL linkage analysis Sib pair analysis multiple regression variation analysis and epi-convergence


Xiong, Momiao; Guo, Sunwei. Multiple regression approach to mapping of quantitative trait loci (QTL) based on sib-pair data: a theoretical analysis. Ann. Statist. 28 (2000), no. 5, 1245--1278. doi:10.1214/aos/1015957393.

Export citation


  • Almasy, L. and Blangero, J. (1998). Multipoint quantitative-trait linkage analysis in general pedigrees. Amer. J. Hum. Genetics 62 1198-1211.
  • Amos, C. I. (1994). Robust variance-components approach for assessing genetic linkage in pedigress. Amer. J. Hum. Genetics 54 535-543.
  • Aubin, J. P. and Frankowska, H. (1990). Set-Valued Analysis. Birkh¨auser, Boston. Begleiter, H., Porjesz, B., Reich, T., Edenberg, H. J., Goate, A., and Blangero, J. et al.
  • (1998). Quantitative trait loci analysis of human event-related brain potentials: P3 voltage. Electroencephologr. Clin. Neurophysiol. 108 244-250.
  • Blangero, J. and Almasy, L. (1997). Multipoint oligogenic linkage analysis of quantitative traits. Genet. Epidemiol. 14 959-964. Comuzzie, A. G., Hixson, J. E., Almasy, L., Mitchell, B. D., Mahaney, M. C., Dyer, T. D., Sten,
  • M. P., MacCluer, J. W. and Blangero, J. (1997). A major quantitative trait locus determining serum leptin levels and fat mass is located on human chromosome 2. Nat. Genetics 15 273-276. Dragani, T. A., Zeng, Z.-B., Canzian, F., Gariboldi, M., Ghilarducci, M. T., Manenti, G.
  • and Pierotti, M. A. (1995). Mapping of body weight loci on mouse chromosome X. Mammalian Genome 6 778-781. Duggirala, R., Blangero, J., Almasy, L., Dyer, T. D., Williams, K. L., Leach, R. J., O'Connell,
  • P. and Stern, M. P. (1999). Linkage of type 2 diabetes mellitus and of age at onset to a genetic location on chromosome 10q in Mexican Americans. Amer. J. Hum. Genetics 64 1127-1140.
  • Dupa cova, J. and Wets, R. (1988). Asymptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems. Ann. Statist. 16 1517-1549.
  • Feingold, S., Brown, O. P. and Siegmund, D. (1993). Gaussian models for genetic linkage analysis using complete high-resolution maps of identity by descent. Amer. J. Hum. Genetics 53 234-251.
  • Fulker, D. W. and Cardon, L. R. (1994). A sib-pair approach to interval mapping of quantitative trait loci. Amer. J. Hum. Genetics 54 1092-1103.
  • Goldgar, D. E. (1990). Multipoint analysis of human quantitative genetic variation. Amer. J. Hum. Genetics 47 957-967.
  • Haley, C. S., Knott, S. A. and Elsen, J. M. (1994). Mapping quantitative trait loci in crosses between outbred lines using least squares. Genetics 136 1195-1207.
  • Haley, C. S. and Knott, S. A. (1992). A simple regression method for mapping quantitative trait loci in line crosses using flanking markers. Heredity 69 315-324.
  • Haseman, J. K. and Elston, R. C. (1972). The investigation of linkage between a quantitative trait and a marker locus. Behavior Genetics 2 3-19.
  • Hoeschele, I. and Vanranden, P. (1993). Bayesian analysis of linkage between genetic markers and quantitative trait loci: II. Combining prior knowledge with experimental evidence. Theor. Appl. Genetics 85 946-952.
  • Jansen, R. C. (1993). Interval mapping of multiple quantitative trait loci. Genetics 135 205-211.
  • Jansen, R. C. (1994). Controlling the Type I and Type II errors in mapping quantitative trait loci. Genetics 138 871-881.
  • Kao, C.-H., Zeng, Z.-B. and Teasdale, R. D. (1999). Multiple interval mapping for quantitative trait loci. Genetics 132 1203-1216.
  • Knott, S. A. and Haley, C. S. (1992). Maximum likelihood mapping of quantitative trait loci using full-sib families. Genetics 132 1211-1222.
  • Kuittinen, H., Sillanp¨a¨a, M. J. and Savolainen, O. (1997). Genetic basis of adaptation, flowering time in Arabidopsis Thaliana. Theor. Appl. Genetics 95 573-583.
  • Lander, E. S. and Botstein, D. (1989). Mapping Mendelian factors underlying quantitative traits using RFLP linkage maps. Genetics 121 185-199.
  • Lander, E. and Schork, N. J. (1994). Genetic dissection of complex traits. Science 265 2037-2048.
  • Liu, J., Mercer, J. M., Stem, L. F., Gibson, G. C., Zeng, Z.-B. and Laurie, C. C. (1996). Genetic analysis of a morphological shape difference in the male genitalia of Drosiphite simulans and D. mauritiana. Genetics 142 1129-1145.
  • Luo, Z. W. and Kearsey, M. J. (1992). Interval mapping of quantitative trait loci in an F2 population. Heredity 69 236-242.
  • Martinez, O. and Curnow, R. N. (1992). Estimating the locations and the sizes of the effects of quantitative trait loci using flanking markers. Theor. Appl. Genetics 85 480-488.
  • Olson, J. M. (1995). Robust multipoint linkage analysis: an extension of the Haseman-Elston method. Genetic Epidemiology 12 177-193.
  • Rodolphe, F. and Lefort, M. (1993). A multi-marker model for detecting chromosomal segments displaying QTL activity. Genetics 134 1277-1288.
  • Satagopan, J. M., Yandell, B. S., Newton, M. A. and Osborn, T. C. (1996). A Bayesian approach to detect quantitative trait loci using Markov chain Monte Carlo. Genetics 144 805-816.
  • Schork, N. J. (1993). Extended multipoint identity-by-descent analysis of human quantitative traits: efficiency, power, and modeling considerations. Amer. J. Hum. Genetics 53 1306-1319.
  • Sillanp¨a¨a, M. J. and Arjas, E. (1998). Bayesian mapping of multiple quantitative trait loci from incomplete line cross data. Genetics 148 1373-1388.
  • Templeton, A. R. (1999). Uses of evolutionary theory in the human genome project. Annu. Rev. Ecol. Syst. 30 23-49.
  • Wang, D. G., Fan, J. B. and Siao, C. J., et al. (1998). Large-scale identification, mapping, and genotyping of single-nucleotide polymorphisms in the human genome. Science 280 1082-1086. Wang, X. L., Mahaney, M. C., Sim, A. S., Wang, J., Wang, J., Blangero, J., Almasy, L.,
  • Badenhop, R. B. and Wilcken, D. E. L. (1997). Genetic contribution of the endothelial constitutive nitric oxide synthase gene to plasma nitric oxide levels. Arterioscler. Thromb. Vasc. Biol. 17 3147-3153.
  • Wright, F. (1994). Asymptotics and robustness for genetic linkage mapping. Ph.D. dissertation, Univ. Chicago.
  • Wets, R. (1991). Constraint estimation: consistency and asymptotics. Appl. Stochastic Models Data Anal. 7 17-32.
  • Xiong, M. and Guo, S. W. (1997). Fine-scale mapping based on linkage disequilibrium: theory and applications. Amer. J. Hum. Genetics 60 1513-1531.
  • Xu, S. and Atchley, W. R. (1995). A random model approach to interval mapping of quantitative trait loci. Genetics 141 1189-1197.
  • Zeng, Z. B. (1993). Theoretical basis for separation of multiple linked gene effects in mapping quantitative trait loci. Proc. Nat. Acad. Sci. U.S.A. 90 10972-10976.
  • Zeng, Z. B. (1994). Precision mapping of quantitative trait loci. Genetics 136 1457-1468.