## The Annals of Statistics

### A score test for linkage using identity by descent data from sibships

#### Abstract

We consider score tests of the null hypothesis $H_0: \theta = 1/2$ against the alternative hypothesis $H_1: 0 \leq \theta < 1/2$, based upon counts multinomially distributed with parameters $n$ and $\rho(\theta,\pi)_{1 \times m} = \pi_{1\times m}T(\theta)_{m \times m}$, where $T(\theta)$ is a transition matrix with $T(0)=I$ , the identity matrix, and $T(1/2)=(1,\dots,1)^T (\alpha_1,\dots,\alpha_m)$. This type of testing problem arises in human genetics when testing the null hypothesis of no linkage between a marker and a disease susceptibility gene, using identity by descent data from families with affected members. In important cases in this genetic context, the score test is independent of the nuisance parameter $\pi$ and based on a widely used test statistic in linkage analysis. The proof of this result involves embedding the states of the multinomial distribution into a continuous-time Markov chain with infinitesimal generator $Q$. The second largest eigenvalue of $Q$ and its multiplicity are key in determining the form of the score statistic. We relate $Q$ to the adjacency matrix of a quotient graph in order to derive its eigenvalues and eigenvectors.

#### Article information

Source
Ann. Statist., Volume 27, Number 3 (1999), 943-986.

Dates
First available in Project Euclid: 5 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1018031264

Digital Object Identifier
doi:10.1214/aos/1018031264

Mathematical Reviews number (MathSciNet)
MR1724037

Zentralblatt MATH identifier
0957.62101

#### Citation

Dudoit, Sandrine; Speed, Terence P. A score test for linkage using identity by descent data from sibships. Ann. Statist. 27 (1999), no. 3, 943--986. doi:10.1214/aos/1018031264. https://projecteuclid.org/euclid.aos/1018031264

#### References

• Blackwelder, W. C. and Elston, R. C. (1985). A comparison of sib-pair linkage tests for disease susceptibility loci. Genet. Epidemiol. 2 85-97.
• Davis, S. and Weeks, D. E. (1997). Comparison of nonparametric statistics for detection of linkage in nuclear families: single-marker evaluation. Amer. J. Hum. Genet. 61 1431-1444.
• Day, N. E. and Simons, M. J. (1976). Disease-susceptibility genes-their identification by multiple case family studies. Tissue Antigens 8 109-119.
• deBruijn, N. G. (1964). P´olya's theory of counting. In Applied Combinatorial Mathematics (E. F. Beckenbach, ed.) 144-184. Wiley, New York.
• Diaconis, P. (1988). Group Representations in Probability and Statistics. IMS, Hayward, CA.
• Donnelly, K. P. (1983). The probability that related individuals share some section of genome identical by descent. Theoret. Population Biol. 23 34-63.
• Dudoit, S. (1999). Linkage analysis of complex human traits using identity by descent data. Ph.D. dissertation, Univ. California, Berkeley.
• Dudoit, S. and Speed, T. P. (1999). Triangle constraints for sib-pair identity by descent probabilities under a general multilocus model for disease susceptibility. In Statistics in Genetics (M. E. Halloran and S. Geisser, eds.). Springer, New York.
• Ethier, S. N. and Hodge, S. E. (1985). Identity-by-descent analysis of sibship configurations. Amer. J. Med. Genet. 22 263-272.
• Faraway, J. J. (1993). Improved sib-pair linkage test for disease susceptibility loci. Genet. Epidemiol. 10 225-233.
• Feingold, E., Brown, P. O. and Siegmund, D. (1993). Gaussian models for genetic linkage analysis using complete high-resolution maps of identity by descent. Amer. J. Hum. Genet. 53 234-251.
• Feingold, E. and Siegmund, D. (1997). Strategies for mapping heterogeneous recessive traits by allele sharing methods. Amer. J. Hum. Genet. 60 965-978.
• Fraleigh, J. B. (1989). A First Course in Abstract Algebra, 4th ed. Addison-Wesley, Reading, MA.
• Godsil, C. D. (1993). Algebraic Combinatorics. Chapman and Hall, New York.
• 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.
• Holmans, P. (1993). Asymptotic properties of affected sib-pair linkage analysis. Amer. J. Hum. Genet. 52 362-374.
• Holmans, P. and Clayton, D. (1995). Efficiency of typing unaffected relatives in an affectedsib-pair linkage study with single-locus and multiple tightly linked markers. Amer. J. Hum. Genet. 57 1221-1232.
• Jacob, B. (1990). Linear Algebra. Freeman, New York. Knapp, M., Seuchter, S. A. and Baur, M. P. (1994a). Linkage analysis in nuclear families I: optimality criteria for affected sib-pair tests. Hum. Hered. 44 37-43. Knapp, M., Seuchter, S. A. and Baur, M. P. (1994b). Linkage analysis in nuclear families II: relationship between affected sib-pair tests and lod score analysis. Hum. Hered. 44 44-51. Knapp, M., Seuchter, S. A. and Baur, M. P. (1994c). Two-locus disease models with two marker loci: the power of affected-sib-pair tests. Amer. J. Hum. Genet. 55 1030-1041.
• Kong, A. and Cox, N. J. (1997). Allele-sharing models: lod scores and accurate linkage tests. Amer. J. Hum. Genet. 61 1179-1188.
• Kruglyak, L., Daly, M. J., Reeve-Daly, M. P. and Lander, E. S. (1996). Parametric and nonparametric linkage analysis: a unified multipoint approach. Amer. J. Hum. Genet. 58 1347-1363.
• Kruglyak, L. and Lander, E. S. (1995). Complete multipoint sib-pair analysis of qualitative and quantitative traits. Amer. J. Hum. Genet. 57 439-454.
• Lander, E. S. and Green, P. (1987). Construction of multilocus genetic maps in humans. Proc. Nat. Acad. Sci. U.S.A. 84 2363-2367.
• McPeek, M. S. (1996). An introduction to recombination and linkage analysis. In Genetic Mapping and DNA Sequencing (T. P. Speed and M. S. Waterman, eds.) 1-14. Springer, New York.
• McPeek, M. S. (1999). Optimal allele-sharing statistics for genetic mapping using affected relatives. Genet. Epidemiol. 16 225-249.
• Ott, J. (1991). Analysis of Human Genetic Linkage, rev. eds. Johns Hopkins Univ. Press, Baltimore. Risch, N. (1990a). Linkage strategies for genetically complex traits II. The power of affected relative pairs. Amer. J. Hum. Genet. 46 229-241. Risch, N. (1990b). Linkage strategies for genetically complex traits III. The effect of marker polymorphism on analysis of affected relative pairs. Amer. J. Hum. Genet. 46 242-253.
• Risch, N. and Zhang, H. (1995). Extreme discordant sib pairs for mapping quantitative trait loci in humans. Science 268 1584-1589.
• Risch, N. and Zhang, H. (1996). Mapping quantitative trait loci with extreme discordant sib pairs: sampling considerations. Amer. J. Hum. Genet. 58 836-843.
• Rosenblatt, M. (1974). Random Processes. Graduate Texts in Mathematics 17, 2nd ed. Springer, New York.
• Schaid, D. J. and Nick, T. G. (1990). Sib-pair linkage tests for disease suscpetibility loci: common tests vs. the asymptotically most powerful test. Genet. Epidemiol. 7 359-370.
• Speed, T. P. (1996). What is a genetic map function? In Genetic Mapping and DNA Sequencing (T. P. Speed and M. S. Waterman, eds.) 65-88. Springer, New York.
• Suarez, B. K. and van Eerdewegh, P. (1984). A comparison of three affected-sib-pair scoring methods to detect HLA-linked disease susceptibility genes. Amer. J. Med. Genet. 18 135-146.
• Teng, J. and Siegmund, D. O. (1997). Combining information within and between pedigrees for mapping complex traits. Amer. J. Hum. Genet. 60 979-992.
• Thompson, E. A. (1997). Conditional gene identity in affected individuals. In Genetic Mapping of Disease Genes (I.-H. Pawlowitzki, J. H. Edwards and E. A. Thompson, eds.) 137-146. Academic Press, San Diego.
• van Lint, J. H. and Wilson, R. M. (1992). A Course in Combinatorics. Cambridge Univ. Press.
• Whittemore, A. S. (1996). Genome scanning for linkage: an overview. Amer. J. Hum. Genet. 59 704-716. Whittemore, A. S. and Halpern, J. (1994a). A class of tests for linkage using affected pedigree members. Biometrics 50 118-127. Whittemore, A. S. and Halpern, J. (1994b). Probability of gene identity by descent: computation and applications. Biometrics 50 109-117.