Abstract
Basic probability theory and statistical models and procedures for the analysis of genetic studies are covered in Chap. 1. This chapter starts with an introduction to basic distribution theory and common distributions that are used in the book, including the uniform, multinomial, normal, t-, F-, Beta, Gamma, chi-squared and hypergeometric distributions. The basic distributions for order statistics are also given. Several types of stochastic convergence used in the book are summarized. Maximum likelihood estimation and its large sample properties are discussed. Various tests, including the efficient Score test, likelihood ratio test and Wald test, are studied with or without nuisance parameters. Multiple testing issues related to testing association with multiple genetic markers and related to hypothesis testing with an unknown genetic model are briefly reviewed. This chapter also covers the Delta method, the EM algorithm, basic concepts of sample size and power calculations, and asymptotic relative efficiency.
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References
Benjamini, Y., Hochberg, Y.: Controlling the false discovery rate: a practical and powerful approach to multiple testing. J. Roy. Stat. Soc. Ser. B 57, 289–300 (1995)
Benjamini, Y., Yekutieli, D.: The control of the false discovery rate in multiple testing under dependency. Ann. Stat. 29, 1165–1188 (2001)
Casella, G., Berger, R.L.: Statistical Inference. Duxbury Press, Belmont (1990)
Ceppellini, R., Siniscalco, M., Smith, C.A.B.: The estimation of gene frequencies in a random mating population. Ann. Hum. Genet. 20, 97–115 (1955)
Cox, D.R., Hinkley, D.V.: Theoretical Statistics. Chapman & Hall/CRC, Boca Raton (1974)
David, H.A., Nagaraja, H.N.: Order Statistics. 3rd edn. Wiley, Hoboken (2003)
Dempster, A., Laird, N.M., Rubin, D.: Maximum likelihood from incomplete data via the EM algorithm (with discussion). J. Roy. Stat. Soc. Ser. B 39, 1–38 (1977)
Dudoit, S., van der Laan, M.J.: Multiple Testing Procedures with Applications to Genomics. Springer, New York (2008)
Elston, R.C., Johnson, W.D.: Basic Biostatistics for Genetists and Epidemiologists. Wiley, West Sussex (2008)
Evans, M., Hastings, N., Peacock, B.: Statistical Distributions. 3rd edn. Wiley, New York (2000)
Freidlin, B., Zheng, G., Li, Z., Gastwirth, J.L.: Trend tests for case-control studies of genetic markers: power, sample size and robustness. Hum. Hered. 53, 146–152 (2002) (Erratum 68, 220 (2009))
Gastwirth, J.L.: On robust procedures. J. Am. Stat. Assoc. 61, 929–948 (1966)
Gastwirth, J.L.: The use of maximin efficiency robust tests in combining contingency tables and survival analysis. J. Am. Stat. Assoc. 80, 380–384 (1985)
Gastwirth, J.L., Freidlin, B.: On power and efficiency robust linkage tests for affected sibs. Ann. Hum. Genet. 64, 443–453 (2000)
Lachin, J.M.: Biostatistical Methods: The Assessment of Relative Risks. Wiley, New York (2000)
Noether, G.E.: On a theorem of Pitman. Ann. Math. Stat. 26, 64–68 (1955)
Robert, C.P., Casella, G.: Monte Carlo Statistical Methods. Springer, New York (2004)
Storey, J.D.: A direct approach to false discovery rates. J. Roy. Stat. Soc. Ser. B 64, 479–498 (2002)
Storey, J.D.: The positive false discovery rate: A Bayesian interpretation and the q-value. Ann. Stat. 31, 2013–2035 (2003)
van der Vaart, A.W.: Asymptotic Statistics. Cambridge University Press, Cambridge (1998)
Zheng, G., Freidlin, B., Gastwirth, J.L.: Robust TDT-type candidate-gene association tests. Ann. Hum. Hered. 66, 145–155 (2002)
Zheng, G., Freidlin, B., Gastwirth, J.L.: Comparison of robust tests for genetic association using case-control studies. In: Rojo, J. (ed.) Optimality: The Second Erich L. Lehmann Symposium. Lecture Notes–Monograph Series, vol. 49, pp. 320–336. Institute of Mathematical Statistics, Beachwood (2006)
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Zheng, G., Yang, Y., Zhu, X., Elston, R.C. (2012). Introduction to Probability Theory and Statistics. In: Analysis of Genetic Association Studies. Statistics for Biology and Health. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-2245-7_1
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