Abstract
In standard interval mapping (IM) of quantitative trait loci (QTL), the QTL effect is described by a normal mixture model. When this assumption of normality is violated, the most commonly adopted strategy is to use the previous model after data transformation. However, an appropriate transformation may not exist or may be difficult to find. Also this approach can raise interpretation issues. An interesting alternative is to consider a skew-normal mixture model in standard IM, and the resulting method is here denoted as skew-normal IM. This flexible model that includes the usual symmetric normal distribution as a special case is important, allowing continuous variation from normality to non-normality. In this paper we briefly introduce the main peculiarities of the skew-normal distribution. The maximum likelihood estimates of parameters of the skew-normal distribution are obtained by the expectation-maximization (EM) algorithm. The proposed model is illustrated with real data from an intercross experiment that shows a significant departure from the normality assumption. The performance of the skew-normal IM is assessed via stochastic simulation. The results indicate that the skew-normal IM has higher power for QTL detection and better precision of QTL location as compared to standard IM and nonparametric IM.
Similar content being viewed by others
References
Arellano-Valle, R.B., Ozan, S., Bolfarine, H., Lachos, V.H., 2005. Skew normal measurement error models. Journal of Multivariate Analysis, 96(2):265–281. [doi:10.1016/j.jmva.2004.11.002]
Azzalini, A., 1985. A class of distributions which includes the normal ones. Scandinavian Journal of Statistics, 12(2): 171–178.
Azzalini, A., 1986. Further results on a class of distributions which includes the normal ones. Statistica, 46(2):199–208.
Azzalini, A., Capitanio, A., 1999. Statistical applications of the multivariate skew-normal distribution. Journal of the Royal Statistical Society, Series B (Statistical Methodology), 61(3):579–602. [doi:10.1111/1467-9868.00194]
Broman, K., 2003. Mapping quantitative trait loci in the case of a spike in the phenotype distribution. Genetics, 163(3):1169–1175.
Churchill, G.A., Doerge, R.W., 1994. Empirical threshold values for quantitative trait mapping. Genetics, 138(3):963–971.
Dalla Valle, A., 2004. The Skew-Normal Distribution. In: Genton, M.G. (Ed.), Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality. Chapman & Hall CRC, Boca Raton, FL.
Dempster, A.P., Laird, N.M., Rubin, D.B., 1977. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39(1):1–38.
Draper, N.R., Smith, H., 1998. Applied Regression Analysis, 3th Ed. John Wiley & Sons Inc., New York.
Henze, N., 1986. A probabilistic representation of the skew-normal distribution. Scandinavian Journal of Statistics, 13(4):271–275.
Ihaka, R., Gentleman, R., 1996. R: a language for data analysis and graphics. Journal of Computational and Graphical Statistics, 5(3):299–314. [doi:10.2307/1390807]
Kao, C.H., Zeng, Z.B., Teasdale, R.D., 1999. Multiple interval mapping for quantitative trait loci. Genetics, 152(3):1203–1216.
Kruglyak, L., Lander, E.S., 1995. A Nonparametric approach for mapping quantitative trait loci. Genetics, 139(3):1421–1428.
Lander, E.S., Botstein, D., 1989. Mapping Mendelian factors underlying quantitative traits using RELP linkage maps. Genetics, 121(1):185–199.
Lynch, M., Walsh, B., 1998. Genetics and Analysis of Quantitative Traits. Sinauer Associates, Inc., Sunderland, MA, Massachusetts, USA.
Morton, N.E., 1984. Trials of Segregation Analysis by Deterministic and Macro Simulation. In: Chakravarti, A. (Ed.), Human Population Genetics: The Pittsburgh Symposium. Van Nostrand Reinhold, New York, p.83–107.
Pewsey, A., 2000. Problems of inference for Azzalini’s skewnormal distribution. Journal of Applied Statistics, 27(7):859–870. [doi:10.1080/02664760050120542]
Pewsey, A., 2006. Modelling asymmetrically distributed circular data using the wrapped skew-normal distribution. Environmental and Ecological Statistics, 13(3):257–269. [doi:10.1007/s10651-005-0010-4]
Roberts, C., 1966. A correlation model useful in the study of twins. Journal of the American Statistical Association, 61(316):1184–1190. [doi:10.2307/2283207]
Rodo, J., Gonçalves, L.A., Demengeot, J., Coutinho, A., Penha-Gonçalves, C., 2006. MHC class II molecules control murine B cell responsiveness to lipopolysaccharide stimulation. The Journal of Immunology, 77(7):4620–4626.
Zeng, Z.B., 1994. Precision mapping of quantitative trait loci. Genetics, 136(4):1457–1468.
Zou, F., Yandell, B.S., Fine, J.P., 2003. Rank-based statistical methodologies for quantitative trait locus mapping. Genetics, 165(3):1599–1605.
Author information
Authors and Affiliations
Corresponding author
Additional information
Project supported in part by Foundation for Science and Technology (FCT) (No. SFRD/BD/5987/2001), and the Operational Program Science, Technology, and Innovation of the FCT, co-financed by the European Regional Development Fund (ERDF)
Rights and permissions
About this article
Cite this article
Fernandes, E., Pacheco, A. & Penha-Gonçalves, C. Mapping of quantitative trait loci using the skew-normal distribution. J. Zhejiang Univ. - Sci. B 8, 792–801 (2007). https://doi.org/10.1631/jzus.2007.B0792
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1631/jzus.2007.B0792
Key words
- Interval mapping (IM)
- Quantitative trait loci (QTL)
- Skew-normal distribution
- Expectation-maximization (EM) algorithm