In this paper, we propose generalized fiducial methods and construct four generalized p-values to test the existence of quantitative trait locus effects under phenotype distributions from a location-scale family. Compared with the likelihood ratio test based on simulation studies, our methods perform better at controlling type I errors while retaining comparable power in cases with small or moderate sample sizes. The four generalized fiducial methods support varied scenarios: two of them are more aggressive and powerful, whereas the other two appear more conservative and robust. A real data example involving mouse blood pressure is used to illustrate our proposed methods.

References

• Agresti, A., & Caffo, B. (2000). Simple and effective confidence intervals for proportions and differences of proportions result from adding two successes and two failures. The American Statistician, 54(4), 280–288. https://doi.org/10.1080/00031305.2000.10474560
• Agresti, A., & Coull, B. A. (1998). Approximate is better than ‘exact’ for interval estimation of binomial proportions. The American Statistician, 52(2), 119–126. https://doi.org/10.2307/2685469
• Bebu, I., Luta, G., Mathew, T., & Agan, B. K. (2016). Generalized confidence intervals and fiducial intervals for some epidemiological measures. International Journal of Environmental Research and Public Health, 13(6), 605. https://doi.org/10.3390/ijerph13060605
• Cai, T. T. (2005). One-sided confidence intervals in discrete distributions. Journal of Statistical Planning and Inference, 131(1), 63–88. https://doi.org/10.1016/j.jspi.2004.01.005
• Chen, Z., & Chen, H. (2005). On some statistical aspects of the interval mapping for QTL detection. Statistica Sinica, 15(4), 909–925.
• Cui, Y., & Hannig, J. (2019). Nonparametric generalized fiducial inference for survival functions under censoring. Biometrika, 106(3), 501–518. https://doi.org/10.1093/biomet/asz016
• Efron, B. (1998). RA Fisher in the 21st century. Statistical Science, 13(2), 95–122. https://doi.org/10.1214/ss/1028905930
• Fisher, R. A. (1930). Inverse probability. In B. J. Green (Ed.), Mathematical proceedings of the Cambridge philosophical society (Vol. 26, pp. 528–535). Cambridge University Press.
• Fisher, R. A. (1932). Statistical methods for research workers (4th ed.). Oliver & Boyd.
• Frühwirth-Schnatter, S. (2006). Finite mixture and Markov switching models. Springer.
• Gelman, A., Stern, H. S., Carlin, J. B., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2014). Bayesian data analysis (3rd ed.). Chapman and Hall/CRC.
• Hannig, J. (2009). On generalized fiducial inference. Statistica Sinica, 19(2), 491–544.
• Hannig, J., Iyer, H., Lai, R. C., & Lee, T. C. (2016). Generalized fiducial inference: A review and new results. Journal of the American Statistical Association, 111(515), 1346–1361. https://doi.org/10.1080/01621459.2016.1165102
• Hannig, J., Iyer, H., & Patterson, P. (2006). Fiducial generalized confidence intervals. Journal of the American Statistical Association, 101(473), 254–269. https://doi.org/10.1198/016214505000000736
• Huang, N., Parco, A., Mew, T., Magpantay, G., McCouch, S., Guiderdoni, E., Xu, J., Subudhi, P., Angeles, E. R., & Khush, G. S. (1997). RFLP mapping of isozymes, RAPD and QTLs for grain shape, brown planthopper resistance in a doubled haploid rice population. Molecular Breeding, 3(2), 105–113. https://doi.org/10.1023/A:1009683603862
• Korol, A., Ronin, Y., Tadmor, Y., Bar-Zur, A., Kirzhner, V., & Nevo, E. (1996). Estimating variance effect of QTL: An important prospect to increase the resolution power of interval mapping. Genetical Research, 67(2), 187–194. https://doi.org/10.1017/S0016672300033632
• Krishnamoorthy, K., & Lee, M. (2010). Inference for functions of parameters in discrete distributions based on fiducial approach: Binomial and Poisson cases. Journal of Statistical Planning and Inference, 140(5), 1182–1192. https://doi.org/10.1016/j.jspi.2009.11.004
• Lai, R. C., Hannig, J., & Lee, T. C. (2015). Generalized fiducial inference for ultrahigh-dimensional regression. Journal of the American Statistical Association, 110(510), 760–772. https://doi.org/10.1080/01621459.2014.931237
• Lander, E. S., & Botstein, D. (1989). Mapping mendelian factors underlying quantitative traits using RFLP linkage maps. Genetics, 121(1), 185–199. https://doi.org/10.1093/genetics/121.1.185
• Li, X., Su, H., & Liang, H. (2018). Fiducial generalized p-values for testing zero-variance components in linear mixed-effects models. Science China Mathematics, 61(7), 1303–1318. https://doi.org/10.1007/s11425-016-9068-8
• Li, X., Xu, X., & Li, G. (2007). A fiducial argument for generalized p-value. Science in China Series A: Mathematics, 50(7), 957–966. https://doi.org/10.1007/s11425-007-0067-7
• Li, X., Zhou, X., & Tian, L. (2013). Interval estimation for the mean of lognormal data with excess zeros. Statistics & Probability Letters, 83(11), 2447–2453. https://doi.org/10.1016/j.spl.2013.07.004
• Liu, G., Li, P., Liu, Y., & Pu, X. (2020). Hypothesis testing for quantitative trait locus effects in both location and scale in genetic backcross studies. Scandinavian Journal of Statistics, 47(4), 1064–1089. https://doi.org/10.1111/sjos.v47.4
• McLachlan, G., & Peel, D. (2000). Finite mixture models. John Wiley & Sons.
• Nkurunziza, S., & Chen, F. (2011). Generalized confidence interval and p-value in location and scale family. Sankhya B, 73(2), 218–240. https://doi.org/10.1007/s13571-011-0026-8
• Perng, S., & Littell, R. C. (1976). A test of equality of two normal population means and variances. Journal of the American Statistical Association, 71(356), 968–971. https://doi.org/10.1080/01621459.1976.10480978
• Rebai, A., Goffinet, B., & Mangin, B. (1994). Approximate thresholds of interval mapping tests for QTL detection. Genetics, 138(1), 235–240. https://doi.org/10.1093/genetics/138.1.235
• Rebai, A., Goffinet, B., & Mangin, B. (1995). Comparing power of different methods for QTL detection. Biometrics, 51(1), 87–99. https://doi.org/10.2307/2533317
• Schaarschmidt, F., Sill, M., & Hothorn, L. A. (2008). Approximate simultaneous confidence intervals for multiple contrasts of binomial proportions. Biometrical Journal, 50(5), 782–792. https://doi.org/10.1002/bimj.v50:5
• Sugiyama, F., Churchill, G. A., Higgins, D. C., Johns, C., Makaritsis, K. P., Gavras, H., & Paigen, B. (2001). Concordance of murine quantitative trait loci for salt-induced hypertension with rat and human loci. Genomics, 71(1), 70–77. https://doi.org/10.1006/geno.2000.6401
• Williams, J. P., & Hannig, J. (2019). Nonpenalized variable selection in high-dimensional linear model settings via generalized fiducial inference. The Annals of Statistics, 47(3), 1723–1753. https://doi.org/10.1214/18-AOS1733
• Wu, R., Ma, C., & Casella, G. (2007). Statistical genetics of quantitative traits: Linkage, maps and QTL. Springer.
• Wu, W. H., & Hsieh, H. N. (2014). Generalized confidence interval estimation for the mean of delta-lognormal distribution: An application to New Zealand trawl survey data. Journal of Applied Statistics, 41(7), 1471–1485. https://doi.org/10.1080/02664763.2014.881780
• Xu, X., & Li, G. (2006). Fiducial inference in the pivotal family of distributions. Science in China Series A, 49(3), 410–432. https://doi.org/10.1007/s11425-006-0410-4
• Zhang, H., Chen, H., & Li, Z. (2008). An explicit representation of the limit of the LRT for interval mapping of quantitative trait loci. Statistics & Probability Letters, 78(3), 207–213. https://doi.org/10.1016/j.spl.2007.05.020

To cite this article: Pengcheng Ren, Guanfu Liu, Xiaolong Pu & Yan Li (2021): Generalized
fiducial methods for testing quantitative trait locus effects in genetic backcross studies, Statistical
Theory and Related Fields, DOI: 10.1080/24754269.2021.1984636
To link to this article: https://doi.org/10.1080/24754269.2021.1984636