Estimation of Parent-Sib Correlations for Quantitative Traits Using the Linear Mixed Regression Model: Applications to Arterial Blood Pressures Data Collected From Nuclear Families


  • Maha Al-Eid Department of Biostatistics, Epidemiology and Scientific Computing, King Faisal SpecialistHospital and Research Center, Riyadh, Saudi Arabia
  • Sarah AL-Gahtani Department of Internal Medicine, King Fahd Medical City, Riyadh, Saudi Arabia
  • Mohamed M. Shoukri Department of Epidemiology and Biostatistics, Schulich of Medicine and Dentistry University of Western Ontario, London, Ontario Canada



Genetic epidemiology, Familial correlations, Heritability, Linear Mixed normal models, Maximum Likelihood estimation, Estimating Ratio of parameters, Bootstrap Confidence interval.


A fundamental question in quantitative genetics is whether observed variation in the phenotypic values of a particular trait is due to environmental or to biological factors. Proportion of variations attributed to genetic factors is known as heritability of the trait. Heritability is a concept that summarizes how much of the variation in a trait is due to variation in genetic factors. Often, this term is used in reference to the resemblance between parents and their offspring. In this context, high heritability implies a strong resemblance between parents and offspring with regard to a specific trait, while low heritability implies a low level of resemblance. While many applications measure the offspring resemblance to their parents using the mid-parental value of a quantitative trait of interest as an input parameter, others focus on estimating maternal and paternal heritability. In this paper we address the problem of estimating parental heritability using the nuclear family as a unit of analysis. We derive moment and maximum likelihood estimators of parental heritability, and test their equality using the likelihood ratio test, the delta method. We also use Fieller’s interval on the ratio of parental heritability to address the question of bioequivalence. The methods are illustrated on published arterial blood pressures data collected from nuclear families.


Mackay TF. The genetic architecture of quantitative traits. Annual Review of Genetics 2001; 35(1): 303-339. DOI:

Mackay TF. Q&A: Genetic analysis of quantitative traits. Journal of Biology 2009; 8(3): 23. DOI:

Pack AI, Keenan BT, Byrne EM, Gehrman PR. Genetics and genomic basis of sleep disorders in humans. In Principles and practice of sleep medicine 2017; pp. 322-339. Elsevier. DOI:

Cloninger CR, Van Eerdewegh P, Goate A, Edenberg HJ, Blangero J, Hesselbrock, Crowe R. Anxiety proneness linked to epistatic loci in genome scan of human personality traits. American Journal of Medical Genetics 1998; 81(4): 313-317.<313::AID-AJMG7>3.0.CO;2-U DOI:<313::AID-AJMG7>3.0.CO;2-U

Abney M, McPeek MS, Ober C. Broad and narrow heritability of quantitative traits in a founder population. The American Journal of Human Genetics 2001; 68(5): 1302-1307. DOI:

Must A, Spadano J, Coakley EH, et al. The disease burden associated with overweight and obesity. JAMA 1999; 282(16): 1523-1529. DOI:

Gu D, He J, Duan X, et al. Body weight and mortality among men and women in China. JAMA 2006; 295(7): 776-783. DOI:

Johnson PC, Logue J, McConnachie A, et al. Intergenerational change and familial aggregation of body mass index. Eur J Epidemiol 2012; 27(1): 53-61. DOI:

Bushnik T, Garriguet D, Colley R. Parent-Child association in body weight status. Health Reports. Statistics Canada, Catalogue no. 82-003-X. June 2017; 28(6): pp. 12-19.

Laird N, Lange C. The Fundamentals of Modern Statistical genetics. Springer, New York 2011. DOI:

Kempthorne O, Tandon O. The estimation of heritability by regression of offspring on parents. Biometrics 9: 90-100. DOI:

Shoukri MM, Asyali MH, Walter SD. Issues of cost and efficiency in the design of reliability studies: Review and new results. Statistical Methods in Medical Research 2004. DOI:

Flynn NT, Whitley E, Peters T. Recruitment strategy in a cluster randomized trial: cost implications. Statistics in Medicine 2002; 21: 397-405. DOI:

Sukhatme PV, Sukhatme BV, Sukhatme S, Asok C. Sampling theory of surveys with applications. Iowa State University Press 1984.

Rao SS. Optimization: theory and applications. New Delhi: Wiley Eastern Limited 1984.

TK, Mak, Ng, KW. Analysis of familial data: Linear model approach. Biometrika 1981; 68: 457-461. DOI:

Shoukri MM, Ward RH. Use of regression models to estimate genetic parameters and measures of familial resemblance in man. The Journal of Royal Statistical Society, series C. 1989; 38(3): 467-479. DOI:

Jacquard A. Heritability: one word, three concepts. Biometrics 1983; 39: 465-477. DOI:

Stuart, Allan, Ord, Keith. Advanced Theory of statistics, Vol. 1 (Distribution theory), 6th edition. J. Wiley, NY, USA 2010.

Fieller EC. “Some Problems in Interval Estimation.” Journal of the Royal Statistical Society. Series, (Methodological). JSTOR 1954; 175-85. DOI:

Davison AC, Hinkley, D. Bootstrap methods and their applications, Cambridge University Press 1997. DOI:

Stuart, Allan, Ord, Keith. Advanced Theory of statistics, Vol. II (Statistical Inference), 5th edition. Hafner publishing company 2009.

Miall WE, Oldham PO. A study of arterial blood pressure and its inheritance in a sample of the general population. Clinical Science 1955; 14: 459-487.

Singer JD. Using SAS PROC MIXED to fit multilevel models, hierarchical models, and individual growth models. Journal of Educational and Behavioral Statistics. Winter 1998; 24(3): 323-355. DOI:

Hu Y, He L, Wu Y, Ma G, Li L, Hu Y. Familial correlation and aggregation of body mass index and blood pressure in Chinese Han Population. BMC Public Health 2013; 13: 686. DOI:

NCD Countdown 2030: worldwide trends in non-communicable disease mortality and progress towards Sustainable Development Goal target 3.4. Lancet 2018; 392: 1072-88. DOI:




How to Cite

Al-Eid, M. ., AL-Gahtani, S. ., & Shoukri, M. M. . (2020). Estimation of Parent-Sib Correlations for Quantitative Traits Using the Linear Mixed Regression Model: Applications to Arterial Blood Pressures Data Collected From Nuclear Families. International Journal of Statistics in Medical Research, 9, 59–68.



General Articles