Hostname: page-component-745bb68f8f-mzp66 Total loading time: 0 Render date: 2025-01-07T18:07:27.408Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Likelihood Ratio Tests in Bivariate ACDE Models

Published online by Cambridge University Press:  01 January 2025

Hao Wu  and
Michael C. Neale
Hao Wu*
Affiliation:
Boston College
Michael C. Neale
Affiliation:
Virginia Commonwealth University
*
Requests for reprints should be sent to Hao Wu, Boston College, 300 McGuinn Hall, 140 Commonwealth Ave, Chestnut Hill, MA, 02467, USA. E-mail: hao.wu.5@bc.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The ACE and ADE models have been heavily exploited in twin studies to identify the genetic and environmental components in phenotypes. However, the validity of the likelihood ratio test (LRT) of the existence of a variance component, a key step in the use of such models, has been doubted because the true values of the parameters lie on the boundary of the parameter space of the alternative model for such tests, violating a regularity condition required for a LRT (e.g., Carey in Behav. Genet. 35:653–665, 2005; Visscher in Twin Res. Hum. Genet. 9:490–495, 2006). Dominicus, Skrondal, Gjessing, Pedersen, and Palmgren (Behav. Genet. 36:331–340, 2006) solve the problem of testing univariate components in ACDE models. Our current work as presented in this paper resolves the issue of LRTs in bivariate ACDE models by exploiting the theoretical frameworks of inequality constrained LRTs based on cone approximations. Our derivation shows that the asymptotic sampling distribution of the test statistic for testing a single bivariate component in an ACE or ADE model is a mixture of χ2 distributions of degrees of freedom (dfs) ranging from 0 to 3, and that for testing both the A and C (or D) components is one of dfs ranging from 0 to 6. These correct distributions are stochastically smaller than the χ2 distributions in traditional LRTs and therefore LRTs based on these distributions are more powerful than those used naively. Formulas for calculating the weights are derived and the sampling distributions are confirmed by simulation studies. Several invariance properties for normal data (at most) missing by person are also proved. Potential generalizations of this work are also discussed.

Keywords

ACE modelvariance componentslikelihood ratio testχ-2 distribution
Type
Original Paper
Information
Psychometrika , Volume 78 , Issue 3 , July 2013 , pp. 441 - 463
Copyright
Copyright © 2012 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Boker, S., Neale, M., Maes, H., Wilde, M., Spiegel, M., Brick, T., et al. (2011). OpenMx: an open source extended structural equation modeling framework. Psychometrika, 76, 306–317CrossRefGoogle ScholarPubMed
Carey, G. (2005). Cholesky problems. Behavior Genetics, 35, 653–665CrossRefGoogle ScholarPubMed
Chernoff, H. (1954). On the distribution of the likelihood ratio. The Annals of Mathematical Statistics, 25, 573–578CrossRefGoogle Scholar
Dominicus, A., Skrondal, A., Gjessing, H.K., Pedersen, N.L., Palmgren, J. (2006). Likelihood ratio tests in behavioral genetics: problems and solutions. Behavior Genetics, 36, 331–340CrossRefGoogle ScholarPubMed
Gupta, A.K., Nagar, D.K. (1999). Matrix variate distributions, London: Chapman & Hall/CRCGoogle Scholar
Han, S.S., Chang, J.T. (2010). Reconsidering the asymptotic null distribution of likelihood ratio tests for genetic linkage in multivariate variance components models under complete pleiotropy. Biostatistics, 11, 226–241CrossRefGoogle ScholarPubMed
Kuriki, S., Takemura, A. (2000). Some geometry of the cone of nonnegative definite matrices and weights of associated chi-bar-squared distribution. Annals of the Institute of Statistical Mathematics, 52, 1–14CrossRefGoogle Scholar
Neale, M.C., Cardon, L.R. (1992). Methodology for genetic studies of twins and families, Dordrecht: Kluwer AcademicCrossRefGoogle Scholar
R Development Core Team (2010). R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org. Google Scholar
Schieken, R.M., Eaves, L.J., Hewitt, J.K., Mosteller, M., Bodurtha, J.M., Moskowitz, W.B., et al. (1989). Univariate genetic analysis of blood pressure in children: the MCV twin study. American Journal of Cardiology, 64, 1333–1337CrossRefGoogle Scholar
Self, S.G., Liang, K.-Y. (1987). Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under non-standard conditions. Journal of the American Statistical Association, 82, 605–610CrossRefGoogle Scholar
Shapiro, A. (1985). Asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints. Biometrika, 72, 133–144CrossRefGoogle Scholar
Shapiro, A. (1987). A conjecture related to chi-rar-squared distributions. The American Mathematical Monthly, 94, 46–48CrossRefGoogle Scholar
Shapiro, A. (1988). Towards a unified theory of inequality constrained testing in multivariate analysis. International Statistical Review, 56, 49–62CrossRefGoogle Scholar
Shapiro, A. (2007). Statistical inference in moment structures. In Lee, S.-Y. (Eds.), Hand book of latent variable and related models, 229260Google Scholar
Silvapulle, M.J., Sen, P.K. (2005). Constrained statistical inference: inequality, order, and shape restrictions, New York: WileyGoogle Scholar
Stram, D.O., Lee, J.W. (1994). Variance components testing in the longitudinal mixed effects model. Biometrics, 50, 1171–1177 Corrected in Biometrics, 51, 1196CrossRefGoogle ScholarPubMed
Visscher, P.M. (2006). A note on the asymptotic distribution of likelihood ratio tests to test variance components. Twin Research and Human Genetics, 9, 490–495CrossRefGoogle ScholarPubMed