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On the Likelihood Ratio Tests in Bivariate ACDE Models

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  • Hao Wu
  • Michael Neale

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 χ 2 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 χ 2 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. Copyright The Psychometric Society 2013

Suggested Citation

  • Hao Wu & Michael Neale, 2013. "On the Likelihood Ratio Tests in Bivariate ACDE Models," Psychometrika, Springer;The Psychometric Society, vol. 78(3), pages 441-463, July.
    • Handle: RePEc:spr:psycho:v:78:y:2013:i:3:p:441-463
    DOI: 10.1007/s11336-012-9304-2
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    References listed on IDEAS

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    1. Steven Boker & Michael Neale & Hermine Maes & Michael Wilde & Michael Spiegel & Timothy Brick & Jeffrey Spies & Ryne Estabrook & Sarah Kenny & Timothy Bates & Paras Mehta & John Fox, 2011. "OpenMx: An Open Source Extended Structural Equation Modeling Framework," Psychometrika, Springer;The Psychometric Society, vol. 76(2), pages 306-317, April.
    2. Satoshi Kuriki & Akimichi Takemura, 2000. "Some Geometry of the Cone of Nonnegative Definite Matrices and Weights of Associated X 2 Distribution," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(1), pages 1-14, March.
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    Cited by:

    1. Chen, Yunxiao & Moustaki, Irini & Zhang, H, 2020. "A note on likelihood ratio tests for models with latent variables," LSE Research Online Documents on Economics 107490, London School of Economics and Political Science, LSE Library.
    2. Hao Wu, 2016. "A Note on the Identifiability of Fixed-Effect 3PL Models," Psychometrika, Springer;The Psychometric Society, vol. 81(4), pages 1093-1097, December.
    3. Yunxiao Chen & Irini Moustaki & Haoran Zhang, 2020. "A Note on Likelihood Ratio Tests for Models with Latent Variables," Psychometrika, Springer;The Psychometric Society, vol. 85(4), pages 996-1012, December.
    4. Hao Wu & Ryne Estabrook, 2016. "Identification of Confirmatory Factor Analysis Models of Different Levels of Invariance for Ordered Categorical Outcomes," Psychometrika, Springer;The Psychometric Society, vol. 81(4), pages 1014-1045, December.

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