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Applying Multivariate Discrete Distributions to Genetically Informative Count Data

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Abstract

We present a novel method of conducting biometric analysis of twin data when the phenotypes are integer-valued counts, which often show an L-shaped distribution. Monte Carlo simulation is used to compare five likelihood-based approaches to modeling: our multivariate discrete method, when its distributional assumptions are correct, when they are incorrect, and three other methods in common use. With data simulated from a skewed discrete distribution, recovery of twin correlations and proportions of additive genetic and common environment variance was generally poor for the Normal, Lognormal and Ordinal models, but good for the two discrete models. Sex-separate applications to substance-use data from twins in the Minnesota Twin Family Study showed superior performance of two discrete models. The new methods are implemented using R and OpenMx and are freely available.

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Notes

  1. This is not the same as the bivariate negative binomial distribution of Kocherlakota andKocherlakota (1992), which is actually a special case of the negative multinomial distribution (Johnson et al. 1997).

  2. We use “monophenotype twin model” to refer to what behavior geneticists commonly refer to as a “univariate twin model.” The latter terminology is rather unfortunate. The independent unit of analysis is the twin pair, and thus, a sample of twin data on a single phenotype is a sample of realizations of random 2-vectors, that is, from a bivariate distribution.

  3. If, say, people’s latent tendency to drink were normally distributed in the population, and number of drinks each day were conditionally Poisson, then daily number of drinks would follow a Hermite distribution in the population (Kemp and Kemp, 1966). The present paper does not consider the Hermite distribution, as it would be most appropriate for variables with multimodal distributions (Johnson et al. 2005; Giles 2010).

  4. We considered basing our quadratic loss metric on \({\mathbf{y}}_{i}\)’s Mahalanobis distance from \({\hat{\mathbf{y}}}_{i}\). But, if the predicted variances are systematically too large, the sum of squared Mahalanobis distances will be smaller than if the variances were accurately predicted—not a desirable property for a loss metric.

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Acknowledgments

The authors were supported by U.S. Public Health Service grant DA026119. William G. Iacono and Matt McGue provided the MTFS dataset, which was supported by U.S. Public Health Service Grants DA05147, AA009367, and DA013240. The first author gives his special thanks to Matt McGue, Niels G. Waller, and Hermine H. Maes for their comments on drafts of the paper.

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Authors and Affiliations

  1. Virginia Institute for Psychiatric & Behavioral Genetics, Virginia Commonwealth University, Richmond, VA, 23298-0126, USA

    Robert M. Kirkpatrick & Michael C. Neale

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  1. Robert M. Kirkpatrick
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  2. Michael C. Neale
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Correspondence to Robert M. Kirkpatrick.

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Robert M. Kirkpatrick and Michael C. Neale declare that they have no conflict of interest.

Human and animal rights and informed consent

The MTFS was reviewed and approved by the Institutional Review Board at the University of Minnesota. Written informed assent or consent was obtained from all participants, with parents providing written consent for their minor children.

Additional information

Edited by Gitta Lubke.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Online Resource 1: Supplementary Appendices. Supplementary material 1 (pdf 150 kb)

10519_2015_9757_MOESM2_ESM.zip

Online Resource 2: 3 text files: a README file for the other two, an example R script from the Monte Carlo simulation, and an R script for producing graphs and summary statistics from the raw simulation data (read in over the web). Supplementary material 2 (zip 15 kb)

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Kirkpatrick, R.M., Neale, M.C. Applying Multivariate Discrete Distributions to Genetically Informative Count Data. Behav Genet 46, 252–268 (2016). https://doi.org/10.1007/s10519-015-9757-z

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