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Negative variance components for non-negative hierarchical data with correlation, over-, and/or underdispersion

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  • I. R. C. Oliveira
  • G. Molenberghs
  • G. Verbeke
  • C. G. B. Demétrio
  • C. T. S. Dias

Abstract

The concept of negative variance components in linear mixed-effects models, while confusing at first sight, has received considerable attention in the literature, for well over half a century, following the early work of Chernoff [7] and Nelder [21]. Broadly, negative variance components in linear mixed models are allowable if inferences are restricted to the implied marginal model. When a hierarchical view-point is adopted, in the sense that outcomes are specified conditionally upon random effects, the variance–covariance matrix of the random effects must be positive-definite (positive-semi-definite is also possible, but raises issues of degenerate distributions). Many contemporary software packages allow for this distinction. Less work has been done for generalized linear mixed models. Here, we study such models, with extension to allow for overdispersion, for non-negative outcomes (counts). Using a study of trichomes counts on tomato plants, it is illustrated how such negative variance components play a natural role in modeling both the correlation between repeated measures on the same experimental unit and over- or underdispersion.

Suggested Citation

  • I. R. C. Oliveira & G. Molenberghs & G. Verbeke & C. G. B. Demétrio & C. T. S. Dias, 2017. "Negative variance components for non-negative hierarchical data with correlation, over-, and/or underdispersion," Journal of Applied Statistics, Taylor & Francis Journals, vol. 44(6), pages 1047-1063, April.
    • Handle: RePEc:taf:japsta:v:44:y:2017:i:6:p:1047-1063
    DOI: 10.1080/02664763.2016.1191624
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    References listed on IDEAS

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    1. Molenberghs, Geert & Kenward, Michael G., 2010. "Semi-parametric marginal models for hierarchical data and their corresponding full models," Computational Statistics & Data Analysis, Elsevier, vol. 54(2), pages 585-597, February.
    2. Pryseley, Assam & Tchonlafi, Clotaire & Verbeke, Geert & Molenberghs, Geert, 2011. "Estimating negative variance components from Gaussian and non-Gaussian data: A mixed models approach," Computational Statistics & Data Analysis, Elsevier, vol. 55(2), pages 1071-1085, February.
    3. Geert Verbeke & Geert Molenberghs, 2003. "The Use of Score Tests for Inference on Variance Components," Biometrics, The International Biometric Society, vol. 59(2), pages 254-262, June.
    4. Tony Vangeneugden & Geert Molenberghs & Geert Verbeke & Clarice G.B. Dem�trio, 2011. "Marginal correlation from an extended random-effects model for repeated and overdispersed counts," Journal of Applied Statistics, Taylor & Francis Journals, vol. 38(2), pages 215-232, September.
    5. Molenberghs, Geert & Verbeke, Geert, 2007. "Likelihood Ratio, Score, and Wald Tests in a Constrained Parameter Space," The American Statistician, American Statistical Association, vol. 61, pages 22-27, February.
    6. Hinde, John & Demetrio, Clarice G. B., 1998. "Overdispersion: Models and estimation," Computational Statistics & Data Analysis, Elsevier, vol. 27(2), pages 151-170, April.
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