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

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 nonnegative 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. ARTICLE HISTORY

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“…Although a few researchers have reported about negative clustering effects (El Leithy et al, 2016;Kenny et al, 2002;Klotzke & Fox, 2019a, 2019bLoeys & Molenberghs, 2013;Molenberghs & Verbeke, 2007Oliveira et al, 2017;Pryseley et al, 2011;Verbeke & Molenberghs, 2003), the effects of ignoring negatively clustered observations has hardly been recognized. Because negative clustering effects are not considered by the majority of the multilevel modelling community, these effects are not well understood.…”

“…However, when modeling the covariance structure with the LME, the τ is restricted to be greater than zero, since it represents the random intercept variance. In the literature, it has been shown that the maximum likelihood estimate of the random effect variance can become negative (El Leithy et al, 2016;Kenny et al, 2002;Klotzke & Fox, 2019a, 2019bLoeys & Molenberghs, 2013;Molenberghs & Verbeke, 2007Oliveira et al, 2017;Pryseley et al, 2011;Verbeke & Molenberghs, 2003). For the (oneway) LME (for balanced groups), two sums of squares are considered to estimate the covariance components τ and σ 2 ,…”

“…For τ > 0, the ICC is often interpreted as the ratio of variance explained by vii 7.2. Method the clustering of observations in comparison to the total variance in the data; ρ = τ /(τ + σ 2 ) (Oliveira et al, 2017;Raudenbush & Bryk, 2002b;Snijders & Bosker, 2012). However, the ICC can also be considered to quantify the degree of resemblance or average similarity of observations within a cluster, or as the 'average correlation' in each cluster (Kenny & Judd, 1986;Kenny et al, 2002).…”

“…Even though negative clustering effects were discussed previously by others (cf. Oliveira et al, 2017;Pryseley et al, 2011), there still appears to be a lack of awareness about these effects. As the LME comes with the restriction that observations need to be positively clustered, several suggestions can be found in the literature that τ should be set to zero, when the ICC estimate becomes negative (see for example Baldwin et al, 2008;Eldridge et al, 2009;Gibson et al, 2015;Krannitz et al, 2015;Langfred, 2007;Maas & Hox, 2005).…”

“…In addition to that, it has been shown that Bayesian Covariance Structure Modelling (BCSM; Klotzke & Fox, 2019a, 2019b) can be used to simulate these negative correlations in clusters. In doing so, a simulation study was developed where observations are negatively clustered and data were simulated that others, most notably Kenny et al (2002) and Oliveira et al (2017), andPryseley et al (2011) described.…”