Convergence analysis of block Gibbs samplers for Bayesian linear mixed models with p>Np>N

Exploration of the intractable posterior distributions associated with Bayesian versions of the general linear mixed model is often performed using Markov chain Monte Carlo. In particular, if a conditionally conjugate prior is used, then there is a simple two-block Gibbs sampler available. Rom\'{a}n and Hobert [Linear Algebra Appl. 473 (2015) 54-77] showed that, when the priors are proper and the $X$ matrix has full column rank, the Markov chains underlying these Gibbs samplers are nearly always geometrically ergodic. In this paper, Rom\'{a}n and Hobert's (2015) result is extended by allowing improper priors on the variance components, and, more importantly, by removing all assumptions on the $X$ matrix. So, not only is $X$ allowed to be (column) rank deficient, which provides additional flexibility in parameterizing the fixed effects, it is also allowed to have more columns than rows, which is necessary in the increasingly important situation where $p>N$. The full rank assumption on $X$ is at the heart of Rom\'{a}n and Hobert's (2015) proof. Consequently, the extension to unrestricted $X$ requires a substantially different analysis.Comment: Published at http://dx.doi.org/10.3150/15-BEJ749 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Alternating hemiplegia of childhood: evolution over time and mouse model corroboration

Alternating hemiplegia of childhood is a rare neurodevelopmental disorder caused by ATP1A3 mutations. Some evidence for disease progression exists, but there are few systematic analyses. Here, we evaluate alternating hemiplegia of childhood progression in humans and in the D801N knock-in alternating hemiplegia of childhood mouse, Mashlool, model. This study performed an ambidirectional (prospective and retrospective data) analysis of an alternating hemiplegia of childhood patient cohort (n = 42, age 10.24 +/- 1.48 years) seen at one US centre. To investigate potential disease progression, we used linear mixed effects models incorporating early and subsequent visits, and Wilcoxon Signed Rank test comparing first and last visits. Potential early-life clinical predictors were determined via multivariable regression. We also compared EEG background at first encounter and at last follow-up. We then performed a retrospective confirmation study on a multicentre cohort of alternating hemiplegia of childhood patients from France (n = 52). To investigate disease progression in the Mashlool mouse, we performed behavioural testing on a cohort of Mashlool' mice at prepubescent and adult ages (n =11). Results: US patients, over time, demonstrated mild worsening of non-paroxysmal disability index scores, but not of paroxysmal disability index scores. Increasing age was a predictor of worse scores: P< 0.0001 for the non-paroxysmal disability index, intellectual disability scale and gross motor scores. Earliest non-paroxysmal disability index score was a predictor of last visit non-paroxysmal disability index score (P= 0.022), and earliest intellectual disability score was a predictor of last intellectual disability score (P= 0.035). More patients with EEG background slowing were noted at last follow-up as compared to initial (P=0.015). Similar worsening of disease with age was also noted in the French cohort: age was a significant predictor of non-paroxysmal disability index score (P= 0.001) and first and last non-paroxysmal disability index score scores significantly differed (P=0.002). In animal studies, adult Mashlool mice had, as compared to younger Mashlool mice, (i) worse balance beam performance; (ii) wider base of support; (iii) higher severity of seizures and resultant mortality; and (iv) no increased predisposition to hemiplegic or dystonic spells. In conclusion, (i) non-paroxysmal alternating hemiplegia of childhood manifestations show, on average over time, progression associated with severity of early-life non-paroxysmal disability and age. (ii) Progression also occurs in Mashlool mice, confirming that ATP1A3 disease can lead to age-related worsening. (iii) Clinical findings provide a basis for counselling patients and for designing therapeutic trials. Animal findings confirm a mouse model for investigation of underlying mechanisms of disease progression, and are also consistent with known mechanisms of ATP1A3-related neurodegeneration