The multivariate linear regression model with errors from a scale mixture of Gaussian densities yields a complex likelihood function. Combining this likelihood with any nontrivial prior distribution leads to a highly intractable posterior density. If a conditionally conjugate prior is used, then there is a well known and easy‐to‐implement data augmentation (DA) algorithm available for exploring the posterior. Hobert et al recently showed that, under an improper conditionally conjugate prior (and weak regularity conditions), the Markov chain that drives the DA algorithm converges at a geometric rate. Unfortunately, the model studied by Hobert et al can only be used in situations where the X matrix has full column rank. In this note, analogous convergence rate results are established for a proper conditionally conjugate prior. An important advantage of using a proper prior is that, not only is the X matrix allowed to be column rank deficient, but it can also have more columns than rows, that is, our model is applicable in cases where p>n. This is an important extension in the era of big data.

Objective A mother–child dyad trajectory model of weight and body composition spanning from conception to adolescence was developed to understand how early life exposures shape childhood body composition. Methods African American (49.3%) and Dominican (50.7%) pregnant mothers (n = 337) were enrolled during pregnancy, and their children (47.5% female) were followed from ages 5 to 14. Gestational weight gain (GWG) was abstracted from medical records. Child weight, height, percentage body fat, and waist circumference were measured. GWG and child body composition trajectories were jointly modeled with a flexible latent class model with a class membership component that included prepregnancy BMI. Results Four prenatal and child body composition trajectory patterns were identified, and sex‐specific patterns were observed for the joint GWG–postnatal body composition trajectories with more distinct patterns among girls but not boys. Girls of mothers with high GWG across gestation had the highest BMI z score, waist circumference, and percentage body fat trajectories from ages 5 to 14; however, boys in this high GWG group did not show similar growth patterns. Conclusions Jointly modeled prenatal weight and child body composition trajectories showed sex‐specific patterns. Growth patterns from childhood though early adolescence appeared to be more profoundly affected by higher GWG patterns in females, suggesting sex differences in developmental programming.

Gibbs sampling is a widely popular Markov chain Monte Carlo algorithm which is often used to analyze intractable posterior distributions associated with Bayesian hierarchical models. The goal of this article is to introduce an alternative to Gibbs sampling that is particularly well suited for Bayesian models which contain latent or missing data. The basic idea of this hybrid algorithm is to update the latent data from its full conditional distribution at every iteration, and then use a random scan to update the parameters of interest. The hybrid algorithm is often easier to analyze from a theoretical standpoint than the deterministic or random scan Gibbs sampler. We highlight a positive result in this direction from Abrahamsen and Hobert (2018), who proved geometric ergodicity of the hybrid algorithm for a Bayesian version of the general linear mixed model with a continuous shrinkage prior. The convergence rate of the Gibbs sampler for this model remains unknown. In addition, we provide new geometric ergodicity results for the hybrid algorithm and the Gibbs sampler for two classes of Bayesian linear regression models with non-Gaussian errors. In both cases, the conditions under which the hybrid algorithm is geometric are much weaker than the corresponding conditions for the Gibbs sampler. Finally, we show that the hybrid algorithm is amenable to a modified version of the sandwich methodology of Hobert and Marchev (2008), which can be used to speed up the convergence rate of the underlying Markov chain while requiring roughly the same computational effort per iteration.

Background: Gestational weight gain (GWG) and anthropometric trajectories may affect foetal programming and are potentially modifiable. Objectives: To assess concomitant patterns of change in weight, circumferences and adiposity across gestation as an integrated prenatal exposure, and determine how they relate to neonatal body composition. Methods: Data are from a prospective cohort of singleton pregnancies (n = 2182) enrolled in United States perinatal centres, 2009-2013. Overall and by prepregnancy BMI group (overweight/obesity and healthy weight), joint latent trajectory models were fit with prenatal weight, mid-upper arm circumference (MUAC), triceps (TSF) and subscapular (SSF) skinfolds. Differences in neonatal body composition by trajectory class were assessed via weighted least squares.

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