Model Selection in Variational Mixed Effects Models

Variational inference is an alternative estimation technique for Bayesian models. Recent work shows that variational methods provide consistent estimation via efficient, deterministic algorithms. Other tools, such as model selection using variational AICs (VAIC) have been developed and studied for the linear regression case. While mixed effects models have enjoyed some study in the variational context, tools for model selection are lacking. One important feature of model selection in mixed effects models, particularly longitudinal models, is the selection of the random effects which in turn determine the covariance structure for the repeatedly sampled outcome. To address this, we derive a VAIC specifically for variational mixed effects (VME) models. We also implement a parameter-efficient VME as part of our study which reduces any general random effects structure down to a single subject-specific score. This model accommodates a wide range of random effect structures including random intercept and slope models as well as random functional effects. Our VAIC can model and perform selection on a variety of VME models including more classic longitudinal models as well as longitudinal scalar-on-function regression. As we demonstrate empirically, our VAIC performs well in discriminating between correctly and incorrectly specified random effects structures. Finally, we illustrate the use of VAICs for VMEs on two datasets: a study of lead levels in children and a study of diffusion tensor imaging

The world economy [December 1992]

In the second quarter of 1992 GDP/GNP growth averaged 0.5% in the main four world economies. The position of the USA, Germany and France improved but growth slowed dramatically in Japan. This brief analysis sets out growth rates for each country and provides a provisional estimate for the major industrial economies during the second quarter

The British economy [March 1992]

The recession in non-oil GDP has continued for six quarters with the real value of output falling by just under 4% since the second quarter of 1990. The present recession is therefore the longest since the Second World War, although the fall is still not as rapid, or as deep, as in the 1979-81 recession where the decline in non-oil GDP from peak to trough occurred over five quarters and amounted to 6.2%. Considerable uncertainty exists over the course of demand over the next few months, with consumption unlikely to exhibit strong growth. The likely outturn for GDP growth during 1992 is for an increase of no more than 1%. Macroeconomic trends, earnings and productivity, and unemployment levels are also analysed