Inferences from multilevel models can be complicated in small samples or complex data structures. When using (restricted) maximum likelihood methods to estimate multilevel models, standard errors and degrees of freedom often need to be adjusted to ensure that inferences for fixed effects are correct. These adjustments do not address problems in estimating variance/covariance components. An alternative to the adjusted likelihood method is to use Bayesian methods, which can produce accurate inferences about fixed effects and variance/covariance parameters. In this article, the authors contrast the benefits and limitations of likelihood and Bayesian methods in the estimation of multilevel models. The issues are discussed in the context of a partially clustered intervention study, a common intervention design that has been shown to require an adjusted likelihood analysis. The authors report a Monte Carlo study that compares the performance of an adjusted restricted maximum likelihood (REML) analysis to a Bayesian analysis. The results suggest that for fixed effects, the models perform equally well with respect to bias, efficiency, and coverage of interval estimates. Bayesian models with a carefully selected gamma prior for the variance components were more biased but also more efficient with respect to estimation of the variance components than the REML model. However, the results also show that the inferences about the variance components in partially clustered studies are sensitive to the prior distribution when sample sizes are small. Finally, the authors compare the results of a Bayesian and adjusted likelihood model using data from a partially clustered intervention trial.
To assess the effect of exercise training on the insulin resistance and impaired pancreatic B-cell function of aging, we studied 13 healthy older men (ages 61-82 yr) before and after 6 mo intensive endurance exercise. An index of insulin sensitivity (SI) was measured using Bergman's minimal model. Intravenous glucose tolerance was quantified using the glucose disappearance constant (KGlc) while oral glucose tolerance was assessed after a 100-g glucose load. B-cell function was evaluated by measuring the acute insulin response (AIR) to glucose injection at fasting glucose (AIRGlc) and the AIR to arginine at multiple clamped glucose levels. Exercise produced an endurance training effect as demonstrated by an 18% increase in maximum O2 consumption (VO2max) [38.2 +/- 1.4 to 45.0 +/- 1.1 (SE) ml.kg fat-free mass-1.min-1, P less than 0.001]. An unchanged fasting glucose (5.3 +/- 0.2 to 5.4 +/- 0.2 mM) despite a reduced fasting insulin (61 +/- 6 to 48 +/- 6 pM, P less than 0.01) suggested exercise training improved insulin sensitivity. This was confirmed by a 36% increase in SI from 3.47 +/- 0.41 to 4.71 +/- 0.42 x 10(-5) min-1/pM (P = 0.01). Intravenous glucose tolerance did not change as measured by KGlc, which was 1.46 +/- 0.09 before and 1.48 +/- 0.16%/min after exercise training. Likewise, the incremental glucose response to oral glucose (633 +/- 49-618 +/- 45 mM.min) was unchanged. B-cell function was decreased as reflected by AIRGlc (351 +/- 73-245 +/- 53 pM, P less than 0.01) and the AIRArg at maximal glycemic potentiation (AIRmax, 1,718 +/- 260-1,228 +/- 191 pM, P less than 0.005).(ABSTRACT TRUNCATED AT 250 WORDS)
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