The concept of explained proportion of variance or modeled proportion of variance is reviewed in the situation of the random effects hierarchical two-level model. It is argued that the proportional reduction in (estimated) variance components is not an attractive parameter to represent the joint importance of the explanatory (independent) variables for modeling the dependent variable. It is preferable instead to work with the proportional reduction in mean squared prediction error for predicting individual values (for the modeled variance at level 1) and the proportional reduction in mean squared prediction error for predicting group averages (for the modeled variance at level 2). It is shown that when predictors are added, the proportion of modeled variance defined in this way cannot go down in the population if the model is correctly specified, but can go down in a sample; the latter situation then points to the possibility of misspecification. This provides a diagnostic means for identifying misspecification.

No abstract

This study is the first to systematically review and quantify the effects of physical activity on brain structure and neurophysiological functioning in children. Electronic data bases were searched for relevant studies. Studies that met the following criteria were included: (1) used an RCT or cross-over design, (2) examined the effects of physical activity on brain structure and/or neurophysiological functioning, (3) included children (5–12 years old) (4) included a control group (RCTs) or control condition (cross-over trials). A total of 26 and 20 studies were included in the systematic review and meta-analysis, respectively, representing and accompanying 973 and 782 unique children. Main analyses were separated for short-term and long-term physical activity and for effects on brain structure and neurophysiological functioning with a distinction between children from healthy and clinical populations. We found evidence for significant beneficial effects of long-term physical activity on neurophysiological functioning ( d = 0.39, p < 0.001). In addition, short-term physical activity may induce changes in neurophysiological functioning ( d = 0.32, p = 0.044), although this evidence showed limited robustness. No meta-analytic evidence was found for positive effects on brain structure. The results underline the importance of physical activity for brain development in children.

No abstract