Spatial extension of mixed models of the analysis of variance

This study proposes a spatial extension of mixed models of the analysis of variance (MANOVA), called mixed spatial ANOVA (MS-ANOVA) models. MS-ANOVA models have been used to evaluate the spatial correlations between random effects in spatial multilevel data in which observations belong to nested clusters. Because the proposed model can be regarded as a Bayesian hierarchical model, we introduce empirical Bayesian estimation methods in which hyperparameters are estimated by quasi-maximum likelihood estimation methods in the first step, and posterior distributions for the parameters are evaluated with the estimated hyperparameters in the second step. Moreover, we justify the asymptotic properties of the first-step estimator. The proposed models were applied to happiness survey data collected in Japan. Empirical results show that social capital, which can be interpreted as 'the beliefs and norms by which a community values collective action and pursues activities worthy for the entire community', significantly increases people's happiness, even after controlling for various individual characteristics and spatial correlations.