#### Abstract

The R^{2} statistic, when used in a regression or ANOVA context, is appealing because it summarizes how well the model explains the data in an easy-to-understand way. R^{2} statistics are also useful to gauge the effect of changing a model. Generalizing R^{2} to mixed models is not obvious when there are correlated errors, as might occur if data are georeferenced or result from a designed experiment with blocking. Such an R^{2} statistic might refer only to the explanation associated with the independent variables, or might capture the explanatory power of the whole model. In the latter case, one might develop an R^{2} statistic from Wald or likelihood ratio statistics, but these can yield different numeric results. Example formulas for these generalizations of R^{2} are given. Two simulated data sets, one based on a randomized complete block design and the other with spatially correlated observations, demonstrate increases in R^{2} as model complexity increases, the result of modeling the covariance structure of the residuals.

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R^{2} STATISTICS FOR MIXED MODELS

The R^{2} statistic, when used in a regression or ANOVA context, is appealing because it summarizes how well the model explains the data in an easy-to-understand way. R^{2} statistics are also useful to gauge the effect of changing a model. Generalizing R^{2} to mixed models is not obvious when there are correlated errors, as might occur if data are georeferenced or result from a designed experiment with blocking. Such an R^{2} statistic might refer only to the explanation associated with the independent variables, or might capture the explanatory power of the whole model. In the latter case, one might develop an R^{2} statistic from Wald or likelihood ratio statistics, but these can yield different numeric results. Example formulas for these generalizations of R^{2} are given. Two simulated data sets, one based on a randomized complete block design and the other with spatially correlated observations, demonstrate increases in R^{2} as model complexity increases, the result of modeling the covariance structure of the residuals.