Data Analysis Using Regression and Multilevel/Hierarchical Models: Understanding and summarizing the fitted models

Now that we can fit multilevel models, we should consider how to understand and summarize the parameters (and important transformations of these parameters) thus estimated. Inferences from classical regression are typically summarized by a table of coefficient estimates and standard errors, sometimes with additional information on residuals and statistical significance (see, for example, the R output on page 39). With multilevel models, however, the sheer number of parameters adds a challenge to interpretation. The coefficient list in a multilevel model can be arbitrarily long (for example, the radon analysis has 85 county-level coefficients for the varying-intercept model, or 170 coefficients if the slope is allowed to vary also), and it is unrealistic to expect even the person who fit the model to be able to interpret each number separately. We prefer graphical displays such as the generic plot of a Bugs object or plots of fitted multilevel models such as displayed in the examples in Part 2A of this book. Our general plan is to follow the same structures when plotting as when modeling. Thus, we plot data with data-level regressions (as in Figure 12.5 on page 266), and estimated group coefficients with group-level regressions (as in Figure 12.6). More complicated plots can be appropriate for non-nested models (for example, Figure 13.10 on page 291 and Figure 13.12 on page 293). More conventional plots of parameter estimates and standard errors (such as Figure 14.1 on page 306) can be helpful in multilevel models too.