Rejoinder.(Discussion Article)

I appreciate Buja's generous comments and will briefly clarify some issues regarding the role of data visualization in model checking, and the relevance of Bayesian inference to model checking.

1. DATA VISUALIZATION

My article presents four general kinds of model-based graphical diagnostics:

1. Displays of raw data, with a model used to simulate the reference distribution of the displays (as in Figure 1). Conversely, Figure 2 illustrates the difficulty, in general, of interpreting data displays without a comparison to a reference distribution.

2. Similar displays of tests of lower-dimensional data summaries (e.g., scalars in Figure 3, and vectors in Figures 4 and 5(a)) compared to simulations from fitted model--again, with discrepancies from the simulations illustrating aspects of the data not explained by the model.

3. Residual plots--put more generally, graphs of differences between data and fitted model which, if the model is true, should follow distributions with invariance properties such as independence and zero mean. Figures 5(b), 6, and 7 illustrate how violations of these invariance properties are visibly apparent without the need for explicit comparisons to simulated replications.

4. Displays of latent data/parameters (as in Figures 8 and 9) or of completed datasets (i.e., combinations of observed and missing or latent data, as in Figure 10).