Graphics Reference
In-Depth Information
Hierarchical Models
and Parameter Naming Conventions
16.2.3
In hierarchical and other structured models, rather than display individual coe
-
cients, we wish to compare the values within batches of parameters. For example, we
might want to compare group-level means together with their uncertainty intervals.
Posterior intervals are easily derived from the matrix of posterior simulations. Tra-
ditional tools for summarizing models (such as looking at coe
cients and analytical
relationships) are too crude to usefully summarize the multiple levels of variation
and uncertainty that arise in such models. hese can be thought of as corresponding
to the “sources of variation” in an ANOVA table.
Hierarchical models feature multiple levels of variation, and hence feature multi-
ple levels of batches of parameters. Hence, the choice of label for the batches is also
important: parameters with similar names can be compared to each other. In this
way, naming can be thought of as a structure analogous to hierarchical modeling.
Instead of using generic θ
,...,θ
k
for all scalar parameters, we would, for example,
name the individual-level regression coe
cients β
=(
β
,...,β
n
)
,andthegroup-
level coe
cients α
,andtheinterceptμ. Figure
.
shows an example
of why this works: parameters with similar names can be compared to each other.
Rather than plotting posterior histograms orkernel density estimates of the parame-
ters, we usually summarize the parameters (at least in a first look for the inferences)
by plotting their posterior intervals.
=(
α
,...,α
J
)
Model-Checking
16.2.4
As stated earlier, we view statistical graphics as implicit or explicit model checks.
Conversely, we view model-checking as a comparison of the data with the replicated
data given bythe model,whichincludesboth exploratory graphics andconfirmatory
calculations suchas p-values. Our goal is not the classical one of identifying whether
themodelfitsornot-anditiscertainlynot theaimtoclassifymodelsintocorrectand
incorrect, which is the focus of the Neyman-Pearson theory for Type
and Type
errors. Instead, we seek to understand the ways in which the data depart from the
fitted model. From this perspective, the two key components of exploratory model-
checking are (
) the graphical display and (
) the reference distribution to which the
data are compared.
he best display to use depends on the aspects of the model being checked, but
in general, graphs of data or functions of data and estimated models (for example,
residual plots) can be visually compared to corresponding graphs of replicated data
sets. his is the fully model-based version of exploratory data analysis. he goal is to
use graphical tools to explore aspects of the data that are not captured by the fitted
model.
