Graphics Reference
In-Depth Information
Figure
.
.
A graph comparing the estimates from a more complex version of the social networks
model, using individual-level regression predictors: log
X
i
ψ
, σ
α
.herows(Nicole,Anthony,
etc.) refer to the groups and the columns refer to the various predictors. Comparisons are e
cient
when coe
cients are rearranged into a “table of graphs” like this
(
a
i
)
N
(
)
Extending the Model by Imposing a Regression Structure
We also fit an extended model with an individual-level regression model, log
(
a
i
)
X
i
ψ
, σ
α
N
(
)
. he predictors include the indicators of female, nonwhite, in-
come
$
,employment,education (high-schoolorhigher).
Figure
.
shows an example of how to visually compare regression coe
cients
ψ
ik
on explanatory variables (a characteristic of the survey respondent, i)fordif-
ferent groups (k). Whenever our goal is to compare estimates, we initially think of
a graph. Figure
.
could have been equivalently summarized by the uncertainty
interval endpoints and the posterior median estimate, but it would not have been
an e
cient tool to use to visualize the coe
cient estimates. When a comparison is
required, draw a graph; for looking up specific numbers, construct a table.
$
,income
<
Posterior Predictive Checks
16.3.1
A natural way to assess the fit of a Bayesian model is to look at how well the pre-
dictions from the model agree with the observed data (Gelman et al.,
). We
do this by comparing the posterior predictive simulations with the data. In our so-
cial networks example, we create a set of predictive simulations by sampling new
data independently from the negative binomial distributions given the parameter
vectors a, b, ω drawn from the posterior simulations already calculated. We draw,
say, L simulations, y
()
ik
,...,y
(
L
)
ik
(
ℓ
)
ik
ℓ
=
is
for each i, k. Each set of simulations
y
an
approximation
of the marginal posterior distribution of y
ik
, denoted by y
rep
ik
,where“rep”
stands for the “replication distribution;” y
rep
stands for the n
K replicated obser-
vation matrix.
It is possible to find a numerical summary (that is, a test statistic) for some feature
of the data (such as standard deviation, mean, etc.) and then compare it to the corre-
sponding summary of y
rep
, but in addition we prefer to draw graphical test statistics,
since a few individual numbers rarely illustrate the complexity of the full dataset. In
