Graphics Reference
In-Depth Information
Example: A Hierarchical Model
of Structure in Social Networks
16.3
As an example, we consider the problem of estimating the sizes of social networks
Zheng et al. (
). he model uses a negative-binomial model with an overdisper-
sion parameter:
y
ik
Negative-binomial
(
mean
=
a
i
b
k
,overdispersion
=
ω
k
)
,
where the groups (subpopulations) are indexed by k (k
,...,K) and the respon-
=
dents are indexed by i (i
.Each respon-
dent is asked how many people he or she knows in each of the K subpopulations.
hesubpopulations areidentified byname(peoplecalledNicole,Michael,Stephanie,
etc.), and by certain characteristics (airline pilots, people with HIV, those in prison,
etc.).
Without going into the details, we remark that a
i
is an individual-level parameter
that indicates the propensity of the person i to know people in other groups (we call
this a“gregariousness” parameter);itismodeled tobe a
i
,...,n). Inthis study, n
and K
=
=
=
e
α
i
where α
i
μ
α
, σ
α
=
N
(
)
;
similarly, b
k
is a group-level prevalency (or group size) parameter modeled as b
k
=
e
β
k
where β
k
μ
β
, σ
β
)
and the hyperparameters are assigned uninformative (or weakly informative) prior
distributions.
he model is fitted using a combination of Gibbs and Metropolis algorithms, so
our inferences for the modeled parameters
N
(
)
. he overdispersion parameter vector ω
=(
ω
,...,ω
K
, and the hyperparameters, (μ
α
,
σ
α
, μ
β
, σ
β
), are obtained as simulated draws from their joint posterior distribution.
(
a, b, ω
)
Model-Informed Exploratory Data Analysis
Figure
.
displays a small portion of an exploratory data analysis, with histograms
of responses for two of the survey questions, along with simulations of what could
appear under three fitted models. he last of the models is the one that we used; as
the EDA shows, the fit is still not perfect.
A First Look at the Estimates
Wecouldsummarizetheestimatedposteriordistributionsofthescalarparametersas
histograms, butin mostcases wefindthat intervals are a moreconcise wayto display
the inferences; our goal here is not just to view estimates, but also to compare the
parameters within each batch.
We display the parameter estimates with their
% and
% posterior intervals
as shown in Fig.
.
. Along with the estimates, the graph summarizes the conver-
gencestatistic graphically. Since thereare over
quantities inthe model,notall of
them can be displayed on one sheet. For smaller models, this graph provides a quick
summary of the results - but of course this is just a starting point.
We are usually satisfied with the convergence of the algorithm if the values of the
R convergence statistic (Gelman et al.,
) are below
.
for all scalar parameters.
