Technical Appendix
Here we provide additional information on the methods used to construct and
analyze physician networks. Our first step involved identifying significant patient
encounters with a physician. We defined a significant encounter with physicians on the
basis of paid physician claims in the Medicare Carrier file. We excluded claims for nondirect patient care specialties or specialties where individual physicians are not selected
(anesthesia, radiology, pathology, emergency medicine). In addition to all evaluation
and management services, we also included procedures with a relative value unit (RVU)
value of at least 2.0 because procedures are often reimbursed via bundled fees that
also include pre- and post-procedure assessments. We excluded claims for lab
services and other claims that do not require a physician visit, even though they might
be billed under a specific physician. Finally, we excluded claims generated from
physicians who practice outside of the defined HRR or who had encounters with fewer
than 30 Medicare patients during 2006. Although patients may see physicians
practicing outside of the geographic area (e.g., when patients live near an adjacent
HRR), our data with regards to these physicians are incomplete.
We then used this information on physician encounters for each patient to
identify physicians with encounters with the same patient and, ultimately, the network
based on such shared patients. The structural backbone from which we discern
physician networks is a so-called bipartite patient-physician network. This may be
represented in a physician-by-patient matrix with entries indicating whether a physician
provided care to a patient within some reference period (in our case, CY 2006). The
term “bipartite” means that nodes in the network can be partitioned into two classes and
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all ties connect nodes in different classes (i.e., physicians and patients).1,2 In its
simplest form, rows of the matrix consist of individual patients (using unique patient
identifiers) and columns identify all physicians with whom patients have had significant
clinical encounters (as defined above). Using matrix multiplication,3,4 these data can
then be used to count the number of patients shared by each pair of physicians.
A key decision in the study involved determining the minimum number of shared
patients that could be used to define a “true connection” between any two doctors. We
previously found that physicians who shared a single patient using these same
Medicare data had just a 20% chance of having a true “information sharing”
relationship.5 This probability increased to over 80% as more patients were shared.
Moreover, this threshold might differ depending upon specialty and the clinical activity of
each individual physician (i.e., smaller numbers of ties will be more meaningful for
physicians who see fewer Medicare patients). We explored both absolute thresholding
(using the same threshold for each physician and specialty) and relative thresholding
(creating a customized threshold for each physician), and we found that using a relative
threshold that maintained the strongest ties for each physician (the top 20% of ties for
each physician) appeared to best maintain intrinsic network characteristics while also
eliminating noise that might result from spurious connections. Although this method
likely eliminates some ties that represent true relationships, it maintains the strongest
ties for each physician and therefore likely maintains the relationships that are most
influential to that physician.
Finally, we will need to uncover the community structure of the network.
Detecting communities in networks is a very active research area of its own, and there
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are many different approaches that have been developed in the last few years.6-8
Detecting and studying network communities is of considerable interest perhaps mainly
because network communities are thought to be associated with functional units in
networks. Network communities are usually understood as sets of densely connected
nodes, and one major family of methods approaches this problem as an optimization
problem. There the idea is to maximize the value of some score, usually called the
quality function, by changing the community assignments of nodes, such that an
assignment of nodes into communities that maximizes the score is considered optimal.
This problem is computationally very difficult, belonging to the class of NP-hard
problems, which in practice means that exhaustive enumeration of possible solutions is
impossible, and one needs to define appropriate heuristics to perform the computation
successfully.
We apply the most commonly used method of modularity maximization and
couple that with the most commonly used null-model which retains the expected degree
distribution of the network. 9 This leads to the maximization of the quantity
𝑄=
𝑘𝑖 𝑘𝑗
1
∑ ∑ [𝐴𝑖𝑗 −
] 𝛿(𝑠𝑖 , 𝑠𝑗 )
2𝑚
2𝑚
𝑖
𝑗
where 𝐴𝑖𝑗 is nonzero if and only if node 𝑖 and 𝑗 are conntected by a tie, and its value
quantifies the number of patients the two physicians share; 𝑘𝑖 is the degree (or strength
in weighted networks) of node 𝑖, 𝑚 is the number of edges in the network (or their total
weight in weighted networks), 𝑠𝑖 is the community assignment of node 𝑖, and 𝛿(∙,∙) is the
Kronecker delta which is equal to 1 if the arguments are identical, otherwise it is zero.
Before carrying out the community detection stage, we extracted the largest connected
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component (LCC) of each network, which is the maximal connected subgraph within a
network. In practice, the patient-sharing networks associated with most HRRs comprise
just one component, in which case that component is by definition the LCC. In case
there are multiple components, the LCC typically contains over 95% of all nodes, so
very few nodes are lost by focusing on the LCC. We then used to so-called Louvain
heuristic to deal with the computational optimization of modularity, which has been
shown to perform well for a variety of networks.10 We retained tie strengths when
performing community detection, i.e. the resulting communities reflect not only network
topology, but are also to some extent driven by network tie strengths.
Finally, since both types of networks displayed care patterns that varied linearly
with network size, we estimated the proportion of care that would occur within each of
the networks if we fixed the network size to be equal to that of the median-sized hospital
network (physician n=96) and the median-sized community network (physician n=150).
These estimates were generated by fitting a linear regression model to each set of
network estimates for each of the measures.
We also present plots below (Appendix Figure A1) of the percent visits by log
number of doctors in the networks in order to provide a visual representation of the
relationship between the proportion of care within each of the types of networks and the
size of the networks. These plots demonstrate that at all sizes, communities always
outperform the hospitals.
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Appendix Figure A1. Scatter Plot of the Percentage of Care Within Hospital-Based and Community-Based
Networks by Network Size.
 Hospital
 Community
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 Hospital
 Community
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