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Health Services and Outcomes Research Methodology

Erschienen in:

01.12.2008

The analysis of social networks

verfasst von: A. James O’Malley, Peter V. Marsden

Erschienen in: Health Services and Outcomes Research Methodology | Ausgabe 4/2008

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Abstract

Many questions about the social organization of medicine and health services involve interdependencies among social actors that may be depicted by networks of relationships. Social network studies have been pursued for some time in social science disciplines, where numerous descriptive methods for analyzing them have been proposed. More recently, interest in the analysis of social network data has grown among statisticians, who have developed more elaborate models and methods for fitting them to network data. This article reviews fundamentals of, and recent innovations in, social network analysis using a physician influence network as an example. After introducing forms of network data, basic network statistics, and common descriptive measures, it describes two distinct types of statistical models for network data: individual-outcome models in which networks enter the construction of explanatory variables, and relational models in which the network itself is a multivariate dependent variable. Complexities in estimating both types of models arise due to the complex correlation structures among outcome measures.

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To aid readers in applying these methods, we provide some references to network software throughout, but our coverage of software is not comprehensive. Huisman and Van Duijn (2005) review software resources available earlier in this decade.

The extent to which distances in a graphical representation correspond to the data on which they rest—dyadic measurements of social distance or proximity—depends on the objective function that serves as a fitting criterion when the plot is constructed. The most widely-used “nonmetric” multidimensional scaling algorithm requires an ordinal correspondence between data and plotted distances; “metric” scaling uses a stronger (linear) criterion. Objective functions used by many spring-embedder methods involve a “node repulsion” term that simplifies the visual representation by discouraging co-location of vertices within a plot, but simultaneously limits the extent to which data and plotted distances correspond. Moreover, a low (ordinarily 2)-dimensional Cartesian plot may do more or less well in representing data on the relationships among N actors, which may in principle be (N − 1)-dimensional.

Note that some or all of the intermediaries along these geodesic paths may be physicians 11–33.

Recall that the undirected physician network is identical to that shown in Fig. 1, except that ties lack directionality.

We calculated centrality scores using the software package UCINET 6 (Borgatti, Everett, and Freeman, 2002).

Eigenvalue centrality can in principle be calculated for a nonsymmetric matrix, but the routine in UCINET 6 handles only the symmetric case.

Because two actors have outdegrees of 0, the associated rows of W sum to 0 as opposed to 1. Therefore, although these actors contribute to the estimation of β and σ², they do not directly contribute any information about the autocorrelation parameters α and ρ. We retained these actors in the analysis because they were cited by other physicians as influencing them and so removing them would omit information about how other actors were influenced.

Computations were performed using the StOCNET software package (Boer et al. 2006).

Under p₁, the estimate of a receiver parameter is infinitely small for actors with indegree 0; likewise, the estimate of a sender parameter is -∞ when the corresponding outdegree is 0.

The p₂ model is closely related to a social relations model developed by Kenny and La Voie (1984) for quantitative network variables.

The large number of terms in κ(θ) complicates the estimation of ERGMs. There are 2^N(N−1)possible directed binary-valued networks; for example, with N = 10, the number of possible networks—hence terms in κ(θ)—is 1.238 × 10²⁷.

For example, an actor with degree 3 contributes 1 3-star, 3 2-stars, and 3 1-stars; 1-stars are equivalent to individual edges.

The set of k-star statistics is equivalent to the set of degree statistics (the number of nodes of degree k, k = 1,2,3,…) in that a bijection exists between the two sets of the statistics (Snijders et al. 2006).

An analogous “sender covariate” statistic allows the density effect to depend on an attribute of the sender (i).

\( {\mathbf{y}}_{ij}^{ + } \)is the realization of the complement network with y _ij = 1, while \( {\mathbf{y}}_{ij}^{ - } \) is the realization of the complement network with y _ij = 0.

An equivalent statistic based on the degree distribution itself is known as the “geometrically weighted degree” statistic; see Hunter and Handcock (2006).

No mutuality term is included, since this is redundant with the edges term in an undirected network. Constraining the value of ρ when fitting the model with the GWESP term is often helpful; attaining adequate convergence is more difficult when it is estimated as a free parameter. We found that setting ρ = 1.2 served well here; the likelihood surface is relatively flat, so that using a value between 1.0 and 1.5 did not affect inferences about other parameters. Note, however, ρ was estimated at 0.93 when we left it as a free parameter in the third model in Table 9.

Although the missing values are replaced with non-missing values during model fitting, the statistics measuring model fit are only evaluated using actors with non-missing values throughout the corresponding interval of time. Thus, standard imputation is not performed.

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Titel: The analysis of social networks
verfasst von: A. James O’Malley
Peter V. Marsden
Publikationsdatum: 01.12.2008
Verlag: Springer US
Erschienen in: Health Services and Outcomes Research Methodology / Ausgabe 4/2008
Print ISSN: 1387-3741
Elektronische ISSN: 1572-9400
DOI: https://doi.org/10.1007/s10742-008-0041-z

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The analysis of social networks

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