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Dear Elly,

I'm in support of the last mentioned possibility by Philip. I think it is
better to see this as trying to model the network than merely trying to
model centrality scores. Potentially you can also see how your variable of
interest (religiosity) influences various positional network features,
e.g., differentiating between incoming and outgoing centrality, homophily,
interaction with reciprocity or with other homophily indicators, etc.

Best wishes,
Tom

=========================================
Tom A.B. Snijders
Professor of Statistics and Methodology, Dept of Sociology, University of
Groningen
Emeritus Fellow, Nuffield College, University of Oxford
http://www.stats.ox.ac.uk/~snijders

On Tue, Mar 17, 2015 at 10:14 AM, Philip Leifeld <
[log in to unmask]> wrote:

> ---------------------- Information from the mail header
> -----------------------
> Sender:       Social Networks Discussion Forum <[log in to unmask]>
> Poster:       Philip Leifeld <[log in to unmask]>
> Subject:      Re: Centrality as the Dependent Variable?
>
> -------------------------------------------------------------------------------
>
> *****  To join INSNA, visit http://www.insna.org  *****
>
> Hi Elly,
>
> The centrality values attached to the nodes are not independent from
> each other because a high centrality value of one node may imply a high
> centrality value of an adjacent node (see measures of degree
> assortativity etc.) -- or merely because the network has a given
> centralization and increasing the centrality of one node must decrease
> the centrality of another node.
>
> If you estimate a regression model, you should therefore specify the
> channels of influence/dependence between the nodes as covariates. For
> example, for each node you can include the cumulated and/or average
> centrality of adjacent nodes (and possibly of indirect friends with path
> length 2). Depending on the centrality measure (e.g., eigenvector
> centrality), it may also make sense to include the aggregated centrality
> scores of structurally similar nodes because being connected to the same
> important other nodes makes both nodes in a dyad similarly important.
> There may be many other ways in which a node's centrality value depends
> on other nodes' values but also potentially just on features of the
> network or the local neighborhood of a node in the network.
>
> Regression models where the dependencies are specified like this go by
> several names:
>
> - If you estimate linear or generalized linear models and include only
> functions of direct neighbors, this is called a "spatial autocorrelation
> model" or a "network autocorrelation model". See the work of Patrick
> Doreian, for example. There is a nice article in Social Networks on
> "Specifying the weight matrix" for these models by Roger Leenders
> (http://dx.doi.org/10.1016/S0378-8733(01)00049-1). There is an R
> implementation for the linear case in the sna package by Carter Butts
> (the lnam function).
>
> - If you include various other dependencies and estimate a binary (=
> logit or probit) model, this was termed "autologistic actor attribute
> model" (ALAAM) by Galina Daraganova and Garry Robins in their chapter 9
> of the ERGM book ("Exponential Random Graph Models for Social Networks")
> by Dean Lusher, Johan Koskinen and Garry Robins. I think there is an
> implementation in PNet or a related program, but others may know better
> than I do. This is kind of a special case of the network autocorrelation
> model but with additional dependencies.
>
> - If you include only basic dependencies (= functions of direct
> neighbors and their attributes), this is called "multiparametric
> spatiotemporal autoregressive model" (m-STAR) in the spatial
> econometrics literature (see an article of Jude Hays, Aya Kachi and Rob
> Franzese here: http://dx.doi.org/10.1016/j.stamet.2009.11.005).
>
> - Finally, I have created a generalized version of all of this. I call
> it "temporal network autocorrelation model" (TNAM) because it's a
> spatial *autocorrelation model* like the one specified above (see first
> bullet point), but it also includes various kinds of dependencies (hence
> the word *"network"*, see second bullet point), and it's possible to
> estimate it with *temporal* data/repeated observations of the network
> and/or the outcome variable (see bullet point 3). More generally, you
> can plug it into any model you would like, including tobit, survival,
> linear mixed models etc. I have implemented this in the tnam function in
> my R package xergm, along with a number of dependency terms to include.
> See here (http://rpackages.ianhowson.com/rforge/xergm/man/tnam.html) and
> here (http://rpackages.ianhowson.com/rforge/xergm/man/tnam-terms.html)
> for a description. A paper will be available in a few weeks.
>
> There are three potential caveats here:
>
> (1) It may be difficult to specify the dependencies appropriately if the
> attributes you are explaining are centralities. It may require you to
> think hard about what is causing them and to what extent you want them
> to be explained by network/influence terms vs. covariates, but I think
> technically it should be a subproblem of the more general models
> outlined above.
>
> (2) A cross-sectional analysis does not really allow you to infer
> causality (this is tricky enough with longitudinal data). It's
> relatively certain that some of your independent variables/model terms
> will be partially caused by the centrality of the nodes.
>
> And (3) centrality scores are usually fairly skewed and often also bound
> between 0 and 1, so it may be inappropriate to use a linear model. But
> this is a general statistical problem, not one that is specific to
> network analysis. You may want to consult the literature on beta
> regression, Box-Cox transformations etc. TNAM should be able to deal
> with this, but you have to find out what model (e.g., GLM with a beta
> distribution) would be appropriate for your data.
>
> As an alternative, you may want to consider modeling the network using
> an exponential random graph model, rather than modeling centrality
> scores, which are merely a function of the network with a huge loss of
> information. By explaining the network, you basically explain the
> structure including who is central.
>
> Best regards
>
> Philip
>
> Am 17.03.2015 um 02:16 schrieb Elly Power:
> > ***** To join INSNA, visit http://www.insna.org *****
> > Hello all,
> >
> > I was hoping I could get some advice on how (or if) I could use
> > centrality measures (e.g., eigenvector centrality) as the /dependent/
> > variable in some analyses.
> >
> > I know that we usually think of centrality as an independent variable,
> > but it seems reasonable that we might want to predict centrality.
> > Personally, I work on religious practice, and I want to understand if
> > the nature of someone's religious practice might influence his/her
> > centrality.
> >
> > The issue, of course, is that centrality measures are not independent.
> > Does anyone know of any ways to deal with this? Is there anyone who has
> > tried to look at this? Any direction would be very much appreciated.
> >
> > Thanks in advance for all of your suggestions.
> >
> > - Elly Power
> >
> > --
> > Eleanor A. Power, PhD Candidate
> > Department of Anthropology
> > Stanford University
> > 450 Serra Mall, Bldg 50
> > Stanford, CA 94305
> > www.stanford.edu/~epower <http://www.stanford.edu/%7Eepower>
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