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Weihua writes: " I think in network case, it may not be necessary to always include nodefactor, as the nodematch term is not exactly like the interaction term between two variables. If nodefactor is one variable, what is the other main variable? How should  we include the main effect for that variable?"





There are various forms of statistics that can be implemented for social selection in an ERGM. In the original social selection paper for ERGMs (Robins, Elliott & Pattison, 2001), for a binary attribute, the main effect statistic (activity) took the form of Yij*Xj (where y is the tie variable and x the attribute variable). The interaction (in effect, homophily) had statistic Yij*Xj*Xi. For a categorical variable, (and I think this is the statistic in Nodematch), Yij*I(Xi=Xj) where I = 1 if the argument is true, 0 otherwise. You can have other versions eg for continuous data Yij*|Xi-Xj|.



In other words, the interaction effect for attributes is expressed either by Xi*Xj or I(Xi=Xj) or |Xi-Xj|, and formally the complete statistic is a three-way interaction of a tie variable and two attributes. That is why in Lusher et al (2013), we often refer to attribute effects as actor-relation effects (e.g. Table 5.1).



This is a little bit different from the usual approach to standard regressions because in the ERGM it appears that the interaction occurs "within" the variable X which makes no sense in regression. However, that is the way of the world with these network models. The difference is the point of imposition of homogeneity. We can think of any measure on an individual i as a variable Xi particular to that individual. Standard regression imposes homogeneity straight away (so that the effects related to Xi are treated the same as for Xj: in other words we have one variable X) - which is why we hardly ever think about the homogeneity issue at all in standard statistics. It is only then, after this 'unseen' homogeneity step, that we form an interaction X*Z in our regression, where Z is another variable. In ERGMs, on the other hand, we calculate the interactions XiXj first and then impose homogeneity, so there is one homophily parameter for the YijXiXj effect. It is that way that we have network configurations parameterised the model.



This comes to the forefront with networks because we have a structure on the domain of measurement (see Brandes et al editorial in Network Science). When we don't have such a structure there is no real question about homogeneity and we are in the world of standard regressions.



In general, if you are interested the attribute effects, then I think it is a good idea to include both main effects and interactions. If, on the other hand, you are interested in endogenous network effects and not so much in the attributes, then maybe it is sufficient just to control the attributes by a single homophily or activity parameter.



Jeff Webb originally raised the question of non-convergence of a tergm with both interaction and main effects. Maybe you've got a lot of collinearity in the changes in attribute statistics. In that case, probably better to include the main effect and not the interaction, if it converges, rather than the other way around.


Professor Garry Robins
Melbourne School of Psychological Sciences
University of Melbourne
http://www.psych.unimelb.edu.au/people/garry-robins

Melnet website: http://www.sna.unimelb.edu.au/

Check out our new book on ERGMs: Lusher, D., Koskinen, J., & Robins, G. (2012). Exponential random graph models for social networks: Theory, methods and applications. Cambridge University Press. (http://www.cambridge.org/us/knowledge/isbn/item6897868/?site_locale=en_US)
Look inside the book: (http://www.amazon.co.uk/Exponential-Random-Models-Social-Networks/dp/0521141389#)





-----Original Message-----
From: Social Networks Discussion Forum [mailto:[log in to unmask]] On Behalf Of Weihua An
Sent: Thursday, 8 August 2013 7:40 AM
To: [log in to unmask]
Subject: Re: nodematch and nodefactor



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Many thanks for the good feedback. Here are last few thoughts from my end.



"When it there is a correlation, you can't just drop the nodefactor terms and interpret the nodematch terms naively.  The interpretation of nodematch terms will be unclear if, for example, there is a difference in the level of homophily for groups that are more (or

less) active."



If nodefactor is not included, the coefficient for nodematch is the log-odds of the presence of ties between actors with same attribute as compared to the presence of ties between actors with differential attributes. We can also allow for differential homophiliy. In either case, the reference category is the presence of ties between actors with differential attributes.



The ambiguity regarding the interpretation of nodematch comes when nodefactor is included. When nodefactor is included, it adds one statistics which is equal to the number of ties that have the active factor attribute, including both 0-1 and 1-1 ties, while nodematch adds a statistics which is equal to the number of ties that both actors share the same attributes including 0-0 and 1-1 ties. The sharing of 1-1 ties in the statistics leads nodefactor and nodematch to be correlated. The coefficients for the two models (with or without

nodefactor) will certainly be different because of the correlation.



The question is whether the correlation makes it illegitimate to specify a model that includes only the nodematch terms and whether we can still properly interpret that model. I think we can, as the reference category is clear. In contrast, in the model with nodefactor included, the reference category seems not to be that clear (ties from/to actors with the passive state of the attribute?). In general, the latter is a more interesting model, as the nodefacor term captures the inequality effect while the nodematch term captures the segregation effect. But this does not imply that an ERGM cannot just have the segregation effect.



In our case, almost all nodefactor terms are not significant, because of the spatial constrain in the friending process. My co-authors and I want to include only just nodematch terms.But we want to hear some good counter-arguments before really deciding to do so. Thanks!





best,



Weihua



P.s. Even if you look at the literature on main effect and interaction effect in regression analysis, there seems not be a good consensus on whether the main effect should always be included. In general, yes, but it also depends. I think in network case, it may not be necessary to always include nodefactor, as the nodematch term is not exactly like the interaction term between two variables. If nodefactor is one variable, what is the other main variable? How should  we include the main effect for that variable?





On Wed, Aug 7, 2013 at 1:44 PM, martina morris <[log in to unmask]<mailto:[log in to unmask]>> wrote:

> On Wed, 7 Aug 2013, Weihua An wrote:

>

>> *****  To join INSNA, visit http://www.insna.org  *****

>>

>> Hi,

>>

>> I have a similar problem. In my networks, by design each person has a

>> fixed (or very close) number of connections. That means no main

>

>

> Nice.  This is a perfect example:  if, by design, your nodes have a

> fixed number of edges, you don't need nodefactor terms.

>

>

>> effects of any covariate will be significant. So our interested is in

>> segregation in friendship networks. Indeed, we find almost no

>> covariates has significant predictive effect, but a lot of the

>> nodematch terms are significant, when both of the terms are included.

>> We are thinking about dropping the nodefactor terms as well, for

>> simpler and clearer interpretation.

>

>

> If the variation in edges by group is insignificant, then this is

> justifiable.

>

>

>> If we keep only the nodematch terms, the base ties will be the

>> asymmetric ties (i.e., the ties that are between people with

>> different binary attributes, assuming the factors are binary).

>

>

> You can make the reference category (base) be either level.

>

>

>> If we keep both nodefactor and nodematch, the base ties seem to be

>> the ties from/to those with the passive attribute (zero in the binary

>> case). So if our understanding is correct, whether to include

>> nodefactor in a model depends on what base ties you want to have and

>> what theoretical questions you want to ask.

>

>

> Not really -- in your case, there is no correlation btwn nodefactor

> and nodematch (b/c there is essentially no variation in nodefactor).

> So dropping the nodefactor term will have no impact on the estimate of

> nodematch (try it and see if this is true).  But this absence of

> correlation will not be true in general.  When it there is a

> correlation, you can't just drop the nodefactor terms and interpret

> the nodematch terms naively.  The interpretation of nodematch terms

> will be unclear if, for example, there is a difference in the level of

> homophily for groups that are more (or less) active.

>

> The kind of interpretability you're pointing to here -- the reference

> category for factor level comparisons -- can be handled by directly

> setting which factor level to use as a reference category.

>

> You haven't mentioned looking at differential homophily, but that's

> another possible option.

>

> best,

> mm

>

>>

>> best,

>>

>> Weihua

>>

>> On Tue, Aug 6, 2013 at 1:16 PM, martina morris

>> <[log in to unmask]<mailto:[log in to unmask]>>

>> wrote:

>>>

>>> *****  To join INSNA, visit http://www.insna.org  *****

>>>

>>> Hi Jeff,

>>>

>>> In general, you probably wouldn't want to fit a model with nodematch

>>> but not nodefactor.  It's a bit like fitting an interaction without

>>> the main effects

>>> -- to the extent that the two are correlated, the interpretation of

>>> the terms changes when one is excluded.  If the nodematch becomes

>>> insignificant when nodefactor is added, it could mean a couple of

>>> different things.

>>>

>>> I'd suggest taking a look at the actual mixing pattern (using the

>>> "mixingmatrix" function), and getting some sense of what is going on

>>> in your data.

>>>

>>> best,

>>> Martina

>>>

>>>

>>> On Tue, 6 Aug 2013, Jeff Webb wrote:

>>>

>>>> ***** To join INSNA, visit http://www.insna.org ***** Dear list

>>>> members,

>>>>

>>>> I'm fitting an ergm model to a small network with 20 actors.  The

>>>> feature of the network in which I'm most interested is homophily

>>>> among members of a subgroup, designated "ea."  I test this with

>>>> nodematch("ea"), which is included in the model along with

>>>> structural terms, edges and GWESP ; the effect is positive and

>>>> significant.  However, Goodreau et al. (2008) note: "if one is

>>>> including nodematch terms in a model, one would typically also

>>>> include nodefactor terms for the same

>>>> attributes."   When I add nodefactor("ea") to the model, the effect of

>>>> nodematch("ea") is no longer significant. Any thoughts on when-if

>>>> ever-one would fit a model using nodematch() without nodefactor()?

>>>>

>>>> A related question.  I would also like to fit a tergm model to two

>>>> waves of the above network.  In this case, unfortunately, the

>>>> formation part of the model will not converge when I include the

>>>> interaction term (node match()) along with the main effect, but it

>>>> will converge if I include only the the interaction term.

>>>>  Would it be reasonable to leave out the main effect in order to

>>>> get a coefficient

>>>> for the interaction?   Thanks in advance for your replies.

>>>> Jeff

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>>

>>

>>

>>

>> --

>> Weihua (Edward) An

>>

>> Assistant Professor of Sociology and Statistics Indiana University

>> Bloomington

>> 752 Ballantine Hall

>> 1020 East Kirkwood Avenue

>> Bloomington, IN 47405-7103

>> http://mypage.iu.edu/~weihuaan/

>>

>> _____________________________________________________________________

>> SOCNET is a service of INSNA, the professional association for social

>> network researchers (http://www.insna.org). To unsubscribe, send an

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>

> ****************************************************************

>  Professor of Sociology and Statistics  Director, UWCFAR

> Sociobehavioral and Prevention Research Core  Box 354322  University

> of Washington  Seattle, WA 98195-4322

>

>  Office:        (206) 685-3402

>  Dept Office:   (206) 543-5882, 543-7237

>  Fax:           (206) 685-7419

>

> [log in to unmask]<mailto:[log in to unmask]>

> http://faculty.washington.edu/morrism/







--

Weihua (Edward) An



Assistant Professor of Sociology and Statistics Indiana University Bloomington

752 Ballantine Hall

1020 East Kirkwood Avenue

Bloomington, IN 47405-7103

http://mypage.iu.edu/~weihuaan/



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