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Weihua,

The "interaction" is between the attributes of node i and the attributes
of node j.  Imagine the ties crosstabulated by these two attributes.  The
"main effects" are the relative row and column mean degrees by subgroup.
The "interaction" effects are the deviations from independence in the
cells. Homophily is one version of this (higher density on the diagonal),
but not the only one.

As Christian and I both noted, widely varying group sizes may influence
the interpretation of the interaction effects if the main effects are
excluded.

Martina

On Wed, 7 Aug 2013, Weihua An wrote:

> 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, 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
>>
>>
>> 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
>>>
>>> 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
>>>>> Jeff
>>>>> _____________________________________________________________________
>>>>> SOCNET is a
>>>>> service of INSNA, the professional association for social network
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>>>>> message.
>>>>>
>>>>
>>>> ****************************************************************
>>>>  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
>>>>
>>>> http://faculty.washington.edu/morrism/
>>>>
>>>>
>>>> _____________________________________________________________________
>>>> SOCNET is a service of INSNA, the professional association for social
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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
>>> UNSUBSCRIBE SOCNET in the body of the message.
>>>
>>
>> ****************************************************************
>>  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
>>
>> 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/
>

****************************************************************
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