Interaction terms using operators: what do they do, and what should they do?

[martinamorris]

Context: Hoping to use the summary() function to calculate mean degree by group.

The term operators in ergm 4 include an "interaction" operator (demonstrated in the Advanced ERGM workshop, section 4.4).

I was hoping to use this to calculate the mean degree by group in a summary call: summary(nw ~ meandeg*nodefactor("Attribute", levels=T). But this doesn't produce the expected result. Reprex below.

library(ergm)
data("faux.mesa.high")
mesa <- faux.mesa.high

# meandeg, overall (works as expected)
## using direct calculation
2*network.edgecount(mesa) / network.size(mesa)
## using summary
summary(mesa ~ meandeg)

# meandeg by grade (not as expected)

## nodes by grade 
nodes.by.grade <- table(mesa %v% "Grade")
## edges by grade (note, sums to 406, so is actually "stubs" by grade)
stubs.by.grade <- summary(mesa ~ nodefactor("Grade", levels=T))
## meandeg by grade using stubs/nodes calculation 
stubs.by.grade/nodes.by.grade
### using summary and interaction term operator with ":"
summary(mesa ~ meandeg:nodefactor("Grade", levels=TRUE))

[martinamorris]

Pavel's initial comment (from email):

The short of it is that it's really not what interaction terms do. In this case, the change statistic for dyad (i,j) will be

(I(i is in Grade k) + I(j is in grade k)) * (1/n*2)

with the first expression coming from the nodefactor and the second from the meandeg. So, for starters, the divisor is wrong: it should be the group size, not n, and I think the multiplier 2 shouldn't be there, since the double-counting of nodefactor already takes care of that.

Interaction terms have always been something I did because I could, without reckoning whether or not I should. That having been said, at Sunbelt, I saw someone actually use the interaction terms to model intersectionality in homophily.

[martinamorris]

@krivit

If I'm following the change statistic logic correctly, a new tie between dyad(i,j) should change the mean degree for the group levels for both i's group and j's group, which may be the same (k) or different groups (k, l). So it seems like the change statistic should contribute either one value (if both are from group k), or two separate values (if they're from different groups).

In any case, I think the question is whether the change stat, as currently defined: (I(i is in Grade k) + I(j is in grade k)) * (1/n*2), is ever a value we would be interested in. And, is it possible to define an interaction for meandeg*Grade that would produce the statistics meandegree by grade.

A somewhat related question - the degree term has an argument "by" for attributes. Would it be possible to provide that same argument for the meandeg term?

[krivit]

In any case, I think the question is whether the change stat, as currently defined: (I(i is in Grade k) + I(j is in grade k)) * (1/n*2), is ever a value we would be interested in. And, is it possible to define an interaction for meandeg*Grade that would produce the statistics meandegree by grade.

What you get out of interaction terms is exactly what you would have gotten if you had reformulated the dyad-independent ERGM as a logistic regression and added an interaction term to that; nothing more, nothing less.

A definition of an interaction needs to be universal and consistent for all terms---at least dyad-independent ones---and I don't think that one that would behave the way you are asking can exist, though I certainly could be wrong.

We can certainly implement a by, levels argument for meandeg().

[martinamorris]

We can certainly implement a by, levels argument for meandeg().

That would be great. I'll file an issue in the ergm repo then.

And I agree that the defn of an interaction needs to be universal and consistent. My intuition about what interaction terms do is mostly drawn from the OLS context, not GLMs, so I'd need to think on that a bit more.

[krivit]

My intuition about what interaction terms do is mostly drawn from the OLS context, not GLMs, so I'd need to think on that a bit more.

The idea is exactly the same: the interaction occurs in the linear predictor (i.e., on the logit scale).