Willard Van Quine, professor of philosophy and mathematics emeritus from Harvard University who is r

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Modeling Peer Influence
Willard Van Quine, professor of philosophy and
mathematics emeritus from Harvard University who
is regarded as one of the four most famous living
philosophers in the world, wrote his doctoral
thesis on a 1927 Remington typewriter, which he
still uses. However, he "had an operation on it"
to change a few keys to accommodate special
symbols. "I found I could do without the second
period, the second comma -- and the question
mark. "You don't miss the question
mark? "Well, you see, I deal in certainties."
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Modeling Peer Influence Issues of Selection
Endogeneity

• Selection
• That some unobserved factor, z, creates both
  friendships and the outcome of interest.
• Endogeneity
• That the causal order of peer relations and
  outcomes is reversed. Peers do not cause Y, but
  Y causes friendship relations

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Selection

• What do we know about how friendships form?

• Opportunity / focal factors
• - Being members of the same group
• - In the same class
• - On the same team
• - Members of the same church
• Structural Relationship factors
• - Reciprocity
• - Social Balance
• Behavior Homophily
• - Smoking
• - Drinking

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Selection
Network Model Coefficients, In school Networks
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Selection
How to correct this problem?

• Essentially, this is an omitted variable problem,
  and my solution has been to identify as many
  potentially relevant alternative variables as I
  can find.

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Endogeneity
Estimated Y b0 b1(P) e where P some
peer function. But the actual model may really
be P b0 b1(Y) e
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Endogeneity
Does it matter?
Algebraically the relation between y and p should
be direct translation of the coefficients
since
The statistical problem of endogeneity is that
when you estimate b1, it does not equal 1/b1,
because of our assumptions about x, and hence e.
(see Joel H. Levine, Exceptions are the Rule, for
a full discussion of this)
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Endogeneity
Fully specified peer influence models
Where W is a matrix of interpersonal weights,
calculated from the friendship adjacency matrix
(the mixed-regressive autoregressive peer
influence model, see Friedkin 1999, chapter 2,
Doreian, 1982, SMR)
These models can be estimated directly with Add
Health data, but again the problem is that W may
be determined by Y.
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Endogeneity
Possible solutions

• Theory Given what we know about how friendships
  form, is it reasonable to assume a bi-directional
  cause? That is, work through the meeting,
  socializing, etc. process and ask whether it
  makes sense that Y is a cause of W.
• Models
• - Time Order. We are on somewhat firmer ground
  if W precedes Y in time. Thus, using the
  in-school friendship structure to predict wave 1
  outcomes is useful.
• - Simultaneous Models. Model both the friendship
  pattern and the outcome of interest
  simultaneously.

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Endogeneity

• Simultaneous models
• One way to do this is with SEMs, but for
  identification, you must find some variables that
  predict friendship that do not also predict Y,
  which given the very nature of the endogeneity
  problem, is hard to do.
• Could also model a mixed network of relations
  and behaviors using dyads and a p style model to
  predict ties within the joint network

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Endogeneity
A mixed selection and influence model
Simultaneous balance on friendship and
behavior. Two linked models a) actors seek
interpersonal balance among friends b) actors
change their opinions / behaviors as a weighted
function of the people they are tied to, with W
weighted by number of transitive ties
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Endogeneity
Mij b0ij bx(Xij)bq(qi) bp(pj) eij
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What are the policy implications?
If we find a relation between peer and behavior,
do these types of problems make a difference for
what we should recommend to policy makers, such
as school officials?