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."

Modeling Peer Influence Issues of Selection & Endogeneity 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

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

Selection Network Model Coefficients, In school Networks -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 Same Race SES GPA Both Smoke College Drinking Fight Reciprocity Same Sex Same Clubs Transitivity Intransitivity Same Grade

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.

where P = some peer function. But the actual model may really be: Endogeneity Estimated: Y = b0 + b1(P) + e where P = some peer function. But the actual model may really be: P = b’0 + b’1(Y) + e

The statistical problem of endogeneity is that when you estimate 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 b’1, 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)

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.

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.

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

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 Positive to attribute Negative to attribute Friendship Attribute Actor

Endogeneity Mij = b0ij + bx(Xij)+bq(qi) + bp(pj) + eij

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?