Peer Influence Background:

Peer Influence Background:
long standing research interest in how our relations shape our attitudes and behaviors. One mechanism is that people, largely through conversation, change each others opinions This implies that position in a communication network should be related to attitudes. Freidkin & Cook: A formal model of influence, based on communication Cohen: An application of a similar peer influence model relating to adolescent college aspirations Haynie: Peer influence among adolescents Topics Covered: Basic Peer influence Selection and influence Dynamic mix of above Dyad models

Addendum A new statistic for determining the number of groups in a network.
The basic point still holds: finding groups takes a good deal of judgement. BUT, some statistics can help. The basic output from PROC CLUSTER A new measure proposed in Molecular Biology: Modularity

Proc cluster gives you a statistic for the basic “fit” of a cluster solution. This statistic varies depending on the method used, but is usually something like an R2. Consider this dendrogram:

Proc cluster gives you a statistic for the basic “fit” of a cluster solution. This statistic varies depending on the method used, but is usually something like an R2. Consider this dendrogram: The SPRSQ and the RSQ are your fit statistics.

A sharp change in the statistic is your best indicator. RSQ SPRSQ

Modularity: M is the modularity score S indexes each group (“module”) ls is the number of lines in group s L is the total number of lines ds is the sum of the degrees of the nodes in s Nm is the number of groups

Modularity:

Modularity: 5 10 20 Pii=.3 Pii=.2 Pii=.1

One piece in a long standing research program. Other cites include:
Friedkin & Cook One piece in a long standing research program. Other cites include: Friedkin, N. E "Structural Cohesion and Equivalence Explanations of Social Homogeneity." Sociological Methods and Research 12: ——— A Structural Theory of Social Influence. Cambridge: Cambridge. Friedkin, N. E. and E. C. Johnsen "Social Influence and Opinions." Journal of Mathematical Sociology 15( ). ——— "Social Positions in Influence Networks." Social Networks 19: See also the supplemental reading on the syllabus. Many particular context examples can be found as well.

Friedkin & Cook Peer influence models assume that individuals’ opinions are formed in a process of interpersonal negotiation and adjustment of opinions. Can result in either consensus or disagreement Looks at interaction among a system of actors In this particular paper, look at the opinion results in an experimental setup.

Basic Peer Influence Model
Attitudes are a function of two sources: a) Individual characteristics Gender, Age, Race, Education, Etc. Standard sociology b) Interpersonal influences Actors negotiate opinions with others

Freidkin claims in his Structural Theory of Social Influence that the theory has four benefits: relaxes the simplifying assumption of actors who must either conform or deviate from a fixed consensus of others (public choice model) Does not necessarily result in consensus, but can have a stable pattern of disagreement Is a multi-level theory: micro level: cognitive theory about how people weigh and combine other’s opinions macro level: concerned with how social structural arrangements enter into and constrain the opinion-formation process Allows an analysis of the systemic consequences of social structures

Formal Model (1) (2) Y(1) = an N x M matrix of initial opinions on M issues for N actors X = an N x K matrix of K exogenous variable that affect Y B = a K x M matrix of coefficients relating X to Y a = a weight of the strength of endogenous interpersonal influences W = an N x N matrix of interpersonal influences

Formal Model (1) This is the standard sociology model for explaining anything: the General Linear Model. It says that a dependent variable (Y) is some function (B) of a set of independent variables (X). At the individual level, the model says that: Usually, one of the X variables is e, the model error term.

(2) This part of the model taps social influence. It says that each person’s final opinion is a weighted average of their own initial opinions And the opinions of those they communicate with (which can include their own current opinions)

The key to the peer influence part of the model is W, a matrix of interpersonal weights. W is a function of the communication structure of the network, and is usually a transformation of the adjacency matrix. In general: Various specifications of the model change the value of wii, the extent to which one weighs their own current opinion and the relative weight of alters.

1 2 Self weight: Even 3 4 2*self degree

Formal Properties of the model (2) When interpersonal influence is complete, model reduces to: When interpersonal influence is absent, model reduces to:

Formal Properties of the model If we allow the model to run over t, we can describe the model as: The model is directly related to spatial econometric models: Where the two coefficients (a and b) are estimated directly (See Doreian, 1982, SMR)

a = .8 Basic Peer Influence Model T: 0 1 2 3 4 5 6 7
Simple example 1 2 Y 1 3 5 7 a = .8 3 4 T:

a = 1.0 Basic Peer Influence Model T: 0 1 2 3 4 5 6 7
Simple example 1 2 Y 1 3 5 7 a = 1.0 3 4 T:

Extended example: building intuition Consider a network with three cohesive groups, and an initially random distribution of opinions: (to run this model, use peerinfl1.sas)

Simulated Peer Influence:
75 actors, 2 initially random opinions, Alpha = .8, 7 iterations

Extended example: building intuition Consider a network with three cohesive groups, and an initially random distribution of opinions: Now weight in-group ties higher than between group ties

Simulated Peer Influence:
75 actors, 2 initially random opinions, Alpha = .8, 7 iterations, in-group tie: 2

Consider the implications for populations of different structures
Consider the implications for populations of different structures. For example, we might have two groups, a large orthodox population and a small heterodox population. We can imagine the groups mixing in various levels: Heterodox: 10 people Orthodox: 100 People Little Mixing Moderate Mixing Heavy Mixing

Light Heavy Moderate

Light mixing

Moderate mixing

High mixing

In an unbalanced situation (small group vs large group) the extent of contact can easily overwhelm the small group. Applications of this idea are evident in: Missionary work (Must be certain to send missionaries out into the world with strong in-group contacts) Overcoming deviant culture (I.e. youth gangs vs. adults) Work by Hyojung Kim (U Washington) focuses on the first of these two processes in social movement models

In recent extensions (Friedkin, 1998), Friedkin generalizes the model so that alpha varies across people. We can extend the basic model by (1) simply changing a to a vector (A), which then changes each person’s opinion directly, and (2) by linking the self weight (wii) to alpha. Were A is a diagonal matrix of endogenous weights, with 0 < aii < 1. A further restriction on the model sets wii = 1-aii This leads to a great deal more flexibility in the theory, and some interesting insights. Consider the case of group opinion leaders with unchanging opinions (I.e. many people have high aii, while a few have low):

Peer Opinion Leaders Group 1 Leaders Group 2 Leaders Group 3 Leaders

Peer Opinion Leaders

Further extensions of the model might:
Time dependent a: people likely value other’s opinions more early than later in a decision context Interact a with XB: people’s self weights are a function of their behaviors & attributes Make W dependent on structure of the network (weight transitive ties greater than intransitive ties, for example) Time dependent W: The network of contacts does not remain constant, but is dynamic, meaning that influence likely moves unevenly through the network And others likely abound….

Testing the fit of the general model. Experimental results
In the Friedkin and Cook paper, they test a version of the general model experimentally in 50 4 person groups. Each person was given time to form an initial opinion on a set of scenarios, and then discuss their opinions with others, based on a given structure. Based on the model, they can predict the relation between people’s initial opinions and the group’s final opinion. They find that the model does predict well, even controlling for the spread of initial opinions, the average opinion, and the structure of the network

Testing the fit of the general model
Identifying peer influence in real data There are two general ways to test for peer influence in an observed network. The first estimates the parameters (a and b) of the peer influence model directly, the second transforms the network into a dyadic model, predicting similarity among actors. Peer influence model: For details, see Doriean, 1982, sociological methods and research. Also Roger Gould (AJS, Paris Commune paper for example)

This is what Doriean calls the QAD estimate of peer influence.
Peer influence model: For details, see Doriean, 1982, sociological methods and research. Also Roger Gould (AJS, Paris Commune paper for example) The basic model says that people’s opinions are a function of the opinions of others and their characteristics. WY = A simple vector which can be added to your model. That is, multiple Y by a W matrix, and run the regression with WY as a new variable, and the regression coefficient is an estimate of a. This is what Doriean calls the QAD estimate of peer influence.

The problem with the above regression is that cases are, by definition, not independent. In fact, WY is also known as the ‘network autocorrelation’ coefficient, since a ‘peer influence’ effect is an autocorrelation effect -- your value is a function of the people you are connected to. In general, OLS is not the best way to estimate this equation. That is, QAD = Quick and Dirty, and your results will not be exact. In practice, the QAD approach (perhaps combined with a GLS estimator) results in empirical estimates that are “virtually indistinguishable” from MLE (Doreian et al, 1984) The proper way to estimate the peer equation is to use maximum likelihood estimates, and Doreian gives the formulas for this in his paper. The other way is to use non-parametric approaches, such as the Quadratic Assignment Procedure, to estimate the effects. Doreian, Patrik, Klaus Teuter and Chi-Hsein Wang “Network Autocorrelation Models: Some monte Carlo Results” Sociological Methods and Research, 13:

An empirical Example: Peer influence in the OSU Graduate Student Network.
Each person was asked to rank their satisfaction with the program, which is the dependent variable in this analysis. I constructed two W matrices, one from HELP the other from Best Friend. I treat relations as symmetric and valued, such that: I also include Race (white/Non-white, Gender and Cohort Year as exogenous variables in the model. (to run the model, see osupeerpi1.sas)

An empirical Example: Peer influence in the OSU Graduate Student Network.
Distribution of Satisfaction with the department.

Parameter Standardized Variable Estimate Pr > |t| Estimate
Parameter Estimates Parameter Standardized Variable Estimate Pr > |t| Estimate Intercept FEMALE NONWHITE y y y y PEER_BF PEER_H If we treat this as an ordered logit instead of a continuous variable, the coefficient for help becomes significant at the .1 level. There are all kinds of statistical problems with this model. It is likely underspecified (there are things left out of the model) and there is no direct correction for autocorrelated errors. Recall, however, that the monte carlo work seems to suggest this is will be a pretty robust measure in the face of different methods. Model R2 = .41, compared to .15 without the peer effects

The most common method for estimating peer effects is to include the mean of ego’s alters in the network. Under certain specifications of the model, this is exactly the same as the QAD analysis sketched above.

Example of mean-peer model: Haynie on Delinquency
Haynie asks whether peers matter for delinquent behavior, focusing on: a) the distinction between selection and influence b) the effect of friendship structure on peer influence Two basic theories underlie her work: a) Hirchi’s Social Control Theory Social bonds constrain otherwise criminal behavior The theory itself is largely ambivalent toward direction of network effects b) Sutherland’s Differential Association Behavior is the result of internalized definitions of the situation The effect of peers is through communication of the appropriateness of particular behaviors Haynie adds to these the idea that the structural context of the network can “boost” the effect of peers: (a) so transmission is more effective in locally dense networks and (b) the effect of peers is stronger on central actors. In Friedkin’s model, (a) is akin to changing w, such that the effect is greater for transitive ties, (b) is akin to making a dependent on centrality.

Example of mean-peer model: Haynie on Delinquency

Peer influence through Dyad Models
Another way to get at peer influence is not through the level of Y, but through the extent to which actors are similar with respect to Y. Recall the simulated example: peer influence is reflected in how close points are to each other.

Peer influence through Dyad Models
The model is now expressed at the dyad level as: Where Y is a matrix of similarities, A is an adjacency matrix, and Xk is a matrix of similarities on attributes

Complete Network Analysis
Network Connections: QAP Comparing multiple networks: QAP The substantive question is how one set of relations (or dyadic attributes) relates to another. For example: Do marriage ties correlate with business ties in the Medici family network? Are friendship relations correlated with joint membership in a club?

Network Connections: QAP Assessing the correlation is straight forward, as we simply correlate each corresponding cell of the two matrices: Dyads: Marriage 1 ACCIAIUOL 2 ALBIZZI 3 BARBADORI 4 BISCHERI 5 CASTELLAN 6 GINORI 7 GUADAGNI 8 LAMBERTES 9 MEDICI PAZZI 11 PERUZZI PUCCI 13 RIDOLFI 14 SALVIATI 15 STROZZI 16 TORNABUON Business Correlation:

Network Connections: QAP But is the observed value statistically significant? Can’t use standard inference, since the assumptions are violated. Instead, we use a permutation approach. Essentially, we are asking whether the observed correlation is large (small) compared to that which we would get if the assignment of variables to nodes were random, but the interdependencies within variables were maintained. Do this by randomly sorting the rows and columns of the matrix, then re-estimating the correlation.

Network Connections: QAP Comparing multiple networks: QAP When you permute, you have to permute both the rows and the columns simultaneously to maintain the interdependencies in the data: ID ORIG A B C D E Sorted A D B C E

Calculate the observed correlation for K iterations do:
Complete Network Analysis Network Connections: QAP Procedure: Calculate the observed correlation for K iterations do: a) randomly sort one of the matrices b) recalculate the correlation c) store the outcome 3. compare the observed correlation to the distribution of correlations created by the random permutations.

Network Connections: QAP

This can be done simply in UCINET
QAP MATRIX CORRELATION Observed matrix: PadgBUS Structure matrix: PadgMAR # of Permutations: Random seed: Univariate statistics PadgBUS PadgMAR 1 Mean 2 Std Dev Sum 4 Variance SSQ 6 MCSSQ 7 Euc Norm 8 Minimum 9 Maximum 10 N of Obs Hubert's gamma: Bivariate Statistics Value Signif Avg SD P(Large) P(Small) NPerm 1 Pearson Correlation: Simple Matching: 3 Jaccard Coefficient: 4 Goodman-Kruskal Gamma: Hamming Distance: This can be done simply in UCINET

Network Connections: QAP Using the same logic,we can estimate alternative models, such as regression, logits, probits, etc. Only complication is that you need to permute all of the independent matrices in the same way each iteration.

Network Connections: QAP Peer-influence results on similarity dyad model, using QAP # of permutations: Diagonal valid? NO Random seed: Dependent variable: EX_SIM Expected values: C:\moody\Classes\soc884\examples\UCINET\mrqap-predicted Independent variables: EX_SSEX EX_SRCE EX_ADJ Number of valid observations among the X variables = 72 N = 72 Number of permutations performed: 1999 MODEL FIT R-square Adj R-Sqr Probability # of Obs REGRESSION COEFFICIENTS Un-stdized Stdized Proportion Proportion Independent Coefficient Coefficient Significance As Large As Small Intercept EX_SSEX EX_SRCE EX_ADJ

If we break the original peer influence model into it’s components, the attribute part of the model suggests that any two people with the same attribute should have the same value for Y. The Peer influence model says that (a) if you and I are tied to each other, then we should have similar opinions and (b) that if we are tied to many of the same people, then we should have similar opinions. We can test both sides of these (and many other dyadic properties) directly at the dyad level.

NODE ADJMAT SAMERCE SAMESEX

Y 0.32 0.59 0.54 0.50 0.04 0.02 0.41 0.01 -0.17 Distance (Dij=abs(Yi-Yj)

Obs SENDER RCVER SIM NOM SAMERCE SAMESEX

Dependent Variable: SIM Analysis of Variance Sum of Mean
The REG Procedure Model: MODEL1 Dependent Variable: SIM Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept <.0001 NOM SAMERCE SAMESEX NCOMFND Note: Because everything in this model is symmetric, I run it with the statement: where sender > rcver; so I only get the top half of the distance matrix.

Like the basic Peer influence model, cases in a dyad model are not independent. However, the non-independence now comes from two sources: the fact that the same person is represented in (n-1) dyads and that i and j are linked through relations. One of the best solutions to this problem is QAP: Quadratic Assignment Procedure. A non-parametric procedure for significance testing. QAP runs the model of interest on the real data, then randomly permutes the rows/cols of the data matrix and estimates the model again. In so doing, it generates an empirical distribution of the coefficients.

MULTIPLE REGRESSION QAP W/ MISSING VALUES
# of permutations: Diagonal valid? NO Random seed: Dependent variable: EX_SIM Expected values: c:\moody\Classes\soc884\examples\UCINET\mrqap-predicted Independent variables: EX_NCOM EX_ADJ EX_SRCE EX_SSEX Number of valid observations among the X variables = 72 N = 72 Number of permutations performed: 1999 MODEL FIT R-square Adj R-Sqr Probability # of Obs REGRESSION COEFFICIENTS Un-stdized Stdized Proportion Proportion Independent Coefficient Coefficient Significance As Large As Small Intercept EX_NCOM EX_ADJ EX_SRCE EX_SSEX

Christakis & Fowler

Christakis & Fowler Used the friend/relative tracking data from a larger heart-health study to identify network contacts, including friends.

The network shows significant evidence of weight-homophily
Christakis & Fowler Used the friend/relative tracking data from a larger heart-health study to identify network contacts, including friends. The network shows significant evidence of weight-homophily

Effects of peer obesity on ego, by peer type
Christakis & Fowler Effects of peer obesity on ego, by peer type Used the friend/relative tracking data from a larger heart-health study to identify network contacts, including friends. Edge-wise regressions of the form: Ego is repeated for all alters; models include random effects on ego id

Christakis & Fowler Effects of peer obesity on ego, by peer type This modeling strategy pools observations on edges and estimates a global effect net of change in ego/alter as a control. Here color is a single ego, number is wave (only 2 egos and 3 waves represented). Peer Effect 3 3 Ego-Current 3 3 3 3 2 2 2 2 2 2 1 1 1 1 1 1 Alter Current

Christakis & Fowler Effects of peer obesity on ego, by peer type Alterative specifications include using change-change models and allowing for a random effect of peers. This allows for greater variability in peer effects, and the potential to model differences. Note none of these models use information from non-ties, other than entering as fixed wave/place effects. b1 b 3 3 Ego-Current 3 3 3 3 2 2 2 b2 2 2 2 1 1 1 1 1 1 Alter Current

Dynamics of Political Polarization
When and why do networks & attitudes polarize? Paradox Problems:

When and why do networks & attitudes polarize? Model Setup:

When and why do networks & attitudes polarize? Model Setup: If actors share the same orientation, interaction reinforces the opinion. If they disagree on an issue, then: -- if they agree on the majority of others, they compromise -- if they disagree they push each other further apart

When and why do networks & attitudes polarize? Results:

Cohen: The problem of Selection and Influence
Well known that cohesive groups tend to be more similar (homogeneous) than the population at large. Why is this so? It may be due either to influence: people change as a function of the people around them or selection: people join groups based on their behaviors Cohen re-analyzed data by Coleman on “Newlawn” a middle-class white suburban school of about 1000 students, and identified cliques of students in the school. (His measure of clique is pretty exclusive: only 9% of the males and 40% of the females in the school fit his definition) He proposes to answer the selection vs. influence question by looking at changes in behavior and changes in group composition over time.

Design: Sf(F) = average std. Dev. Of fall members in the fall Sf(S) = average std. Dev. Of spring members in the fall Ss(F) = average std. Dev. Of Fall members in the Spring Ss(S) = average std. Dev. Of Spring members in the Spring Ss(S) - Sf(F) = change in group homogeneity over time. Not clear whether it is due to changes in behavior or changes in members Ss(F) - Sf(F) = change in group homogeneity over time, for only those people who are in the same group both times. Removes the selection effect.

Homophily due to selection is equal to overall uniformity (U) minus conformity (C). C are the differences in table 2, and he estimates selection effects here. (Finding that the selection effect is “less substantial than conformity effects” (p.233)) By comparing newly formed spring cliques, he concludes that the majority of overall conformity is due to initial selection.

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

Peer Influence Background:

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