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13 An Introduction to Stochastic Actor-Oriented Models (2016)

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13 An Introduction to Stochastic Actor-Oriented Models (2016)

  1. 1. An Introduc+on to Stochas+c Actor-Oriented Models (aka SIENA) Dr. David R. Schaefer Arizona State University Social Networks & Health Training Workshop Duke University June 20, 2016
  2. 2. 1 Dyad independent models 2 R (sna) = lnam Outcome Figure adapted from jimi adams Modeled Interdependencies None w/in Dyad1 Dyad+ A;ribute General Linear Model Actor-Partner Interdependence (APIM) Network Autoregressive2 Stochas+c Actor- Oriented Model (SAOM) Network Erdös-Renyi (MR)QAP Exponen+al Random Graph (ERGM/TERGM), Rela+onal Event May 20, 2016 Duke Social Networks & Health Workshop 2 Sta?s?cal Modeling & SNA
  3. 3. When to use a SAOM May 20, 2016 Duke Social Networks & Health Workshop 3 •  Ques+ons about changes in network structure over +me –  Including mul+ple networks –  Including two-mode networks (selec+ng into foci) •  Ques+ons about how networks affect individual “behaviors,” such as through peer influence –  Including mul+ple behaviors and possible reciprocal effects •  Ques+ons about the endogenous associa+on between networks and behavior
  4. 4. Stochas?c Actor-Oriented Model •  Also called Stochas+c Actor-Based Model (SABM), or “SIENA” based on the sobware used to es+mate the model –  Simula'on Inves'ga'on for Empirical Network Analysis –  Currently es+mable in R (RSiena) •  Recogni+on that networks and behavior are interdependent –  Behaviors can affect network structure –  Network structure can affect behavior –  Thus, both “outcomes” are endogenous –  Complicates adempts to answer important theore+cal ques+ons (e.g., peer influence) May 20, 2016 Duke Social Networks & Health Workshop 4
  5. 5. Network Homogeneity on Smoking Peer Influence or Friend Selection time t time t-1 A C D B A C D B A C D B May 20, 2016 Duke Social Networks & Health Workshop 5
  6. 6. Smoking-Related Popularity Popularity leads to smoking or Smoking enhances popularity time t time t-1 C D BA C D BA C D BA May 20, 2016 Duke Social Networks & Health Workshop 6
  7. 7. Inferring Network → Behavior Requires controlling for network selec?on based on: 1.  Pre-exis+ng similarity in the behavior 2.  Similarity on adributes correlated with the behavior 3.  Network processes, such as triad closure •  Can amplify network-behavior paderns (see below) May 20, 2016 Duke Social Networks & Health Workshop 7 C D BA I ♥ Homophily! C D BA C D BA Homophily through Reciprocity Homophily through Transi?vity
  8. 8. Overview of Model Presenta?on 1.  The general form of the model –  Network func+on for rela+onship change –  Behavior func+on for “behavior” change –  Rate func+ons 2.  Model es+ma+on procedure –  Model assump+ons –  MCMC es+ma+on algorithm –  Goodness of Fit 3.  Empirical example 4.  Extensions & Miscellany May 20, 2016 Duke Social Networks & Health Workshop 8
  9. 9. 1. General SAOM Form May 20, 2016 Duke Social Networks & Health Workshop 9
  10. 10. •  Discrete change is modeled as occurring in con+nuous +me (between observa+ons) through a sequence of micro steps •  Actors control their outgoing +es and behavior –  Func+ons specify when and how they change SAOM Components Decision Timing (when changes occur) Decision Rules (how changes occur) Network Evolu+on Network rate func+on Network objec+ve func+on Behavior Evolu+on Behavior rate func+on Behavior objec+ve func+on May 20, 2016 Duke Social Networks & Health Workshop 10
  11. 11. Network Objec?ve Func?on •  Network change is modeled by allowing actors to select +es (by adding or dropping them) based upon: fi(β,x) is the value of the network objec+ve func+on for actor i, given: •  the current set of parameter es+mates (β) •  state of the network (x) •  For k effects, represented as ski, which may be based on –  the network (x), or individual adributes (z) •  Es+mated with random disturbance (ε) associated with x, z, t and j •  Goal of model fimng is to es+mate each βk May 20, 2016 Duke Social Networks & Health Workshop 11 fi (β, x) = βkski k ∑ (x)+ε(x, z,t, j)
  12. 12. j3 ego j4 j2 j1 Network Decision € fego(β,x) = -2 € xij j ∑ + 1.8 € xij x ji j ∑ outdegree reciprocity € fego(β,x) = -2 € xij j ∑ + 1.8 € xij x ji j ∑fego(β,x) = -2 € xij j ∑ + 1.8 € xij x ji j ∑ During a micro step, an actor evaluates how changing its outgoing +e in each dyad would affect the value of the objec+ve func+on (goal is to maximize the value of the func+on) ego j1 j2 j3 j4 ego - 1 1 0 0 j1 1 - 0 0 0 j2 0 0 - 0 0 j3 1 0 0 - 0 j4 0 0 0 0 - May 20, 2016 Duke Social Networks & Health Workshop 12 If… outdegree reciprocity sum No change -2 * 2 = -4 1.8 * 1 = 1.8 -2.2 Drop j1 -2 * 1 = -2 1.8 * 0 = 0 -2 Drop j2 -2 * 1 = -2 1.8 * 1 = 1.8 -.2 Add j3 -2 * 3 = -6 1.8 * 3 = 3.6 -2.4 Add j4 -2 * 3 = -6 1.8 * 1 = 1.8 -4.2 Given the current state of the network, ego is most likely to drop the ?e to j2, because that decision maximizes the objec+ve func+on
  13. 13. •  Outdegree always present •  Network processes (e.g., reciprocity, transi+vity) •  Adribute based: –  Sociality: effect of adribute on outgoing +es –  Popularity: effect of behavior on incoming +es –  Homophily: ego-alter similarity –  Note: adributes may be stable or +me-changing (exogenous or endogenously modeled) •  Dyadic adributes (e.g., co-membership) May 20, 2016 Duke Social Networks & Health Workshop 13 Network Objec?ve Func?on Effects
  14. 14. •  Predict change in “behavior,” which is the generic term for an individual adribute –  Refers to any amtude, belief, health factor, etc. •  Op+onal: SAOMs don’t require one and they’re not relevant for many ques+ons •  Ordinal measurement required (~2-10 levels best) •  Goal is to es+mate effect of network on behavior change May 20, 2016 Duke Social Networks & Health Workshop 14 Behavior Objec?ve Func?on
  15. 15. Behavior Objec?ve Func?on •  Choice probabili+es take the form of a mul+nomial logit model instan+ated by the objec+ve func+on where z represents the behavior •  The func+on dictates which level of the behavior actors adopt –  Actors evaluate all possible changes •  Increase/decrease by one unit, or no change –  Op+on with highest evalua+on most likely May 20, 2016 Duke Social Networks & Health Workshop 15 fi z (β, x, z) = βk z ski z k ∑ (x, z)+ε(x, z,t,δ) Figure adapted from C. Steglich
  16. 16. •  Linear term to control for distribu+on (quadra+c term if the behavior has 3+ levels) •  Predictors of peer influence –  Alters’ value on the behavior, or another adribute or behavior •  Mul+ple specifica+ons, including mean, minimum, maximum… •  Ego’s other behaviors or adributes (e.g., gender, age) –  Ego’s network posi+on (e.g., degree) –  Interac+ons with reciprocity May 20, 2016 Duke Social Networks & Health Workshop 16 Behavior Objec?ve Func?on Effects
  17. 17. Behavior Decision May 20, 2016 Duke Social Networks & Health Workshop 17 Linear effect Quadra+c effect Adribute effect (e.g. age) Similarity effect How adrac+ve is each level of the behavior based on these effects?
  18. 18. Ego, j1 1 - | 1 - 1 | / 2 = .5 1 (.5 - .05) = .45 Ego, j2 1 - | 1 - 1 | / 2 = .5 1 (.5 - .05) = .45 Ego, j3 1 - | 1 - 0 | / 2 = 0 1 (0 - .05) = .05 Ego, j4 1 - | 1 - 2 | / 2 = 0 0 (0 - .05) = 0 Similarity sta?s?c = .95 Behavior Decision* May 20, 2016 Duke Social Networks & Health Workshop 18 J3(0) Ego (1) J4(2) J1(1) € xij (simij Z − simZ ) j ∑ € simij Z =1−| € zi−z j| € /ΔZ € ΔZ = maxij | € zi−z j| = 2where = € simZ = similarity expected by chance= similarity expected by chance = .05 simij Z xij (simij Z − simZ ) j2(1) Maintain z=1 First, calculate similarity for each of ego’s possible decisions * Assume covariates uncentered
  19. 19. Ego, j1 1 - | 0 - 1 | / 2 = 0 1 (0 - .05) = -.05 Ego, j2 1 - | 0 - 1 | / 2 = 0 1 (0 - .05) = -.05 Ego, j3 1 - | 0 - 0 | / 2 = .5 1 (.5 - .05) = .45 Ego, j4 1 - | 0 - 2 | / 2 = -.5 0 (-.5 - .05) = 0 Similarity sta?s?c = .35 Behavior Decision* May 20, 2016 Duke Social Networks & Health Workshop 19 J3(0) Ego (1) J4(2) J1(1) First, calculate similarity for each of ego’s possible decisions € xij (simij Z − simZ ) j ∑ € simij Z =1−| € zi−z j| € /ΔZ € ΔZ = maxij | € zi−z j| = 2= € simZ = similarity expected by chance= similarity expected by chance = .05 simij Z xij (simij Z − simZ ) j2(1) Decrease to z=0 * Assume covariates uncentered where
  20. 20. Ego, j1 1 - | 2 - 1 | / 2 = 0 1 (0 - .05) = -.05 Ego, j2 1 - | 2 - 1 | / 2 = 0 1 (0 - .05) = -.05 Ego, j3 1 - | 2 - 0 | / 2 = -.5 1 (-.5 - .05) = -.45 Ego, j4 1 - | 2 - 2 | / 2 = .5 0 (.5 - .05) = 0 Similarity sta?s?c = -.55 Behavior Decision* May 20, 2016 Duke Social Networks & Health Workshop 20 J3(0) Ego (1) J4(2) J1(1) € xij (simij Z − simZ ) j ∑ € simij Z =1−| € zi−z j| € /ΔZ € ΔZ = maxij | € zi−z j| = 2= € simZ = similarity expected by chance= similarity expected by chance = .05 simij Z xij (simij Z − simZ ) j2(1) Increase to z=2 First, calculate similarity for each of ego’s possible decisions * Assume covariates uncentered where
  21. 21. Behavior Decision* May 20, 2016 Duke Social Networks & Health Workshop 21 If… linear quad age similarity sum Drop to 0 -.5 * 0 = 0 .25 * 0 = 0 .1 * 10 * 0 = 0 1 * .35 = .35 .35 Stay at 1 -.5 * 1 = -.5 .25 * 1 = .25 .1 * 10 * 1 = 1 1 * .95 = .95 1.7 Up to 2 -.5 * 2 = -1 .25 * 4 = 1 .1 * 10 * 2 = 2 1 * -.55 = -.55 1.45 * Assume covariates uncentered Second, calculate the contribu+ons for each of the other effects
  22. 22. Behavior Decision* May 20, 2016 Duke Social Networks & Health Workshop 22 If… linear quad age similarity sum Drop to 0 -.5 * 0 = 0 .25 * 0 = 0 .1 * 10 * 0 = 0 1 * .35 = .35 .35 Stay at 1 -.5 * 1 = -.5 .25 * 1 = .25 .1 * 10 * 1 = 1 1 * .95 = .95 1.7 Up to 2 -.5 * 2 = -1 .25 * 4 = 1 .1 * 10 * 2 = 2 1 * -.55 = -.55 1.45 * Assume covariates uncentered These effects pull ego toward the extremes The posi+ve age b pushes ego’s behavior upward Similarity pushes ego to stay the same Altogether, the greatest contribu+on to the behavior func+on comes from ego choosing to maintain the same behavior level
  23. 23. •  Necessary for both network and behavior •  Determine the wai+ng +me un+l actor’s chance to make decisions •  Func+on of observed changes –  But not the same as the number of changes observed –  Separate rate parameter for each period between observa+ons •  Wai+ng +me distributed uniformly by default, but differences can be modeled based on: •  Actor adributes: do some types of actors experience more or less change •  Degree: do actors with more/fewer +es experience a different volume of change May 20, 2016 Duke Social Networks & Health Workshop 23 Rate Func?ons
  24. 24. 2. SAOM Es+ma+on May 20, 2016 Duke Social Networks & Health Workshop 24
  25. 25. SAOM Es?ma?on •  Goal during es+ma+on is to iden+fy parameter values (i.e., a model) that produce networks whose sta+s+cs are centered on target sta+s+cs –  Same as modeled effects measured at t1+ •  Robbins-Monro algorithm in three phases 1.  Ini+alize parameter star+ng values 2.  Use simula+ons to refine parameter es+mates (next slide) •  A large number of simula+on itera+ons, nested in 4+ subphases •  Actor decisions and +ming based on objec+ve and rate func+ons •  Update parameter es+mates aber each simula+on itera+on –  Adempt to minimize devia+on of ending state from target 3.  Addi+onal simula+ons (2,000+) to calculate standard errors based on parameter es+mates from phase 2 May 20, 2016 Duke Social Networks & Health Workshop 25
  26. 26. Markov Chain Algorithm May 20, 2016 Duke Social Networks & Health Workshop 26 Ini+alize at first observa+on Actors draw: 1)  Wai+ng +me for network 2)  Wai+ng +me for behavior Determined by rate func+ons Shortest wai+ng +me/type iden+fied Time up? Actor changes +e|behavior Determined by objec+ve func+ons Update +me (next micro step) “STOP” Yes No For each step in a Markov chain: Max itera+ons? No Yes If Phase 2, update parameters Store ending network & behavior
  27. 27. Post-Es?ma?on 1 •  Check for Convergence •  Convergence achieved when model is able to reproduce observed network & behavior at +me 2+ –  For each effect, t-ra+o to compare target sta+s+cs with distribu+on (t should be < .10) –  Maximum t-ra+o for convergence (tconv.max) should be less than .25 –  If convergence not reached, rerun with using es+mates as new star+ng values; may need to respecify model May 20, 2016 Duke Social Networks & Health Workshop 27
  28. 28. Post-Es?ma?on 2 •  Goodness of Fit •  Use simula+ons to compare networks generated by model to sta+s+cs NOT explicitly in the model –  Typical candidates: •  In- & Out-degree distribu+ons •  Triad Census •  Geodesic distribu+on •  Behavior distribu+on •  Behavior network associa+ons May 20, 2016 Duke Social Networks & Health Workshop 28
  29. 29. 3. SAOM Example May 20, 2016 Duke Social Networks & Health Workshop 29
  30. 30. An Empirical Example with Adolescent Smoking •  Na+onal Longitudinal Study of Adolescent Health (Add Health) •  In-home surveys conducted 1994-1995 (2 waves) –  Earlier in-school survey has network data but limited behavior data •  Students nominated up to 5 male and 5 female friends (directed network) –  Friendships coded present (1) or absent (0) for each dyad May 20, 2016 Duke Social Networks & Health Workshop 30
  31. 31. 30-day smoking None (0) 1-11 days (1) 12+ days (2) Jefferson High (Add Health, 1995) May 20, 2016 Duke Social Networks & Health Workshop 31
  32. 32. •  Helpful to imagine the network func+on as a logis+c regression –  Unit of analysis: dyad –  Outcome: +e presence (keeping or adding) vs. absence (dissolving or failing to add) –  Each effect represents how a one-unit change in the effect sta+s+c affects the log-odds of a +e, all else being equal •  Some effects interpretable using odds ra+os, but – One-unit changes may not be meaningful – All else is never equal (any change also affects the outdegree count, at a minimum) •  Behavior func+on specifies how a one-unit change in the effect sta+s+c affects the odds of increasing behavior one unit May 20, 2016 Duke Social Networks & Health Workshop 32 Interpre?ng Results
  33. 33. Network func?on b SE Rate 10.26 *** .49 Outdegree -3.91 *** .08 Reciprocity 1.91 *** .09 Transi+ve triplets .52 *** .04 Popularity .29 *** .04 Extracurric. act. overlap .28 *** .06 Smoke similarity .68 *** .12 Smoke alter .14 ** .05 Smoke ego -.04 .05 Female similarity .24 *** .04 Female alter -.11 * .05 Female ego -.04 .05 Age similarity 1.00 *** .13 Age alter -.01 .03 Age ego -.04 .03 Delinquency similarity .15 .08 Delinquency alter -.04 .04 Delinquency ego .02 .04 Alcohol similarity .27 ** .10 Alcohol alter -.03 .03 Alcohol ego -.03 .04 GPA similarity .70 *** .13 GPA alter -.05 .04 GPA ego -.02 .04 From Schaefer, D.R. S.A. Haas, and N. Bishop. 2012. “A Dynamic Model of US Adolescents’ Smoking and Friendship Networks.” American Journal of Public Health, 102:e12-e18. May 20, 2016 Duke Social Networks & Health Workshop 33 Rate: Each actor is given ~10 micro steps in which to make a change to its network •  Add a +e, drop a +e, or make no change Rate
  34. 34. Network func?on b SE Rate 10.26 *** .49 Outdegree -3.91 *** .08 Reciprocity 1.91 *** .09 Transi+ve triplets .52 *** .04 Popularity .29 *** .04 Extracurric. act. overlap .28 *** .06 Smoke similarity .68 *** .12 Smoke alter .14 ** .05 Smoke ego -.04 .05 Female similarity .24 *** .04 Female alter -.11 * .05 Female ego -.04 .05 Age similarity 1.00 *** .13 Age alter -.01 .03 Age ego -.04 .03 Delinquency similarity .15 .08 Delinquency alter -.04 .04 Delinquency ego .02 .04 Alcohol similarity .27 ** .10 Alcohol alter -.03 .03 Alcohol ego -.03 .04 GPA similarity .70 *** .13 GPA alter -.05 .04 GPA ego -.02 .04 From Schaefer, D.R. S.A. Haas, and N. Bishop. 2012. “A Dynamic Model of US Adolescents’ Smoking and Friendship Networks.” American Journal of Public Health, 102:e12-e18. May 20, 2016 Duke Social Networks & Health Workshop 34 Outdegree: The nega+ve sign is typical. It means that +es are unlikely, unless other effects in the model make a posi+ve contribu+on to the network func+on. density
  35. 35. Network func?on b SE Rate 10.26 *** .49 Outdegree -3.91 *** .08 Reciprocity 1.91 *** .09 Transi+ve triplets .52 *** .04 Popularity .29 *** .04 Extracurric. act. overlap .28 *** .06 Smoke similarity .68 *** .12 Smoke alter .14 ** .05 Smoke ego -.04 .05 Female similarity .24 *** .04 Female alter -.11 * .05 Female ego -.04 .05 Age similarity 1.00 *** .13 Age alter -.01 .03 Age ego -.04 .03 Delinquency similarity .15 .08 Delinquency alter -.04 .04 Delinquency ego .02 .04 Alcohol similarity .27 ** .10 Alcohol alter -.03 .03 Alcohol ego -.03 .04 GPA similarity .70 *** .13 GPA alter -.05 .04 GPA ego -.02 .04 From Schaefer, D.R. S.A. Haas, and N. Bishop. 2012. “A Dynamic Model of US Adolescents’ Smoking and Friendship Networks.” American Journal of Public Health, 102:e12-e18. May 20, 2016 Duke Social Networks & Health Workshop 35 Reciprocity: Ties that create a reciprocated +e are more likely to be added or maintained. This effect hovers around 2 in friendship-type network. recip
  36. 36. Network func?on b SE Rate 10.26 *** .49 Outdegree -3.91 *** .08 Reciprocity 1.91 *** .09 Transi+ve triplets .52 *** .04 Popularity .29 *** .04 Extracurric. act. overlap .28 *** .06 Smoke similarity .68 *** .12 Smoke alter .14 ** .05 Smoke ego -.04 .05 Female similarity .24 *** .04 Female alter -.11 * .05 Female ego -.04 .05 Age similarity 1.00 *** .13 Age alter -.01 .03 Age ego -.04 .03 Delinquency similarity .15 .08 Delinquency alter -.04 .04 Delinquency ego .02 .04 Alcohol similarity .27 ** .10 Alcohol alter -.03 .03 Alcohol ego -.03 .04 GPA similarity .70 *** .13 GPA alter -.05 .04 GPA ego -.02 .04 From Schaefer, D.R. S.A. Haas, and N. Bishop. 2012. “A Dynamic Model of US Adolescents’ Smoking and Friendship Networks.” American Journal of Public Health, 102:e12-e18. May 20, 2016 Duke Social Networks & Health Workshop 36 Transi?ve triplets: Ties that create more transi+ve triads have a greater likelihood. •  Should also test interac+on with Reciprocity (usually nega+ve) transTrip
  37. 37. Network func?on b SE Rate 10.26 *** .49 Outdegree -3.91 *** .08 Reciprocity 1.91 *** .09 Transi+ve triplets .52 *** .04 Popularity .29 *** .04 Extracurric. act. overlap .28 *** .06 Smoke similarity .68 *** .12 Smoke alter .14 ** .05 Smoke ego -.04 .05 Female similarity .24 *** .04 Female alter -.11 * .05 Female ego -.04 .05 Age similarity 1.00 *** .13 Age alter -.01 .03 Age ego -.04 .03 Delinquency similarity .15 .08 Delinquency alter -.04 .04 Delinquency ego .02 .04 Alcohol similarity .27 ** .10 Alcohol alter -.03 .03 Alcohol ego -.03 .04 GPA similarity .70 *** .13 GPA alter -.05 .04 GPA ego -.02 .04 From Schaefer, D.R. S.A. Haas, and N. Bishop. 2012. “A Dynamic Model of US Adolescents’ Smoking and Friendship Networks.” American Journal of Public Health, 102:e12-e18. May 20, 2016 Duke Social Networks & Health Workshop 37 Indegree Popularity: Actors with more incoming +es have a greater likelihood of receiving future +es inPop
  38. 38. Network func?on b SE Rate 10.26 *** .49 Outdegree -3.91 *** .08 Reciprocity 1.91 *** .09 Transi+ve triplets .52 *** .04 Popularity .29 *** .04 Extracurric. act. overlap .28 *** .06 Smoke similarity .68 *** .12 Smoke alter .14 ** .05 Smoke ego -.04 .05 Female similarity .24 *** .04 Female alter -.11 * .05 Female ego -.04 .05 Age similarity 1.00 *** .13 Age alter -.01 .03 Age ego -.04 .03 Delinquency similarity .15 .08 Delinquency alter -.04 .04 Delinquency ego .02 .04 Alcohol similarity .27 ** .10 Alcohol alter -.03 .03 Alcohol ego -.03 .04 GPA similarity .70 *** .13 GPA alter -.05 .04 GPA ego -.02 .04 From Schaefer, D.R. S.A. Haas, and N. Bishop. 2012. “A Dynamic Model of US Adolescents’ Smoking and Friendship Networks.” American Journal of Public Health, 102:e12-e18. May 20, 2016 Duke Social Networks & Health Workshop 38 Dyadic Covariate: Actors who share an extracurricular ac+vity (coded 1) are more likely to have a friendship +e X
  39. 39. Network func?on b SE Rate 10.26 *** .49 Outdegree -3.91 *** .08 Reciprocity 1.91 *** .09 Transi+ve triplets .52 *** .04 Popularity .29 *** .04 Extracurric. act. overlap .28 *** .06 Smoke similarity .68 *** .12 Smoke alter .14 ** .05 Smoke ego -.04 .05 Female similarity .24 *** .04 Female alter -.11 * .05 Female ego -.04 .05 Age similarity 1.00 *** .13 Age alter -.01 .03 Age ego -.04 .03 Delinquency similarity .15 .08 Delinquency alter -.04 .04 Delinquency ego .02 .04 Alcohol similarity .27 ** .10 Alcohol alter -.03 .03 Alcohol ego -.03 .04 GPA similarity .70 *** .13 GPA alter -.05 .04 GPA ego -.02 .04 Ties driven by similarity on: Gender (could use “same” effect) Age Alcohol use GPA Females less adrac+ve as friends than males. From Schaefer, D.R. S.A. Haas, and N. Bishop. 2012. “A Dynamic Model of US Adolescents’ Smoking and Friendship Networks.” American Journal of Public Health, 102:e12-e18. altX egoX simX May 20, 2016 Duke Social Networks & Health Workshop 39
  40. 40. Network func?on b SE Rate 10.26 *** .49 Outdegree -3.91 *** .08 Reciprocity 1.91 *** .09 Transi+ve triplets .52 *** .04 Popularity .29 *** .04 Extracurric. act. overlap .28 *** .06 Smoke similarity .68 *** .12 Smoke alter .14 ** .05 Smoke ego -.04 .05 Female similarity .24 *** .04 Female alter -.11 * .05 Female ego -.04 .05 Age similarity 1.00 *** .13 Age alter -.01 .03 Age ego -.04 .03 Delinquency similarity .15 .08 Delinquency alter -.04 .04 Delinquency ego .02 .04 Alcohol similarity .27 ** .10 Alcohol alter -.03 .03 Alcohol ego -.03 .04 GPA similarity .70 *** .13 GPA alter -.05 .04 GPA ego -.02 .04 From Schaefer, D.R. S.A. Haas, and N. Bishop. 2012. “A Dynamic Model of US Adolescents’ Smoking and Friendship Networks.” American Journal of Public Health, 102:e12-e18. Ties driven by similarity on smoking behavior. Smokers more adrac+ve as friends than non-smokers. Alter Nonsmoker Smoker Ego Nonsmoker .25 -.19 Smoker -.51 .41 Similarity is an “interac+on” between ego and alter, thus interpreta+on requires considering the main effects Ego-alter selec+on: Contribu+ons to network objec+ve func+on by dyad type May 20, 2016 Duke Social Networks & Health Workshop 40
  41. 41. From Schaefer, D.R. S.A. Haas, and N. Bishop. 2012. “A Dynamic Model of US Adolescents’ Smoking and Friendship Networks.” American Journal of Public Health, 102:e12-e18. Smoking func?on b SE Rate 2.06 *** .26 Linear shape -.11 .22 Quadra+c shape 1.17 *** .16 Female .16 .19 Age -.00 .10 Parent Smoking .01 .23 Delinquency .44 ** .16 Alcohol -.10 .14 GPA -.09 .13 Average similarity 2.89 *** .91 In-degree -.04 .11 In-degree squared .00 .01 May 20, 2016 Duke Social Networks & Health Workshop 41 Rate: Students have around 2 chances on average (micro steps) to change their smoking behavior Rate
  42. 42. From Schaefer, D.R. S.A. Haas, and N. Bishop. 2012. “A Dynamic Model of US Adolescents’ Smoking and Friendship Networks.” American Journal of Public Health, 102:e12-e18. Smoking func?on b SE Rate 2.06 *** .26 Linear shape -.11 .22 Quadra+c shape 1.17 *** .16 Female .16 .19 Age -.00 .10 Parent Smoking .01 .23 Delinquency .44 ** .16 Alcohol -.10 .14 GPA -.09 .13 Average similarity 2.89 *** .91 In-degree -.04 .11 In-degree squared .00 .01 May 20, 2016 Duke Social Networks & Health Workshop 42 linear quad
  43. 43. From Schaefer, D.R. S.A. Haas, and N. Bishop. 2012. “A Dynamic Model of US Adolescents’ Smoking and Friendship Networks.” American Journal of Public Health, 102:e12-e18. Smoking func?on b SE Rate 2.06 *** .26 Linear shape -.11 .22 Quadra+c shape 1.17 *** .16 Female .16 .19 Age -.00 .10 Parent Smoking .01 .23 Delinquency .44 ** .16 Alcohol -.10 .14 GPA -.09 .13 Average similarity 2.89 *** .91 In-degree -.04 .11 In-degree squared .00 .01 May 20, 2016 Duke Social Networks & Health Workshop 43 Smoking (z, M=.9) Linear Quad Raw Centered b = -.11 b = 1.17 Sum 0 -.90 .099 .948 1.047 1 .10 -.011 .012 .001 2 1.10 -.121 1.416 1.295 Smoking Level Summed Effects In combina+on, the linear and quad effects represent the U- shaped smoking distribu+on. •  Kids either don’t smoke or smoke 12+ days/month. .0 .2 .4 .6 .8 1.0 1.2 1.4 0 1 2 Contribu?on to Behavior Func?on + = + = + =
  44. 44. From Schaefer, D.R. S.A. Haas, and N. Bishop. 2012. “A Dynamic Model of US Adolescents’ Smoking and Friendship Networks.” American Journal of Public Health, 102:e12-e18. Smoking func?on b SE Rate 2.06 *** .26 Linear shape -.11 .22 Quadra+c shape 1.17 *** .16 Female .16 .19 Age -.00 .10 Parent Smoking .01 .23 Delinquency .44 ** .16 Alcohol -.10 .14 GPA -.09 .13 Average similarity 2.89 *** .91 In-degree -.04 .11 In-degree squared .00 .01 Ego Covariate: Delinquency leads to higher levels of smoking May 20, 2016 Duke Social Networks & Health Workshop 44 effFrom
  45. 45. From Schaefer, D.R. S.A. Haas, and N. Bishop. 2012. “A Dynamic Model of US Adolescents’ Smoking and Friendship Networks.” American Journal of Public Health, 102:e12-e18. Smoking func?on b SE Rate 2.06 *** .26 Linear shape -.11 .22 Quadra+c shape 1.17 *** .16 Female .16 .19 Age -.00 .10 Parent Smoking .01 .23 Delinquency .44 ** .16 Alcohol -.10 .14 GPA -.09 .13 Average similarity 2.89 *** .91 In-degree -.04 .11 In-degree squared .00 .01 Average Similarity: Students adopt smoking levels that bring them closer to the average of their friends xi+ −1 xijj ∑ (simij z − simz ) Δ −−Δ = ji ij zz sim jiij zz −=Δ max May 20, 2016 Duke Social Networks & Health Workshop 45 avSim
  46. 46. •  How well is the es+mated model able to reproduce features of the observed data that were not explicitly modeled? –  Network •  Degree distribu+on •  Geodesic distribu+on •  Triad census –  Behavior distribu?on Lots of room to improve GOF measures, especially behavior May 20, 2016 Duke Social Networks & Health Workshop 46 Goodness of Fit (GOF)
  47. 47. Cumula?ve Indegree Distribu?on Goodness of Fit of IndegreeDistribution p: 0 Statistic 0 1 2 3 4 5 6 7 8 139 193 282 343 401 437 459 483 491 May 20, 2016 Duke Social Networks & Health Workshop 47
  48. 48. Geodesic Distribu?on Goodness of Fit of GeodesicDistribution p: 0.001 Statistic 1 2 3 4 5 6 7 1381 2795 5014 7772 10598 12081 11892 May 20, 2016 Duke Social Networks & Health Workshop 48
  49. 49. Triad Census Goodness of Fit of TriadCensus p: 0.114 Statistic(centeredandscaled) 003 012 102 021D 021U 021C 111D 111U 030T 030C 201 120D 120U 120C 210 300 21286492 428358 129429 693 1141 1052 923 625 108 4 171 114 58 39 91 36 May 20, 2016 Duke Social Networks & Health Workshop 49
  50. 50. Smoking Distribu?on Goodness of Fit of BehaviorDistribution p: 1 Statistic 0 1 2 222 98 182 May 20, 2016 Duke Social Networks & Health Workshop 50
  51. 51. 4. Extensions & Miscellany May 20, 2016 Duke Social Networks & Health Workshop 51
  52. 52. Extensions to Basic Model May 20, 2016 Duke Social Networks & Health Workshop 52 •  interac+ons •  event history outcomes •  mul+ple behaviors •  mul+ple network op+ons •  valued +es •  mul+level networks •  two mode networks •  increase vs. decrease in +es and/or behavior •  +me heterogeneity •  simula+ons (test interven+ons) •  ML, Bayes es+ma+on
  53. 53. Asymmetric Peer Influence •  Implicit assump+on that effects work the same for: –  Tie forma+on vs. dissolu+on –  Behavior increase vs. decrease •  Unrealis+c for smoking –  Physical/psychological dependence, social learning •  Easy to relax this assump+on –  Separate behavior objec+ve func+on into: •  Crea?on func?on: only considers increases •  Maintenance func?on: only considers decreases –  Could make similar dis+nc+on in the network func+on May 20, 2016 Duke Social Networks & Health Workshop 53
  54. 54. Contribu?ons to the Smoking Func?on Contribu+on Prospec+ve Smoking Nonsmoking Alters J = Jefferson High School S = Sunshine High School From Haas, Steven A. and David R. Schaefer. 2014. “With a Lidle Help from My Friends? Asymmetrical Social Influence on Adolescent Smoking Ini+a+on and Cessa+on.” Journal of Health and Social Behavior, 55:126-143. Smoking level with greatest contribu+on most likely to be adopted (with caveat that actors can only move behavior one level during a given micro step) -3-113 Current Smoking Util. 0 1 2 J J J S S S A -3-113 Current Smoking Util. 0 1 2 J J J S S S B -3-113 Current Smoking Util. 0 1 2 J J J S S S C -3-113 Util. J J J S S S D -3-113 Util. J J J S S S E -3-113 Util. J J J S S S F Contribu+on Prospec+ve Smoking Smoking Alters -3-11 Current Smoking Util. 0 1 2 J J J S S S -3-11 Current Smoking Util. 0 1 2 J J J S S S -3-11 Util. -3-113 Util. 0 1 2 J J J S S S G -3-113 Util. 0 1 2 J J J S S S H -3-113 Util. Ego is currently a moderate smoker (1) May 20, 2016 Duke Social Networks & Health Workshop 54
  55. 55. SIENA as an ABM •  Useful to evaluate goodness-of-fit, decompose network- behavior associa+ons, evaluate interven+ons •  Uses the same algorithm as model fimng 1.  Fit model to empirical data (op+onal) 2.  Simulate network evolu+on using es+mated parameters or manipula+ons of them •  Can also manipulate ini+al condi+ons (e.g., network structure, behavior distribu+on, etc.) 3.  Measure simulated network/behavior proper+es of interest May 20, 2016 Duke Social Networks & Health Workshop 55
  56. 56. Decomposing Network Homogeneity Source Selec?on (%) Influence (%) Sample Schaefer et al. 2012 40 34 U.S. Mercken et al. 2009 17-47 6-23 Europe (6 countries) Mercken et al. 2010 31-46 15-22 Finland Steglich et al. 2010 25-34 20-37 Scotland •  How much network homogeneity on smoking is due to selec?on vs. influence? –  Systema+cally set selec+on and influence parameters to zero and simulate network-behavior co-evolu+on (see Steglich et al. 2010) May 20, 2016 Duke Social Networks & Health Workshop 56
  57. 57. Evalua?ng Interven?ons How do smoking/friendship dynamics affect smoking prevalence? •  Manipulate model parameters related to key “interven+on levers” –  Peer influence (absent…strong) –  Smoker popularity (unpopular…absent…popular) •  Remaining model parameters from fided model •  Ini+al condi+ons = observed wave 1 data May 20, 2016 Duke Social Networks & Health Workshop 57
  58. 58. Results of Manipula?ng Peer Influence (PI) and Smoking-based Popularity (smoke alter) Schaefer DR, adams j, Haas SA. 2013. Social Networks and Smoking: Exploring the Effects of Peer Influence and Smoker Popularity through Simula+ons. Health Educa'on & Behavior, 40(S1):24-32. May 20, 2016 Duke Social Networks & Health Workshop 58 Independent Manipula+ons Joint Manipula+on Stronger peer influence increases smoking prevalence, but only when smokers are popular (nega+ve effects when smokers unpopular)
  59. 59. Context Effects How do these effects depend upon context? •  Randomly manipulate ini+al smoking prevalence –  25% ini+al smokers up to 75% •  Randomly distribute smokers and nonsmokers across the network –  Similar results with empirical and model-based manipula+ons •  Full results in adams, jimi & David R. Schaefer. 2016. “How Ini+al Prevalence Moderates Network-Based Smoking Change: Es+ma+ng Contextual Effects with Stochas+c Actor Based Models.” Journal of Health & Social Behavior 57(1): 22-38. May 20, 2016 Duke Social Networks & Health Workshop 59
  60. 60. May 20, 2016 Duke Social Networks & Health Workshop 60 Smoking Distribu+on: Empirically-Based, Model-Based, Random
  61. 61. May 20, 2016 Duke Social Networks & Health Workshop 61 PI Parameter0 1 2 3 4 5 6 SmokeAlterParameter -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 ChangeinSmokers -0.2 -0.1 0.0 0.1 0.2 25% Ini+al Smokers 75% Ini+al Smokers PI Parameter0 1 2 3 4 5 6 SmokeAlterParameter -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 ChangeinSmokers -0.2 -0.1 0.0 0.1 0.2 Contras?ng Contexts
  62. 62. •  Ties are more or less enduring states –  Plausible for friendship or collabora+ons –  Not useful for “event” data (e.g. phone calls) •  Change occurs in con+nuous +me •  Markov process: future state only a func+on of current state –  No lagged effects, “grudges” •  Actors control outgoing +es and behavior •  One change at a +me –  No coordinated or simultaneous changes May 20, 2016 Duke Social Networks & Health Workshop 62 Assump?ons
  63. 63. •  Up to 10% probably ok, more than 20% likely a problem •  Endogenous network & behavior imputa+on –  Missing values at t0 set to 0 (network) or mode (behavior) –  Missing values at t1+ imputed with last valid value if possible, otherwise 0 •  Covariates imputed with the mean –  Other values can be specified •  Imputed values are treated as non-informa+ve, thus not used in calcula+ng target sta+s+cs –  Convergence and fit are determined based only upon observed cases May 20, 2016 Duke Social Networks & Health Workshop 63 Missing Data
  64. 64. Good Sources of Informa?on May 20, 2016 Duke Social Networks & Health Workshop 64 •  RSiena manual •  Snijders, van de Bunt & Steglich, 2010 •  Steglich, Snijders & Pearson, 2010 •  Tom Snijders’ SIENA website www.stats.ox.ac.uk/siena/ –  Workshops –  Scripts –  Applica+ons in the literature –  Latest version of RSiena –  Link to stocnet listserv – important updates announced here –  “Siena_algorithms.pdf”
  65. 65. End of Lecture May 20, 2016 Duke Social Networks & Health Workshop 65
  66. 66. SAOM Lab If you haven’t done so already: •  Download the “RSiena lab.R” script from dropbox •  Install the RSiena library – See “RSiena lab.R” sec+on 1 or – Type: install.packages("RSiena”) May 20, 2016 Duke Social Networks & Health Workshop 66
  67. 67. •  One mode or two mode network with at least two observa+ons, each represented as a matrix –  Ties coded 0, 1, 10 (structural 0), 11 (structural 1), or NA •  For each “period” between adjacent waves, stability measured by the Jaccard coefficient should be at least .25 –  Ties persisted / (+es formed + +es dissolved + +es persisted) •  “Complete network data” all actors w/in bounded semng –  Some turnover in set of actors allowed but same actors in the data for each wave (even if not observed during wave) –  See manual for how to deal with composi+on change •  Recommended N: 30-2000 May 20, 2016 Duke Social Networks & Health Workshop 67 Data Structure: Network
  68. 68. •  Dependent behaviors –  Time-varying adributes used as dependent variable(s) –  Coded as integer (e.g., 1-10) –  Last +me point is used •  Changing actor covariates –  Time-varying adributes used as independent variables –  Last +me point not used (only applicable for 3+ waves) •  Constant covariates –  Ex: age, sex, race/ethnicity, behavior •  Dyadic covariates –  Ex: semngs that drive contact NOTE: Covariates are centered by default May 20, 2016 Duke Social Networks & Health Workshop 68 Addi?onal Data Structures

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