Hierarchical Models for Social Networks

Transcription

1 Hierarchical Models for Social Networks Tracy M. Sweet University of Maryland Innovative Assessment Collaboration November 4, 2014 Acknowledgements Program for Interdisciplinary Education Research (PIER) supported by IES Predoctoral Interdisciplinary Research Training Programs in the Education Sciences Grant R305B Hierarchical Network Models for Education Research supported by IES Statistical and Research Methodology in Education Grant R305D

2 Outline 1 Motivation 2 Estimation of Node/Dyad Attributes Effect of Teaching the Same Grade with HLSM 3 Estimation of Network Attributes Effect of Network Covariate with HMMSBM 2

3 Motivation Social Networks in Education Independent network replications Teacher networks within schools (Frank et al., 2004; Moolenaar et al., 2010; Spillane et al., 2012; Weinbaum et al., 2008) Student networks within classes/schools (Gest and Rodkin, 2011; Harris et al., 2008) 3

4 Motivation Social Networks in Education Independent network replications Teacher networks within schools (Frank et al., 2004; Moolenaar et al., 2010; Spillane et al., 2012; Weinbaum et al., 2008) Student networks within classes/schools (Gest and Rodkin, 2011; Harris et al., 2008) How can we accommodate multiple networks using social network models? Modeling multiple networks simultaneously Estimating treatment effects 3

5 Hierarchical Network Models (Sweet et al., 2013, JEBS) POPULATION Θ 1 Θ K Θ 2... Θ K-1 4

6 Hierarchical Network Models K P(Y X,Θ) = P(Y k X k = (X 1k,..., X Pk ), Θ k = (θ 1k,..., θ Qk )) k=1 (Θ 1,..., Θ K ) F (Θ 1,..., Θ K W 1,..., W K, ψ), P(Y k X k, Θ k ) is a model for a single network Y k with covariates X k and parameters Θ k W k can model a variety of dependence assumptions across networks ψ may specify additional hierarchical structure on Θ 5

7 Hierarchical Latent Space Model (HLSM) Latent Social Space Z 4 Z 3 Z 2 Z 1 Z 5 Z Network Level 1 Level 2 logit P[Y ijk = 1] = β 0k + β 1k X ijk Z ik Z jk β 0k N(µ 0, σ 2 0 ) β 1k N(µ 1, σ 2 1 ) X ijk is a node-, tie-, or network- level covariate Z ik is the latent space position for actor i in network k 6

8 An Example Teacher Advice Network Data (Pitts and Spillane, 2009) 15 Elementary/K-8 schools 2D latent space positions Network size of teachers Coded model fitting algorithm (MCMC) in R Fitting the Model Level 1 logit P[Y ijk = 1] = β 0k + β 1k X ijk Z ik Z jk Y ijk = 1 if teacher i asked j for advice X ijk = 1 if teachers i and j teach the same grade β 1k is the effect of teaching the same grade for school k 7

9 Level 1 Parameters β 1 : Effect of Teaching the Same Grade β Separate Network Models * * * * HLSM * β School 8

10 Level 2 (µ 1, σ 1 2 ) Effect of Teaching the Same Grade Fitting the Model Level 1 logit P[Y ijk = 1] = β 0k + β 1k X ijk Z ik Z jk Level 2 β 0k N(µ 0, σ 2 0) β 1k N(µ 1, σ 2 1) µ 1 is the mean effect of teaching the same grade σ 2 1 is the variance for the effect of teaching the same grade µ σ

11 Mixed Membership Stochastic Blockmodels Models are for networks with subgroup structure Each individual belongs to one of n groups with some probability Probability of a tie within subgroups is much higher than between subgroups Typical blockmodels assume each individual belongs to one subgroup Mixed membership allows individuals to belong to multiple subgroups 10

12 Mixed Membership in Networks Network from Blockmodel Network from MM Blockmodel 11

13 Mixed Membership Stochastic Blockmodel (Airoldi et al., 2008) T Y ij Ber (S ij BR ji ) S ij Multi (θ i ) R ji Multi (θ j ) θ i Dir (ξγ) B em Beta (a em, b em ) B sr is the probability of a tie from group s to group r S ij is the group membership of node i when sending a tie to person j R ji is the group membership of node j when receiving a tie from person i θ i is the membership probability vector for person i Small values of γ generate extreme the membership probabilities 12

14 γ: the Amount of Mixed Membership , 3, 3 Figure : ξ = { }, γ = 0.09 Figure : ξ = {,, 3 }, γ =

15 I Higher e ciency in educational research Is the Amount of Mixing Related to Network Attributes? I Make multiple networks comparable Figure: Selected 5th Grade Cognitive Friendship Networks 14

16 The Hierarchical Mixed Membership Stochastic Blockmodel with Network-level Covariates T Y ijk Ber (S ijk B k R jik ) S ijk Multi (θ ik ) R jik Multi (θ jk ) θ ik Dir (ξ k γ k ) γ k = exp (β ' X k ) B emk Beta (a emk, b emk ) ξ k Dir (1) β MVN(µ, σ 2 ) X = (X 1,..., X K ) is a set of network-level covariates 15

17 Real Data Analysis with 5th Grade Friendship Networks (Gest and Rodkin, 2011) Data: Cognitive friendship adjacency matrices in 5th grade, 17 classrooms Average classroom size (network nodes): 20 Teacher Management Hypothesis: Teachers with higher social management have friendship networks with less isolated subgroups Model: 4-group HMMSBM with Network-level Covariates 16

18 Posterior Density: β Effect of Cognitive Facilitation Observed Cognitive Facilitation β = β Significantly positive β: Teachers with higher Cognitive Facilitation have friendship networks with higher probability of mixing 17

19 Positive Relationship between Teacher Management and Mixed Membership CF = 0.56 CF = 0.03 CF =

20 Airoldi, E., Blei, D., Fienberg, S., and Xing, E. (2008), Mixed membership stochastic blockmodels, The Journal of Machine Learning Research, 9, Bradley, R. and Gart, J. (1962), The asymptotic properties of ML estimators when sampling from associated populations, Biometrika, 49, Cramér, H. (1946), Mathematical methods of statistics, Princeton, NJ: Princeton University Press. Daly, A., Moolenaar, N., Bolivar, J., and Burke, P. (2010), Relationships in reform: The role of teachers social networks, Journal of educational administration, 48, Ennett, S. and Bauman, K. (1993), Peer group structure and adolescent cigarette smoking: a social network analysis, Journal of Health and Social Behavior, Espelage, D. and Low, S. (2009), Multi-site Evaluation of Second Step: Student Success Through Prevention (Second Step SSTP) in Preventing Bullying and Sexual Violence, CDC Grant. Frank, K. A., Zhao, Y., and Borman, K. (2004), Social Capital and the Diffusion of Innovations Within Organizations: The Case of Computer Technology in Schools, Sociology of Education, 77, Gest, S. D. and Rodkin, P. C. (2011), Teaching practices and elementary classroom peer ecologies, Journal of Applied Developmental Psychology, 32, Harris, K. M., Florey, F., Tabor, J., Bearman, P. S., Jones, J., Udry, J. R., of Adolescent Health, N. L. S., et al. (2008), Research design, Carolina Population Center, University of North Carolina at Chapel Hill. Hoff, P. D., Raftery, A. E., and Handcock, M. S. (2002), Latent Space Approaches to Social Network Analysis, Journal of the American Statistical Association, 97, Krackhardt, D. and Porter, L. (1986), The snowball effect: Turnover embedded in communication networks. Journal of Applied Psychology; Journal of Applied Psychology, 71, 50. Lawrence, J., Sweet, T., and Junker, B. (2014), Professional learning communities and the adoption of novel instructional practices: An evaluation of the School Reform Initiative, IES Grant Proposal. Lewis, K., Kaufman, J., Gonzalez, M., Wimmer, A., and Christakis, N. (2008), Tastes, ties, and time: A new social network dataset using Facebook. com, Social Networks, 30, Moolenaar, N., Daly, A., and Sleegers, P. (2010), Occupying the principal position: Examining relationships between transformational leadership, social network position, and schools innovative climate, Educational Administration Quarterly, 46,

21 Pitts, V. and Spillane, J. (2009), Using social network methods to study school leadership, International Journal of Research & Method in Education, 32, Smith, D. and White, D. (1992), Structure and dynamics of the global economy: Network analysis of international trade , Social Forces, 70, Snijders, T. and Nowicki, K. (1997), Estimation and prediction for stochastic blockmodels for graphs with latent block structure, Journal of Classification, 14, Spillane, J., Kim, C., and Frank, K. (2011), Instructional Advice and Information Providing and Receiving Behavior in Elementary Schools: Exploring Tie Formation as a Building Block in Social Capital Development, Institute for Policy Research Northwester University Working Paper Series. Spillane, J. P. and Hopkins, M. (2013), Organizing for instruction in education systems and school organizations: how the subject matters, Journal of Curriculum Studies, 45, Spillane, J. P., Kim, C. M., and Frank, K. A. (2012), Instructional Advice and Information Providing and Receiving Behavior in Elementary Schools Exploring Tie Formation as a Building Block in Social Capital Development, American Educational Research Journal. Sweet, T. M., Thomas, A. C., and Junker, B. W. (2013), Hierarchical Network Models for Education Research: Hierarchical Latent Space Models, Journal of Educational and Behavioral Statistics, 38, Wasserman, S. and Pattison, P. (1996), Logit models and logistic regressions for social networks: I. An introduction to Markov graphs and p*, Psychometrika, 61, , /BF Weinbaum, E., Cole, R., Weiss, M., and Supovitz, J. (2008), Going with the flow: Communication and reform in high schools, in The implementation gap: understanding reform in high schools, eds. Supovitz, J. and Weinbaum, E., Teachers College Press, pp

22 For More Information: or Current Work HLSMs for covariate effects and interventions Power Analysis for HLSMs Blockmodels and Mixed Membership Other ways to incorporate covariates into blockmodels HNMs for Mediation and Influence Future Work Develop new models and extensions (longitudinal models) Operating characteristics Identifiability and estimation issues Applications with educational data Valued ties 21

23 HLSM for Influence logit P[Y ij = 1] = β t X ij d(z i, Z j ) W i = φ 0 W i t + φn(z ) i W i t + ν + E i W i is the covariate for node i at time 2 φ denotes the effect of the network W ; i is the covariate for node i at time 1 N(Z ) i is the weight of the network distances 1

24 Extra Slides 2

25 Fix Group Number in the Model across Networks 24 nodes in each network. True γ values are the same for all networks Fit all networks with 4-group MMSBM. γ^ = 0.07 γ^ = 0.09 γ^ = 0.13 Figure : 1 Figure : 2 Figure : 3 increased by 29% increased by 44% γˆ = 0.07 γˆ = 0.09 γˆ =

26 Fitting the HLSM for Interventions Fitted Model logit P[Y ijk = 1] = β 0 + β 1k X ijk Z ik Z jk + αt k Z ik MVN, β 0 N(0, 100) β 1k N(0, 100) α N(0, 100) X ijk is the indicator that teacher i and j in network k teach the same grade Coded model fitting algorithm (MCMC) in R 4

27 Fitting a HLSM logit P[Y ijk = 1] =β 0k + β 1k X 1ijk Z ik Z jk, 0 a 0 Z ik MVN,, i = 1,..., n k, 0 0 a β 0k N(µ 0, σ 0 2 ), k = 1,..., K, β 1k N(µ 1, σ 1 2 ), k = 1,..., K, µ 0 N(b 1, b 2 ), µ 1 N(c 1, c 2 ), σ 0 2 Inv Gamma(d 1, d 2 ), σ 1 2 Inv Gamma(e 1, e 2 ), X ijk is the indicator that teacher i and j in network k teach the same grade 5

28 Mixed Membership Stochastic Blockmodels Models are for networks with subgroup structure Each individual belongs to one of n groups with some probability Probability of a tie within subgroups is much higher than between subgroups Typical blockmodels assume each individual belongs to one subgroup Mixed membership allows individuals to belong to multiple subgroups 6

29 The Hierarchical Mixed Membership Stochastic Blockmodel (HMMSBM) Y ijk T Bernoulli(S ijk B k R jik ) S ijk Multinomial(θ ik, 1) R jik Multinomial(θ jk, 1) θ ik Dirichlet(λ k ) B emk Beta(a emk, b emk ) B k [S, R] is the probability of a tie from group S to group R in network k S ijk is the group membership of person i when sending a tie to person j R jik is the group membership of person j when receiving a tie from person i θ isk is the probability that person i in network k belongs to group s Small values of λ k generate extreme the membership probabilities 7

30 A HMMSBM for an Intervention Suppose an intervention is hypothesized to change isolated subgroups... P(Y ijk = 1) = B k [S ijk, R jik ] P(S ijk = s) = θ isk P(R jik = r) = θ jrk θ ik Dirichlet(λ k ) λ k = λ 0 + T k (l1 λ 0 )(1 α) T k is treatment indicator l1 is (1,...1) with length equal to number of groups α is the treatment effect λ 0 generates the control group s membership probabilities 8

31 Simulated Data from 20 Networks of Size 20; α = 0.53 θ ik Dirichlet(λ k ) λ k = λ 0 + T k (l1 λ 0)(1 α) λ k = T k Control Treatment 9

32 A HMMSBM for an Intervention Fitted Model Y ijk Bernoulli(B k [S ijk, R jik ]) S ijk Multinomial(θ ik, 1) R jik Multinomial(θ jk, 1) θ ik Dirichlet(λ k ) B eek Beta(3, 1) B emk Beta(1, 10), = m λ k = λ 0 + T k ( 1 λ 0 )(1 α) α Uniform(0, 1) Model fit using MCMC algorithm coded in R λ 0 = subgroups assumed a priori 10

33 Treatment Effect Parameter (α) Recovery α

34 Tie Probability Recovery