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Missing data in social networks - Problems and prospects for model-based inference Johan Koskinen The Social Statistics Discipline Area, School of Social Sciences Mitchell Centre for Network Analysis Tuesday, 20 December

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A relational perspective – networks matter Vegetarian partner x Ethical Economics Health Taste Dr D eats (predominantly) vegetarian food... Dr Dean Lushers relational take

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A relational perspective – networks matter Someone close to you is unhappy will you remain unaffected?

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A relational perspective – networks matter Equal opportunities based on our individual qualities......

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A relational perspective – networks matter... bowl alone others bowl in leagues Some people bowl alone

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Part 1 Network representations

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Social networks mary paul We conceive of a network as a Relation defined on a collection of individuals relates to … go to for advice…

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Social networks mary paul We conceive of a network as a Relation defined on a collection of individuals relates to … consider a friend…

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Social networks mary paul We conceive of a network as a Relation defined on a collection of individuals relates to on off Generally binary Tie present Tie absent

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Network representations A non-directed graph A social network of tertiary students – Kalish (2003)

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Network representations

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Police training squad: Confiding network (Pane, 2003)

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Network representations World Trade in 1992 Plümper, 2003, JOSS

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Network representations: attributes The actors (nodes) in the network are individuals with –attitudes, behaviours, and attributes These may –guide them in their choices of partners –be shaped (influenced) by their partners The actors may have individual and collective outcomes

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Network representations: attributes A non-directed graph A social network of tertiary students – Kalish (2003)

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Network representations: attributes A non-directed graph A social network of tertiary students – Kalish (2003) JewishArab

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Network representations: attributes High School friendship, Moody, 2001 white black other

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Network representations: attributes Romantic/sexual relationships at a US high school (Bearman, Moody & Stovel, 2004) Guess the blue and pink

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Network representations: attributes detachedteam orientedpositive Team structures in training squads (Pane, 2003) (friendship network in 12 th week of training)

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Multiple relations – entrailment, exchange, and generalized exchange Physical violence Violence & attitudes among school boys (Lusher, 2003)

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Social networks We conceive of the Graph as a collection of Tie variables: {X ij : i,j V} john pete mary paul i - x ij x ik x il jx ji -x jl kx ki x kj - x kl lx li x lj x lk - x = i j k l =

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Social networks We conceive of the Graph as a collection of Tie variables: {X ij : i,j V} i - x ij x ik x il jx ji -x jl kx ki x kj - x kl lx li x lj x lk - x = i j k l = l i j k

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Social networks The Adjacency matrix: The matrix of the collection Tie var. {X ij : i,j V} i - x ij x ik x il jx ji -x jl kx ki x kj - x kl lx li x lj x lk - x =

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Social networks: adjacency matrix Read Highland tribes

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Social networks: adjacency matrix Read Highland tribes

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Social networks: adjacency matrix Read Highland tribes

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Social networks: adjacency matrix Read Highland tribes Symmetric for a non-directed network

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Social networks: adjacency matrix Read Highland tribes

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Social networks: adjacency matrix Read Highland tribes Zeroes along the diagonal – self ties not permitted

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Part 2 Analysing social networks – Putting the building blocks of networks together using ERGM

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Do we need to analyse networks? -Is the network a unique narrative? -stick to an ethnography? Possible answers -Detecting systematic tendencies -Social mechanisms -lift the description to describe network in generalizable terms

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Networks matter – ERGMS matter 6018 grade 6 children 1966 FEMALE Male

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Networks matter – ERGMS matter 6018 grade 6 children 1966 – 300 schools Stockholm

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Networks matter – ERGMS matter 6018 grade 6 children 1966 – 200 schools Stockholm Koskinen and Stenberg (in press) JEBS

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Networks matter – ERGMS matter 6018 grade 6 children 1966 – 200 schools Stockholm Koskinen and Stenberg (in press) JEBS

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Networks matter – ERGMS matter 6018 grade 6 children 1966 – 200 schools Stockholm Koskinen and Stenberg (in press) JEBS

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Networks matter – ERGMS matter 6018 grade 6 children 1966 – 200 schools Stockholm Koskinen and Stenberg (in press) JEBS

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Do we need to analyse networks? -Is the network a unique narrative? -stick to an ethnography? Possible answers -Detecting systematic tendencies -Social mechanisms -lift the description to describe network in generalizable terms Conceptualising the network as a graph is what enables this

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ERGMS – modelling graphs

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ERGMS – modelling graphs: example Marriage network of Padgetts Florentine families

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ERGMS – modelling graphs: example Marriage network of Padgetts Florentine families Model this as combination of 4 local structures Their importance measured by their parameters

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ERGMS – modelling graphs: example effectMLES.E. Edge star star Triangle

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ERGMS – modelling graphs: example effectMLES.E. Edge star star Triangle

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Part 3 Modelling graphs – deriving building blocks out of dependencies

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Independence - Deriving the ERGM l i j k m n heads tails l i l i heads tails i k i k

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Independence - Deriving the ERGM 0.25 AUD 0.5 SEK 0.5 l i k l i k Knowledge of AUD, e.g. does not help us predict SEK e.g. whetheror

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Independence - Deriving the ERGM i i k Knowledge of AUD, e.g.does not help us predict SEK e.g. whetheror even though dyad { i,l } l i and dyad { i,k } have vertex i in common

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Independence - Deriving the ERGM AUD 0.5 SEK 0.5 l i k l i k May we find model such that knowledge of AUD, e.g. does help us predict SEK e.g. whetheror?

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Deriving the ERGM: From Markov graph to Dependence graph john pete mary paul Consider the tie-variables that have Mary in common How may we make these dependent?

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Deriving the ERGM: From Markov graph to Dependence graph john pete mary paul pete mary

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Deriving the ERGM: From Markov graph to Dependence graph john pete mary paul john pete mary

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Deriving the ERGM: From Markov graph to Dependence graph john pete mary paul john pete mary paul mary

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Deriving the ERGM: From Markov graph to Dependence graph john pete mary paul john pete mary paul mary pete john

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Deriving the ERGM: From Markov graph to Dependence graph john pete mary paul john pete mary paul mary paul john pete john

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Deriving the ERGM: From Markov graph to Dependence graph john pete mary paul john pete mary paul mary paul pete paul john pete john

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Deriving the ERGM: From Markov graph to Dependence graph john pete mary paul m,pa pa,pe pa,j m,pe pe,j m,j The probability structure of a Markov graph is described by cliques of the dependence graph (Hammersley-Clifford)….

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Deriving the ERGM: From Markov graph to Dependence graph pete mary paul m,pa pa,pe pa,j m,pe pe,j m,j

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Deriving the ERGM: From Markov graph to Dependence graph john pete mary paul m,pe

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Deriving the ERGM: From Markov graph to Dependence graph john pete mary paul m,pa pa,pe pa,j m,pe pe,j m,j

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Deriving the ERGM: From Markov graph to Dependence graph mary john pete paul m,pa pa,pe pa,j m,pe pe,j m,j

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Deriving the ERGM: From Markov graph to Dependence graph mary john pete paul m,pe m,j

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Deriving the ERGM: From Markov graph to Dependence graph mary john pete paul m,pa pa,pe pa,j m,pe pe,j m,j

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Deriving the ERGM: From Markov graph to Dependence graph mary john pete paul m,pa pa,pe pa,j m,pe pe,j m,j

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Deriving the ERGM: From Markov graph to Dependence graph mary john pete paul m,pe pe,j m,j

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Deriving the ERGM: From Markov graph to Dependence graph mary john pete paul m,pa pa,pe pa,j m,pe pe,j m,j

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Deriving the ERGM: From Markov graph to Dependence graph mary john pete paul m,pa pa,pe pa,j m,pe pe,j m,j

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Deriving the ERGM: From Markov graph to Dependence graph mary john pete paul m,pa pa,j m,pe pe,j

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Deriving the ERGM: From Markov graph to Dependence graph mary john pete paul m,pa pa,pe pa,j m,pe pe,j m,j

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From Markov graph to Dependence graph – distinct subgraphs? too many statistics (parameters)

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The homogeneity assumption = = = =

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A log-linear model (ERGM) for ties Aggregated to a joint model for entire adjacency matrix Interaction terms in log-linear model of types

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A log-linear model (ERGM) for ties By definition of (in-) dependence E.g. andco-occuring i j i j k i k Main effectsinteraction term More than is explained by margins

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Part 4 Estimation of ERGM

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Likelihood equations for exponential fam Aggregated to a joint model for entire adjacency matrix X Sum over all 2 n(n-1)/2 graphs The MLE solves the equation (cf. Lehmann, 1983):

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Likelihood equations for exponential fam Solving Using the cumulant generating function (Corander, Dahmström, and Dahmström, 1998) Stochastic approximation (Snijders, 2002, based on Robbins-Monro, 1951) Importance sampling (Handcock, 2003; Hunter and Handcock, 2006, based on Geyer- Thompson 1992)

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Robbins-Monro algorithm Solving Snijders, 2002, algorithm - Initialisation phase - Main estimation - convergence check and cal. of standard errors MAIN: Draw using MCMC

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Geyer-Thompson Solving Handcock, 2003, approximate Fisher scoring MAIN: Approximated using importance sample from MCMC

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Bayes: dealing with likelihood The normalising constant of the posterior not essential for Bayesian inference, all we need is: … but Sum over all 2 n(n-1)/2 graphs

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Bayes: MCMC? Consequently, in e.g. Metropolis-Hastings, acceptance probability of move to θ … which contains

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Bayes: Linked Importance Sampler Auxiliary Variable MCMC LISA (Koskinen, 2008; Koskinen, Robins & Pattison, 2010): Based on Møller et al. (2006), we define an auxiliary variable And produce draws from the joint posterior using the proposal distributions and

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Bayes: alternative auxiliary variable LISA (Koskinen, 2008; Koskinen, Robins & Pattison, 2010): Based on Møller et al. (2006), we define an auxiliary variable Improvement: use exchange algorithm (Murray et al. 2006) Many linked chains: - Computation time - storage (memory and time issues) and Accept θ* with log-probability: Caimo & Friel, 2011

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Bayes: Implications of using alternative auxiliary variable Improvement: use exchange algorithm (Murray et al. 2006) and Accept θ* with log-probability: Caimo & Friel, 2011 Storing only parameters No pre tuning – no need for good initial values Standard MCMC properties of sampler Less sensitive to near degeneracy in estimation Easier than anything else to implement QUICK and ROBUST

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Part 5 Types of missing data

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Sampling in/on networks

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missing data observed data Sampling in/on networks

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= 0 x = - 1 1

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Sampling in/on networks = x =

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Sampling in/on networks = x =

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Sampling in/on networks = x =

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Sampling in/on networks = x =

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Sampling in/on networks = ? 0 ? 0 ? ? ? ? x = ? ? ? ? ? ? - 1 ? ? 1 -

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Ignoring non-sampled? = ? 0 ? 0 ? ? ? ? x = ? ? ? ? ? ? - 1 ? ? 1 -

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What about alter – alter across ego? = ? 0 ? 0 ? ? ? ? x = ? ? ? ? ? ? - ? ? ? ? -

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School classes

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Multilevel attribute models If network like another level: Groups: Group indicators: Networks in groups (scaled): With random intercepts:

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Empirical setup =

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=

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Problem of boundary specification By design – children do not nominate alters outside of school class

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Problem of boundary specification By design – children do not nominate alters outside of school class Out of school To other school class

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Problem of boundary specification By design – children do not nominate alters outside of school class Out of school To other school class

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Multilevel autocorrelation/nef models =

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= ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

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Part 6 Estimation of ERGM with missing data

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Model assisted treatment of missing network data missing data observed data If you dont have a model for what you have observed How are you going to be able to say something about what you have not observed using what you have observed

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Model assisted treatment of missing network data Importance sampling (Handcock & Gile 2010; Koskinen, Robins & Pattison, 2010) Stochastic approximation and the missing data principle (Orchard & Woodbury,1972) (Koskinen & Snijders, forthcoming) Bayesian data augmentation (Koskinen, Robins & Pattison, 2010)

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What about alter – alter across ego? missing data observed data Available case analysis: pretend missing does not exist

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The principled approach in ERGM framework missing data observed data We have to simulate the missing (complement) and pool our inferences

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Subgraph of ERGM not ERGM i j k Dependence in ERGMWe may also have dependence i j l k But if k ? j We should include counts of: Marginalisation (Snijders, 2010; Koskinen et al, 2010)

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Bayesian Data Augmentation With missing data: Simulate parameters In each iteration simulate graphs missing Bayesian Data Augmentation

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Simulate parameters With missing data: In each iteration simulate graphs missing Most likely missing given current Bayesian Data Augmentation

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Simulate parameters With missing data: In each iteration simulate graphs missing Most likely given current missing Bayesian Data Augmentation

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Simulate parameters With missing data: In each iteration simulate graphs missing Most likely missing given current Bayesian Data Augmentation

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Simulate parameters With missing data: In each iteration simulate graphs missing Most likely given current missing Bayesian Data Augmentation

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Simulate parameters With missing data: In each iteration simulate graphs missing Most likely missing given current Bayesian Data Augmentation

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Simulate parameters With missing data: In each iteration simulate graphs missing and so on… Bayesian Data Augmentation

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Simulate parameters With missing data: In each iteration simulate graphs missing … until Bayesian Data Augmentation

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What does it give us? Distribution of parameters Distribution of missing data Subtle point Missing data does not depend on the parameters (we dont have to choose parameters to simulate missing) missing Bayesian Data Augmentation

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What does it give us? Distribution of parameters Distribution of missing data Subtle point Missing data does not depend on the parameters (we dont have to choose parameters to simulate missing) missing Bayesian Data Augmentation

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Part 7 Estimation of ERGM with missing data - Example Missing ties

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Bayesian Data AugmentationLazegas (2001) Lawyers Collaboration network among 36 lawyers in a New England law firm (Lazega, 2001) Boston office: Hartford office: Providence off.: least senior: most senior:

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Bayesian Data AugmentationLazegas (2001) Lawyers 133 Edges: Seniority: Practice: Homophily Sex: Office: GWESP: with 8 = log( ) Practice: Main effect t 1 : t 2 : etc. ( b i = 1, if i corporate, 0 litigation) t 3 :

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Bayesian Data Augmentation Lazegas (2001) Lawyers – ERGM posteriors (Koskinen, 2008)

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Bayesian Data Augmentation Cross validation (Koskinen, Robins & Pattison, 2010) Remove 200 of the 630 dyads at random Fit inhomogeneous Bernoulli model obtain the posterior predictive tie-probabilities for the missing tie-variables Fit ERGM and obtain the posterior predictive tie- probabilities for the missing tie-variables (Koskinen et al., in press) Fit Hoffs (2008) latent variable probit model with linear predictor T z(x ij ) + w i w j T Repeat many times

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Bayesian Data Augmentation ROC curve for predictive probabilities combined over 20 replications (Koskinen et al. 2010)

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Bayesian Data Augmentation ROC curve for predictive probabilities combined over 20 replications (Koskinen et al. 2010)

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Bayesian Data Augmentation ROC curve for predictive probabilities combined over 20 replications (Koskinen et al. 2010)

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Part 8 Estimation of ERGM with missing data - Sampled data and covert actors

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Bayesian Data AugmentationSnowball sampling Snowball sampling design ignorable for ERGM (Thompson and Frank, 2000, Handcock & Gile 2010; Koskinen, Robins & Pattison, 2010)... but snowball sampling rarely used when population size is known... Using the Sageman (2004) clandestine network as test-bed for unknown N

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Bayesian Data AugmentationSnowball sampling Snowball sampling design ignorable for ERGM (Thompson and Frank, 2000, Handcock & Gile 2010; Koskinen, Robins & Pattison, 2010)... but snowball sampling rarely used when population size is known... Using the Sageman (2004) clandestine network as test-bed for unknown N

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Bayesian Data Augmentationthe Sageman (2004) N = 366 network

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Bayesian Data Augmentationthe Sageman (2004) N = 366 network

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Bayesian Data Augmentationthe Sageman (2004) N = 366 network

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Bayesian Data Augmentationthe Sageman (2004) N = 366 network

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Bayesian Data Augmentationthe Sageman (2004) N = 366 network Take seed of size n = 120 Snowball out 1 wave. Additional nodes m = 160

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Bayesian Data Augmentationthe Sageman (2004) N = 366 network Seed n = 120, first wave m = 160, N 280

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Bayesian Data Augmentationthe Sageman (2004) N = 366 network Seed n = 120, first wave m = 160, N 280 Assume N = 281

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Bayesian Data Augmentationthe Sageman (2004) N = 366 network Seed n = 120, first wave m = 160, N 280 Assume N = 281 N = 291

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Bayesian Data Augmentationthe Sageman (2004) N = 366 network Seed n = 120, first wave m = 160, N 280 Assume N = 281 N = 291 N = 301

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Bayesian Data Augmentationthe Sageman (2004) N = 366 network Seed n = 120, first wave m = 160, N 280 Assume N = 281 N = 291 N = 301 N = 311

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Bayesian Data Augmentationthe Sageman (2004) N = 366 network Seed n = 120, first wave m = 160, N 280 Assume N = 281 N = 291 N = 301 N = N = 391 N = 396 N = 399

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Bayesian Data Augmentationthe Sageman (2004) N = 366 network Seed n = 120, first wave m = 160, N 280 Assume N = 281 N = 291 N = 301 N = N = 391 N = 396 N = credibility intervals N

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Bayesian Data Augmentationthe Sageman (2004) N = 366 network Seed n = 120, first wave m = 160, N 280 Assume N = 281 N = 291 N = 301 N = N = 391 N = 396 N = credibility intervals

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Bayesian Data Augmentationthe Sageman (2004) N = 366 network Seed n = 120, first wave m = 160, N 280 Assume N = 281 N = 291 N = 301 N = N = 391 N = 396 N = credibility intervals

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Bayesian Data Augmentationthe Sageman (2004) N = 366 network Seed n = 120, first wave m = 160, N prediction intervals Bernoulli ERGM obs

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Bayesian Data Augmentationthe Sageman (2004) N = 366 network Seed n = 120, first wave m = 160, N prediction intervals Bernoulli ERGM obs

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Bayesian Data AugmentationSnowball sampling – Next steps We can fit and predict missing conditional on N Next: Marginalise with respect to N, and Estimate N -Use path sampler -Take combinatorics of zero block into account

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Part 9 Further issues

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How large networks can we allow for? Large N ERGMs do not scale up (cp missing data experiments) Lot of unobserved data – lot of unobserved covariates Computational issues – time and memory Heterogeneity…

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How large networks can we allow for? ERGMs typically assume homogeneity (A)Block modelling and ERGM (Koskinen, 2009) (B) Latent class ERGM (Schweingberger & Handcock)

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Solutions and future directions Ignoring unknown N : - Conditional MLE for snowball sample does not require knowledge of N (sic!) (Pattison et al., in preparation) Estimating N : - Bernoulli assumptions (Frank and Snijders, 1994 JOS) - Using ERGM and Bayes factors? (Koskinen et al., in preparation) - Using heuristic GOF; posterior predictive distributions, re-sampling and copula (?)

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Wrap-up ERGMs - Increasingly being used - Increasingly being understood - I ncreasingly being able to handle imperfect data (also missing link prediction) Methods -Plenty of open issues -Bayes is the way of the future Legitimacy and dissemination - e.g. Lusher, Koskinen, Robins ERGMs for SN, CUP, 2011

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