Outline Introduction Social Networks Linear Models that Use Social Network Data Simulation Study Conclusions References Using Social Network Information In Survey Estimation Thomas S¨ uße and Raymond Chambers National Institute for Applied Statistics Research Australia (NIASRA) University of Wollongong 2013 Graybill Conference, Fort Collins, Colorado 11 June 2013 1/36
Outline Introduction Social Networks Linear Models that Use Social Network Data Simulation Study Conclusions References Outline Introduction 1 2/36
Outline Introduction Social Networks Linear Models that Use Social Network Data Simulation Study Conclusions References Outline Introduction 1 Social Networks 2 2/36
Outline Introduction Social Networks Linear Models that Use Social Network Data Simulation Study Conclusions References Outline Introduction 1 Social Networks 2 Linear Models that Use Social Network Data 3 2/36
Outline Introduction Social Networks Linear Models that Use Social Network Data Simulation Study Conclusions References Outline Introduction 1 Social Networks 2 Linear Models that Use Social Network Data 3 Simulation Study 4 2/36
Outline Introduction Social Networks Linear Models that Use Social Network Data Simulation Study Conclusions References Outline Introduction 1 Social Networks 2 Linear Models that Use Social Network Data 3 Simulation Study 4 Conclusions 5 2/36
Outline Introduction Social Networks Linear Models that Use Social Network Data Simulation Study Conclusions References Introduction Population U of size N Sample s of size n , remainder of population r := U \ s of size N − n Survey variable Y with realisations y i , i ∈ U Focus on estimating population total t y = ∑ i ∈ U y i Auxiliary variables X 1 ,..., X p Non-informative sampling method given population values of auxiliaries Model-based approach 3/36
Outline Introduction Social Networks Linear Models that Use Social Network Data Simulation Study Conclusions References A Place To Start Simple linear model for Y in terms of X 1 ,..., X p y i = x 1 i β 1 + ··· + x pi β p + ε i ε i ∼ ( 0 , σ 2 ) In matrix terms y i = X T i β + ε i or Y U = X U β + ε U Best linear unbiased predictor (BLUP) for population total t y t y = ∑ r ( X r ˆ ˆ y i = 1 T s Y s + 1 T y i + ∑ ˆ β ) i ∈ s i ∈ r � � � � Y s X s ˆ β = ( X T s X s ) − 1 X T s Y s , Y U = , X U = Y r X r 4/36
Outline Introduction Social Networks Linear Models that Use Social Network Data Simulation Study Conclusions References A More Complex Reality: Hierarchical Data Data available at enumeration district (ED) and ward level Individuals i (level 1); EDs j (level 2); wards k (level 3) Multilevel model: ijk β + u ( 3 ) + u ( 2 ) jk + u ( 1 ) y ijk = X T k ijk with � 0 , τ ( 3 ) � � 0 , τ ( 2 ) � � 0 , τ ( 1 ) � u ( 3 ) , u ( 2 ) , u ( 1 ) ∼ jk ∼ ijk ∼ k 5/36
Outline Introduction Social Networks Linear Models that Use Social Network Data Simulation Study Conclusions References A Patterned Covariance Structure Var ( y ijk ) = σ 2 = τ ( 3 ) + τ ( 2 ) + τ ( 1 ) τ ( 2 ) + τ ( 3 ) different people, same ED τ ( 3 ) Cov ( y ijk , y lmn ) = different EDs, same ward 0 different wards Linear model for population has the form Y U = X U β + ε U , ε U ∼ ( 0 , σ 2 V U ) where V U has a nested block diagonal structure 6/36
Outline Introduction Social Networks Linear Models that Use Social Network Data Simulation Study Conclusions References The General BLUP BLUP for dependent responses � � X r ˆ ss ( Y s − X s ˆ ˆ t y = 1 T s Y s + 1 T β + V rs V − 1 β ) r with best linear unbiased estimator (BLUE) ˆ s V − 1 ss X s ) − 1 X T s V − 1 β s = ( X T ss Y s and � � V ss V rs V U = V rs V rr 7/36
Outline Introduction Social Networks Linear Models that Use Social Network Data Simulation Study Conclusions References Using Social Networks to Characterise Non-Hierarchical Dependence Widespread (Facebook, Linkedin, Google, family, friends, colleagues, etc.) N actors or nodes Simplest characterisation via adjacency matrix Z U = [ Z ij ] N i , j = 1 with Z ij = 1 if relationship (’edge’) exists between i and j ; Z ij = 0 otherwise Z U has zero main diagonal and is symmetric (undirected network) or asymmetric (directed network) Extensions exist for multiple types of relationships and count or continuous values for Z ij , e.g. level/strength of communication between two nodes 8/36
Outline Introduction Social Networks Linear Models that Use Social Network Data Simulation Study Conclusions References Example: Law Firm Collaborations Working relations among N = 36 partners in a law firm (Lazega, 2001) An edge exists between two partners if, and only if, both indicate that they collaborate with the other Undirected network Numbers of edges (row and column sums) associated with each of the N = 36 nodes range from 0 to 16, with an average of 6.4 Node attributes (covariates collected on each partner) include seniority (rank number of entry into the firm), gender, office (three offices in different cities), and practice (litigation = 0, and corporate law = 1) 9/36
Outline Introduction Social Networks Linear Models that Use Social Network Data Simulation Study Conclusions References Example: Adjacency Matrix for Law Firm Collaborations 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 1 2 3 4 5 6 7 8 9 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1 0 0 0 1 0 1 1 0 1 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 1 1 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 0 0 0 0 7 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 15 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 0 0 1 0 0 1 0 0 1 1 16 0 1 0 0 0 0 0 0 1 0 0 1 0 1 1 0 1 0 0 0 0 1 0 0 0 1 1 0 1 0 0 1 0 1 0 1 17 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 1 0 0 1 0 0 1 0 1 1 1 0 1 1 0 0 0 0 1 0 0 18 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 19 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 20 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 22 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 24 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 25 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 26 0 1 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 0 1 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 28 0 0 1 1 1 1 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 1 1 0 0 1 0 29 0 1 0 1 0 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 30 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 31 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 1 0 1 0 32 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 1 0 33 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 34 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 35 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 10/36
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