Using Social Network Information In Survey Estimation Thomas S ue - - PowerPoint PPT Presentation

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

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Outline Introduction Social Networks Linear Models that Use Social Network Data Simulation Study Conclusions References

Outline

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Introduction

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Outline Introduction Social Networks Linear Models that Use Social Network Data Simulation Study Conclusions References

Outline

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Introduction

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Social Networks

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Outline Introduction Social Networks Linear Models that Use Social Network Data Simulation Study Conclusions References

Outline

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Introduction

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Social Networks

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Linear Models that Use Social Network Data

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Outline Introduction Social Networks Linear Models that Use Social Network Data Simulation Study Conclusions References

Outline

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Introduction

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Social Networks

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Linear Models that Use Social Network Data

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Simulation Study

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Outline Introduction Social Networks Linear Models that Use Social Network Data Simulation Study Conclusions References

Outline

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Introduction

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Social Networks

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Linear Models that Use Social Network Data

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Simulation Study

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Conclusions

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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 yi,i ∈ U Focus on estimating population total ty = ∑i∈U yi Auxiliary variables X1,...,Xp Non-informative sampling method given population values of auxiliaries Model-based approach

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A Place To Start Simple linear model for Y in terms of X1,...,Xp yi = x1iβ1 +···+xpiβp +εi εi ∼ (0,σ2) In matrix terms yi = XT

i β +εi

• r YU = XUβ +εU

Best linear unbiased predictor (BLUP) for population total ty ˆ ty = ∑

i∈s

yi +∑

i∈r

ˆ yi = 1T

s Ys +1T r (Xr ˆ

β) ˆ β = (XT

s Xs)−1XT s Ys,

YU =

• Ys

Yr

• ,

XU =

• Xs

Xr

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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: yijk = XT

ijkβ +u(3) k

+u(2)

jk +u(1) ijk

with u(3)

k

∼

• 0,τ(3)

,u(2)

jk ∼

• 0,τ(2)

,u(1)

ijk ∼

• 0,τ(1)

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A Patterned Covariance Structure Var(yijk) = σ2 = τ(3) +τ(2) +τ(1) Cov(yijk,ylmn) =    τ(2) +τ(3) different people, same ED τ(3) different EDs, same ward different wards Linear model for population has the form YU = XUβ +εU,εU ∼ (0,σ2VU) where VU has a nested block diagonal structure

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The General BLUP BLUP for dependent responses ˆ ty = 1T

s Ys +1T r

• Xr ˆ

β +VrsV−1

ss (Ys −Xs ˆ

β)

• with best linear unbiased estimator (BLUE)

ˆ β s = (XT

s V−1 ss Xs)−1XT s V−1 ss Ys

and VU =

• Vss

Vrs Vrs Vrr

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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 ZU = [Zij]N

i,j=1 with

Zij = 1 if relationship (’edge’) exists between i and j; Zij = 0

• therwise

ZU 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 Zij, e.g. level/strength of communication between two nodes

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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

• f 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)

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Example: Adjacency Matrix for Law Firm Collaborations

1 2 3 4 5 6 7 8 9 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 1 2 1 1 1 1 1 1 3 1 1 1 4 1 1 1 1 1 1 1 1 1 5 1 1 1 1 1 1 6 1 1 1 1 1 7 1 1 8 9 1 1 1 10 1 1 1 1 1 11 1 12 1 1 1 1 1 1 1 1 1 13 1 1 14 1 1 1 1 1 1 15 1 1 1 1 1 1 1 1 1 1 1 16 1 1 1 1 1 1 1 1 1 1 1 1 1 17 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 18 1 1 1 1 1 1 1 1 19 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 21 1 22 1 1 1 1 1 1 1 1 1 23 24 1 1 1 1 1 1 1 1 1 25 1 1 1 1 1 26 1 1 1 1 1 1 1 1 1 1 1 1 27 1 1 1 28 1 1 1 1 1 1 1 1 1 1 1 1 1 29 1 1 1 1 1 1 1 1 1 30 1 1 1 1 31 1 1 1 1 1 1 1 1 1 1 1 1 1 32 1 1 1 1 1 1 1 1 1 1 1 1 33 1 1 1 1 1 34 1 1 1 1 1 1 35 1 1 1 1 1 1 1 36 1 1 1 10/36

Outline Introduction Social Networks Linear Models that Use Social Network Data Simulation Study Conclusions References

Example: Graph of Law Firm Collaborations

1 2 3 4 5 6 7 8 9 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

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Modelling ZU: Exponential Random Graph Models Widely used family of models for network data Probability distribution generated by an ERGM is Pr(ZU = z) = exp

• ηT g(z)−κ(η)
• =

exp

• ηT g(z)
• ∑ζ∈Z exp(ηT g(ζ))

η vector of model parameters g(z) vector of network statistics κ is the normalising constant κ(η) = log

• ∑

ζ∈Z

exp(ηT g(ζ))

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Examples of Network Statistics Edges Statistic A B Two-Star Statistic A B C Edgewise Shared-Partner Statistic A B C D E Triangle Statistic A B C

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The GWESP Statistic EPk(ZU) is the number of edges (Zij = 1 with i < j) that share exactly k neighbors in common EP0 +···+EPN−2 = number of edges Geometrically weighted edgewise shared partner (GWESP) statistic defined as GWESP(ZU,θ) = exp(θ)

N−2

∑

k=1

• 1−(1−exp(−θ))k

EPk(ZU) Geometrically weighted sum of EPk(ZU) values, with parameter θ controlling distribution of weights

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Fitting ERGMs ERGM’s are difficult to fit because the normalising constant κ cannot be calculated explicitly in any realistic application MCMC techniques are typically used to approximate the log-likelihood Geometrically weighted statistics (e.g. GWESP) generate MCMC samples that are degenerate less often

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Three Questions (a) Is embedding social network information into linear models useful for survey estimation based on these models? (b) If the answer to (a) is yes, then

(b1) Which network-based linear models are potentially useful? (b2) How much network data needs to be collected in order to obtain potentially higher precision for survey estimation?

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Linear Models with Embedded Social Network Information There are basically three types of linear models that use the information in the adjacency matrix ZU generated by a social network

• 1. Contextual Network models
• 2. Autocorrelation models
• 3. Network Disturbance models

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Contextual Network (CN) Models Basic idea is to add one or more network-based contextual covariates to the model Motivation: Student academic performance (AP) as a function of socio-economic status (SES) Network: Student friendship network Model student’s AP as a function of his/her SES and average SES of his/her friends (Friedkin, 1990)

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Contextual Network (CN) Models CN model can be written as YU = XUβ +WUTUγ +εU where

TU is the population matrix of covariates measured on the network WU is a row-normalised version of ZU, i.e. the rows of WU sum to

• ne

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Autocorrelation (AR) Models The matrix TU can be any set of measurements on the individuals in the network, and in particular it can be YU Autocorrelation (AR) models, also known as network effects models (Ord, 1975; Doreian et al., 1984; Duke, 1993; Leenders, 2002), are defined by YU = Xβ +λWUYU +εU where λ ∈ (−1,+1) The conditional (on XU) mean and variance of YU are µ = D−1

U XUβ and VU = σ2(DT UDU)−1, where DU = IU −λWU

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Network Disturbance (ND) Models The linear model errors are assumed to have an AR structure, i.e. YU = XUβ +εU, where εU = λWUεU +vU and vU ∼ (0,σ2IU) Conditional (on XU) the mean and variance of YU are µ = XUβ and VU = σ2(DT

UDU)−1 respectively. That is, the network induces

correlation structure but does not affect mean structure (Ord, 1975; Leenders, 2002) AR and ND models are similar to conditional autoregressive (CAR) and simultaneous autoregressive (SAR) models commonly used for spatial data (Banerjee et al., 2004)

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BLUP/EBLUP Specification In order to calculate the BLUP , we generally need to specify

A design matrix HU such that the conditional mean µ of YU given XU satisfies µ = HUξ A positive definite matrix VU proportional to the conditional variance of YU given XU

When these quantities themselves depend on unknown parameters, we first estimate these parameters from the sample data and then substitute in HU and VU before calculating the BLUP . This is the ‘plug-in’ version of the EBLUP

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Model Specification Standard HU = XU and VU = σ2IU. The residual mean squared error is an unbiased estimator of σ2 CN HU = [XU,WUTU] and VU = σ2IU. Again, the residual mean squared error is an unbiased estimator of σ2 AR HU = D−1

U XU and VU = σ2(DT UDU)−1 with

DU = IU −λWU. Estimates of σ2 and λ can be

• btained by maximum likelihood (ML)

ND HU = XU and VU = σ2(DT

UDU)−1. Both σ2 and λ can be

estimated via ML

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Imputation of Missing Network Information Calculation of the EBLUPs defined by the CN, AR and ND models assumes that the population network ZU is known In practice this is extremely unlikely, and it is more realistic to consider situations where ZU is partially known

SS We only know Zss, i.e. the sub-network of relationships between the n sampled individuals in s SS+SR We also know the links between the sampled individuals and the remaining N −n non-sampled individuals in the population, i.e. we know Zsr. Note that for an undirected network this means that we know Zrs as well

We use model-based imputation to ‘fill in’ the rest of ZU in either case

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Optimum Imputation An optimal model-based approach is to assume that ZU can be adequately modelled via an ERGM and to use the minimum mean squared error predictor E(Zmis

U |Zobs U

= zobs) In this case the conditional distribution of Zmis

U

is defined by Pr(Zmis

U

= zmis|Zobs

U

= zobs) = exp

• ηT g(zmis,zobs;θ)
• ∑ζ mis∈Z mis exp
• ηT g(ζ mis,zobs;θ)
• where Z mis is the sample space of Zmis

U

In theory, MCMC techniques can be used to sample from this conditional distribution, with η and θ replaced by estimates based on the observed network. However, this is impractical at present

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Practical Imputation

Method 1

Suppose conditionally on zobs that Z mis

ij

and Z mis

kl

are conditionally independent for any two distinct pairs ij and kl, i.e. Pr(Zmis = zmis|Zobs = zobs) = ∏ij Pr(Z mis

ij

|Zobs = zobs) This leads to Pr(Z mis

ij

= 1|Zobs = zobs) Pr(Z mis

ij

= 0|Zobs = zobs) = exp(ηT ∆gmis

ij

) where ∆gmis

ij

is the change statistic, i.e. the difference in g between (zmis

ij

,zobs) = (1,zobs) and (zmis

ij

,zobs) = (0,zobs)

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Practical Imputation

Method 1

Re-arranging this equation gives the MMSEP under conditional independence, E(Zij = 1|Zobs = zobs) = Pr(Zij = 1|Zobs = zobs) = expit(ηT ∆gmis

ij

) with expit(x) = exp(x)/(1+exp(x)) It is only necessary to compute ∆gmis

ij

in order to obtain this MMSEP for any distinct pair ij ∈ mis Since the conditional independence assumption is generally unwarranted, this approach can only be considered as defining an approximation to Pr(Zmis|Zobs = zobs) However, it is computationally feasible for realistic sample and population sizes

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Practical Imputation

Method 2

A very simple approach is to calculate the proportion of Zij = 1 in zobs and use this proportion (the network density) to impute Zmis This corresponds to imputing on the basis of an ERGM model defined by just the EDGES statistic, i.e. the number of edges in the network

Equivalent to assuming that each Zij in the network matrix ZU is an independent Bernoulli variable with a common probability of a ‘success’

If the network model also contains exogenous effects, then this simple approach corresponds to imputation on the basis of the logistic regression model defined by these effects

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Simulation Study - Model Specification Standard Yi = β0 +β1Xi +εi, εi ∼ N(0,1) β0 = 40, β1 = 5 and Xi is drawn randomly from 1,...,9 CN Yi = β0 +Xiβ1 +Uiγ +εi, εi ∼ N(0,1) γ = 2 and Ui is the contextual variable defined by average value of X for all individuals in the network that are linked to individual i AR Yi = β0 +Xiβ1 +Uiλ +εi, εi ∼ N(0,1) λ = 0.5 and Ui is the average value of Y for all individuals in the network that are linked to individual i ND Yi = β0 +β1Xi +εi, with εi = Uiλ +vi vi ∼ N(0,1) and Ui is the average value of ε for all individuals in the network that are linked to individual i

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Simulation Study - Network Specification Two types of networks were simulated

An ERGM network. Here ZU was generated as a random draw from an ERGM with a density of about 15 network links for each subject (EDGES statistic equal to −4.18 on the logit scale) and a weight parameter of θ = 1 for the GWESP statistic A Gang network, where ZU defined a network of 100 ‘gangs’, each

• f size 10. In this network each gang member only knows every
• ther member of his/her gang, so Z, after re-ordering rows, is block
• diagonal. This is analogous to the network defined by members of

the same household.

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Simulation Study - Characteristics & Notation Population of size N = 1,000 was independently simulated 2,000 times Independent simple random samples of size n = 100 and n = 200 were independently selected without replacement from each simulated population SS denotes the case where only Zss is observed SS+SR/1 denotes where Zss and Zsr are observed and imputation method 1 is used SS+SR/2 denotes where Zss and Zsr are observed and imputation method 2 is used

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Table: Monte Carlo Bias of EBLUP (average population total ≈ 65K); n = 100

ERGM network Gang Network True Model True Model Prediction Based On CN AR ND CN AR ND BLUP 1.81 2.19 2.27 2.14 −2.11 −1.93 full network known 1.81 2.50 2.14 2.14 −1.84 −1.83 SS 1.96 3.06 2.22 2.90 0.93 −1.12 CN SS+SR/1 0.92 2.11 2.07 – – – SS+SR/2 1.05 2.37 2.12 2.10 −1.24 −1.17 SS 0.51 2.29 2.08 2.34 −1.57 −0.84 AR SS+SR/1 1.94 2.71 2.08 – – – SS+SR/2 1.26 1.27 2.05 2.24 −3.47 −1.15 SS 0.78 1.48 2.20 2.39 −1.54 −1.63 ND SS+SR/1 0.80 1.79 2.23 – – – SS+SR/2 0.85 1.49 2.13 3.09 14.1 −1.83 standard model 0.78 1.59 2.13 2.98 1.21 −1.06

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Table: Monte Carlo MSE of EBLUP relative to MSE of BLUP; n = 100

ERGM network Gang Network True Model True Model Prediction Based On CN AR ND CN AR ND BLUP - actual MSE 9,390 8,739 8,736 9,421 12,330 12,315 full network known 1.00 1.10 1.02 1.00 1.08 1.01 CN SS 2.28 3.38 1.00 2.66 9.57 1.05 SS+SR/1 1.19 1.39 1.00 – – – SS+SR/2 1.14 1.31 1.00 1.01 1.07 1.06 AR SS 2.30 3.34 1.01 2.57 8.67 1.05 SS+SR/1 1.45 1.42 1.00 – – – SS+SR/2 1.31 1.30 1.00 1.24 1.10 1.06 ND SS 2.30 3.52 1.02 2.54 8.79 1.01 SS+SR/1 2.30 3.48 1.02 – – – SS+SR/2 2.30 3.49 1.02 3.14 11.2 1.01 standard model 2.30 3.50 1.00 2.87 10.5 1.04

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Table: Monte Carlo average length of nominal 95% Gaussian confidence interval generated by EBLUP relative to that generated by BLUP; n = 100; corresponding coverage (%) shown in subscript

ERGM Network Gang Network True Model True Model Prediction Based On CN AR ND CN AR ND BLUP - average length 37094.5 38296.2 38296.4 37094.0 41893.9 41893.4 full network known 1.0094.4 0.9995.0 0.9895.3 1.0094.0 1.0092.8 1.2794.1 CN SS 1.5493.5 1.7995.4 0.9995.5 1.6594.4 3.0791.8 1.0093.2 SS+SR/1 1.0092.3 1.0192.2 0.9995.6 – – – SS+SR/2 1.0092.9 1.0192.7 0.9995.8 1.0093.7 1.0193.0 1.0193.7 AR SS 1.5494.1 1.7795.3 0.9995.4 1.6294.2 2.8992.4 1.0093.4 SS+SR/1 1.1092.2 1.0192.2 0.9995.5 – – – SS+SR/2 1.1093.1 1.0192.8 0.9995.6 1.0693.7 0.9992.4 1.0093.4 ND SS 1.5994.8 1.8395.7 0.9895.1 1.5993.5 2.9392.3 0.9893.2 SS+SR/1 1.5994.8 1.7794.4 0.9794.4 – – – SS+SR/2 1.6094.8 1.8095.2 0.9894.9 2.0190.1 2.8889.5 1.2794.1 standard model 1.5994.8 1.8496.0 0.9995.7 1.7293.8 3.2392.8 1.0193.6

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Tentative Answers to Three Questions (a) Is embedding social network information into linear models useful for survey estimation based on these models?

Yes

(b1) Which network-based linear models are potentially useful?

CN and AR models are useful when either model is true, since in both cases the mean of the response depends on the network Ignoring the network does not result in a significant loss of efficiency when the ND model is true

(b2) How much network data needs to be collected in order to obtain potentially higher precision for survey estimation?

Both Zss and Zsr must be available in order to obtain efficiency

• gains. Knowledge of Zss alone is not enough

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A Recommendation & A Caution The AR model can be difficult to fit, see Suesse (2012), so we recommend that the CN model be used if it is a reasonable fit to the data and relevant population level auxiliary network data are

• available. Otherwise ignoring the network might be the best
• ption

Note that we have assumed that the method of sampling is independent of the network structure given the available population auxiliary information

There are important applications, see Thompson and Seber (1996), where inclusion in sample depends on being linked to another sampled individual via a network In these cases we cannot treat the observed network structure in Zss and Zsr as ancillary (as we have here), and this ‘informative’ method of sampling needs to be taken into account when we impute the unknown components of ZU

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Banerjee S., Carlin B. P . and Gelfand A. E. (2004) Hierarchical modelling and analysis for spatial data Boca Raton, Fla.: Chapman & Hall/CRC Press. Doreian, P ., Teuter, K. and Wang, C. H. (1984) Network auto-correlation models - some monte-carlo results. Sociological Methods & Research 13, 155–200. Duke, J. B. (1993) Estimation of the network effects model in a large data set. Sociological Methods & Research 21, 465–481. Friedkin, N. E. (1990) Social networks in structural equation models. Social Psychology Quarterly 53, 316–328. Lazega, E. (2001) The Collegial Phenomenon: The Social Mechanism of Cooperation Among Peers in a Corporate Law Partnership. Oxford: Oxford University Press. Leenders, R. (2002) Modeling social influence through network autocorrelation: constructing the weight matrix. Social Networks 24, 21–47. Ord, K. (1975) Estimation methods for models of spatial interaction. Journal of the American Statistical Association 70, 120–126. Suesse, T. (2012) Estimation in autoregressive population models. In Proceedings of Fifth Annual ASEARC Research Conference. University of Wollongong: ASEARC. 2-3 February 2012. Thompson, S. K. and Seber, G. A. F. (1996) Adaptive sampling. Wiley series in probability and mathematical statistics. New York: Wiley.

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