Network data in regression framework

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1 13-14 July 2009 University of Salerno (Italy) Network data in regression framework Maria ProsperinaVitale Department of Economics and Statistics University of Salerno (Italy)

2 - Theoretical Framework - Research hypothesis and Aims Network data in regression framework - Performance Assessment and Interpersonal influence Outline - Interdependent data points: Linear Regression and Structural Equation Model - Network autocorrelation model W weight matrix -A case study: students performance assessment in a second level degree at the University of Salerno

3 Theoretical Framework In the assessment of student performance in a learning environment, both background variables and relational data can be considered (Braun and Mohler, 2003; Capò et al., 2007; Thompson, MacDonald, 2005 ). Background variables are related to past experiences, sociodemographic and educational data. Relational data measure the relevance of collaboration, advice, social support and friendship that can facilitate learning processes. Performance could be embedded within networks of interdependencies Interpersonal influence

4 Research Hypothesis What is the relationship between socio-demographic background, attitudinal variables, academic background, classroom community collaboration and performance for university student? How important are collaboration among students? Characteristics of actors (e.g. job satisfaction, performance) are influenced by the networks surrounding the actor. Looking for statistical models where actors adjust their own attitudes and behavior to that of others they are connected with Testing hypotheses about network effects on individual attributes.

5 Social Influence Social or interpersonal influence occurs when an actor adapts his behavior, attitude, or belief, to the behaviors, attitudes of the other actors in the social system (influence or contagion process). (Leenders, 2002) Aims - to present a review of models proposed in the literature to deal with interpersonal influence in classical regression approach in order to analyze the network effects on actor performance; - to discuss the presence of interdependent data points in the frameworks of Linear Regression and Structural Equation Model.

6 Statistical models to identify network effects on individual outcomes Relational or dyad-level models outcome variable is relational - dyads as units of analysis the interest is on the analysis of network structure, using both network statistics and covariates, to explain a dependent variable with individual linkages (ties) as its elements Individual outcome regression models (e.g. Robins et al., 2007a,b) outcome variable is actors behavior/attitude - individuals as units of analysis describes how a dependent variable y is related to independent variables in presence of interpersonal influence (see Bramoullé, Fortin, 2008; O'Malley, Marsden, 2008)

7 Linear Regression model and Interdependent data The objective is to focus on models proposed in the literature to deal with interpersonal influence in classical regression approach introducing network effects. Regression models seem particularly vulnerable when there are interdependent individual units embedded within social structures network effect (Doreian, 1996, Friedkin, 2003). Spatial effects model (Cressie, 1993) could be extended to settings in which the autocorrelation stems from social, rather than purely physical, proximity (White et al., 1981). Hence, the interpersonal influence on individual s outcome has been dealt with the specification of Network Autocorrelation Models (see Doreian et al., 1984; Leenders, 2002)

8 Network Autocorrelation Model: specification Spatial effects model or spatial lag model y = ρ Wy + Xβ + ε [1] Spatial errors model or disturbance model y = Xβ + ε ε = ρwε + ν Mixed network autocorrelation model y = ρ 1W1y + Xβ + ε ε = ρ W 2 2ε + ν [2] [3] W is an N N matrix representing the social distances between the observations while ρ is a scalar estimating the extent to which an actor s outcome is affected by the behavior of those to whom is socially proximate (autocorrelation effect). Lagged terms, Wy or Wε, can be interpreted as a form of social influence, and thus provides a bridge between statistical analysis and social theories.

9 Network data in regression framework Network Autocorrelation Model: Weight matrix (Leenders, 2002) Units of analysis: individual social entities linked to other individuals by affective, professional and other types of social ties. Autocorrelation is a result of processes that include social influence, imitation and other similar or related behaviors. Social influence enters network autocorrelation model through the weight matrix W, where w ij represents the extent to which y i is dependent on y j. Different ways of formulating the weights (binary, power, inverse distance weights, contiguity, etc.). For spatial and network autocorrelation models is recommended normalization of the W matrix, such that the sum of each row equals 1. A normalized W is in nearly all cases asymmetric (Cliff and Ord, 1981; Anselin, 1988) The dominant procedure for estimating the coefficients β and ρ in the network (spatial) autocorrelation models is maximum likelihood (ML). Estimation LNAM R software (Butts, 2007)

10 Structural Equation Model and Interdependent data In Structural Equation Model, when observed and latent variables could be involved to describe complex social phenomena, the presence of interpersonal influence to individual action also violates the independent assumption. Friedkin (1990) considered the mixed regressive-autoregressive model above described to embed automatically social network effects in a causal scheme. But the introduction of an autocorrelation term in presence of latent variables are not discussed. In analogy with spatial dependence used recently in geographical framework (i.e. when W includes contiguity information), the aim is dealing with spatial dependence when W represents social distance and network data could be enter in SEM model through an autocorrelation term. (Oud, Folmer, 2008)

11 Structural Equation Model and Interdependent data Observed spatial error model Observed level obs. unit form ~ ~ y = Xγ + ~ ε ~ ε or ~ ~ y = λwy ~ + Xγ SEM notation y = λy (Oud, Folmer, 2008) Spatial Dependence (in geographical framework) + γ x ~ = λw~ ε + ζ ~ ~ λwxγ + ζ SEM notation λγ w x w + ζ latent spatial lag model Latent level obs. unit form ~ ~ ~ η = ρwη ~ + Xγ + ζ ~ ~ y = ΛWη ~ + ~ ε y w η = ρη w = w Λη + ε w w + γ x + ζ w

In university framework, it makes sense to promote initiatives that encourage the

12 Case study: Student performance Research hypothesis Does the collaboration and community formation influence the student performance at University? In university framework, it makes sense to promote initiatives that encourage the emergence of communities of students and collaboration among peers in order to improve student performance?

13 Student performance: hypothesized NAM models Social-demographic characteristics Individual characteristics Performance of other students What is the relationship between sociodemographic background, attitudinal variables, academic background, classroom community collaboration and performance for university student? 1. yperf Students performance Weight Matrix W = ρ Wperf + Xβ + ε Collaboration dynamics (information exchange, exams preparation, classmates) Community formation (friendship, advice, support) 2. yperf = Xβ + ε ε = ρwε + ν 3. y 1 1 = ρ W y + Xβ + ε ε ρ ε + ν = W 2 2

14 Student s performance: data collection and variables Students enrolled at the first academic year of the second level degree of Sociology at University of Salerno 2008/09: Web survey: Questionnaire to collect individual characteristics and relational data about collaboration dynamics and community formation during the first level degree. 60% answers (48 students out of 81 enrolled students) Exogenous variables - Socio-demographic (father education years; residence; ) - Individual characteristics (work, enrollment age, grade and type secondary school; sociability, ) Endogenous variables Performance in learning environment can be measured by: Indicators related to the Dublin descriptors [Perceived performance] ( ss/p!ekljfab) Final grade of first level degree

15 Temporal relational dynamics Relational data - Relationships before the enrollment at the first level at University (friendship, secondary school, classmate, ) Collaboration Network Formal contacts W (or W 1 ) - Relationships during the first level degree (advice, classmate, exams, information exchange, ) -Information exchange -Classmate - Exams preparation - Relationships during the second level degree (advice, classmate, exams, information exchange, ) Community Network Informal contacts W 2 - Support - Advice - Friendship

Collaboration Network Formal contacts Classmate Exams

symmetric adjacency matrices W (W 1 ) 48x48 students where w ij

to one binary symmetric adjacency matrix W 48x48 obtained by

16 Collaboration Network Formal contacts Classmate Exams preparation Information exchange 0,309 0,064 0,296 From 3 binary symmetric adjacency matrices W (W 1 ) 48x48 students where w ij =1 if students i is connected to student j and 0 otherwise 0,443 to one binary symmetric adjacency matrix W 48x48 obtained by joining the 3 above described matrices W has been normalized by row marginals

17 Network data in regression framework Community formation Informal contacts Support Advice Friendship 0,004 Informal Network 0,084 0,003 0,005 From 3 binary direct adjacency matrices W 2 48x48 students where w ij =1 if students i is connected to student j and 0 otherwise to one binary direct adjacency matrix W 2 48x48 obtained by joining the 3 above described matrices W 2 is normalized by row marginals

18 Student performance: NAM model definition 1. yperf = ρ Wperf + Xβ + ε 2. yperf = Xβ + ε ε = ρwε + ν 3. y 1 1 = ρ W y + Xβ + ε ε = ρ2w2ε + ν y= Performance in regression model W= Formal network W= Formal network W 1 (=W) = Formal network W 2 = Informal network - Final Grade of first level degree - Objective Performance Final Grade and Years to obtain the first degree (compound indicator) - Perceived Performance Dublin s Descriptors (compound indicator) - Performance Objective and perceived performance (compound indicator)

19 Some results

20 OLS model y performance Network effects and disturbances model y performance

21 Concluding remarks It seems that doesn t exist a network effect on student performance. Only the model with y as compound indicator has a low significant rho coefficient but with values that do not respect the boundaries of correlation coefficient.this happens for low density of formal and informal networks, for small population size, for a misspecification of performance indicator Further developments - Network autocorrelation model in SEM framework: weight social distance matrix W in latent variable

22 Some References Bollen, Kenneth A Structural Equation with Latent Variables. New York: John Wiley. Bramoullé Y., Fortin B. (2009), The Econometrics of Social Networks, Working Paper 09-13, CIRPEE, Université Laval. Doreian P. (1996), When the Data Points are not Indipendent, in: A. Ferligoj and A. Kramberger (Eds), Developments in Data Analysis, Medoloski svezki, 12, Ljubljana: FDV, Doreian P. (2001), Causality in Social Network Analysis, Sociological Methods and Research, 30, Friedkin, Noah E. (1990), Social Networks in Structural Equation Models, Social Psychology Quarterly, 53, Leenders, Roger Th.A.J. (2002), Modeling social influence through network autocorrelation: constructing the weight matrix, Social Networks, 24, O'Malley, A.J., Marsden, P.V. (2008), The Analysis of Social Networks, Health Services and Outcomes Research Methodology, 8, Oud J.H. L., Folmer (2008), A Structural Equation Approach to Models with Spatial Dependence, Geographical Analysis,40,

W-BASED VS LATENT VARIABLES SPATIAL AUTOREGRESSIVE MODELS: EVIDENCE FROM MONTE CARLO SIMULATIONS

1 W-BASED VS LATENT VARIABLES SPATIAL AUTOREGRESSIVE MODELS: EVIDENCE FROM MONTE CARLO SIMULATIONS An Liu University of Groningen Henk Folmer University of Groningen Wageningen University Han Oud Radboud