The statistical analysis of personal network data

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The statistical analysis of personal network data

• I. Cross-sectional analysis
• II. Dynamic analysis
• Miranda Lubbers,
• Autonomous University of Barcelona

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Sociocentric networks
Sociocentric or complete networks consist of
the set of relations among the actors of a
defined group (e.g., a school class, a firm)
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Personal networks
A personal network consists of the set of
relations a focal person (ego) has with an
unconstrained set of others (alters) and the
relations among them.
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Egonet, software to aid the collection of
personal network data

• Information about the respondent (ego e.g., age,
  sex, nationality)
• Information about the associates (alters) to whom
  ego is connected (e.g., alters age, sex,
  nationality)
• Information about the ego-alter pairs (e.g.,
  closeness, frequency and / or means of contact,
  time of knowing, geographic distance, whether
  they discuss a certain topic, type of relation
  e.g., family, friend, neighbour, workmate )
• Information about the relations among alters as
  perceived by ego (simply whether they are related
  or not, or strong/weak/no relation)

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The statistical analysis of personal versus
sociocentric networks what are the differences?

• Whereas sociocentric network researchers often
  (yet not always) concentrate on a single network,
  personal network researchers typically
  investigate a sample of networks (ideally a
  random, representative sample).
• The dependency structure of sociocentric networks
  is complex, therefore leading to the need of
  specialized social network software, but personal
  network researchers, as they have up till now
  hardly used the data on alter-alter relations,
  have a simpler dependency structure...

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Personal network data have a multilevel
structure

• E.g. sample of 100 respondents for each
  respondent, data of 45 alters were collected, so
  in total a collection of 4500 alters

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For cross-sectional analysis, three types of
analysis have been used in past research

• Type I Aggregated analysis
• Type II Disaggregated analysis
• (not okay, forget about it quickly!)
• Type III Multilevel analysis

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Type 1 Aggregated analysis

• First, aggregate all information to the ego-level
  (this can be exported directly from Egonet)
• Compositional variables (aggregated
  characteristics of alters or ego-alter
  relations) e.g., percentage of women, average
  closeness, average distance between ego and his
  nominees,...)
• Then use standard statistical procedures to e.g.
• Describe the network size and / or composition or
  compare it across populations
• Explain the size and / or composition of the
  networks (network as a dependent variable) with
  for example regression analysis (e.g., in SPSS, R)

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Regression analysis

• In simple linear regression, the model that
  describes the relation between a single dependent
  variable y and a single explanatory variable x is
  
• yi ß0 ß1xi ei
• ß0 and ß1 are referred to as the model
  parameters, and e is a probabilistic error term
  that accounts for the variability in y that
  cannot be explained by the linear relationship
  with x.

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Regression analysis

• Simple linear regression
• yi ß0 ß1xi ei
• More explanatory variables can be added
• yi ß0 ?ßpxip ei

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Illustration aggregate analysis

• S. G. B. Roberts, R. I. M. Dunbar, T. V. Pollet,
  T. Kuppens (2009). Exploring variation in active
  network size Constraints and ego
  characteristics. Social Networks, 31, 138-146.

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Illustration explaining personal network size
1. Explaining unrelated network size
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Illustration explaining personal network size
2. Explaining related network size
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Regression analysis at the aggregate level

• Is statistically correct provided that you do not
  make any cross-level inferences ( ecological
  fallacy)

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Hypothetical illustration of the statement to not
make cross-level inferences on the basis of
aggregate results

• I ask three persons to name ten friends each
• I further ask what the sex of each friend is and
  how close they feel with each friend on a scale
  from 0 (not close at all) to 4 (very close).
• My question is Do persons who have many women in
  their networks feel closer with their network
  members?

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Example Statistical relation at aggregate level
cannot be interpreted at tie level
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Example Statistical relation at aggregate level
cannot be interpreted at tie level
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Example Statistical relation at aggregate level
cannot be interpreted at tie level
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Type 2 Disaggregate analysis

• Disaggregated analysis of dyadic relations (e.g.,
  a linear regression analysis on the 4500 alters)
  is statistically not correct even though it has
  been done (e.g. Wellman et al., 1997, Suitor et
  al., 1997)
• Observations of alters are not statistically
  independent as is assumed by standard statistical
  procedures
• If observations of one respondent are correlated,
  standard errors will be underestimated, and
  consequently significance will be overestimated

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Type 3 Multilevel analysis

• Multilevel analysis is a generalization of linear
  regression, where the variance in outcome
  variables can be analyzed at multiple
  hierarchical levels. In our case, alters (level
  1) are nested within egos / networks (level 2),
  hence the variance is decomposed in variance
  between and within networks.
• The regression equation yi ß0 ß1xi Ri
  is now extended to yij ß0j ß1jxij
  Rij,
• where ß0j
  ?00 U0j

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Type 3 Multilevel analysis

• Dependent variable Some characteristic of the
  dyadic relationships (e.g., strength of tie).
• Note Special multilevel models have been
  developed for discrete dependent variables.
• Explanatory variables can be (among others)
• characteristics of egos (level 2),
• characteristics of alters (level 1),
• characteristics of the ego-alter pairs (level 1).
• Software e.g., R, MLwiN, HLM, VarCL

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Illustrations of multilevel analysis for personal
networks

• G. Mollenhorst, B. Völker, H. Flap (2008). Social
  contexts and personal relationships The effect
  of meeting opportunities on similarity for
  relationships of different strength. Social
  Networks, 30, 60-68.
• Mok, D., Carrasco, J.-A., Wellman, B. (2009).
  Does Distance Still Matter in the Age of the
  Internet? Urban Studies, forthcoming.

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The effect of the context where people meet on
the amount of similarity between them
(Mollenhorst, Völker, Flap)
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Illustration Analysis of the importance of
distance for overall contact frequency (Mok,
Carrasco Wellman)

• LnDist is the natural logarithm of residential
  distance between ego and alter, RIMM is a dummy
  variable indicating whether ego is an immigrant.
  Bold figures are significant at p lt .05, bold and
  italic at p lt .10.

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See for a good article about the possibilities of
multilevel analysis of personal networks

• Van Duijn, M. A. J., Van Busschbach, J. T.,
  Snijders, T. A. B. (1999). Multilevel analysis of
  personal networks as dependent variables. Social
  Networks, 21, 187-209.

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In summary, cross-sectional analysis of personal
networks...
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... but what about the relationships among alters?

• So far, we have only looked at the relationships
  a person (ego) has with his or her network
  members (alters)

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e.g., we ask people to nominate 45 others and to
report about their relationships with them
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But data can also be collected on the
relationships among network members
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... but what about the relationships among alters?

• Most researchers are only interested in
  alter-alter relations to say something about the
  structure of personal networks at the network
  level only

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... but what about the relations among alters?

• Most researchers are only interested in
  alter-alter relations to say something about the
  structure of personal networks at the network
  level only
• Compute structural measures at the aggregate
  level (e.g., density, betweenness centralization,
  number of cliques)
• Predict the structure of the networks in an
  aggregated analysis using for example regression
  analysis

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... but what about the relations among alters?

• It may however be interesting to analyze which
  alters are related (at the tie level)
• What predicts transitivity in personal relations?
  Or, as Louch expressed it, what predicts network
  integration?

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Exponential Random Graph Models (ERGMs)

• The class of ERGMs is a class of statistical
  models for the state of a social network at one
  time point.
• The presence or absence of a tie between any pair
  of actors in the network is modeled as a function
  of structural tendencies (e.g., transitivity,
  popularity), individual and dyadic covariates
  (e.g., similarity).

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Exponential Random Graph Models (ERGMs)

• ERGMs can be estimated in, among others, the
  software SIENA (up to version 3), statnet, pnet
  (e.g., in R)
• Dependent variable whether pairs of alters are
  related or not
• Explanatory variables
• characteristics of alters,
• characteristics of the relation alters have with
  ego,
• characteristics of the alter-alter pair,
• endogenous network characteristics such as
  transitivity
• (in a meta-analysis, characteristics of ego can
  be added as well)
• Type of analysis Apply a common ERGM to each
  network, then run a meta-analysis (cf. Lubbers,
  2003 Snijders Baerveldt, 2003 Lubbers
  Snijders, 2007).

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Ego influences parameter estimates strongly
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so we tend to leave ego out
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Example ERGM Predicting relations among alters
in the personal networks of immigrants
p lt .05, p lt .01. Conditioned on degree.
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In summary, cross-sectional analysis of personal
networks...
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Part II. Dynamic analysis

• How do personal networks change over time?
• Studies that collect data on personal networks in
  two or more waves in a panel study

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Interest in dynamic analysis

• Networks at one point in time are snapshots, the
  results of an untraceable history (Snijders)
• E.g., personal communities in Toronto (Wellman et
  al.)
• Changes following a focal life event (individual
  level)
• E.g., transition from high school to university
  (Degenne Lebeaux, 2005) childbearing, moving,
  return to school in midlife (Suitor Keeton,
  1997) retirement (Van Tilburg, 1992) marriage
  (Kalmijn et al., 2003) divorce (Terhell, Broese
  Van Groenou, Van Tilburg, 2007) widowhood
  (Morgan, Neal, Carder, 2000) migration
  (Lubbers, Molina, Lerner, Ávila, Brandes
  McCarty, 2009)
• Broader studies of social change Social and
  cultural changes in countries with dramatic
  institutional changes
• E.g., post-communism in Finland, Russia (Lonkila,
  1998), Eastern Germany (Völker Flap, 1995),
  Hungary (Angelusz Tardos, 2001), China (Ruan,
  Freeman, Dai, Pan, Zhang, 1997),

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Sources of change in (personal) networks

• Unreliability due to measurement error
• Inherent instability
• Systemic change
• External change
• Leik Chalkley (1997), Social Networks 19, 63-74

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Sources of change in (personal) networks

• Unreliability due to measurement error
• Inherent instability
• Systemic change
• External change
• Leik Chalkley (1997), Social Networks 19, 63-74

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Personal networks are layered
Personal network (
150)
Close / active network ( 50)
Sympathy group ( 15)
Support clique ( 5)
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Dependent variables in dynamic personal network
studies
Typology Feld, Suitor, Gartner Hoegh, 2007,
Field Methods, 19, 218-236.
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Type 1 Persistence of ties with alters across
time

• Dependent variable whether a tie persists or not
  to a subsequent time (dichotomous)
• Explanatory variables
• characteristics of ego at t1 (gender, job
  situation)
• change characteristics of ego t1-t2 (e.g., change
  in marital status)
• characteristics of alter at t1 (e.g., educational
  level)
• characteristics of the ego-alter pair at t1
  (e.g., tie strength)
• cross-level interactions (e.g., egos marital
  status kin)
• Type of analysis Logistic multilevel analysis
  (e.g., MLwin, Mixno)

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Type 1 Persistence of ties with alters across
time

• Logistic regression is used to predict the log
  odds that a tie persists over time (log odds
  log (p / q)).
• Logistic regression is in reality ordinary
  regression using the log odds as the response
  variable.
• The coefficients B in a logistic regression model
  are in terms of the log odds
• A unit increase in the explanatory variable x1
  will multiply the log odds for having a tie with
  eß1

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Illustration type 1 Explaining persistence of
ties for immigrants
p lt .05, p lt .01. Excluded Sex, employment
status, marital status, recent visits to country
of origin, changes in employment marital
status, tie duration, kin
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Type 2 Changes in characteristics of persistent
ties across time

• Dependent variable change in some characteristic
  of the relationship (e.g., change in strength of
  tie) or characteristic at t2, and use same
  characteristic at t1 as covariate
  (auto-correlation approach)
• Explanatory variables
• characteristics of ego at t1 (gender, job
  situation)
• change characteristics of ego t1-t2 (e.g., change
  in marital status)
• characteristics of alter at t1 (e.g., educational
  level)
• characteristics of the ego-alter pair at t1
  (e.g., tie strength)
• cross-level interactions (e.g., egos marital
  status kin)
• Type of analysis Multilevel analysis

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Example

• Change in contact frequency (visits and telephone
  calls) after an important life event
• Two time points shortly after the life event
  took place and four years later
• Van Duijn, M. A. J., Van Busschbach, J. T.,
  Snijders, T. A. B. (1999).

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Type 3 Changes in the size of the network across
time

• Dependent variable change in number of ties in
  the personal network
• Explanatory variables
• characteristics of ego at t1 (gender, job
  situation)
• change characteristics of ego t1-t2 (e.g., change
  in marital status)
• characteristics of the set of alters at t1
• Type of analysis Regression analysis at the
  aggregate level

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Illustration of the analysis of the stability of
personal networks over time (East York studies,
Wellman et al.)
Multiple regression predicting network turnover
(n 33)
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Type 4 Changes in overall network
characteristics across time

• Dependent variable change in compositional or
  structural variable (e.g., percentage of alters
  with higher education, density of the network)
• Explanatory variables, e.g.
• Characteristics of ego at t1
• Characteristics of the network at t1
• Type of analysis Regression analysis at the
  aggregate level

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Dynamic personal network analysis More than two
observations

• Add an extra level to the analysis representing
  the observation
• One-level models become two-level models
• Two-level models become three-level

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Dynamic personal network analysis More than two
observations

• Example of type 2 analysis with multiple
  observations Changes in contact after widowhood

Guiaux, M., van Tilburg, T. Broese van Groenou,
M. (2007). Changes in contact and support
exchange in personal networks after widowhood.
Personal Relationships, 14, 457-473
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More than two observations example of
alternative way (type 3 analysis)
E. L. Terhell, M. I. Broese van Groenou T. van
Tilburg (2004). Network dynamics in the long-term
period after divorce. Journal of Social and
Personal Relationships, 21, 719-738
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More than two observations example of
alternative way (type 3 analysis) contd
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See for example the chapter on longitudinal data
in this book

• T. A. B. Snijders R. J. Bosker (1999).
  Multilevel analysis. An introduction to basic and
  advanced multilevel modeling. London Sage
  Publications.

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In summary, dynamic analysis of personal networks
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... but what about the dynamics of alter-alter
relations?

• ??

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Time 1
An example of a changing personal network
Node color Stable alters are dark blue temporal
alters light blue Edge color Relations among
stable alters are dark blue among / with
temporal alters light blue Node size Egos
closeness with alter Labels Spanish, Fellow
Migrants, Originals, TransNationals
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An example of a changing personal network
Node color Stable alters are dark blue temporal
alters light blue Edge color Relations among
stable alters are dark blue among / with
temporal alters light blue Node size Egos
closeness with alter Labels Spanish, Fellow
Migrants, Originals, TransNationals
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Dependent variables in dynamic personal network
studies Composition and structure
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Type 5 Changes in ties among alters across time

• Dependent variable whether alters make new ties
  or break existing ties with other alters across
  time
• Independent variables
• characteristics of alters,
• characteristics of the relation alters have with
  ego,
• characteristics of the alter-alter pair,
• endogenous network characteristics such as
  transitivity
• (in a meta-analysis, characteristics of ego can
  be added as well)
• Type of analysis Apply a common SIENA model to
  each network (leaving ego out), then run a
  meta-analysis (cf. Lubbers, 2003 Snijders
  Baerveldt, 2003 Lubbers Snijders, 2007). A
  multilevel version of SIENA is on the agenda.

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Just a few thoughts about the use of SIENA for
personal networks

• Ego influences parameter estimates considerably,
  therefore, ego should be left out or
  alternatively, his or her relations can be given
  structural ones (to model that ego is by
  definition related to everyone else)
• As ego reports about the relationships between
  his or her alters, relations tend to be
  symmetric, so non-directed model type for SIENA
• Smaller networks or networks that have only a few
  changes per network (less than 40) can be
  combined into one or multiple multigroup
  project(s)

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Example Predicting the changes in ties among
alters in immigrant networks
p lt .01. N 44 respondents
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In summary, dynamic analysis of personal networks
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Conclusion

• Multiple statistical methods for personal network
  research, depending on your research interest
• Combining several methods probably gives the
  greatest insight into your data

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• Thanks!
• My e-mail address MirandaJessica.Lubbers_at_uab.es