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### The statistical analysis of personal network data

I. Cross-sectional analysis

II. Dynamic analysis

Miranda Lubbers,

Autonomous University of Barcelona

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)

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.

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., alter’s age, sex, nationality)
- Information about the ego-alterpairs (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)

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...

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

Type I: Aggregated analysis

Type II: Disaggregated analysis

(not okay, forget about it quickly!)

Type III: Multilevel analysis

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

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 = β0 + β1xi + εi

- β0and β1are referred to as the model parameters, and ε is a probabilistic error term that accounts for the variability in y that cannot be explained by the linear relationship with x.

Regression analysis

- Simple linear regression:

yi = β0 + β1xi + εi

- More explanatory variables can be added:

yi = β0 + ∑βpxip + εi

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.

Illustration: explaining personal network size

1. Explaining unrelated network size

Illustration: explaining personal network size

2. Explaining related network size

Regression analysis at the aggregate level…

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

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?”

Example: Statistical relation at aggregate level cannot be interpreted at tie level

Example: Statistical relation at aggregate level cannot be interpreted at tie level

Example: Statistical relation at aggregate level cannot be interpreted at tie level

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

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 ego’s / networks (level 2), hence the variance is decomposed in variance between and within networks.
- The regression equation yi = β0 + β1xi+ Ri is now extended to yij= β0j + β1jxij + Rij,

where β0j= γ00 + U0j

- 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 ego’s (level 2),
- characteristics of alters (level 1),
- characteristics of the ego-alter pairs (level 1).
- Software: e.g., R, MLwiN, HLM, VarCL

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.

The effect of the context where people meet on the amount of similarity between them (Mollenhorst, Völker, Flap)

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 < .05, bold and italic at p <.10.

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.

... 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)…

e.g., we ask people to nominate 45 others and to report about their relationships with them…

But data can also be collected on the relationships among network members…

... 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

... 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

... 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?

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).

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).

Example ERGM: Predicting relations among alters in the personal networks of immigrants

* p < .05, ** p < .01. Conditioned on degree.

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

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),

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

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

Dependent variables in dynamic personal network studies

Typology: Feld, Suitor, & Gartner Hoegh, 2007, Field Methods, 19, 218-236.

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., ego’s marital status × kin)
- Type of analysis: Logistic multilevel analysis (e.g., MLwin, Mixno)

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 eβ1

Illustration type 1: Explaining persistence of ties for immigrants

* p < .05, ** p < .01.Excluded: Sex, employment status, marital status, recent visits to country of origin, changes in employment & marital status, tie duration, kin

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., ego’s marital status × kin)
- Type of analysis: Multilevel analysis

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).

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

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)

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

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

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

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

More than two observations: example of alternative way (type 3 analysis) – cont´d

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.

An example of a changing personal network

Time 1Node 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: Ego’s closeness with alter

Labels:Spanish, Fellow Migrants, Originals, TransNationals

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: Ego’s closeness with alter

Labels:Spanish, Fellow Migrants, Originals, TransNationals

Dependent variables in dynamic personal network studies: Composition and structure

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.

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)

Example: Predicting the changes in ties among alters in immigrant networks

* p < .01. N = 44 respondents

Conclusion

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

Thanks!

My e-mail address: [email protected]

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