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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., 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)

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

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

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)

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.

Regression analysis

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

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?

Network A Network A Network B Network B Network C Network C

F 1.0 M 0.5 F 0.5 M 0.5 F 0.5 M 3.0

F 2.0 M 0.5 F 1.0 M 1.0 F 0.5 M 4.0

M 1.0 F 1.5 M 1.5 F 1.0

M 1.0 F 2.0 M 2.0 F 1.0

M 1.0 F 2.0 M 2.0 F 1.0

M 1.5 F 1.5

M 1.5 F 1.5

M 2.0 F 2.0

Example Statistical relation at aggregate level

cannot be interpreted at tie level

Network A Network A Network B Network B Network C Network C

F 1.0 M 0.5 F 0.5 M 0.5 F 0.5 M 3.0

F 2.0 M 0.5 F 1.0 M 1.0 F 0.5 M 4.0

M 1.0 F 1.5 M 1.5 F 1.0

M 1.0 F 2.0 M 2.0 F 1.0

M 1.0 F 2.0 M 2.0 F 1.0

M 1.5 F 1.5

M 1.5 F 1.5

M 2.0 F 2.0

20 female 20 female 50 female 50 female 80 female 80 female

Av. tie strength 1.2 Av. tie strength 1.2 Av. tie strength 1.4 Av. tie strength 1.4 Av. tie strength 1.6 Av. tie strength 1.6

Example Statistical relation at aggregate level

cannot be interpreted at tie level

Network A Network A Network B Network B Network C Network C

F 1.0 M 0.5 F 0.5 M 0.5 F 0.5 M 3.0

F 2.0 M 0.5 F 1.0 M 1.0 F 0.5 M 4.0

M 1.0 F 1.5 M 1.5 F 1.0

M 1.0 F 2.0 M 2.0 F 1.0

M 1.0 F 2.0 M 2.0 F 1.0

M 1.5 F 1.5

M 1.5 F 1.5

M 2.0 F 2.0

20 female 20 female 50 female 50 female 80 female 80 female

Av. tie strength 1.2 Av. tie strength 1.2 Av. tie strength 1.4 Av. tie strength 1.4 Av. tie strength 1.6 Av. tie strength 1.6

At tie level 50 female, 50 male, av. tie strength women 1.3, av. tie strength men 1.5 At tie level 50 female, 50 male, av. tie strength women 1.3, av. tie strength men 1.5 At tie level 50 female, 50 male, av. tie strength women 1.3, av. tie strength men 1.5 At tie level 50 female, 50 male, av. tie strength women 1.3, av. tie strength men 1.5 At tie level 50 female, 50 male, av. tie strength women 1.3, av. tie strength men 1.5 At tie level 50 female, 50 male, av. tie strength women 1.3, av. tie strength men 1.5

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 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

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

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 lt .05, bold and

italic at p lt .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.

In summary, cross-sectional analysis of personal

networks...

Unit of analysis Focus of analysis Focus of analysis

Unit of analysis Existence of ties Content of ties

Ties - What predicts the contents of ties? Multilevel analysis

Personal networks What predicts the size of the network? Regression analysis at aggregate level What predicts the composition of networks? Regression analysis at aggregate level

... 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).

Ego influences parameter estimates strongly

so we tend to leave ego out

Example ERGM Predicting relations among alters

in the personal networks of immigrants

Parameter s.e. Q

Alternating 2-stars (degree) -0.17 0.20 0.81 181.74

Alternating 2-triangles (transitivity) 2.36 0.34 1.36 233.03

Alter is Spanish (vs. from country of origin) Alter is a fellow migrant (vs. ,,) Two alters have same country of residence and origin -0.01 0.07 0.51 0.04 0.07 0.10 0.10 0.36 0.34 53.43 86.40 40.56

Two alters have shared group membership 0.54 0.11 0.44 95.44

Egos feelings of closeness with alter 0.05 0.02 0.06 50.78

p lt .05, p lt .01. Conditioned on degree.

In summary, cross-sectional analysis of personal

networks...

Unit of analysis Focus of analysis Focus of analysis

Unit of analysis Existence of ties Contents of ties

Ego-alter ties - What predicts the contents of ties? Multilevel analysis

Alter-alter ties What predicts whether there are ties among alters? ERGM What predicts the contents of ties among alters? Social Relation Model

Personal networks What predicts the size of the network? Regression analysis at aggregate level What predicts the composition / structure of networks? Regression analysis at aggregate level

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

Personal networks are layered

Personal network (

150)

Close / active network ( 50)

Sympathy group ( 15)

Support clique ( 5)

Dependent variables in dynamic personal network

studies

Focus Level Existence of ties (dichotomous) Contents of ties (valued)

Ego-alter ties Persistence of ties with alters Changing contents of ties with alters

Networks Expansion / contraction of networks Changing composition of networks

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., egos 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

eß1

Illustration type 1 Explaining persistence of

ties for immigrants

Fixed effects B SE (B)

Constant -3.256 0.520

Egos length of residence in Spain 0.192 0.109

Personal network density -3.251 1.380

Egos frequency of contact with alter 0.323 0.048

Egos emotional closeness with alter 0.508 0.073

Alter is Spanish 0.915 0.513

Alter is a fellow migrant -0.626 0.227

Alter is a transnational -0.498 0.235

Alters degree centrality 0.073 0.014

Egos length of residence alter is Spanish -0.365 0.122

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

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

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).

(No Transcript)

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

(No Transcript)

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) contd

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.

In summary, dynamic analysis of personal networks

Focus Level Existence of ties (dichotomous) Contents of ties (valued)

Ego-alter ties Persistence of ties with alters Logistic multilevel analysis Changing contents of ties with alters Multilevel analysis

Networks Expansion / contraction of networks Regression analysis at the aggregate level Changing composition of networks Regression analysis at the aggregate level

... but what about the dynamics of alter-alter

relations?

- ??

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

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

Dependent variables in dynamic personal network

studies Composition and structure

Focus Units Existence of ties (dichotomous) Contents of ties (valued)

Ego-alter ties Persistence of ties with alters Changing contents of ties with alters

Alter-alter ties Formation / decay of ties among alters Changing contents of ties among alters

Networks Expansion / contraction of networks changing structure Changing composition of networks

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

Parameter µ s.e. Q

Rate 6.83 0.74 2.48 86.51

Degree -0.94 0.30 1.49 320.08

Degree-related popularity (sqrt) -0.20 0.02 0.0 259.01

Transitivity 0.48 0.12 0.75 1371.63

Alter is Spanish Alter is a fellow migrant Same country residence / origin 0.29 0.57 0.69 0.16 0.13 0.05 0.59 0.50 0.0 66.81 155.07 126.39

Shared group membership 0.73 0.05 0.0 79.74

Closeness alter 0.18 0.03 0.0 139.23

Closeness alter 1 alter 2 0.01 0.02 0.0 56.45

p lt .01. N 44 respondents

In summary, dynamic analysis of personal networks

Focus Level Existence of ties (dichotomous) Contents of ties (valued)

Ego-alter ties Persistence of ties with alters Logistic multilevel analysis Changing contents of ties with alters Multilevel analysis

Alter-alter ties Formation / decay of ties among alters SIENA Changing contents of ties among alters SIENA valued data

Networks Expansion / contraction of networks Regression analysis at the aggregate level Changing composition of networks Regression analysis at the aggregate level

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 MirandaJessica.Lubbers_at_uab.es

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