references references (continued) anomalies risk-taking network density paradox apparent...

DYNAMIC EFFECTS IN DRIFTING SMOKE RINGS

MICHAEL PEARSON

CENTRE FOR MATHEMATICS & STATISTICS

MANAGEMENT SCHOOL

NAPIER UNIVERSITY

EDINBURGH EH11 4BN

E-mail : [email protected]

References

E&B Ennett, S.T. ; Bauman, K.E.(1993)

Peer group structure and adolescent cigarette smoking: a social network

analysis. Journal of Health and Social Behavior, 34, 226-236.

Haynie D.L. (2001) Delinquent peers revisited : Does network structure

matter? American Journal of Sociology, 106(4), 1013-57.

K&L Kalbfleisch, J.D. & Lawless, J.F. (1985) The Analysis of Panel Data under a

Markov Assumption. Journal of the American Statistical Association.

80(392), 863-871

O&D Oetting, E.R. & Donnermeyer J.F. (1998) Primary Socialisation Theory : The

Etiology of Drug Use and Deviance. Substance Use & Misuse, 33(4), 995-

1026.

P&M Pearson, M.A. & Michell, L. (2000) Smoke Rings : Social network analysis of

friendship groups, smoking and drug-taking. Drugs: education, prevention

and policy, Vol 7, No. 1 p 21-37.

P&W Pearson, M.A. & West, P. (2003) Drifting Smoke Rings : Social network

analysis and Markov processes in a longitudinal study of friendship groups,

risk-taking. Connections 25(2):59-76.

Richards, W. W. (1989) The NEGOPY network analysis program.

Department of Communications, Simon Fraser University, Burnaby, BC.

Singer, B. & Spilerman, S. (1976) The representation of social processes by

Markov models. American Journal of Sociology 82(1) 1-54.

Snijders, T.A.B. 1996. Stochastic Actor-Oriented Models for Network Change.

Journal of Mathematical Sociology 21, 149-72.

Wasserman, S. & Faust, K. (1994). Social Network Analysis : Methods and

Applications. Cambridge University Press.

West, P. & Sweeting, H. (1996) Background rationale and design of the West

of Scotland 11-16 Study. MRC Medical Sociology Unit Working Paper

No.52. Glasgow.

References (continued)

ANOMALIESRisk-taking Network Density Paradox

• Apparent contradictions in research findings• Network density is an important moderator of peer

delinquency, defined as a range of behaviour patterns (Haynie, 2001)

• Higher density implies higher delinquency• Higher smoking among liaisons and isolates than

among group members (Ennett & Bauman,1994)• Higher smoking among popular pupils (Abel et al)

ANOMALIESRisk-taking Network Density Paradox

• Researchers use differing methodologies

• Network density defined as ego-centric measure (Urdry & Bearman, Haynie) when limited data available

• Ego-centric network density is NOT an ideal measure of peer cohesion


• Longitudinal Social Network Study selected from the sample frame of the West of Scotland 11-16 study into teenage lifestyle, health behaviour and friendships

• Three time points selected from 1995 till 1997 in one average school in Glasgow

• We measured risk-taking (smoking and cannabis use) behaviour and also social network position

• We identified three main social positions : Group member, peripheral to group and relative isolate

SMOKE RINGS(P&M) METHODS

• Primary socialisation theory highlights the central part played by peer groups for the socialisation issues of selection and influence (O&D)

• Cohesive peer groups are central to the study, since a (near) complete data set is available (95% of year group)

• Group peripherals considered to be an important target for selection and influence surrounding risk-taking and non risk-taking behaviour

• The remaining pupils were categorised as relative isolates

SMOKE RINGS METHODOLOGY

• Three Cohesive Network Positions

• Peer Group Member

• Peripheral to Peer Group

• Relative Isolate

• Two Behavioural Characteristics

• Risk-Taker (smoker or cannabis)

• Non Risk-Taker

DEFINITION OF COHESION

• Peer cohesion defined as• Mutuality of ties• Closeness or reachability of subgroup

members• Frequency of ties among members• Relative frequency of ties among subgroup

members compared to non members (Wasserman & Faust)

CHOICE OF SOFTWARE

• NEGOPY defines cohesive groups as a set of at least 3 people who :

• Have more than 50% of their linkage with one another (closeness & frequency)

• Are connected by some path lying entirely within the group to each of the other members in the group (reachability)

• Who remain so connected when up to 10% of the group is removed (relative frequency)

10

11

14

189

158 T2

8356

70

79194 136 T2

96

TN

88 T2

90 T2

134 T272

61

6269

74

32 T2

91 T2

152 T2

195

112

46

153 TN

156 T2

42 TN

157 T2

146 TN

60 T2

81 T2

Smoke Rings : Male Groups and Peripherals (time point 2)

Smokes occasionally/regularly

Tried/uses cannabis

Tried/uses glue

Tried/uses other drugs

KEY

Figure 3 Time Point 1(S2) Top girls and ‘peripherals’

Group 3

107

99

38

77

51

44

37 41

40

Tree Node Isolate 2

98

84 Isolate 2 Tree Node

smokers marijuana alcohol weekly glue, acid, speed, pills

51 LF84

44

142

147

139

Group 5 All Girls Group 1

All Girls Group 13 All Girls

11

107

38

37

43

98

41

202 57

Drifting Smoke Rings : Top Girls and Peripherals (time point 2)

Figure 6Time Point 3 Top girls

Group 1

107

98

38

43

11

44

3739

Isolate 2

99Isolate 2

smokers marijuana alcohol weekly glue, acid, speed, pills

Group 7All Girls 51

84

142201

147

26

DRIFTING SMOKE RINGSLONGITUDINAL METHODOLOGY

• Panel Data Collected• Behavioural effect (risk-taking or non risk-taking)

together with network effect (peer group, peripheral, isolate) give 6 states

• Extension to two time points gives rise to 36 Markov transitional states

• In Drifting Smoke Rings we studied the Markov transitional matrices for time points 1 to 2 and for time points 2 to 3.

MARKOV METHODS

• Singer & Spilerman determined whether observations on an empirical process arise via the evolution of a continuous time Markov model (Embeddability)

• Kalbfleisch & Lawless avoid complexity of embeddability by using a Maximum Likelihood estimator for the intensity matrix rather than the transitional matrix

SMOKE RINGS AND DRIFTING SMOKE RINGS

KEY FINDINGS(PERIPHERALS)

• The Markov process is non-stationary. More peripherals than expected move to Group Risk-Taking at the transition from age 14 to 15

• The expected time spent in the peripheral states (PENRT and PERT) is less than that spent in other states (unstable)

• At all time points of the study the risk-taking behaviour of the pupils on the periphery of peer groups significantly reflected the behaviour of the groups themselves (gullible)

Transitional Table for Sociometric States (Age 13 to 14) TP1 to TP2 S3NUM S2NUM GPRT PERT ISRT ISNRT PENRT GPNRT OTHER Total GPRT 9 3 4 2 3 21 PERT 4 2 2 1 9 ISRT 1 1 1 1 1 5 ISNRT 1 4 2 5 8 2 2 24 PENRT 1 4 3 11 12 1 32 GPNRT 13 1 5 5 10 23 4 61 OTHER 1 1 5 7 Total 29 11 18 15 31 41 14 159 NB An empty space implies no pupils made that transition

Table 1 Key : GP = Group PE = Peripheral IS = Relative Isolate RT = Risk-Taker NRT = Non Risk-Taker

Transition Table for Sociometric States(Age 14 to 15) TP2 to TP3 S4NUM S3NUM GPRT PERT ISRT ISNRT PENRT GPNRT OTHER Total GPRT 19 6 1 3 29 PERT 5 1 3 2 11 ISRT 4 1 9 1 3 18 ISNRT 1 1 4 2 3 4 15 PENRT 6 3 3 6 2 8 3 31 GPNRT 12 2 3 6 17 1 41 OTHER 2 4 8 14 Total 47 15 21 13 11 30 22 159 NB An empty space implies no pupils made that transition

Table 2 Key : GP = Group PE = Peripheral IS = Relative Isolate RT = Risk-Taker NRT = Non Risk-Taker

Transition Matrix (23P) for Sociometric States (Age 14 to 15) TP2-TP3 GPRT PERT ISRT ISNRT PENRT GPNRT OTHER

GPRT 0.655 0.207 0.035 0 0 0 0.103 PERT 0.455 0.091 0.273 0 0 0.182 0 ISRT 0.222 0.056 0.5 0 0.056 0 0.167

ISNRT 0.067 0 0.067 0.267 0.133 0.2 0.267 PENRT 0.193 0.097 0.097 0.194 0.065 0.258 0.097 GPNRT 0.293 0.049 0 0.073 0.146 0.415 0.024 OTHER 0 0.143 0.286 0 0 0 0.571

Key : GP = Group PE = Peripheral IS = Relative Isolate RT = Risk-Taker NRT = Non Risk-Taker

EXPECTED SOJOURN TIMES

• Maximum Likelihood Approach (K & L)

• Algorithm implemented using MATLAB

• Search for a solution, Q, to

• Where P is the transitional matrix and Q is the intensity matrix

QtetP )(

SOJOURN TIMES

• Once Q is identified then the expected waiting (sojourn) times spent in each state (i) during a transitional period are given by :

Expected time (i) =

• Find an initial approximation for Q as :

iiq/1

)ln( iiii pq

SOJOURN TIMES

• Assign other values using :

And expm(Q) = P (since t=1)

Where expm( ) is the MATLAB operator for

matrix exponentiation

07

1

j

ijq

SOJOURN TIMES

• Choose a basis :

For the intensity matrix , Q, such that

We tested models with b=12,18 and 22 and identified an improved value of using the K&L algorithm.

],....[ 10 b

),....,( 1 bij fq )7,....,1;7,...,1( ji

Q̂

Maximum Likelihood Estimator of Expected Time in Each Transition State and Observed Mean Value GPRT PERT ISRT ISNRT PENRT GPNRT Age 13-14 12 7.3 8.5 7.6 11.2 11.2 Age 14-15 28.4 6.3 20.2 9.8 5.6 15.4 Average 16.9 6.4 12 8.6 7 12.9 Observed Age 13-15

13.4 7.8 10.8 9.5 9.7 12.9

Time in Months

SOCIOMETRIC POSITIONS 16.9 6.4 12 8.6 7 12.9

Average Waiting Time between Throws of the Die(months)

Group Risk-Takers

Peripheral Risk-Takers

Isolate Risk-Takers

Isolate Non Risk-Takers

Peripheral Non Risk-Takers

Group Non Risk-Takers

LOW RISK-TAKERS HIGH RISK-TAKERS

GPRT GPRT

INFLUENCE

TIME


INFLUENCE WITHIN A GROUP

Group Non Risk-Taker matches Group behaviour and becomes a Group Risk-Taker

GPNRT GPRT

Expected time for GPNRT to make transition is 12.9 months

Expected time for GPRT to make transition is 16.9 months

PENRT

GPRT

PERT

GPRT

GPRT

INFLUENCE SELECTION

TIME


INFLUENCE FOLLOWED BY SELECTION

(EVOLUTIONARY)

Peripheral Non Risk-Taker changes behaviour to match that of the Group

Peripheral Risk-Taker is selected by the Group and becomes a Group Risk-Taker

GPRT

Expected time for PENRT to make transition is 7 months

Expected time for PERT to make transition is 6.4 months

Total = 13.4 Months

PENRT

GPRT GPRT

GPRT

SELECTION INFLUENCE

TIME


SELECTION FOLLOWED BY INFLUENCE

(NON-EVOLUTIONARY)

Peripheral Non Risk-Taker is selected by the Group and becomes a Group Non Risk-Taker

Group Non Risk-Taker matches Group behaviour and becomes a Group Risk-Taker

GPNRT


Expected time for GPNRT to make transition is 12.9 months

Total = 19.9 Months

GPRT

PENRT

GPRT

PERT

GPRT GPRT

ISRT

INFLUENCE REJECTION

TIME


INFLUENCE FOLLOWED BY REJECTION

EVOLUTIONARY RISK


Peripheral Risk-Taker is rejected by the Group and becomes an Isolate



Expected time for ISRT to make transition is 12 months

ISNRT

GPRT

PENRT

GPRT GPRT

PERT

SELECTION INFLUENCE

TIME


Isolate Non Risk-Taker selects friend in the Group


Expected time for ISNRT to make transition is 8.6 months



ASYMMETRICAL SELECTION/INFLUENCE

(EVOLUTIONARY)

Total = 22 Months

SELECTION

ISNRT

GPRT

ISRT

GPRT GPRT

PERT

INFLUENCE SELECTION

TIME


Isolate Non Risk-Taker changes behaviour to match that of the Group

Isolate Risk-Taker is selected by the Group and becomes a Peripheral




Total = 27 Months

SYMMETRICAL INFLUENCE /SELECTION

(NON-EVOLUTIONARY)

SELECTION

PERT

GPNRT

PENRT

GPNRT

GPNRT

INFLUENCE SELECTION

TIME


INFLUENCE FOLLOWED BY SELECTION

(EVOLUTIONARY)

Peripheral Risk-Taker changes behaviour to match that of the Group

Peripheral Non Risk-Taker is selected by the Group and becomes a Group Non Risk-Taker

GPNRT



Total = 13.4 Months

PERT

GPNRT GPNRT

GPNRT

SELECTION INFLUENCE

TIME


SELECTION FOLLOWED BY INFLUENCE

(NON-EVOLUTIONARY)

Peripheral Risk-Taker is selected by the Group and becomes a Group Risk-Taker

Group Risk-Taker matches Group behaviour and becomes a Group Non Risk-Taker

GPRT


Expected time for GPRT to make transition is 16.9 months

Total = 23.3 Months

GPNRT

ISRT

GPNRT

PERT

GPNRT GPNRT

PENRT

SELECTION INFLUENCE

TIME


Isolate Risk-Taker selects friend to become a peripheral

Peripheral Risk-Taker changes behaviour to match that of the Group




ASYMMETRICAL SELECTION/INFLUENCE

(EVOLUTIONARY)

Total = 25.6 Months

SELECTION

ISRT

GPNRT

ISNRT

GPNRT GPNRT

PENRT

INFLUENCE SELECTION

TIME


Isolate Risk-Taker changes behaviour to match that of the Group

Isolate Non Risk-Taker selects friend in the Group and becomes a Peripheral




SYMMETRICAL SELECTION/INFLUENCE

(NON-EVOLUTIONARY)

Total = 27.6 Months

SELECTION

Evolutionary Network Paths

• Existing Link with Another • Change behaviour to match other (influence)• Selection into Group (or rejection) follows• No Existing Link with Another (isolate)• Establish link (selection)• Match behaviour (influence)• Selection into Group (or rejection) follows

Transitional Paths from Ages 13 to 14 to 15 by Sociometric States Path Frequency (Gender) GPNRT-GPNRT-GPNRT (6-6-6) 11 (6M,5F) GPNRT-GPRT-GPRT (6-1-1) 11 (9M,2F) GPRT-GPRT-GPRT (1-1-1) 6 (2M,4F) PENRT-GPNRT-GPNRT (5-6-6) 4 (3M,1F) GPRT-GPRT-PERT (1-1-2) 3 GPRT-PERT-GPRT (1-2-1) 3 GPRT-ISRT-ISRT (1-3-3) 3 GPRT-GPNRT-GPRT (1-6-1) 3 GPNRT-GPNRT-PENRT (6-5-5) 3 PENRT-PENRT-ISNRT (5-5-4) 3 PENRT-PENRT-GPNRT (5-5-6) 3 PENRT-GPNRT-PENRT (5-6-5) 3 GPNRT-ISRT-ISRT (6-3-3) 3 PENRT-ISRT-GPRT (5-3-1) 2 PENRT-GPNRT-GPRT (2-6-1) 2 Table 2

Anomalies Revisited: Possible Explanations

• Stagnating effect of isolate risk-taking compared with isolate non risk-taking reflected in higher sojourn times

• Confusion between network density and popularity (measured by in-degree)

• The anomaly of smoking and risk-taking associated with sociometric position and popularity (in-degree) is largely explained by Socio-Economic Status (West of Scotland THiS Study)

OTHER FINDINGS

• Abel et al. support the findings of Pearson & Michell concerning high-status ‘top girls’, who are popular and smoke together in small groups

• low-status peripheral ‘try-hards’, who smoke in an effort to be included in a group

references references (continued) anomalies risk-taking network density paradox apparent...

Documents

girls group

risktaker slide

relative frequency slide

network effect peer

year group group peripherals

group reachability

relative isolate slide

peer group relative