references references (continued) anomalies risk-taking network density paradox apparent...
Post on 21-Dec-2015
218 views
TRANSCRIPT
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
DYNAMIC EFFECTS IN DRIFTING SMOKE RINGS
• 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
DYNAMIC EFFECTS IN DRIFTING SMOKE RINGS
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
DYNAMIC EFFECTS IN DRIFTING SMOKE RINGS
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
DYNAMIC EFFECTS IN DRIFTING SMOKE RINGS
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 PENRT to make transition is 7 months
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
DYNAMIC EFFECTS IN DRIFTING SMOKE RINGS
INFLUENCE FOLLOWED BY REJECTION
EVOLUTIONARY RISK
Peripheral Non Risk-Taker changes behaviour to match that of the Group
Peripheral Risk-Taker is rejected by the Group and becomes an Isolate
Expected time for PENRT to make transition is 7 months
Expected time for PERT to make transition is 6.4 months
Expected time for ISRT to make transition is 12 months
ISNRT
GPRT
PENRT
GPRT GPRT
PERT
SELECTION INFLUENCE
TIME
DYNAMIC EFFECTS IN DRIFTING SMOKE RINGS
Isolate Non Risk-Taker selects friend in the Group
Peripheral Non Risk-Taker changes behaviour to match that of the Group
Expected time for ISNRT to make transition is 8.6 months
Expected time for PENRT to make transition is 7 months
Expected time for PERT to make transition is 6.4 months
ASYMMETRICAL SELECTION/INFLUENCE
(EVOLUTIONARY)
Total = 22 Months
SELECTION
ISNRT
GPRT
ISRT
GPRT GPRT
PERT
INFLUENCE SELECTION
TIME
DYNAMIC EFFECTS IN DRIFTING SMOKE RINGS
Isolate Non Risk-Taker changes behaviour to match that of the Group
Isolate Risk-Taker is selected by the Group and becomes a Peripheral
Expected time for ISNRT to make transition is 8.6 months
Expected time for ISRT to make transition is 12 months
Expected time for PERT to make transition is 6.4 months
Total = 27 Months
SYMMETRICAL INFLUENCE /SELECTION
(NON-EVOLUTIONARY)
SELECTION
PERT
GPNRT
PENRT
GPNRT
GPNRT
INFLUENCE SELECTION
TIME
DYNAMIC EFFECTS IN DRIFTING SMOKE RINGS
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
Expected time for PERT to make transition is 6.4 months
Expected time for PENRT to make transition is 7 months
Total = 13.4 Months
PERT
GPNRT GPNRT
GPNRT
SELECTION INFLUENCE
TIME
DYNAMIC EFFECTS IN DRIFTING SMOKE RINGS
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 PERT to make transition is 6.4 months
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
DYNAMIC EFFECTS IN DRIFTING SMOKE RINGS
Isolate Risk-Taker selects friend to become a peripheral
Peripheral Risk-Taker changes behaviour to match that of the Group
Expected time for ISRT to make transition is 12 months
Expected time for PERT to make transition is 6.4 months
Expected time for PENRT to make transition is 7 months
ASYMMETRICAL SELECTION/INFLUENCE
(EVOLUTIONARY)
Total = 25.6 Months
SELECTION
ISRT
GPNRT
ISNRT
GPNRT GPNRT
PENRT
INFLUENCE SELECTION
TIME
DYNAMIC EFFECTS IN DRIFTING SMOKE RINGS
Isolate Risk-Taker changes behaviour to match that of the Group
Isolate Non Risk-Taker selects friend in the Group and becomes a Peripheral
Expected time for ISRT to make transition is 12 months
Expected time for ISNRT to make transition is 8.6 months
Expected time for PENRT to make transition is 7 months
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