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), 9951026.
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.
References (continued)
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.
ANOMALIES
Risk-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)
ANOMALIES
Risk-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 risktaking 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)
Smoke Rings : Male Groups and Peripherals (time point 2)
158 T2
10
56
83
14
79
136 T2
194
195
189
70
91 T2
46
152 T2
32 T2
88 T2
96
TN
90 T2
112
11
134 T2
72
153 TN
157 T2
61
69
62
156 T2
146 TN
KEY
Smokes occasionally/regularly
42 TN
Tried/uses cannabis
81 T2
74
Tried/uses glue
Tried/uses other drugs
60 T2
Figure 3 Time Point 1(S2)
Top girls and ‘peripherals’
Group 3
41
38
37
44
Tree
Node
107
51
99
77
smokers
marijuana
alcohol weekly
glue, acid,
speed, pills
98
Tree
Node
84 Isolate 2
40
Isolate 2
Drifting Smoke Rings : Top Girls and Peripherals (time point 2)
38
51 LF
84
44
37
41
11
142
43
107
139
202
Group 5
All Girls
147
98
Group 1
Group 13
All Girls
All Girls
57
Figure 6
Time Point 3 Top girls
Group 1
107
44
Isolate 2
11
98
43
smokers
marijuana
alcohol weekly
glue, acid,
speed, pills
39
38
37
Group 7
All Girls
51
26
99
Isolate 2
84
147
142
201
DRIFTING SMOKE RINGS
LONGITUDINAL 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 RiskTaking 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 ( P23 ) for Sociometric States (Age 14 to 15)
TP2-TP3
GPRT
PERT
ISRT
ISNRT
PENRT
GPNRT
OTHER
GPRT
0.655
0.455
0.222
0.067
0.193
0.293
0
PERT
0.207
0.091
0.056
0
0.097
0.049
0.143
ISRT
0.035
0.273
0.5
0.067
0.097
0
0.286
ISNRT
0
0
0
0.267
0.194
0.073
0
PENRT
0
0
0.056
0.133
0.065
0.146
0
Key : GP = Group PE = Peripheral IS = Relative Isolate
RT = Risk-Taker NRT = Non Risk-Taker
GPNRT
0
0.182
0
0.2
0.258
0.415
0
OTHER
0.103
0
0.167
0.267
0.097
0.024
0.571
EXPECTED SOJOURN TIMES
• Maximum Likelihood Approach (K & L)
• Algorithm implemented using MATLAB
• Search for a solution, Q, to
P(t )  e
• Where P is the transitional matrix and Q is
the intensity matrix
Qt
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) = 1 / qii
• Find an initial approximation for Q as :
qii  ln( pii )
SOJOURN TIMES
• Assign other values using :
7
q ij

j 1
0
And
expm(Q) = P (since t=1)
Where expm( ) is the MATLAB operator for
matrix exponentiation
SOJOURN TIMES
• Choose a basis :
 0  [1 ,.... b ]
For the intensity matrix , Q, such that
qij  f (1 ,...., b )
(i  1,..., 7; j  1,...., 7)
We tested models with b=12,18 and 22 and
identified an improved value of Q̂ using the K&L
algorithm.
Maximum Likelihood Estimator of Expected Time in Each
Transition State and Observed Mean Value
Age 13-14
Age 14-15
Average
Observed
Age 13-15
GPRT
12
28.4
16.9
13.4
PERT
7.3
6.3
6.4
7.8
Time in Months
ISRT
8.5
20.2
12
10.8
ISNRT
7.6
9.8
8.6
9.5
PENRT
11.2
5.6
7
9.7
GPNRT
11.2
15.4
12.9
12.9
SOCIOMETRIC
POSITIONS
HIGH RISK-TAKERS
Group
Risk-Takers
16.9
Peripheral
Risk-Takers
6.4
LOW RISK-TAKERS
Isolate
Risk-Takers
12
Isolate Non
Risk-Takers
8.6
Peripheral Non
Risk-Takers
7
Group Non
Risk-Takers
12.9
Average Waiting Time between Throws of the Die(months)
DYNAMIC EFFECTS IN DRIFTING SMOKE RINGS
INFLUENCE WITHIN A GROUP
Group Non Risk-Taker matches Group
behaviour and becomes a Group Risk-Taker
INFLUENCE
GPNRT
GPRT
GPRT
GPRT
Expected time for GPNRT to
make transition is 12.9 months
TIME
Expected time for GPRT to
make transition is 16.9 months
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
INFLUENCE
SELECTION
GPRT
PENRT
PERT
GPRT
Expected time for PENRT to
make transition is 7 months
GPRT
GPRT
Expected time for PERT to
make transition is 6.4 months
TIME
Total = 13.4 Months
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
SELECTION
INFLUENCE
GPRT
GPNRT
PENRT
GPRT
Expected time for PENRT to
make transition is 7 months
GPRT
GPRT
Expected time for GPNRT to
make transition is 12.9 months
TIME
Total = 19.9 Months
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
INFLUENCE
REJECTION
GPRT
PENRT
GPRT
PERT
Expected time for PENRT to
make transition is 7 months
GPRT
ISRT
Expected time for PERT to
Expected time for ISRT to
make transition is 6.4 months make transition is 12 months
TIME
DYNAMIC EFFECTS IN DRIFTING SMOKE RINGS
ASYMMETRICAL SELECTION/INFLUENCE
(EVOLUTIONARY)
Isolate Non Risk-Taker
selects friend in the Group
Peripheral Non Risk-Taker changes
behaviour to match that of the Group
SELECTION
ISNRT
INFLUENCE
PENRT
GPRT
GPRT
Expected time for ISNRT to make
Expected time for PENRT to
transition is 8.6 months
make transition is 7 months
TIME
SELECTION
PERT
GPRT
Expected time for PERT to
make transition is 6.4 months
Total = 22 Months
DYNAMIC EFFECTS IN DRIFTING SMOKE RINGS
SYMMETRICAL INFLUENCE /SELECTION
(NON-EVOLUTIONARY)
Isolate Non Risk-Taker changes
behaviour to match that of the Group
Isolate Risk-Taker is selected by the
Group and becomes a Peripheral
INFLUENCE
ISNRT
SELECTION
ISRT
GPRT
SELECTION
PERT
GPRT
GPRT
Expected time for ISNRT to makeExpected time for ISRT to
Expected time for PERT to
transition is 8.6 months
make transition is 12 months make transition is 6.4 months
TIME
Total = 27 Months
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
INFLUENCE
SELECTION
GPNRT
PERT
PENRT
GPNRT
Expected time for PERT to
make transition is 6.4 months
GPNRT
GPNRT
Expected time for PENRT to
make transition is 7 months
TIME
Total = 13.4 Months
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
SELECTION
INFLUENCE
GPNRT
GPRT
PERT
GPNRT
Expected time for PERT to
make transition is 6.4 months
GPNRT
GPNRT
Expected time for GPRT to
make transition is 16.9 months
TIME
Total = 23.3 Months
DYNAMIC EFFECTS IN DRIFTING SMOKE RINGS
ASYMMETRICAL SELECTION/INFLUENCE
(EVOLUTIONARY)
Isolate Risk-Taker selects friend to
become a peripheral
Peripheral Risk-Taker changes
behaviour to match that of the Group
SELECTION
ISRT
INFLUENCE
PERT
GPNRT
Expected time for ISRT to
make transition is 12 months
SELECTION
PENRT
GPNRT
Expected time for PERT to
make transition is 6.4 months
TIME
GPNRT
Expected time for PENRT to
make transition is 7 months
Total = 25.6 Months
DYNAMIC EFFECTS IN DRIFTING SMOKE RINGS
SYMMETRICAL SELECTION/INFLUENCE
(NON-EVOLUTIONARY)
Isolate Risk-Taker changes behaviour
to match that of the Group
INFLUENCE
ISRT
Isolate Non Risk-Taker selects friend in
the Group and becomes a Peripheral
SELECTION
ISNRT
GPNRT
Expected time for ISRT to
make transition is 12 months
SELECTION
PENRT
GPNRT
Expected time for ISNRT to
make transition is 8.6 months
TIME
GPNRT
Expected time for PENRT to
make transition is 7 months
Total = 27.6 Months
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)
GPNRT-GPRT-GPRT (6-1-1)
GPRT-GPRT-GPRT (1-1-1)
PENRT-GPNRT-GPNRT (5-6-6)
GPRT-GPRT-PERT (1-1-2)
GPRT-PERT-GPRT (1-2-1)
GPRT-ISRT-ISRT (1-3-3)
GPRT-GPNRT-GPRT (1-6-1)
GPNRT-GPNRT-PENRT (6-5-5)
PENRT-PENRT-ISNRT (5-5-4)
PENRT-PENRT-GPNRT (5-5-6)
PENRT-GPNRT-PENRT (5-6-5)
GPNRT-ISRT-ISRT (6-3-3)
PENRT-ISRT-GPRT (5-3-1)
PENRT-GPNRT-GPRT (2-6-1)
11
11
6
4
3
3
3
3
3
3
3
3
3
2
2
Table 2
(6M,5F)
(9M,2F)
(2M,4F)
(3M,1F)
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