~~c~~~erw#rks, 1 (1978/79) 193-210
OElsevier Sequoia S.A., Lausanne - Printed in the Netherlands
193
The Process of Friendship Formation*
Maureen T. Hallinan
University of ~~sco~s~~,Madison
This paper poses two questions about the process of fr~ends~~i~formation:
what is the relative stability of as~)mmetri~ and mutual friendship dyads and
what is the nature of change in asymmetric dyads over time? These questions
are examined in longitudinal sociometric data from five elementary classes.
Change in friendship choices is shown to be at least partially embeddable as
a continuous time, stationary Markov process and the unique Q matrices
governing the process are determined. The findings sl~ow that unreciprocated
friendship choices of the chi~dre?l in the sample are less stable than reciprocated choices and that their unreciprocated choices tend to be withdrawn
rather than reciprocated over time.
Introduction
Most empirical studies of friendship focus on the friendship choices of
ind~iduals. These studies generaity examine the distribution of friendship
choices across a group or class and identify dete~inants
of popularity or
social isolation. They also relate characteristics of individuals to their
friendliness and social adjustment. Most of this research is cross-sectional,
although some studies examine correlates of popularity or friendliness at
two or more points in time.
In contrast to the research on individual’s friendship choices, only a few
studies focus on the friendship dyad, that is, the sentiment bond linking two
individuals. Rarely are studies found that examine stability and change in
dyadic ties over time. (Among the exceptions are Katz and Proctor 1959;
Hailinan 1976; Wasserman 1977a, b.) The paucity of research in this area is
unfortunate since info~ation
about the structure of dyads and the nature
of structural change in dyads is essential to an understanding of the friendship
process. The present study will focus on stability and change in children’s
friendship dyads and examine how friendships form and dissolve over time.
*The author expresses gratitude to Edwin Bridges, Burton Singer, Aage S$rensen, and Seymour
Spilerman for advice on conceptual and methodoIogicaI issues, to Donna Eder and Diane Felmlee for
help in data collection and to Bob Kaufman and Ken Kunen for programming assistance.
194
M. T. Hallinan
Two questions will be investigated:
(1) are asymmetric dyads more or less
stable than mutual dyads, and (2) when asymmetric dyads change, do they
become mutual or null?’
A related set of questions concerns W/II) friendships are established and
dissolve. The theoretical
literature
suggests that similarity and status are
major determinants
of friendship choice and empirical evidence supports this
proposition
(see Rainio 1966; Tuma and Hallinan 1978).’ Since the present
study does not aim to test a causal model of friendship formation, questions
about why friendships change will not be addressed here. Once the nature of
change in friendship dyads has been described, the underlying mechanisms
of the friendship process can more easily be explained.
The model of friendship to be adopted here assumes that friendship formation is a sequential process having four elements. First, P must desire to
have 0 as a friend (attraction).
Second, P must initiate a move to establish a
friendship with 0. Third, 0 must recognize P’s overture of friendship. Fourth,
0 must reciprocate P’s offer of friendship.
Considerable theoretical support for this friendship model can be found in
the literature. The process of making friends has been conceptualized
as a
series of tentative moves on the part of one person toward another. The
initiator of the friendship offer generally has lower status than the recipient.
Each response of the higher status person generates the next move in the
process. A favorable response leads to another overture of friendship by the
lower status person while an unfavorable
response or series of responses
results in a withdrawal of the friendship offer (Goffman 1963). Similarly, it
is argued that friendship begins when a higher status person accepts an offer
of friendship by a lower status person (Wailer 1938). Exchange theorists
claim that the friendship process involves a cost benefit analysis: after a
friendship
offer is made the recipient compares the advantages and disadvantages of the possible friendship and responds positively if the relationship is perceived to be rewarding and negatively if it is viewed as too costly
(Homans
196 1; Thibaut
and Kelley 1959; Gouldner
1960; Blau 1964).
These theoretical
notions of friendship formation imply that establishing
a friendship
involves an offer and its acceptance.
The dissolution
of an
existing friendship or the failure to make a new one involves the rejection of
an offer followed by its withdrawal. This suggests that an asymmetric dyad
represents an unfinished
interaction
that will change to a mutual or a null
dyad when the friendship offer is responded to.3 Consequently,
asymmetric
dyads are expected to be relatively unstable and to have a short duration.
‘An asymmetric
dyad is one in which a person P chooses another person 0 as a friend while 0 does
not choose P as a friend. A mutual dyad is one in which P and 0 choose each other as friends. In a null
dyad, neither P nor 0 choose each other as friends.
‘Rainio’s
(1966) work is one of the more theoretical
studies employing
Markov models to analyze
friendship
data. He derives a theoretical
sociomatriv
based on assumptions
about frequency
of contact
and friendship.
30f course asymmetric
dyads may also arise from admiration
or esteem. In this case, no response is
expected
and the asymmetric
dyad is likely to remain stable over time. However, in the present study,
only best-friend
choices will be considered
and these are not likely to be based primarily
on esteem.
Mutual dyads, on the other hand, are expected to remain stable over time.
The main factor affecting the stability of mutual dyads is the extent to
which a f~endship is rewarding to both members. Mural friends develop
similar attitudes and interests that provide reciprocal rewards over time
(Newcomb 1956). A norm of reciprocity governing dyadic interactions
brings about equal gratifications (Gouldner 1960). Mutual friends reward
each other by eonsensual validation (Byrne 1971) and by providing opportunities for social comparison (Schacter 1959). Withdraws from a mutual
friendship is avoided because the resulting imbalance is u~comfortab1~
(Heider 1958), whereas the ongoing interaction of mutual friends is likely to
deepen the friendship (Homans 1950). These theoretical propositions lead to
the predication that mutual dyads are stable over time. In particular, it is
expected that mutual dyads are more stable than asymmetric dyads.
A second question about the friendship process concerns whether friendship offers are typicahy accepted or rejected. That is, when asymmet~c
dyads change, do they tend to become mutual or null dyads? Before a
friendship offer can develop into a mutual friendship, the person selected
must be aware of the offer and desire to reciprocate. The likelihood that a
friendship offer is recognized by the recipient depends on the directness and
clarity of the communication. The initiator of a friendship takes less risk by
giving an unclear message than a direct one because he can avoid the embarrassment of rejection by claiming that the offer was never intended in the
first place. Goffman’s desc~ption of “tentative” moves toward the other is
consistent with the concept of ambiguous friendship offers. If friendship
overtures are unclear, they are likely to go unnoticed by the person to
whom they are directed. Friendship offers that receive no response are
likely to be withdrawn.
When a f~endship offer is recognized, the l~elihood of a positive response
depends on the selected person’s perception of the rewards he will obtain
from the relationship. A friendship offer is usually initiated by a lower status
person possessing fewer valued resources than the recipient. If the discrepancy
between the resources of the two members of the dyad is large, the higher
status person is IikeIy to reject the offer because the exchange is perceived
as too unbalanced (Blau 1964). Even when esteem and admiration are given
to the higher status person to make the exchange more attractive, large
status differences are difficult to overcome in interpersonal interactions
(Newcomb 1956; Thibaut and Kelley 1959). Moreover, since higher status
persons generally receive many friendship offers, time constraints make it
impossible to accept all of them. A friendship offer to a person close in
status is more likely to be accepted, but this too depends on the attractiveness of the initiator, time factors, and other commitments of the recipient
(Hargreaves 1972). In general, friendship offers are expected to be rejected
more often than they are accepted. This leads to the hypothesis that asymmetric dyads are more likely to change to null dyads than to mutual dyads.
How friendships form and dissolve will be examined in the present paper
by empIoying a continuous time Markov model to analyze a longitudinal
data set. A continous time Markov model is a more a~?propriate tool than a
discrete time model to analyze friendship ties for several reasons. First, a
continuoL~s time model permits change to occur at any point in time rather
than only at discrete time intervals and friendship formation is a process that
is not restricted to fixed time periods. Second, the model yields parameters
that are amenable to substantive interpretation
which cannot be obtained
from a discrete time model. Measures of the duration of mutual and asymmetric dyads are essential to test the hypotheses outlined above and can only
be obtained from a continuous
time model. Finally, unlike a discrete time
Markov chain, a contin~]oLls time model permits an estimatio~l of the effects
of fixed observation intervals.
A few examples of the use of Markov models to study change in social ties
are found in the literature.
In an early empirical study, Katz and Proctor
(1959) showed that change in children’s preferences
for seating partners
could be represented
by a first-order,
stationary,
discrete time Markov
model. Wasserman (1977a, b, c) applied a continuous time Markov model to
change in dyads and estimated popularity
and reciprocity
parameters for a
number of data sets. Sg(rensen and Hallinan (1976) used a continuous time
model to estimate the stability of various triad types and showed that triadic
choices tended to move away from intransitivity
over time. The mathematical and statistical properties of these models are discussed by Holland and
Leinhardt (1977) and Wasserman (1977b, c). The present application of a
continL~o~ls time Markov model to change in social ties differs from Sorensen
and lialfinan’s study by focusing on dyads rather than triads and from
Wasse~~an’s work by analyzing the stability and direction
of change in
dyads rather than decomposin, * change parameters and examining effects of
reciprocity and popularity on dyadic choice.
The model proposed here is a continuous
time, stationary, homogeneous
Markov model. Empirical evidence in support of the stationarity
assumption
can be found in Katz and Proctor’s (1959) study of change in children’s
seating partner preferences.
They found that in a class of 25 students choice
of seating partners could be modeled as a discrete time, stationary,
firstorder Markov chain. The assumption of stationarity
appears reasonable in
the present study for two reasons. First, nearly all the children in the sample
had been in the same class since first grade so that they knew each other well
at the beginniIlg of the data collection. This increased the likelihood that the
tendency
to form certain kinds of dyadic choices would be constant
throughout
the year. Second, a month of the school year was allowed to
elapse before the first data collection, giving students an opportunity
to readjust to their social environment
after their summer vacation. This avoided
the effects of a new environment on their initial choices.
Two studies present data that challenge the assumption of homogeneity.
Katz and Proctor (1959) observed a sex cleavage in their data, indicating that
dyadic choices between members of the same sex may change at a different
rate than cross-sex dyadic choices. Sdrensen and Hatlinan (1976) found that
various triadic con~g~~rations in which dyads are embedded change at dif-
The process of friendship formation
197
ferent rates, suggesting that some dyads are more likely to change than
others. Despite these findings the assumption of homogeneity
seems appropriate to the present study because of certain characteristics
of the sample.
Two possible sources of heterogeneity,
differences in socioeconomic
status
and academic achievement,
occurred to only a small degree in the data. The
children came from similar backgrounds and varied little in ability. Another
obvious basis for non-homogeneity,
sex differences, seemed to have only a
small effect on the children’s friendship choices.4 Between 25% and 35%
of the friendship dyads in the classes were cross-sex choices. This was considerably more than expected by chance as measured by Freeman’s (1978)
index of segregation
in social networks.
Nevertheless,
the data will be
examined for evidence of stationarity
and homogeneity
before determining
whether
friendship choices are embeddable
as a continous
time Markov
process.
Sociometric
data
Longitudinal
sociometric
data were collected from four sixth-grade classes
and one fourth-grade
class in five white, rural elementary schools in the Midwest. The classes ranged in size from 18 to 30 pupils. One month after the
beginning of the academic year, the students were given a sociometric
questionnaire
asking them to name their best friends and their friends. The
children were allowed to choose as many best friends and friends as they
wished. This modified free choice sociometric technique has the advantage
of minimizing response error (Hallinan 1974). The same procedure was repeated in the sixth grades six additional times over the school year at sixweek intervals. The questionnaire was administered a total of six times in the
fourth-grade
class at unequal time intervals ranging from 7 to 38 days. The
responses to the questionnaires
were arranged as sociomatrices
or crossclassification tables in which the categories (or states) were null, asymmetric
and mutual dyads.5 The cell entries were the number of dyads that changed
from one state to another between two consecutive data collections.
A continuous
time model of the friendship
process
A Markov model specifies the probabilities
of moving from one discrete
system state to another. The process ordinarily assumes (1) time independence (stationarity),
that is, the transition probabilities
are constant over
4This may have been due to the age of the children (8 to 11 years) and to their rural environment
since attending a rural school is frequently
associated with slower maturation.
50riginally,
four by four tables were constructed.
In addition to the mutual and null states, two
asymmetric
states were included to represent the two asymmetric
configurations
i -+j, j j+i, and i /+j,
j + i. These two states were subsequently
collapsed since very few instances of mobility from one
asymmetric
state to the other were observed in the data.
The process of friendship formation
199
If the three conditions governing the elements of the Q matrix are not
satisfied, then although Q may be a true solution of equation (l), it is not a
transition matrix and the process cannot be represented by a continuous
time Markov model (Singer and Spilerman 1976a). When the conditions
are satisfied, the elements qii E Q are the transition rates for the process,
that is, the instantan~ons rates of change in the dyads. The following
parameters are given a substantive inte~retation:
(i) qii/-qii
(ii) -l/qji
= the conditional probability that a dyad in state i will move
to statej, given that a transition occurs
= th e expected duration of a dyad in state i
Test of Markov assumptions
In attempting to fit a discrete or continuous time Markov model to empirical data it is necessary first to determine whether the data satisfy the Markov
property
i)t4,t*&,tc1
= Ll,t+l
(6)
that is, the product of the observed matrices over two consecutive time intervals is equal to the observed transition matrix for the whole interval. If equation (6) does not hoId, then the data fail to satisfy one or more of the
assumptions of first order, stationarity, and homogeneity. An additional
reason for deviation from (6) could be the presence of response errar in the
data.
Equation (6) was c~culat~d for the five classes in the sampie at consecutive time intervals8 A x2 g~odnes~of-fit test with six degrees of freedom
was performed. The values of x2 ranged from 20 to 29 with x2 = 16.81
indicating a good fit at p > 0.01. Consequently, the data deviated somewhat from the Markov property. To determine which assumptions were
violated, tests of the first-order and stationarity properties were performed.
No direct test of homogeneity of transition probabilities is available.
First-order a~u~ptio~
To test the hypothesis that a Markov chain is of the first order against the
assumption that it is a second-order chain, the likelihood ratio criterion
‘Ideally, data collections should be spaced unequally to avoid the effects of fixed observation
intervals on the estimation of Q. Collection of the data at unequal time intervals was possible only
for Class 5. Since the results for this class did not differ substantially from those of the other classes
and since there is no reason to believe that six weeks is associated with periodicity in children’s
friendship relationships, it seems unlikely that the Q estimates were affected by the length of time
between observations.
h
=
(jj.kIp.‘k)~ijk
fi
i,j,k=
1
(7)
u
I
is used where $ijk is the maximum likelihood estimate of @jk and njjk is the
observed frequency of the states i, j, and k at t -- 2, t -- 1, and t, respectively
(Anderson
1954). Under the null hypothesis of first order, -2 log X has an
asymptotic
x2 distribution
with m(m - l)2 degress of freedom. The values
for the several sets of time intervals for the sociometric data were all significant (p > O.Ol), indicating that the data for each class show no evidence of
being a second-order chain. Thus, no knowledge of past distributions
of dyad
types is necessary to predict future dist~b~~tions.
Stationarity
and homogeneity
assumptions
The stationarity
assumption states that pii
= pii for t = 1, 2, .,. 7”. This
means that the transition probability
for each cell of the transition matrix
remains constant over time. The stationarity
assumption can be tested by
calculating
(8)
X
with m(m - 1) degrees of freedom (Allderson 1954). An estimate Of pij is
obtained by averaging the values of pij(t) over the several time intervals
or by calculating the equivalent formula
ii
t= 1
riij(t)l
g
5 flfj(t)
(9)
t=O i=l
where nii(t) is the observed frequency in state i at time t ~ 1 and state j at
time t.
In testing the stationarity
assumption on the sociometric
data, the pijs
for each class were estimated by averaging transition probabilities
over all
but the last time interval. Data from the last interval were reserved for a
goodness-of-fit
test of the friendsl~ip model. The values of x2 for the five
classes are given in Table I. Since x2 < 16.81 for p > 0.01, the stationarity
Table
1.
Values vf x2 for test of stationarity for five classes
Time
Class 1
Class 2
Class 3
Class 4
Class 5
1-2
2-3
3-4
4-s
5-6
7.90
2.53
8.11
6.82
2.08
27.61*
5.73
7.24
8.56
15.61
19.69*
2.41
7.00
1.06
8.28
5.13
36.91*
20.24*
**
5.72
7.81
1.75
5.36
11.56
*p > 0.01,
indicating
rejection of null hypothesis
for this time interval.
**P(t) was not embeddabk
of stationarity.
The process of friendship fornation
201
assumption was upheld for all of the time intervais in Classes 1 and 2 and in
all except the first time interval in Classes 3 and 4. In Class 5, the stationarity
assumption was violated in the second and third time intervals, but the
deviation was large only at the second interval. In general, the assumption
that constant probabilities govern the process of friendship formation and
change is upheld.
Support for the first-order and stationarity assumptions in the data indicates that the failure of the transition matrices to satisfy equation (6) arises
from another source. Since heterogeneity and nonstationarity are difficult
to distinguish in empirical data (McFarland 1970; Ginsberg 197 I), evidence
of stationarity in the data implies that not much heterogeneity is present in
the population of dyads. Moreover, the assumption that all dyads are
governed by the same transition probabilities appears reasonable given the
simila~ti~s of the students. Even if characteristics of individual children or
structural characteristics of friendship networks did exert some influence on
dyadic change, it seems unlikely that the effects would outweigh the effects
of dyadic structure. Consequently, in the absence of a direct test, homogeneity will be assumed in the present analysis.
On the other hand, sociometric data are particularly vulnerable to
response error regardtess of the technique employed to collect them. Thus it
is reasonable to attribute the failure of the data to fit equation (6) to noise.
To obtain more stable estimates, the data were averaged over the several time
intervals (excluding the last interval) and a single transition matrix was
obtained for each class. Since noise in the average transition matrices is Iikely
to be small, the pooled matrices were used for the analysis.
EmbeddabiIity
and uniqueness
After reasonable evidence has been obtained that a transition matrix satisfies
the assumptions of a Markov process, it is necessary to dete~i~e whether
the observed P(t) matrix is uniquely embeddable as a continuous time
Markov process9 The embedding problem refers to finding the conditions
under which an observed P(t) can be written in the form of equation (2),
where Q is an intensity matrix.
Much con$.tsion exists in the literature concerning the embeddability of
an observed P(a’)matrix, In the past, Q ordinarily was obtained by evaluating
log P(r) using the series expansion
cm (-I)[P(t) - I3k
Q= f;logP(t):
f &
k!
‘The reader is referred to Singer and Spilerman 11976a) for a detailed rna~ernat~c~ exposition
of the probIem of em~ddab~ity
and its solution. The authors derive necessary and sufficient conditions for the compatibility of any empirically determined matrix with a continuous time Markov
formulation. A less mathematical discussion of embeddability and identification problems is found
in Singer and Spilennan (1976b).
If the expansion converged, it was thought that the observed P(t) was embeddabJe. Singer and Spilerrnan (1976a) have shown that this is not the case.
Some P(t) are coinpatible
with a Markov structure even when the power
series does not converge, while not all P(t) can be written in the form of
equation (2) even when the power series converges.
A more appropriate
way of obtaining Q is first to write P(r) in diagonal
form
E’=HI\H-’
(11)
where
and H is a nonsingular
Q=logP
=HlogAH-’
similarity
transformation.
Then calculate
(12)
where
Equation (12) yields muttiple values of Q, each of which can be examined
for adherence to the conditions
of an intensity matrix. If the elements qij
for one or more Qs satisfy these conditions,
then P(r) is embeddable as a
continuous time Markov process described by those Qs.
A second problem concerns the uniqueness of a continuous
time Markov
process, that is, the uniqueness of the Q matrix. This is known as the identification problem. Equation (12) yields a number of Q matrices which may
satisfy the conditions
for an intensity matrix. It may not be known which
intensity
matrix governs an observed empirical process. This problem is
avoided if it can be shown that an intensity matrix satisfying (I) is unique.
The most straightforward
sufficiency -condition for the solution Q of (1) to
be unique is that the eigenvalues of P(t) be distinct, real, and nonnegative.
When the eigenvalues satisfy this condition, there can be at most one branch
of log P(t) that is an intensity matrix and the Q obtained from (12) is
unique. If the eigenvalues do not satisfy this ~o~idition, more compIi~ated
procedures
are required to determine uniqueness (see Singer and Spilerman
1976a). It is necessary to employ these tests to dete~ine
whether the sociometric data being analyzed are embeddable
as a continuous
time Markov
process and, if so, whether they generate a unique Q solution.
Implications
uniqueness
of measurement
error. for determining
embeddability
and
Response error which results in misclassification of dyads in the_ system
states may have important consequences for the embeddability of P(t) and
the uniqueness of its solution. Singer and Spilerman (1976a) show that an
observed P(t) containing measurement error may be embeddable as a continuous time Markov process although the true P(t) may not be compatible
with a Markov structure. Similarly, an observed P(t) with error may not have
an in_tensity matrix for a solution while the error free P(t) may. Moreover,
two P(t) matrices whose corresponding elements are within error distance of
each other may yield two significantly different Q solutions.
The error problem is reduced when data are available for more than two
time points if the process is stationary. The embeddability problem concerns
the compatibility of a single observed matrix with a continuous time Markov
process. If the observed P(t) matrix is believed to contain noise, then one
may have little basis for concluding that the process is Markovian. However,
the transition matrices from several time intervals may be pooled to obtain
a more accurate estimate ofthe matrix which governs the process. If the Q
obtained from the pooled P(t) is an intensity matrix, the process may be
assumed embeddable. This Q matrix then may be tested for uniqueness.
Even if P(t) is embeddable in a continuous time Markov process and Q
is determined to be its unique solution, a-second P(r) matrix may exist
whose elements are within error distance of P(t) and whose solution is a different Markov process. When this occurs, it is unclear which Q matrix accurately describes the social process. If several intensity matrices are obtained
for a process over a number of time periods and the corresponding elements
of the Qs are close in magnitude, then the estimate of Q based on the pooled
P(r) may be assumed to represent the correct Markov structure.
Test of embeddab~~iry and ~~j~ue~e~~
To determine whether frie_ndship formation is embeddable as a continuous
time Markov process, the P(t) matrices for each class were obtained. These
were averaged over the first five time periods (first four time intervals for
Class 5) to minimize measurement error. The Q matrices based on the pooled
P(t) matrices were derived and examined for adherence to the conditions
(5) of an intensity matrix. The Q estimates based on the pooled @(t)s appear
in Table 2. The conditions for an intensity matrix were satisfied in all five
cases, implying embeddab~i~.
The uniqueness of the five Q matrices may be established by examining
the eigenvalues of the P(t) matrices. The eigenvalues of the pooled P(t)
matrices were all rest; unequal, and positive, satisfying the sufficiency condition for uniqueness. As a consequence, the Q matrix for each class may be
regarded as the unique structure compatible with the Markovian fo~ulation
of the f~endship process.
Table 2.
ultimates of Q matrixes f~~~ye classes
t, .-,\ t,
class 1
N= 26
Class 2
N= 28
Class 3
N= 30
Class 4
N = 27
Class 5
N= 18
Predictive
Null
--_-
Asymmetric
._
N
A
M
-0.0053
0.0108
0.0001
0.0053
--0.0181
0.0100
0.0001
0.0066
- 0.0100
N
A
M
-0.0039
0.0086
0.0000
0.0039
-0.0162
0.0075
0.0000
0.0075
-0.0075
N
A
M
-0.0025
0.0084
0.0000
0.0025
--0.0179
0.0095
0.0000
0.0095
0.0095
N
A
M
-0.0033
0.0130
0.0000
0.0033
- 0.0247
0.0041
0.0000
0.0117
-0.004 1
N
A
M
PO.0066
0.0162
0.0000
0.0066
-0.0260
0.0265
0.0000
0.0098
-0.0265
power of the model
To detemrine
whether the Markov model of friendship accurately predicts
change in dyadic structure over time, the model was tested on the last wave
of sociometric
data for each of the five classes. These data covered the time
interval ~immediately
subsequent to the last interval used to calculate the
pooled P(t)s. The expected matrix of transition probabilities,
P(t), was obtained from equation (2), where Q is the intensity matrix and t is the last
time interval. The goodness of tit of the model can be tested with the
statistic
(13)
with Yli equal to the number of dyads in state i at the initial time point. The
with m(wl - 1) degrees of freedom
G2 has an asymmetric x2 distribution
under the null hypothesis (Fienberg 1977). Since G2 has only six degrees of
freedom for a 3 X 3 table, p = 0.01 was chosen as the level of significance to
test the fit of the model. .
Table 3 presents the expected and observed transition matrices for each of
the five classes. The G2 values are not signi~cant (P > 0.01) for all of the
classes, indicating a reasonably good fit of the model to the data. Thus the
The process offriendshi~ formation
Table 3.
205
Observed and expected transition matrices and values of G2 for five classes
tn_ t\ t,,
Null
Asymmetric
Mutual
Expected F(f)
~
Asymmetric
Null
N
A
M
0.875
0.306
0.057
0.095
0.5 14
0.321
0.030
0.181
0.623
0.836
0.290
0.063
0.143
0.529
0.246
0.018
0.160
0.687
t = 42 days
Class 2
N
A
M
0.900
0.320
0.094
0.100
0.553
0.250
0.000
0.126
0.656
0.865
0.247
0.050
0.108
0.553
0.193
0.025
0.198
0.753
t = 42 days
G2 = 9.67*
Class 3
N
A
M
0.955
0.253
0.029
0.038
0.453
0.217
0.007
0.293
0.754
0.914
0.250
0.057
0.071
0.5 15
0.228
0.014
0.235
0.719
t = 42 days
G2 = 6.17*
Class4
N
A
M
0.912
0.464
0.182
0.068
0.286
0.159
0.020
0.250
0.659
0.895
0.329
0.075
0.085
0.416
0.198
0.020
0.256
0.726
t = 42 days
N
A
M
0.941
0.303
0.105
0.040
0.576
0.368
0.020
0.121
0.526
0.861
0.237
0.115
0.109
0.589
0.263
0.030
0.174
0.622
t = 27 days
G2 = 3.53*
Observed
Class
1
Class 5
P(r)
Mutual
G2 = 5.34*
G2 = 4.50*
*p > 0.01, indicating
acceptance
of nult hypothesis
of a good fit.
parameters of the model can be used to describe and predict the friendship
process. Despite the statistical significance of the results, however, deviations
of the model from the data do appear. These are likely due to the presence
of some heterogeneity
in the data, small effects of network properties, and
noise. The best fit of the model was obtained in Class 5, the fourth-grade
class. This is not surprising since age is likely to influence a child’s awareness
of differences among peers and to sensitize children to structural properties
of groups. In particular, fourth graders may be less concerned about sex differences or status when selecting their friends than sixth graders.
The expected and observed distributions of dyads across the system states
at t, are presented in Table 4 (t6 for Class 5). The two distributions for each
class are fairly similar as expected given the reasonably
good fit of the
model. All five classes contained considerably more null dyads than asymmetric or mutual dyads with little between class variance in the percentages.
The preponderance
of null dyads means that most of the children did not
choose each other as best friend. Classes I, 2, and 5 had more asymmetric
than mutual dyads, while Classes 3 and 4 had more mutual dyads. Class 5,
the only fourth grade in the sample, had the smallest percentage of mutual
dyads. This result suggests that friendship formation is part of a developmental process and that fourth-grade
students are still learning the social skills
needed to establish close friendships. Table 4 provides no information
about
stability or change in the dyads; the parameters of the continuous
time
Markov model, discussed in the following section, are needed to reveal the
dynamics of friendship formation.
Class
~~1
2
3
4
S
~I~
Table 5.
Asymmetric
Mutual
-
Null
Table 4.
200 (6 2%)
73 (22%)
52 (16%)
Obs rl
-
212
92
74
EXP t?
Class 2
229 (60%)
94 (25%)
55 (15%)
Obs t7
189 days
256
400
303
152
Null
___~.-
52 days
62
56
40
38
Asymmetric
100 days
133
105
110
38
Mutual
Expected duration (-1 /qii) of dyad types for
five classes
I92
80
53
EXP 17
Class 1
289
75
71
Obs t7
i
2
3
4
5
-.---
Class
Table 6.
299 (69%)
60 (14%)
76 (17%)
_-^---
EXP t-l
Class 3
Observed and expected distributions of dyads for five classes
252
49
50
EXP t7
Class 4
263 (75%)
40 (11%)
48 (14%)
---
~.Obs t7
82
48
23
EXP 16
Class 5
91 (59%)
44 (29%)
18 (12%)
.lll
-~Obs t6
I_____
0.633
0.541
0.470
0.526
0.623
^_~_.____
NUll
--
0.367
0.459
0.531
0.474
0.311
Mutual
~_
Conditional probabilities
(qijJ_qii) of entering
null and mutual statesgiven exit from asymmetn’c
state for five classes
-
The process of friendship formation
207
Relative stability of asymmetric and mutual dyads
It was hypothesized
that mutual dyads are more stable than asymmetric
dyads over time. The parameters of the Markov model permit a test of this
hypothesis. The qii parameter is the instantaneous
rate of change in the ith
system state while - 1/qii is the expected duration in state i. A large expected
duration indicates stability.
Table 5 presents the values of -l/qii
obtained from the intensity matrix
for each class. In the four sixth grades, mutual dyads lasted from three to
live months on the average. The expected duration of the asymmetric dyad
in these classes was between one and two months, less than half as long as
the mutual dyads. Little variance in the stability of either dyad type was
found across the classes. Thus the mutual dyads in the sixth-grade classes
were considerably more stable than the asymmetric dyads.
In Class 5, the fourth grade, the duration of both asymmetric and mutual
dyads was 38 days. This finding fails to support the hypothesis and may be
due to the age of the children. Table 5 shows that the social relationships of
the fourth-grade
children changed more frequently than those of the older
students since all three dyad types had a shorter duration in Class 5 than in
the other four classes. With the exception
of Class 5, the results provide
empirical evidence that mutual dyads are more stable than asymmetric dyads,
implying that reciprocated
friendship choices are more likely to endure than
unreciprocated
choices.
Direction of change in asymmetric dyads
The second hypothesis
stated that asymmetric
dyads are more likely to
change to null dyads than to mutual dyads over time. The conditional probability of entrance into the null or mutual state given exit from the asymmetric state is given by the parameter qii/-qii.
Table 6 shows that asymmetric dyads were more likely to become null than to become mutual in
four of the five classes. In particular, in Classes 1 and 5, the probability
of
change from asymmetric to null was about twice as high as from asymmetric
to mutual. Thus in these four classes, an asymmetric
choice or friendship
offer was more likely to be withdrawn than reciprocated.
In Class 3, asymmetric dyads had a slightly higher probability of becoming mutual than null.
This finding was not anticipated and cannot be explained. It may be related
to the fact that the average stability of the three dyad types was greater in
Class 3 than in the other four classes in the sample, as seen in Table 5. Since
little change occurred in this class, friendship offers may have been more
readily perceived and therefore reciprocated.
In general, the findings show
support for the hypothesis and indicate that friendship offers are more likely
to be withdrawn than to lead to mutual friendships.
208
M. T. Hallinan
Conclusions
The central concern of the present paper was to examine the nature of
change in the structure
of friendship dyads. The analysis revealed two
aspects of the dynamic process of friendship formation. First, mutual bestfriend dyads were found to be more stable than asymmetric dyads. Second,
asymmetric dyads tended to change to null rather than to mutual dyads over
time. Both of these results are consistent
with much of the theoretical
literature on friendship.
A major finding of the study was that among young children in suburban
elementary
schools, friendship
is a dynamic, interactive process in which
change occurs frequently.
While the children’s mutual best friendships were
twice as stable as their unreciprocated
best friendship choices, they endured
on the average only about three months. Characteristics
of the institutional
setting may have influenced
the stability of the students’ friendship ties.
For example, structural properties of schools such as ability grouping are
likely to separate some friends and place nonfriends
in close proximity.
The short duration
of the friendships
observed in this study raises the
question
of whether friendships developed outside an institutional
setting
would show the same qualities as those formed within the constraints of a
classroom. In particular, would a less restricted setting than a partial institution make establishing and maintaining close friendships easier by providing
greater opportunities
for friends to interact
and to enjoy each other’s
company.
Future research should take into account the institutional
or
environmental
setting as a possible factor affecting the stability of friendships.
The study also found that the best friendships of sixth graders lasted considerably longer than those of fourth-grade
students. This gives some hint
that friendships become more enduring as persons grow older. It may be that
friendship takes on a different meaning or set of meanings with age. If this
is so, then the results of the present study may not be generalizable to older
youth or adults. On the other hand, older students may simply have more
opportunities
to foster and maintain friendships outside the classroom than
younger students. The present research should be replicated across age levels
to determine how age influences stability and change in friendships.
This research leaves open questions about other influences on change in
dyads over time. In this study, an attempt was made to control for characteristics of children, such as age, race, socioeconomic
status, and achievement, that could create differences in the tendency of dyads to change. This
permitted the simplifying assumption of homogeneity
and the use of a fairly
general continuous
time model to analyze the friendship process. The model
obtained a fairly good fit and showed that dyadic structure explains a considerable part of the change in dyads. Nevertheless,
deviations from the
model indicate that other factors were influencing structural change. These
factors are likely to include differences in the attributes of dyad members,
that is heterogeneity,
and properties
of the social network in which the
dyads are embedded. Future research should examine the effects of these
factors on dyadic change directly by studying networks whose members
differ in ascribed and achieved characteristics and by taking into account the
structural con~guration of the network.
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