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The process of friendship formation
~~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the chi~dre?l in the sample are less stable than reciprocated choices and that their unreciprocated choices tend to be withdrawn rather than reciprocated over time.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.
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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.
References Anderson, T. W “Probability 1954
models for analyzing time changes in attitudes”.
In P. F. Lazarsfeld ted.),
Mathematical Thinking in the Social Sciences. New York: Free Press.
Blau, P. M. 1964 Exchange and Power in Social Life. New York: Wiley. Byrne, D. The Attraction Paradigm. New York: Academic Press. 1971 Fienberg, S. E. The Analysis of Crass-Classified Categorical Data. Cambridge, MA: MIT Press. 1977 Freeman, L. C. “Segregation in social networks”. Sociological Methods and Research 6 :41 l-429. 1978 Ginsberg, R. “Semi-Markov processes and mobility”. Journal of ~at~e~atieQ~ ~cio~ogy I :233-263. 1971 Goffman, E. 1963 Behavbur in Public Piaces. New York: Free Press. Gouldner, A. W. “The norm of reciprocity: a preliminary statement”. American Sociological Review 25: 1960 161-179. Hallinan, M. T. The Structure of Positive Sentiment. New York: Elsevier. 1974 1976 “Friendship formation: a continuous time Markov process”. Working Paper 76-S. University of Wisconsin, Madison: Center for Demography and Ecology. Hargreaves, D. H. 1972 Inte~e~onalRelations and Education. Boston: Routledge and Kegan Paul. Heider, F. The PsycholoB? of rnte~ersona~ Relations. New York :WiIey. 1958 Holland, I’. and S. Leinhardt “A dynamic model for social networks”. Jozzrnal of ~athema~~al Sociology 55-20. 1971 Homans, G. C. The Human Group. New York: Harcourt. 1950 196 1 Social Behavior: Ifs Elementary Forms New York: Harcourt, Brace and World. Katz, L. and C. Proctor 1959 “The concept of con~guration of inter~rson~ relations in a group as a time~ependent stochastic process”. Psychornet~ka 24:3 17-327. McFarland, D. 1970 “Intragenerational social mobility as a Markov process: including a time-stationary Markovian model that explains observed declines in mobility rates”. American Sociological Review 35:463476.
Newcomb, T. “The prediction of interpersonal attraction”. The American Psychologist II 575-581. 1956 Rainio, K 1966 “A study of sociometric group structure: an application of a stochastic theory of social interaction”. In I. Berger, M. Zelditch and 3. Anderson, SocioZog&~ Theories in Progress, Vol. 1. Boston: Moulton Mifflin. Schachter, S. 1959 The Psychoio~ ofdffiiation. Stanford, CA: Stanford University Press. Singer, B. and S. Spilerman 1976a “Representation of social processes by Markov models”. American Journal of Sociology 82~1-54. 1976b
“Some methodological and Socia~~ea~renzent
issues in the analysis of longitudinal surveys”. Annals of Economic 5:447-474.
Sdrensen, A. B. and M. T. Halhnan “A stochastic model for change in group structure”. Social Science Research 5 :43-6 1. 1976 Thibaut, J..W. and H. H. Kelley The Social Psychology of Groups. New York : Wiley. 1959 Tuma, N. B. and M. T. HaiIinan “The effects of sex, race and achievement on schoolchildren’s friendships”. Social Ebrcrs, 1978 in press. Wailer, W. The Family: A Dynamic r~te~retation. NW York: Drydcn. 1938 Wasserman, S. 1977a “Anatyzing social networks as stochastic process”. Technical Report No. 314, University of Minnesota, School of Statistics. 197733 “Stochastic models for directed graphs”. Ph.D. dissertation. Harvard University: Department of Statistics. “A stochastic model for directed graphs with transition rates determined by reciprocity”. 197lc Technical Report No. 305. University of Minnesota: School of Statistics.
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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