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. Author manuscript; available in PMC: 2017 Mar 1.
Published in final edited form as: Group Dyn. 2016 Jan 28;20(1):51–64. doi: 10.1037/gdn0000040

A Mathematical Framework for the Complex System Approach to Group Dynamics: The Case of Recovery House Social Integration

John M Light 1,✉, Leonard A Jason 2, Edward B Stevens 3, Sarah Callahan 4, Ariel Stone 5
PMCID: PMC4821464  NIHMSID: NIHMS739553  PMID: 27065722

Abstract

The complex system conception of group social dynamics often involves not only changing individual characteristics, but also changing within-group relationships. Recent advances in stochastic dynamic network modeling allow these interdependencies to be modeled from data. This methodology is discussed within a context of other mathematical and statistical approaches that have been or could be applied to study the temporal evolution of relationships and behaviors within small- to medium-sized groups. An example model is presented, based on a pilot study of five Oxford House recovery homes, sober living environments for individuals following release from acute substance abuse treatment. This model demonstrates how dynamic network modeling can be applied to such systems, examines and discusses several options for pooling, and shows how results are interpreted in line with complex system concepts. Results suggest that this approach (a) is a credible modeling framework for studying group dynamics even with limited data, (b) improves upon the most common alternatives, and (c) is especially well-suited to complex system conceptions. Continuing improvements in stochastic models and associated software may finally lead to mainstream use of these techniques for the study of group dynamics, a shift already occurring in related fields of behavioral science.

Keywords: group dynamics, group processes, complex systems, social network, stochastic process

Group dynamics research was once a central area of social psychological research and theory (e.g., Homans, 1950; Thibaut & Kelley, 1959), but since the 1970s has been more prominent as an applied discipline, for instance, in education, organizational psychology, and political science (McGrath, 1997). In the past few decades, a promising empirical and theoretical emphasis on group dynamics as a complex system has begun to emerge (Arrow, 1997; Arrow, McGrath, & Berdahl, 2000; Burlingame & Fuhriman, 1997; McGrath, 1997).

Although there are good arguments for the systems approach (McGrath, 1997), it is quite different from much extant theoretical and empirical work, with attendant conceptual and methodological challenges (e.g., Burlingame & Fuhriman, 1997). Statistical methods consistent with a systems approach have been slower to develop, but recent advances in statistics and computing have led to increasingly sophisticated options. Because some of these developments have occurred in fields related to but distinct from social psychology, many social psychologists may not be fully aware of them.

Wölfer, Faber, and Hewstone (2015) recently proposed applying dynamic social network analysis to the study of group dynamics, in the course of a broader discussion about the relevance of social network concepts to group processes. In this paper we discuss how such an approach fits naturally within the complex system conception of group dynamics (Arrow et al., 2000; McGrath, 1997), and present an example application, examining the natural coevolution of social relationships and behavioral change in a small sample of recovery homes—post treatment residences for individuals recovering from substance abuse—based on the stochastic actor-oriented model (SAOM; Snijders, van de Bunt, & Steglich, 2010) originally developed for longitudinal social network modeling. This presentation focuses on statistical issues such as estimation methods and data pooling that are likely to arise in applying SAOM with small groups.

Group Dynamics and Social Networks

The potential benefits of the social network perspective for studying group dynamics have been discussed from several points of view. Burke (2006) noted various historical connections between these fields, for example, studies of leadership and network exchange (see also Wölfer et al., 2015). McGrath (1997) recognized that network concepts could be used to unpack relationship structure and dynamics within groups. Historically, within-group structure is overwhelmingly conceptualized and measured in terms of linkages between individuals and the rest of the group-as-a-whole (Lewin, 1947; Thayer & Burlingame, 2014), for instance, how supported one feels by the rest of the group. Although instruments designed to measure such linkages have been studied and validated (Thayer & Burlingame, 2014), and group membership has been found to explain as much as 25% of the variance in, for instance, the Group Questionnaire (Krogel et al., 2013)—a fairly high number, by social psychological standards—still, a good deal is unexplained. This could suggest that the concept of group-as-a-whole may be a tractable approximation to relevant within-group relationship structures and their dynamics. More likely, in small- to medium-sized groups, the effects of individual-level characteristics on such dynamics, for example, baseline attitudes, experience, and knowledge, personality factors, or aspects of embedding in the group's network of social relationship are appreciable, and such factors cannot be studied in the aggregate.

Wölfer et al. (2015) suggested SAOM as a superior modeling framework for group studies, in the course of a broad discussion of the applicability of network-based methods. Our purpose is to follow through on their recommended dynamic network approach with an empirical example, and to deal with some modeling issues specifically likely to arise when attempting to apply SAOM to relatively small groups, that is, about 5–10 members. Although several not-so-recent articles (e.g., Burke, 2006; Dunbar, 2008; McGrath, 1997) have examined the potential of network methods for studying traditional topics in group dynamics such as leadership, group therapy processes, group communication and productivity, remarkably little such work has appeared since. This may partially reflect the somewhat higher technical demands of both the system perspective and its application with methods like SAOM (Felmlee, 2006). In the course of presenting this study of small group dynamics, we attempt to address practical conceptual and technical issues that others are likely to encounter with similar studies, in an attempt to overcome some of these hurdles.

Modeling Frameworks for Group Dynamics

The empirical study of group dynamics is complicated in several respects. First, since such groups are often too small to model individually (often 10 or fewer individuals), many group studies, be they naturally or experimentally constituted, require observations on multiple groups (Wölfer et al., 2015). When individual outcomes and predictors are also of interest, as is typically the case, there is group-level clustering in the data (Kenny, Mannetti, Pierro, Livi, & Kashy, 2002). Multilevel models (Snijders & Bosker, 2012) can effectively deal with such dependency (e.g., Tasca, Illing, Ogrodniczuk, & Joyce, 2009), but have one other significant current limitation: feedback loops cannot be modeled. Both individual- and group-level variables can be outcomes (Croon & van Veldhoven, 2007), but no variable can be endogenous, that is, both a predictor and an outcome. Group processes are understood to routinely contain feedback loops, a point emphasized in the complex system perspective (Arrow et al., 2000). Substantive examples include leadership dynamics (Eberly, Johnson, Hernandez, & Avolio, 2013), deviance and social control (Heerdink, van Kleef, Homan, & Fischer, 2013; Pinto, Marques, Levine, & Abrams, 2010), and group therapy (Paquin, Kivlighan, & Drogosz, 2013).

Traditional structural equation models (SEM) specify a “nomological network” of constructs (Cronbach & Meehl, 1955). Although feedback loops are possible, such models are not naturally dynamic; time is not directly represented, and temporal evolution cannot be simulated (Snijders & Steglich, 2015). Rather, the feedback appears in the form of “causal” loops among sets of variables. Time can be built into such models by designing it into the data collection, as, for instance, with the cross-lagged panel approach (Kessler & Greenberg, 1981). Discrete-time models are difficult to interpret and compare across studies, because parameters depend on wave intervals (Oud, 2007).

More recently, statistical modeling frameworks capable of representing endogenous group dynamics have begun to appear. For instance, the Actor Partner Interaction Model (APIM) has been proposed as one such framework (Kenny et al., 2002), and has been applied in such areas as the evaluation of group counseling (Paquin et al., 2013) and family dynamics (Baiocchi-Wagner & Talley, 2013). This approach accounts for group-level clustering on many member characteristics and can incorporate some types of feedback loops (Cook & Kenny, 2005), but as the name suggests, these causal loops are essentially dyadic in nature (an actor and a partner). Dyads are embedded in a broader relational structure, however, and this wider embedding may be important (Kenny et al., 2002; McGrath, 1997). For instance, personal relationships often develop through transitivity (friends of friends tend to become friends), a triadic effect that can be confounded with homophily, for example, behavioral similarity (Snijders et al., 2010).

Differential equations (DEs) directly represent change in one or more outcomes as a continuous or discrete function of time. Systems of linked DEs involve an equation for each of several outcomes f(t), g(t), h(t),… and interrelate changes in these state variables and their derivatives and the other state variables. Originally inspired by physical science applications, this is a natural approach for modeling all kinds of change, including group dynamics (Arrow et al., 2000; McGrath, 1997). Relevant statistical methods continue to develop (e.g., Boker & Nesselroade, 2002; Delsing & Oud, 2011; Oud & Singer, 2008) in a way that applies SEM to estimate state variable coefficients as latent constructs. This application is conceptually quite distinct from the above-noted use of SEM in nomological networks. Extensions to stochastic systems (Oud & Singer, 2008; Voelkle, Oud, Davidov, & Schmidt, 2012) may provide an even more realistic representation of social processes. Such models involve a set of deterministic relationships plus “noise” (Øksendal, 2000).

Stochastic Actor-Oriented Modeling

The stochastic actor-oriented model (SAOM; Snijders et al., 2010, also sometimes called the stochastic actor-based model or SABM; Wölfer et al., 2015) is, as the name suggests, an example of a stochastic framework, adapted for modeling social network dynamics. The goal of SAOM is to describe interactive change in individual behaviors and other mutable individual characteristics, along with interindividual relationships, within relatively closed social ecologies, that is, “whole networks.” Consistent with systems approaches to groups (Arrow et al., 2000), neither behaviors nor relationships are taken as fixed. Individual behaviors adapt to one's social surroundings, while the social surroundings—an individual's social relationships—also change and adapt. Out of this dynamic interplay, trajectories of interest emerge, that is, changes in relationships (equivalently, social embedding) as well as changes in individual behaviors. The mathematical specification of the model (see Snijders, Steglich, & Schweinberger, 2007, and Snijders et al., 2010, for details) is a Markov chain, a type of stochastic process describing how system states change probabilistically in continuous time. The continuous time specification means that state variables may change at any time and, in general, randomly, implying that the temporal occurrence of changes in any time interval t0,…, ∞ is given by waiting times between changes having the exponential distribution e-λt with rate parameter λ. The rate parameter can depend on covariates. By contrast, discrete time stochastic processes only change at fixed time points. A Markov chain is limited to discrete, as opposed to continuous, outcome states. The SAOM allows nominal or ordered categorical values for state variables, but continuous state variables are not possible at this writing. For behaviors only, continuous state dynamics can be estimated other ways (Voelkle et al., 2012). The Markov property implies that the probability distribution of the next transition is independent of its history, or equivalently, depends only on the state variables in the moment just prior to the new transition. Transition probabilities in SAOM can be made conditional on the state variables themselves (endogenous effects), in addition to exogenous covariates. Covariates (“effects”) are linearly combined in an “evaluation function,” which represents change probabilities using a multinomial link function (Ripley, Snijders, Boda, Voros, & Preciado 2015).

Rather than defining the stochastic model on a state space of one or several single-dimensional variables only, the SAOM defines a set of n × n matrices X(1), X(2),…,X(t)…,X(M) as part of the process. The elements xij(t) are 1 if person i is considered to “direct a tie” of relationship type X (e.g., friendship) to person j, and 0 otherwise (self-ties, where i = j, are not modeled). Hence, even if only one type of relationship is being modeled, a single network model could contain as many as n(n-2) relationships (ignoring self-relationships), each of which has its own complex dependency on the rest of the network and members' behaviors. Although the SAOM satisfies mathematical conditions guaranteeing a solution (Karlin & Taylor, 1975), model complexity precludes explicit solution formulas for any but very simple, substantively implausible models (Snijders, 2005). Therefore, estimates are obtained using iterative estimation methods. The methods currently available in the R package RSiena (Ripley et al., 2015) comprise a method-of-moments (MOM) approach, which is relatively fast, maximum likelihood (ML), which is much (about 100 times) slower but also more efficient with smaller samples, and a Bayesian approach, implemented for specific types of modeling. For further details consult Snijders et al., 2010, and Ripley et al., 2015.

Applying SAOM to Group Dynamics

An overview of the general format for application of SAOM to group dynamics is given in Wölfer et al. (2015), and the RSiena manual (Ripley et al., 2015) provides full instructions. We focus here on issues specific to studying group dynamics.

With group research, many groups will be on the small side, about 5–10 individuals. In the larger groups more typical of standard social network research, one can often estimate meaningful stand-alone group-level models, which can then be pooled via meta-analysis. With small groups, however, model estimation will usually require a pooling strategy from the outset. SAOM offers three such methods. Complete pooling can be accomplished by defining a relationship in a single adjacency matrix, with the (i,j)th entry set to a “structural zero” if actors i and j are not in the same group. Structural zeros are always zero by definition, to reflect the constraint that two people can never be linked if they are in different groups. Once this matrix is created, groups are not distinguished from each other in any way, and the same model is assumed to apply to all of them.

The rate homogeneity assumption may be loosened in the second approach, a multigroup model where, in addition to group specific rates of change from one wave to another, group specific dummy variables may be defined and interacted with other effects. This essentially fixed-effect approach can be further extended in a third approach, random coefficient multilevel modeling. These models are similar to multigroup models, except that parameters other than rates may be set to vary randomly across groups according to a multivariate normal distribution, as with ordinary multilevel models. Rate parameters are automatically variable across groups. In this paper we will discuss our experience with structural zero and multigroup models. Random coefficient modeling is currently still in a testing phase, but deserves consideration in the future.

An Example: Trust and Mentorship in Recovery Houses

In a previous study (Jason, Light, Stevens, & Beers, 2014), we examined the contingent development of trust and confident relationships among 31 individuals living in five recovery houses, over a three-month period. While effective recovery-supportive social environments might take various forms depending on the individual and his or her available resources, sober living environments—recovery houses—are one such form that have been shown to improve long-term treatment outcomes (Jason, Olsen, Ferrari, & Lo Sasso, 2006). Previous studies suggest that this residential experience can help maintain sobriety, if the individual remains in residence for at least three months (Jason, Stevens, Ferrari, Thompson, & Legler, 2012; Stevens, Ram, Jason, & Light, in press). About 50% of residents drop out prior to that time, however. It is natural to suggest that dropout has something do with social integration difficulties (Humphreys, Manikowski, Moos, & Finney, 1999; Vaillant, 1983).

That model employed the “structural zeros” approach described earlier. In that study, we found that trust relationships tend to be reciprocated, which is typical of symmetrical friendship-like relationships. On the other hand, confidant relationships—the highest rating of “relationship closeness”—showed no significant tendency towards reciprocation. This pattern is typical of role specialization, for example, in confidant relationships there is a confider and a listener. Friendships, by contrast, tend to become symmetrical over time (Snijders, 2001). This pattern led us to reconceptualize confidant relationships as being more like mentoring relationships than friendships. Additional discussion of rationale, methods, and results can be found in Jason et al. (2014).

A New Model

Here we report on a model based on our earlier work (Jason et al., 2014), with several innovations. First, in the current example we include three levels of trust; in the previous study it was binary-coded. Low trust is defined as being willing to lend alter less than $50, medium trust as $50–$100, and high trust as $100 or more. In RSiena, relationship closeness can be modeled ordinally. Two trust matrices are defined, X for medium trust and Y for high trust. More precisely, “medium trust” should be called “at least medium trust”; for brevity we use the former terminology. For medium trust, xij = 1 if person i named person j as someone they would loan $50 or more, and zero if less than $50; and for high trust, yij = 1 if person i named person j as someone they would loan more than $100, and zero otherwise. Thus, both medium trust and high trust appear as separate dependent variables in SAOMs, but change is constrained to occur only between adjacent ordinal levels of trust. As a third network we also model the development of putatively close-friend relationships described in the survey instrument as “confidants”; more is said about confidants further on. Individual characteristics used as predictors included a 9-item inventory of 12-step (Alcoholic Anonymous) activities (AAA; Humphreys et al., 1998, current sample alpha = 0.82), scored from 0 to 9, which asked questions such as “Do you now have an AA sponsor” and “Have you ever called an AA member for help”. We also used length of current house residence (“time in house”, TIH) as a predictor, categorized into three classes: 1 = 6 months or less, 2 = 7–12 months, and 3 = 13 or more months. Descriptive statistics for these measures are shown in Table 1. Additional measure details are given in Jason et al. (2014).

Table 1. Descriptive Statistics of Recovery House Networks by Wave b,d.

House: 1 2 3 4 5
Wave: 1 2 1 2 1 2 1 2 1 2
Number of Residentsa 4 5 6 7 5
Percent female 100% 0% 0% 0% 0%
Mean Age 46.5 53.0 47.8 46.4 47.8
Mean AA Activities 7.6 7.2 5.2 8.0 7.2
Mean Time In Residence 2.3 2.4 2.8 1.9 1.2
Percent non-White 100% 100% 0% 0% 0%
Percent HS Grad/GED 75% 80% 67% 86% 100%
Confidant Outdegree 1.00 1.25 0.60 1.00 0.33 1.33 1.14 1.71 2.00 0.60
Confidant Reciprocity 0.33 0.25 0.00 0.00 0.00 0.14 0.00 0.09 0.25 0.00
Confidant Transitivityc 1.00 1.00 1.00 1.00 0.00 0.43 1.00 0.78 0.75 1.00
Medium Trust Outdegree 2.75 2.25 1.60 2.40 4.33 4.33 4.71 5.29 0.60 1.20
Medium Trust Reciprocity 0.83 0.50 0.14 0.71 0.73 0.73 0.65 0.76 0.00 0.50
Medium Trust Transitivityc 0.90 0.69 0.86 0.82 0.91 0.86 0.92 0.97 1.00 0.00
High Trust Outdegree 2.00 2.00 0.20 1.00 2.33 2.16 3.43 4.00 0.40 1.20
High Trust Reciprocity 0.33 0.33 0.00 0.00 0.27 0.18 0.50 0.56 0.00 0.05
High Trust Transitivityc 0.70 0.70 1.00 1.00 0.74 0.88 0.78 0.93 1.00 0.00
Confidant Jaccardd 0.50 0.60 0.25 0.33 0.29
Medium Trust Jaccard 0.82 0.42 0.86 0.75 0.07
High Trust Jaccard 0.60 0.20 0.93 0.67 0.09
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a

Includes residents present for both waves.

b

Data are provided for residents who completed both waves of the study (n = 27).

c

“Weak” transitivity, where a- > b and b- > c implies a- > c, and “- >” means “chooses.”

d

Outdegree, reciprocity, and transitivity were calculated using the R package ‘sna’ (Butts, 2010), v. 2.2-0.

d

The Jaccard coefficient measures the amount of stability in network linkages between waves 1 and 2: 1 means no changes, 0 means all linkages changed.

Additionally, we experimented with several approaches for loosening the strong homogeneity assumption of the “structural zeros” model. Because of its practical interest to group researchers, we report in some detail on this process. First, we attempted to estimate a multigroup model with all five available groups (described earlier; see also Ripley et al., 2015). Results of this effort were not completely successful, for reasons discussed further on. The final model, shown in Table 2, had a hybrid structure: (a) one group (group 1) was eliminated because of very low structural change which also differed from any of the other groups, as well as the fact that the house was the only all-female residence, and might reasonably have different dynamics than an all-male residence, (b) the data were reorganized into two “sets”, each set comprising two recovery houses with similar levels of tie density and change rates, and (c) those two sets were then analyzed in a two-“group” multigroup model. This structure yielded findings similar to what had been found with a structural zeros model, and is reported because the extra generality lends some credibility to the findings. Time homogeneity and goodness-of-fit statistics are reported from an approximation model estimated by MOM, because these functions are not currently available for models estimated by ML, a more accurate estimation method for small samples (Snijders et al., 2010).

Table 2. Stochastic Actor-Oriented Model Results—Maximum Likelihood Estimation, 2-Set Model.

Effect Parameter Estimate SE p-Value 95% Confidence Interval Convergence t-ratioa,b
Medium Trust Rate: Set 1 11.97 3.47 < .001 (5.2, 18.8) 0.13
Medium Trust Rate: Set 2 7.18 2.91 .013 (1.5,12.9) 0.05
Medium Trust: Outdegree -1.07 0.38 .005 (-1.8,-0.3) 0.01
Medium Trust: Reciprocity 1.82 0.59 .002 (0.7,3.0) 0.02
Medium Trust: 12-Step (AAA) Alter -0.26 0.12 .030 (-0.5.,-0.0) 0.01
Medium Trust: 12-Step (AAA) Ego 0.19 0.08 .033 (0.0,0.3) 0.04
High Trust Rate: Set 1 1.52 0.56 .007 (0.4,2.6) 0.03
High Trust Rate: Set 2 3.59 1.22 .003 (1.2,6.0) 0.06
High Trust: Outdegree 1.91 0.78 .014 (0.4,3.4) 0.03
High Trust: House Time Ego 2.37 0.89 .008 (0.6,4.1) 0.01
Confidant Rate: Set 1 1.80 0.54 < .001 (1.0,3.3) 0.01
Confidant Rate: Set 2 7.38 2.74 .007 (2.0,12.8) 0.04
Confidant: Outdegree -1.02 0.31 .001 (-1.6,-0.4) 0.01
Confidant: Reciprocity -1.17 1.09 .283 (-3.3,1.0) 0.03
Confidant: 12-Step (AAA) Alter 0.06 0.15 .689 (-0.2,0.4) 0.03
Confidant: High Trust 2.33 0.75 .002 (0.9,3.8) 0.01
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a

Ratio of deviations of simulated vs. observed statistics for each effect, calculated in Phase 3 of the RSiena model estimation procedure. Conventionally, a value of less than 0.10 indicates good convergence (Ripley et al., 2015).

b

Overall maximum convergence ratio = 0.132; < 0.20 is recommended (Ripley et al., 2015)

Results

We applied the multigroup specification to data from groups 2-5, with rates free to vary across groups, and other effect parameters assumed equal. The recommended (Ripley et al., 2015) forward selection process was used, starting with model of just rates, outdegree, and reciprocity (Ripley et al., 2015; Wölfer et al., 2015) for all three outcomes, that is, medium trust, high trust, and confidant relationships, and adding ego and alter effects of AA activities and time in house. MOM estimation was used to eliminate highly insignificant effects, as its greater speed compared to ML was convenient for screening purposes; however, because our experience (see also Ripley et al., 2015) had shown MOM to be considerably less efficient with small samples, we kept effects with t-values in the vicinity of 1 to 1.2 or greater. Subsequent models were re-estimated using ML to determine statistical significance of effects.

Model Selection and Evaluation

A four-group model failed to produce reliable rate estimates; apparently, insufficient information was available to estimate rates independently for each group, which is not unexpected when group sizes range from five to seven individuals and there are only two waves of longitudinal data. However, descriptive information by group (Table1) suggested that both the number of linkages and the amount and type of changes in them differed enough across groups so that simply assuming a fully pooled structural zeros model could be misleading. Such a model might accurately reflect average mechanisms among the groups, but sometimes heterogeneity can produce biased and misleading estimates (Lospinoso et al., 2011; this cannot, of course, be tested directly in a structural zeros-based model, where groups are not individually represented). However a third possibility exists: create subsets of structural-zero-pooled groups by matching them on the basis of a reasonable set of descriptive statistics. Each such “matched set” can then be treated as a “group” in a multigroup model, which will lead to separate rate parameters for each matched set, rather than each individual group. Accordingly, we matched groups 2 and 5 as Set 1, and groups 3 and 4 as Set 2, and estimated the model shown in Table 2 using this approach.

The resulting model is shown in Table 2. Model convergence was very good; the overall maximum convergence ratio (a summary measure across effects), was .132, well below the recommended threshold value of .20 (Ripley et al., 2015), and all individual parameter convergence t-ratios (the autocorrelation between successive iterative estimates, which ideally are near zero) were .06 or less except for the medium trust group set 1 rate parameter (.13).

In SAOMs, there is no natural null model (Schweinberger, 2012). Fit is assessed by testing how well the model reproduces fundamental structural or behavioral features of the data. The standard suite of structural features includes outdegree distribution (total number of others chosen by ego), indegree distribution (total number of others who choose ego), geodesic distribution (minimum tie distance from ego to alter), and the triad census (all possible triadic structures). A well-fitting model should do a reasonable job of predicting all of these frequency distributions for each type of relationship modeled (Ripley et al., 2015; Schweinberger, 2012). A family of these tests, related to a score test (Neyman, 1959) was proposed by Schweinberger (2012) and are available in RSiena for MOM-estimated, but not ML-estimated, models. At present the fit of ML models can only be assessed from MOM-based approximations.

Table 4 shows goodness-of-fit p-values on the above-noted structural features for the 2-set model (Table 2), and also the same model estimated using a structural zeros approach. The 2-set model is only slightly better: 7 of the 12 fits differ nonsignificantly from the null hypothesis of equal predicted and actual values, compared to 6 of the 12 fits for the structural zeros model. Yet these marginal results need to be interpreted in light of the less efficient MOM approximations to an ML-based model. The 2-set MOM model is shown in Table 3, and it is evident that there are some important differences between that model and its ML counterpart from Table 2. First, 7 of the 16 model parameters in the MOM model are statistically significant, whereas 14 of 16 are in the ML model. This reflects the known (cf. Ripley et al., 2015) efficiency advantage of ML with small samples. But not only is the ML model more precisely estimated, it also arrived at materially different parameters for several important effects, namely high trust outdegree, high trust house time ego, confidant outdegree, confidant reciprocity, and confidant effect from high trust. Moreover, these likely inferior estimates from MOM are concentrated among high trust and confidant fits for outdegrees and the triad census. The latter comprise two very general characteristics of networks, which one would not expect to be well fit in the absence of sound estimates of outdegree and reciprocity parameters.

Table 4. Goodness-of-Fit p-Values--Four Standard, Unmodeled Network Statistics, Based on MOM Approximations to 2-Set and Structural Zeros Model)a.

Medium Trust High Trust Confidant
a. 2-Set Model
Indegrees .22 .91 .15
Outdegrees .07 <.01 <.01
Geodesic Distances .03 .78 .92
Triad Census .12 .01 <.01
b. Structural Zeros
Indegrees .10 .85 .41
Outdegrees <.01 <.01 .01
Geodesic Distances .18 .76 .44
Triad Census .05 .03 .05
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a

p-values indicate the confidence with which good fit can be rejected for the given effect.

Table 3. Stochastic Actor-Oriented Model Results—Method of Moments Estimation, 2-Set Model.

Effect Parameter Estimate SE p-Value 95% Confidence Interval Convergence t-ratioa,b
Medium Trust Rate: Set 1 5.09 3.64 .162 (-2.0, 12.2) 0.00
Medium Trust Rate: Set 2 8.73 6.21 .160 (-3.4,20.9) 0.02
Medium Trust: Outdegree -1.11 0.51 .030 (-2.1,-0.1) 0.00
Medium Trust: Reciprocity 1.84 0.77 .017 (0.3,3.3) 0.02
Medium Trust: 12-Step (AAA) Alter -0.31 0.17 .068 (-0.6.,0.0) 0.03
Medium Trust: 12-Step (AAA) Ego 0.31 0.15 .038 (0.0,0.6) 0.02
High Trust Rate: Set 1 1.43 0.54 .008 (0.4,2.5) 0.05
High Trust Rate: Set 2 3.97 2.03 .050 (0.0,8.0) 0.01
High Trust: Outdegree 0.72 0.46 .118 (-0.2,3.4) 0.01
High Trust: House Time Ego 0.77 0.52 .139 (-0.2,1.8) 0.03
Confidant Rate: Set 1 2.97 1.29 .021 (0.4,5.5) 0.01
Confidant Rate: Set 2 2.77 1.02 .007 (0.8,4.8) 0.03
Confidant: Outdegree -0.73 0.39 .061 (0.0,1.5) 0.06
Confidant: Reciprocity -0.85 0.61 .163 (-2.0,0.3) 0.06
Confidant: 12-Step (AAA) Alter 0.10 0.11 .363 (-0.1,0.3) 0.00
Confidant: High Trust 0.45 0.61 .461 (-0.7,1.6) 0.03
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a

Ratio of deviations of simulated vs. observed statistics for each effect, calculated in Phase 3 of the RSiena model estimation procedure. Conventionally, a value of less than 0.10 indicates good convergence (Ripley et al., 2015).

b

Overall maximum convergence ratio = 0.25; < 0.20 is recommended (Ripley et al., 2015)

Time homogeneity test results are shown in Table 5, again based on the MOM approximation (Table 3). In a two wave multigroup model, this test assesses homogeneity of results across subgroups, because the RSiena implementation of multigroup models treats each group's changes as a separate wave. Thus with two groups of two waves each, there are effectively four “pseudo-waves,” restricted so that there can be no transitions from wave 2 to wave 3 (since this amounts to a change of group, and is not meaningful). Therefore the RSiena time test can be used to calculate the probability that parameters other than rate parameters, which always are group-specific, are credibly homogeneous between the two pairs of waves 1–2 and 3–4, that is, between the two groups in question. It is evident from these results that the homogeneity assumption is tenuous—overall joint significance is low enough to reject the homogeneity null hypothesis, and this is also the case for medium trust parameters for outdegree, reciprocity, and 12-step ego, and almost (p = .06) for confidant outdegree. Attempts to generalize this specification by interacting set-specific dummy variables with the most heterogeneous parameters resulted in persistently unstable ML-based estimations, suggesting that even with the additional pooling of the 2-set model, there was insufficient data to model this level of detail. This left only the MOM-based tests to evaluate pooling of these parameters. However, since the MOM model was an approximation to the clearly preferred ML model, it is possible that an ML-based homogeneity test would produce better results.

Table 5. Time Homogeneity Test p-Values--MOM Approximation to 2-Set Model.

p-valuea
Medium Trust: Outdegree .03
Medium Trust: Reciprocity .02
Medium Trust: 12 Step (AAA) Alter .08
Medium Trust: 12 Step (AAA) Ego .04
High Trust: Outdegree .12
High Trust: House Time Ego .14
Confidant: Outdegree .06
Confidant: Reciprocity .17
Confidant: 12 Step (AAA) Alter .58
Confidant: High Trust .65
Joint Significance Test: .03
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a

p-values indicate the confidence with which the null hypothesis of no difference between group set 1 (groups 3 and 4) and group set 2 (groups 2 and 5) can be rejected.

Turning to substantive results, we had previous hypothesized (Jason et al., 2014) that trust and confidant relationships would mutually reinforce each other, in a positive feedback loop, but that analysis examined only two levels of trust. We were interested to see whether perhaps such a loop would be formed for confidant and medium trust relationships. We found in the present study that confidant relationships were indeed more likely if ego had high trust for alter (β = 2.33, p < .01), consistent with our previous result. However, neither medium nor high trust was affected by a preexisting confidant relationship.

Structural effects of outdegree, transitivity, 3-cycles, and reciprocity are important to include in most network models, either because they are of substantive interest (e.g., if two pairs of a triad are friends, is a friendship likely to form between the third pair?), or because they provide important controls (e.g., for proximity effects) that could be confounded with, for instance, individual behavioral similarities. These effects may, however, operate differently in small groups, compared to the larger groups normally examined in network studies. Indeed, of these so-called “network closure” effects, only reciprocity was consistently found in our models, and only for medium trust. For confidants, in fact, there was a nonsignificant trend in the direction of anti-reciprocity—if a chooses b as a confidant, it is somewhat less likely that b also chooses a as a confidant. This pattern is thus more consistent with a hierarchical relationship such as mentoring than it is like a friendship, where reciprocation is typical.

Another interesting finding, also noted in Jason et al. (2014), is the significantly larger parameter for transitions from medium to high trust, compared to the parameter for low to medium trust (.72, vs. -1.11). This comparison implicitly ignores differing rate parameters for each level of trust, thus assuming that the overall amount of change in each level of trust relationship is the same, and addressing only what specific change (up, down, unchanged) is most likely to occur, given that any change occurs. With this caveat, the parameter differential implies that medium to high trust transitions occur more readily than the low to medium transitions, that is, medium trust is an unstable state that tends to increase to high trust. One might also call medium trust a “threshold” which, once achieved in a relationship, is more likely to increase (especially if House Time of the sender of the relationship is positive). This logic points to predictors of the formation of medium trust relationships as, perhaps, particularly important to house social integration processes.

Overall, the 2-set ML model (Table 2) was quite similar to our previously published model (Jason et al., 2014) with a few modest but still interesting exceptions. For medium trust, a significant positive effect of 12-step ego was found, and a significant positive effect of time-in-house ego replaced a positive but borderline-significant time-in-house alter effect. Other parameters were similar in both level and significance, despite the slightly different specification (strictly speaking, one cannot directly compare coefficients in different samples or models if the link function is nonlinear, as with SAOMs; Ripley et al., 2015). From this, we infer that pooling strategy can indeed affect parameter estimation and model-related decisions in unpredictable ways; thus, attention to a defensible approach is important.

Other network closure effects (transitivity and 3-cycles) capture social or spatial proximity effects and hierarchies extending beyond dyads, respectively. The absence of such effects could reflect the relatively small group sizes and similar levels of interindividual exposure in the house environment. These effects are typical in larger networks where the relationship in question is friendship (Snijders et al., 2010), though, probably because they act as proximity effects—friends of friends are more likely to meet and interact as a result of these relationships-in-common. Of course, even in small groups, some individuals may interact with certain alters relatively more often, for instance, because of involvement in the same project at work or having adjoining offices. Thus, network closure effects should be routinely examined, as ignoring them can lead to inferences that homophily (preference for someone “like ego” in some way) is associated with relationship preference, when it is actually the (often-correlated) effect of proximity.

The probability of a relationship forming can depend on the value of a predictor for either ego (the chooser) or alter (the chosen). The 12-step involvement effects for ego and alter were examined as predictors for medium and high trust. A positive effect for ego indicated that individuals who participated more in AA activities were more likely to extend medium trust ties (0.19, p < .04). A negative effect for alters was also found, meaning that trusted alters tended to be less involved with AA activities. Although no interaction between this effect and time in house was found, the sample might have been too small to reliably estimate such subtle effects. Thus the negative relationship between medium trust of alters and their AA activities could reflect that fact that these tended to be longer-time residents, whose 12-step activity might have stabilized at a lower level than is typical for newcomers, but who nevertheless were generally trusted. For high trust, there was no significant alter or ego effect for AA activities.

We had also hypothesized that alter's time in house would predict formation of trust relationships. We found evidence to this effect for high trust (β = 2.37, p < .01), but not for medium trust or confidants. Perhaps high trust requires more time to get to know the person in question more deeply.

Discussion

In this paper we have presented and discussed a variety of issues specific to applying dynamic network modeling (SAOM, specifically) to the study of group dynamics. These issues have not received much attention to date, because network applications of SAOM have typically involved larger groups, for example, the size of school classrooms (Snijders & Baerveldt, 2003) at a minimum. Summarizing this study, we might begin by noting that even with quite a small sample—five recovery houses and 27 individuals, reduced to four houses and 23 individuals in an effort to increase homogeneity, and only two longitudinal waves—it is possible to estimate a plausible SAOM. The model seems plausible because (a) although the sample was small, the number of observable relationship ties (network linkages) is quite large: 67 per relationship type, for a total of 201. Network models of within-group relationships thus are based on more information than individual-only survey data, for instance, and more than is necessarily obvious. (b) Similar results were obtained from several different modeling approaches. For instance, a statistically significant effect of high trust on formation of confidant relationships appeared in all models, whether estimated by ML or MOM, and across pooling methods, from structural zeros to full multigroup, even when the latter failed to produce stable estimates of other, typically rate, parameters (result not shown). The same is true for the larger outdegree effect on high (vs. medium) trust relationships. Although other effects varied in magnitude and significance across modeling methods and specifications, these two seemed ubiquitous. Further (c), time (group homogeneity) tests and goodness-of-fit tests indicated that the model specification predicted important auxiliary features of the data fairly well, when one considers that these tests came from MOM approximations to more accurate ML estimates. Undoubtedly, these tests will eventually be available for ML models; theoretically, they apply in either case. Thus in the future, this additional conjecture will not be necessary. Nevertheless the differences in MOM vs. ML estimates at least partially explain MOM-based evidence of cross-group heterogeneity and poor model fit, suggesting that useful results are possible even with limited data.

The hybrid model, moreover, may be particularly useful for small group studies. Our analysis suggested that full multigroup models may not necessarily be estimable with group sizes on the order of 5–10 individuals. This logic would of course also rule out meta-analysis, which relies even more on stable within-group models. The hybrid approach allows researchers to fully pool effects for groups judged to be similar either on a priori theoretical grounds (for instance, we excluded the only all-female group for this reason, but if we had several all-female residences, they might have been combined into a single fully pooled “set”), or on the basis of descriptive data. Regarding the latter, we took a somewhat informal approach, reflecting in part the fact that with only a few groups, analyzing similarity more formally using, for example, clustering methods, would not be especially meaningful. Nevertheless, hybrid models constructed with the help of cluster analysis offer a way to effectively model data from a significant number of groups, even if many of them are relatively small. Special attention should be paid to tie density and tie stability in such analyses, as these tend to be correlated with other model effects, and should be as homogeneous as possible within fully pooled sets of groups.

In sum, SAOM provides a flexible framework for modeling group dynamics, including potentially quite small groups in those cases where pooled dynamics are plausible. SAOM has several inherent advantages over extant methods of modeling group dynamics, including a more process or mechanism-oriented framework compared to traditional regression-based approaches (Snijders & Steglich, 2013), and a capability for representing relationships and relationship change, the ability to represent interacting relationship and behavior change over time, among others (Wölfer et al., 2015). In this article we have discussed several approaches to pooling, and have proposed that “hybrid” pooling may be especially relevant. We have offered evidence that even with a small sample of small groups, interesting and useful models can be developed, although this point should not obscure the ambiguities likely to ensue, or the need to carefully consider how much data is needed to draw conclusions appropriate to the research questions at hand. For larger studies, in particular, more definitive pooling choices and model evaluations are possible, and obviously preferable. Space limitations prevented a correspondingly detailed evaluation of random effects models, but it seems likely that such models will also have particular advantages and disadvantages for small group studies, and should be investigated further. This and other options are constantly being revised and improved, with the most recent experience regularly incorporated into the latest version of the RSiena manual (Ripley et al., 2015), and reported on the Siena website at http://www.stats.ox.ac.uk/∼snijders/siena/.

Aside from being relatively new and quite different from standard social science modeling methods, a problem we hope that our article and other recent literature has begun to address, SAOM has some potential weaknesses. One of these is the Markov assumption, that is, that changes in relationships and behaviors are conditional only on the system's state at the time a choice is made, and how the system got to that point can be ignored. It remains to be seen how restrictive this assumption is in practice, but it seems reasonable to wonder whether, for example, the reformation of relationships is based on the same factors that caused them to form in the first place (McGrath, 1997). On the other hand, in behavioral science, temporally proximal predictors are usually stronger than more distal predictors, so even in systems where the Markov assumption is an approximation, it may nevertheless be a useful approximation. Further, time-lagged predictors can be included in cases where historical effects are known or suspected. Also, because of the time-consuming iterative estimation algorithm, formal power calculations have yet to appear. Future work will likely address this, as well as time and fit tests for ML and Bayesian estimation methods, and models for using continuous state space variables, among others. Clearly, SAOM is an increasingly flexible and mature modeling platform worthy of attention from group dynamics researchers.

Acknowledgments

Dr. Light was supported by award HD052887 from the Eunice Kennedy Shriver National Institute of Child Health & Human Development. Dr. Jason was supported by award DA19935 from the National Institute on Drug Abuse. The content is solely the responsibility of the authors and does not necessarily represent the official views of the Eunice Kennedy Shriver National Institute of Child Health & Human Development, the National Institute on Drug Abuse, or the National Institutes of Health. We thank Tom A.B. Snijders, the editors, and an anonymous reviewer for their helpful comments; the authors are solely responsible for any remaining flaws.

Contributor Information

John M. Light, Email: jlight@ori.org, Oregon Research Institute.

Leonard A. Jason, De Paul University

Edward B. Stevens, De Paul University

Sarah Callahan, De Paul University.

Ariel Stone, De Paul University.

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