Modeling Organizational Positions Chapter 2
2.1 Social Structure as a Theoretical and Methodological Problem Chapter 1 outlines how organizational theorists have engaged a set of issues entailed in the problem of assessing the social structure of economic exchange. These theorists identify two critical components of positional accounts of social structure the boundaries of the positional contexts in which organizations act, and a way of assessing the similarity of organizations assigned to a given positional context. In the previous chapter I argue that the blockmodeling approach proposed by White, Boorman, and Breiger (1976) is a methodology that can be applied to the analysis of organizational relations in a way that begins to incorporate both of these ideas. While this blockmodeling approach addresses the issue of how contexts shape the behavior of actors within their boundaries, it does not assess the relationship between an actor and its context in a way that is meaningfully interpretable. Consider as an example the analysis by White, Boorman, and Breiger (1976) of the pattern of relations of positive affect in a monastery. White and his coauthors suggest that the monks in this study might be meaningfully divided into a number of groups, based on their expression of positive affect toward one another. To the extent that a group is a social context that is important in determining emotional orientations, group memberships should be able to provide some information about whether or not a monk might like another monk, based on their respective group memberships. Following Sampson s (1969) original analysis, White and his coauthors identify a three-group model, and labeled these groups as the Loyal Opposition, the Young Turks, and the Outcasts. The results of their blockmodel analysis suggest, for example, that the Outcasts p. 23
express positive affect towards the Young Turks, but that the Young Turks do not reciprocate this affection. One objective of this analysis would be to use a set of social contexts and their associated boundaries to identify the ideal typical behavior of the actors embedded in a given context. This objective can complicate the interpretation of the results of a structural analysis. The three-group model proposed by White and his co-authors essentially claims that there are three social contexts that actors can belong to, and it makes specific claims about interaction within these contexts. For example, one claim of this model is that monks acting in the context of the Outcast group will express positive affect towards monks acting in the context of the Young Turks group. While this claim is true in very broad terms, it is certainly not descriptive of the relationship between every monk in the two groups. Models generated by these analyses are rarely perfect in this sense, and as a result, empirical social scientists are left with the task of choosing between alternative models. In assessing the appropriateness of blockmodels of the social network of these monks, this choice is based on two questions. One question concerns whether the three contexts identified are the three contexts that best describe the social structure of these monks. Alternatively, a structural sociologist particularly interested in defining the boundaries of the contexts of interaction might ask as White, Boorman, and Breiger did (1976: 751-2) whether or not the three-group model proposed by Sampson might be usefully decomposed into a more fine-grained partition. A somewhat different objective of this analysis would be to develop a model intended to predict individual behavior within the particular contexts identified as a part of a given social structure. Such a model might be used to characterize the ways in which p. 24
the behavior of actors in a particular social context is similar to the ideal-typical behavior expected of actors within that context. One advantage of this approach is that it allows a claim about social structure to be qualified in such a way that it might be empirically assessed. Rather than simply claiming that the Outcasts express positive affect toward the Young Turks, a model derived using this approach might express the claim that monks from the Outcast group have a certain probability of expressing positive affect toward the members of the Young Turks group. These two objectives represent qualitatively different approaches to modeling social structure. The first represents what might be labeled as a descriptive modeling approach, in that its objective is to describe the structural regularities in individual behavior. Descriptive models, in turn, can be evaluated with respect to their ability to effectively describe the behavior they are based on. The second objective is consistent with what might be termed a predictive modeling approach. Predictive models describe structural regularities in individual behavior as well, but they also make explicit predictions about the behavior of the actors that they model. While the notion of assessing how well a model describes a given set of behavior is relatively flexible, the assessment of how well a model predicts behavior is a somewhat more narrowly defined metric. As I will argue, this allows predictive models to be assessed in a manner that is less arbitrary and more powerful than methods that are frequently used to assess descriptive models. While these two modeling approaches differ in significant ways, they both address two fundamental measures of the quality of the relationship between a structural model and an observed set of behavior. The first of these is the accuracy of the claims p. 25
made by a given model. A descriptive model is accurate to the extent that it provides a good description of the behavior it is based on. In common sense terms, a descriptive model that claimed that the Young Turks expressed positive affect toward the Outcasts would be less accurate than one that claimed no such expression had been observed. Similarly, a predictive model that estimated the probability of observing a bond of positive affect from an Outcast directed at a Young Turk as 0.14 would be more accurate than a model that estimated this probability as 0.86. In addition to being evaluated in terms of their accuracy, structural models can be evaluated in terms of their complexity. An in-depth discussion of assessing the complexity of a model in the general case will be addressed in the following chapter, but in the case of Sampson s monks, most measures would assess a model with more groups as more complex than a model with fewer groups. In this chapter, I review the descriptive and predictive modeling approaches used by organizational researchers interested in assessing social structure, and how each of these approaches engages (or fails to engage) the ideas of accuracy and complexity in their determination of the most appropriate structural model. In particular, I show how the meaning of closeness in a predictive model is tied to a specific claim grounded in probability theory, while the assessment of closeness in a descriptive model is based on less powerful and essentially arbitrary measures. 2.2 Descriptive Modeling Approaches One of the objectives of any structural analysis is to narrow the field of possible structural models down to the smallest number of models ideally just one. This process p. 26
inevitably involves some explicit or implicit measure of goodness-of-fit that can be used to determine which model or models should be selected. Goodness-of-fit measures for structural models typically take the accuracy of a model with respect to the data into account, and are sometimes affected by the complexity of a model. The choice of accuracy and complexity measures, therefore, can fundamentally affect the way variations in the data affect the way models are chosen. A review of empirical research done using the descriptive modeling approach will demonstrate that the goodness-of-fit measures used are generally not explicit with respect to this issue. As a result, it is difficult to establish whether the model selection rationales implied by these studies are consistent with the theoretical frameworks they are intended to empirically support. A significant amount of methodological work has been done in the area of blockmodel analyses of social networks. The distinction between descriptive and predictive modeling approaches is made particularly clear in this body of work, because of the relative clarity with which goodness-of-fit measures are made explicit. Wasserman and Faust (1992) provide a comprehensive review of these goodness-of-fit measures, some of which are considered here. In order to discuss these measures in detail, it is necessary to briefly review some of the mechanics of blockmodel analysis, and to introduce some terminology. Social networks are frequently characterized by a matrix X, where the elements of the matrix x ij correspond to social ties. The assignment x ij = 1 corresponds to a case in which an actor i sends a tie to an actor j. A descriptive blockmodel θ is composed of a mapping of actors to positions and an image matrix. The mapping φ( ) maps every actor i to a position B r, typically on the basis of some measure of structural equivalence, such p. 27
that if φ (i) = φ (j), then actors i and j are structurally equivalent. The image matrix B is comprised of elements b rs that represent hypotheses about the proposed social structure of the network. In the strictest terms, the assignment b rs = 0 represents the hypothesis that there are no ties from actors in position B r to actors in position B s. A descriptive blockmodel is completely characterized by the mapping φ( ) and the image matrix B. Measures that characterize a position or the relation between two positions figure centrally into goodness-of-fit measures for blockmodels of social networks. Wasserman and Faust (1992) denote the number of actors in a position B r as g r, and define g rs as the number of possible ties from actors in position B r to actors in position B s as g rs = g g r s if g r g s g r (g r 1) if g r = g s, (2.1) corresponding to the assumption that there are no ties between actors and themselves. They go on to define the density of ties from a position B r to a position B s as Δ rs = x i Br, j B ij s g rs. (2.2) Assessing the goodness-of-fit of a blockmodel also requires some notion of the pattern of ties predicted by the model. Wasserman and Faust (1994) refer to this as a target blockmodel denoted X (t). In general, the network of ties predicted by a blockmodel p. 28
can be referred to as the estimated network ˆ X. The entries ˆ x ij of this model can be computed as ˆ x ij = b φ(i)φ( j ). (2.3) where ˆ x ij = 1 means that the blockmodel θ = (B, φ) predicts that there should be a tie between actor i and actor j. Wasserman and Faust use these definitions to outline several different goodnessof-fit measures for descriptive blockmodels. Each of these measures has slightly different implications for the way in which the accuracy of a model and the complexity of a model contribute to determining which models are selected. Two of these measures are relatively straightforward, effectively taking the form of a city-block distance measure. The measure δ x1 measures the distance between the observed tie densities and the predicted tie densities, while δ b1 measures the distance between the observed block densities and the predicted block densities. δ x1 = x ˆ ij x ij (2.4) i j δ b1 = b ˆ rs b rs. (2.5) r s While these two expressions are very similar, the measures reflect a subtly different balance of accuracy and complexity. The actor-level measure δ x1 is effectively a direct measure of model accuracy every deviation of the predicted set of tie values is treated p. 29
with the same weight. The complexity of the model, as a function of the number of blocks, does not influence this measure either directly or indirectly, net of the ability of a more complicated model to predict tie values with greater accuracy. The block-level measure δ b1 differs from the actor-level measure in that it weights deviations from predicted ties by the size of the block. While this is a subtle distinction, it indirectly causes the measure to factor in complexity effects, albeit in a less than straightforward way. With a bit of algebraic manipulation, Equation 2.5 can be rewritten as i Br, j B s r,s g rs x ˆ ij x ij δ b1 =. (2.6) In a given block, the value ˆ x ij x ij will never change sign. In a block where b rs = 0, this value will either be 0 or 1, and in a block where b rs = 1, this value will either be 0 or 1. Either way, taking the absolute value of this expression inside the interior summation is equivalent to taking it outside of the summation, as this interior sum is simply a measure of the absolute value of the difference between the predicted and actual density in a block. As a result, Equation 2.6 can be rewritten as i Br, j B s r,s g rs x ˆ ij x ij δ b1 = = x ˆ ij x ij. (2.7) i j g φ(i)φ( j ) p. 30
This formulation of the block density measure makes the weighting of tie-level deviations by the size of the block explicit. Moreover, it begins to show how the complexity of the model can affect this measure of goodness-of-fit. If we assume that actors are evenly distributed into positions, and denote the number of positions in a blockmodel as B, then the average position B r has g/ B actors in it, and the value g φ(i)φ(j) can be estimated as g 2 / B 2. This means that Equation 2.7 can be estimated as δ b1 ˆ δ b1 = B 2 g 2 x ˆ ij x ij = B 2 g δ. (2.8) 2 x1 i j A descriptive blockmodel with B positions is parameterized by B 2 b rs values. Accordingly, this measure assesses the complexity of a model as a function of the number of parameters B 2, and as a function of the total number of possible ties g 2. Blockmodels of higher complexity in this sense are penalized in the complexity term, balancing out the extent to which they are positively evaluated for being accurate. If measure δ x1 is taken to be a measure of model accuracy, then the measure δ b1 does, in some sense, incorporate both accuracy and complexity. In addition to these two relatively straightforward measures, Wasserman and Faust review other measures of descriptive blockmodel goodness-of-fit. The Carrington- Heil-Berkowitz measure (Carrington, Heil and Berkowitz 1979; Carrington and Heil 1981), denoted here as δ b2, is conceptually similar to a χ 2 measure. It assesses the fit of a blockmodel based on an α-fit criterion, such that a block is assigned to be a zero-block p. 31
only if the density in that block is less than α. Carrington, Heil and Berkowitz define a quantity t rs as t rs = 1 if Δ < α rs 1 α otherwise. α (2.9) This quantity, in turn, can be used to define the goodness-of-fit measure δ b2 as δ b 2 = r,s 2 g rs Δ rs α. (2.10) g(g 1) αt rs The second term in the summand in Equation 2.10 is, like the measure δ x1, a measure of the accuracy of a model. If the actual set of ties in a social network matches the predictions of the blockmodel exactly, this term is equal to 1, and it is minimized to the extent that the ties do not match the hypothesized pattern. Given the assumption that the average position B r has g/ B actors in it proposed above, the first term in the summand effectively becomes B 2, a count of the number of parameters in the model. However, rather than balancing accuracy and complexity, increased complexity and increased accuracy both raise the value of this measure. As such, it cannot be used to choose a model that balances accuracy and complexity. A final measure Wasserman and Faust review is an actor-level matrix correlation measure δ x3 (Panning 1982). If we define x ij as the mean of all ties x ij, and ˆ x ij as the mean of all predicted ties ˆ x ij, then the matrix correlation measure is defined as p. 32
δ x 3 = (x ij x ij )( ˆ x ij ˆ x ij ) i j ( (x ij x ij ) 2 i j ) 1/ 2 ( ( x ˆ x ˆ ) 2 i j ). (2.11) 1/ 2 ij ij The matrix correlation δ x3 effectively measures the pair-wise correlation between the actual tie values in a network and the ties values predicted by a blockmodel. As such, this measure is, like the measure δ x3, basically a measure of the accuracy of a blockmodel. Net of the ability of a more complex blockmodel to more accurately predict ties, this measure does not incorporate the complexity of a blockmodel. The performance of these four measures can be demonstrated by considering the network of expressed affect between the monks studied by Sampson (1969). Table 2.1 shows how each of these measures assesses the fit of four candidate blockmodels. The subscript of each model corresponds to the number of blocks in the model. The models θ 3 and θ 5 correspond to the three and five-position models proposed by White, Boorman, and Breiger (1976), respectively. The model θ 1 is a blockmodel with a single position, and the model θ 18 is a blockmodel with each actor assigned to his own position. Model δ x1 δ x3 δ b1 δ b2 θ 1 250 n/a 0.82 0.01 θ 3 78 0.49 2.41 0.39 θ 5 71 0.52 6.35 0.48 θ 18 0 1.00 0.00 1.00 Table 2.1: Descriptive Blockmodel Goodness of Fit Measures p. 33
The results in the first two columns of Table 2.1 demonstrate how the two actorlevel goodness-of-fit measures δ x1 and δ x3 both behave as measures of model accuracy. As the models become successively more fine-grained, they become more accurate, as these measures indicate. As the effects of complexity and accuracy work in the same direction as incorporated in the Carrington-Heil-Berkowitz measure δ b2, it is impossible to determine from these data the extent to which complexity and accuracy independently affect the measure. It is clear, however, that the measure rewards models that are complex and accurate relative to models that are simple but inaccurate. Of these four measures, only the block-density measure δ b1 balances complexity and accuracy. For instance, it evaluates the three-position model θ 5 more favorably than it does the fiveposition model θ 3, even though the five-position model is more accurate. Still, all four measures evaluate the fully saturated eighteen-position model θ 18 as the one that fits the data best. That the results of these analyses do not reflect Sampson s intuitive insights about the structure of this group does not alone imply that these measures should be dismissed as inadequate. A measure that cannot produce a result that suggests there is no structure in a social system would clearly be problematic. Rather, all four of these measures are inappropriate for determining structure in this way because they will never evaluate a fully saturated model less favorably than a model of lesser complexity. The examples of descriptive modeling approaches presented here all relate to modeling group structure using social networks. While the critique presented here is directed at descriptive blockmodeling approaches, it can be directed at any descriptive modeling analysis in which the way that the accuracy and complexity implications of the model selection criterion are not made explicit. For example, in their analysis of career p. 34
systems, Stovel, Savage, and Bearman (1996) use a blockmodeling approach to cluster career paths, and argue that a career structure based on five groups is appropriate given their data. Their empirical support for this choice (1996: fn. 18) is based on the fact that the mean within-block distances at this level of analysis are smaller than the mean between-block distances, at a statistically significant level. This metric essentially assesses the accuracy of the model of career systems without considering their complexity. If the set of careers was partitioned into a successively larger number of subgroups, the statistical significance of the difference between within-block and between-block distances would grow monotonically until each career path were partitioned into its own group. The fact that Stovel, Savage, and Bearman choose a fiveblock model rather than a fully-saturated 80-block model is at least consistent with the possibility that they sought to balance the complexity of their model against the accuracy of their measure. 2.3 Predictive Modeling Approaches Predictive modeling approaches are distinguished from descriptive modeling approaches in that they provide explicit accounts for the mechanisms by which actors diverge from these structural regularities. Many descriptive models of social structure can be straightforwardly transformed into predictive models by making the assumptions embodied in a descriptive model explicit. Predictive models of exchange in a social network are generally referred to as stochastic network models, and predictive models that take into account the assignment of actors to categories are termed stochastic blockmodels. In this section I discuss these stochastic blockmodels and compare them to p. 35
descriptive blockmodels with respect to their ability to assess the structure of exchange in a network. 2.3.1 Stochastic Blockmodeling The term stochastic blockmodeling can be used to refer to an entire class of models that assign a probability p(x θ) to the observation of a particular pattern of network ties x given a set of model parameters θ. While there are a number of researchers who have presented stochastic blockmodeling approaches (Wasserman and Pattison 1996; van Duijn, Snijders, and Zijlstra 2004), I focus here on two models that are particularly germane to the kinds of exchange in the networks that will be empirically investigated in this dissertation. Both of these models are based on a set of ideas drawn out of the basic p 1 stochastic graph model (Feinberg and Wasserman 1981; Holland and Leinhardt 1981), the details of which I present below. The p 1 stochastic graph model is an extension of a basic Bernoulli graph that attempts to take into account the fact that some actors are relatively more likely to engage in exchange than others, and that in some cases, actors may be likely to reciprocate the exchange behavior that is directed at them from other actors. Holland and Leinhardt base their model on the assumption that, net of a set of structural parameters, the exchange behavior in a dyad D ij = (x ij, x ji ) is independent of the behavior in all other dyads in a network. They derive an expression for the probability of the observation of a given pattern of dyadic exchange based on the likelihood that the dyad reflects a mutual, asymmetric, or null pattern of exchange. This distribution is termed the MAN distribution, such that for a given dyad D ij : p. 36
m ij = P(D ij = (1,1)) i < j, (2.12) a ij = P(D ij = (1,0)) i < j, (2.13) a ji = P(D ij = (0,1)) i < j, (2.14) n ij = P(D ij = (0,0)) i < j, (2.15) and m ij + a ij + a ji + n ij = 1, for all i < j. (2.16) The authors use this formulation to show that the probability of the observation of a given network of ties can be expressed as: x P(X = x) = m ij x ji x ij a ij (1 x ji ) ij n (1 x )(1 x ) ij ji ij, (2.17) i< j i j i< j which can be expressed in an exponential form as: P(X = x) = exp{ ρ ij x ij x ji + θ ij x ij } n ij, (2.18) i< j i j i< j where ρ ij = log((m ij n ij )/(a ij a ji )) i < j (2.19) and θ ij = log(a ij /n ij ) i j. (2.20) Holland and Leinhardt explain that the parameter ρ ij governs what they term the force of reciprocation, that is, the increase in the log-odds of the likelihood that a tie will be sent from an actor i to an actor j (x ij = 1) if there is a tie sent from the actor j to the p. 37
actor i (x ji = 1). Similarly, they explain that the parameter θ ij measures the increase in the log-odds of the likelihood that a tie will be sent from an actor i to an actor j (x ij = 1) in the absence of a tie from the actor j to the actor i (x ji = 0). The family of networks described by a full set of these parameters is not estimable, so Holland and Leinhardt propose a model based on a restricted set of parameters such that and ρ ij = ρ (2.21) θ ij = θ + α i + β j. (2.22) In other words, they restrict reciprocity to act in a constant way across all dyads, and force the asymmetric choice parameter θ ij to be a function of the productivity of the sending actor α i, the attractiveness of the receiving actor β j, and the mean choice tendency θ. It is also worth noting that the expected value of the logit is determined by this function as well, such that where E(log(p ij /(1-p ij )) = θ + α i + β j, (2.23) p ij = P(x ij = 1). (2.24) Collectively, these formulations can be summarized by noting that, for a binaryvalued network with no reciprocity, the p 1 model predicts that the expected value of the logit of tie values is an additive function of the overall tendency of ties to exist in the p. 38
network θ, the productivity of the sender of the tie α i, and the attractiveness of the tie recipient β j. In networks where the reciprocity ρ diverges from zero, the likelihood of a tie to be sent from an actor i to and actor j will be increasingly determined by whether or a not a tie is sent in the opposite direction, to the extreme case where ρ = ±, and a network becomes completely symmetric (or asymmetric), wherein the tendency of a tie is completely determined by this reciprocal behavior. 2.3.1 The p 1 Stochastic Blockmodel The p 1 distribution is useful for characterizing the probabilistic structure of graphs and network in a general sense, but it does not provide a mechanism for explicitly modeling the influence of the group structure of actors on the likelihood of exchange behavior. While a variety of stochastic blockmodeling approaches have been proposed to achieve this aim (Holland, Laskey, and Leinhardt 1983, Anderson, Wasserman, and Faust 1992) the p 1 stochastic blockmodel proposed by Wang and Wong (1987) most fully achieves this objective in the context of the p 1 random network distribution. The principal contribution of the p 1 stochastic blockmodel to the basic p 1 distribution is that it allows the asymmetric choice parameter to be determined in part by the group memberships of the sending and receiving actors involved in a dyadic exchange. If the sending actor i is a member of a block labeled r, and the receiving actor j is a member of the block labeled s, then Equation 2.22 above can be expanded as θ ij = θ + θ rs + α i + β j, (2.25) p. 39
where θ rs corresponds to the relative excess tendency for actors in block r to direct choices toward actors in block s. This simple extension allows a p 1 stochastic blockmodel to capture the effect of different assignments of actors to groups on the likelihood of observing a particular pattern of network exchange. 2.3.2 The p 1R Stochastic Blockmodel The exchange behavior that many stochastic network analytic approaches attempt to model is essentially dichotomous the outcome of interest is simply whether a focal actor chooses a particular actor or not. While many kinds of social exchange behavior can be reasonably modeled as dichotomous outcomes, there are clearly some kinds of behavior for which reduction to a dichotomy would represent a fairly severe limiting of the expressive range of the phenomena. There are surprisingly few stochastic network models that can be used to measure non-dichotomous exchange behavior. The principal analytic strategy taken by these models has been to move from only considering the likelihood of an exchange taking on a single (dichotomous) response level to considering the likelihood of an exchange taking on one of a number of response levels. Wasserman and Iacobucci (1986) introduce an early model along these lines that expands the p 1 model to the analysis of networks where relations take on one of C discrete values. Anderson and Wasserman (1995) generalize this model by considering the interactions between response levels in addition to their first-order effects. There are a number of empirical phenomena that might effectively be analyzed using a model based on ordinal or categorical relations. As an example, Wasserman and p. 40
Iacobucci (1986) analyze networks of behavior expression in which the frequency between two actors is characterized as rarely, sometimes or frequently. While the models proposed by Wasserman and co-authors can straightforwardly be applied to these phenomena, there are other relational behaviors that are not so easily reduced to ordinal or categorical responses. In particular, these categorical models do not correspond well to networks that represent resource flows. Networks that represent the flow of individual migrants between cities or nations, investments between firms or nations, or goods and money between industries (Burt 1983) exemplify these resource exchange networks. In many of these cases it would be difficult to generate the theoretical logic that would support modeling a level of exchange that is fundamentally continuous as a categorical variable. The p 1R stochastic network model presented here departs from these categorical models in that it explicitly models network exchange as a continuous variable. One of the most significant differences between binary and real-valued networks, of course, is that ties in binary networks can only take on two values, while exchange levels in real-valued networks can take on any of a continuous range of values. As a result, the distribution of tie values in a random real-valued graph that underlies such a network is a bit more complex than the relatively simple one-parameter Bernoulli graph that underlies a binary network. For positive real-valued exchange networks 1, a relatively simple approach would be to assume that the tie values are log-normally distributed with mean θ ij and 1 This approach can also be used for non-negative real-valued exchange networks if all zero-valued exchanges are assigned some value smaller than the lowest observed nonzero tie value. This approach may be particularly valid in those cases where zeroes in the data represent exchange levels that were too low to report, rather than exchanges that were actually zero. p. 41
variance σ 2 ij. The model can be further simplified by assuming that the variance of this distribution is constant across the network, such that σ 2 ij = σ 2. Given these assumptions, it is possible to expand the governing equations of the p 1 model to a model for real-valued exchange networks. Equation 2.23 above represents the expected value of exchange between two actors in a dichotomous network. This equation can be expanded to model real-valued exchange as follows: E(log(x ij )) = θ + (α i + β j ) + ρ(α j + β i ), (2.26) 1+ ρ where the reciprocity parameter ρ ranges from 0 to 1. This formulation allows Equation 2.26 to reduce to Equation 2.23 when there is no reciprocity in an exchange network, and it forces the expected value of an exchange E(x ij ) to be equal to the expected value of exchange E(x ji ) when reciprocity is at its maximum of 1. Equation 2.26 can be expanded to a governing equation for a p 1R stochastic blockmodel by introducing a block parameter θ rs as follows: E(log(x ij )) = θ + θ rs + (α i + β j ) + ρ(α j + β i ) 1+ ρ. (2.27) This formulation allows the p 1R stochastic blockmodel to capture the expressive range of models that the basic p 1 stochastic blockmodel does in the context of real-valued network exchange. To the extent that p 1 stochastic blockmodels are appropriate for empirical investigations of the organizational position metaphor in the context of p. 42
dichotomous exchange, p 1R stochastic blockmodels should therefore be appropriate for studying organizational positions as defined by real-valued exchange. 2.3.3 Stochastic Blockmodel Goodness of Fit Measures Comparatively speaking, there are many fewer goodness-of-fit measures for stochastic blockmodels than there are for descriptive blockmodels. The p 1 and p 1R stochastic blockmodels presented here are like many stochastic blockmodels in that they can be used to produce a set of predicted tie values ˆ x ij in addition to assigning a probability p(x θ) to any observed pattern of ties. Wasserman and Faust (1992) argue that the likelihood-ratio statistic G 2 is an appropriate goodness-of-fit measure for stochastic blockmodels that can be characterized in this way. They determine the value of this statistic in the context of a stochastic blockmodel θ as G 2 θ = 2 x ij log(x ij / x ˆ ij ). (2.28) i, j Wasserman and Faust argue that this is a reasonable goodness-of-fit metric for the assessment of stochastic blockmodels that assume the dyadic independence of tie values net of the structural parameters of the model. Under this assumption, they argue that this G 2 θ metric is distributed as χ 2, and as such can be used to compare the goodness of fit of models of different complexity, as long as these models are nested by evaluating the p- value of the G 2 θ measure given the degrees of freedom in each model. The implication of this approach is that the model that should be chosen for a given network is the most complex one for which the p-value is still insignificant. Wasserman and Faust argue that p. 43
an alternative to this approach is to use a normalized G 2 θ metric, where the measure is simply divided by the degrees of freedom. Following this logic, the model that should be selected is the one with the lowest normalized G 2 θ. Both of these proscriptions position the G 2 θ metric as a goodness-of-fit measure that balances the accuracy of the data with respect to a model against the complexity of the evaluated model. The statistical rationale supporting the use of this measure for the purpose of model selection highlights the distinction between descriptive and predictive modeling approaches. While the predictive modeling approach can bring the power of statistical analysis to bear upon the problem of model selection, statistical measures such as the G 2 θ metric cannot comprehensively address all of the issues this problem presents. One inherent problem with this approach is that the probability theory underlying these measures is based on the assumption that only one model is being evaluated. As Wasserman and Faust (1992: 703) note, This theory should be applied only to a priori stochastic blockmodels, because the data mucking that must be done to fit their a posteriori counterparts invalidates the use of a statistical theory. If the objective of the model selection task is to compare a wide range of models to determine the one that is the best representation of the data, then these approaches cannot be used. 2.4 Conclusions The blockmodeling approaches outlined in this chapter provide formal methods for addressing some of the issues raised by the organizational position metaphor. Descriptive models and their associated goodness-of-fit measures are useful for beginning to think about how to assign actors to organizational positions on the basis of p. 44
their degree of structural equivalence. Given a particular level of aggregation, these methods can be useful in identifying good ways to partition actors into their respective contexts, and as such identify the boundaries of these positions. Descriptive blockmodel goodness-of-fit measures can be helpful in evaluating the relative fit of one partitioning relative to another, but the theoretical meaning of these measures is not precisely clear. While descriptive blockmodeling approaches provide some purchase on the problem of identifying the boundaries of organizational positions, predictive approaches provide a way of thinking about the idea of closeness as implicated by the organizational position metaphor. Specifically, in assessing the likelihood of observing a particular exchange between organizations, a predictive model allows a researcher to directly assess the extent to which the behavior of a given organization is close to the aggregate behavior of other organizations located in the same position. Under this modeling approach, organizations embody the positional idea of closeness explicitly to the extent that they are likely to engage in a particular pattern of exchange behavior. The conclusions that can be drawn from a predictive blockmodel may, in fact, be richer and more informative about structural processes than those reached through a descriptive modeling approach. The logic that Wang and Wong (1987) apply to analyzing the impact of gender on the production of friendship relationships could straightforwardly be extended to the analysis of the impact of industry structure on the exchange of goods and resources between firms. The p 1R stochastic blockmodel introduced in this chapter aims to extend the analysis in exactly this way. A descriptive blockmodel analysis of this exchange network might produce a set of ways to partition organizations into industrial positions based on the similarity of their patterns of p. 45
exchange behavior. A stochastic blockmodel analysis, on the other hand, would use one of these industrial classification schemes to indicate the extent to which the economic exchange behavior of individual firms is related to the aggregate exchange behavior of other firms within their respective industry position. While a predictive blockmodel analysis might be useful in assessing how well a given set of exchanges between organizations corresponds to a particular industry structure, it would not by itself be able to unequivocally identify the boundaries of a set of organizational positions. The fundamental problem with both the descriptive and the predictive modeling approaches is that neither provides a transparent facility for directly assessing the complexity of a model of social structure. Predictive models of social structure like the p 1R stochastic blockmodel are more useful than descriptive models because they define the accuracy of a model in terms of the probability that it will predict the observed pattern of behavior, rather than in terms of an arbitrary metric. While predictive models provide this statistically grounded rationale for evaluating the accuracy of models, they do not provide such a rationale for the direct assessment of model complexity. In the following chapter, I introduce a method drawn from the field of information science that specifically can be used to answer this question. This method allows the problem of assessing model complexity to be laid out in terms of a formal and transparent probabilistic theory. When taken in combination with the predictive models of social structure outlined in this chapter, this method can be used to empirically identify organizational positions, and move the organizational position metaphor in the direction of an organizational position construct. p. 46