Comment on Harrison White, "Chance Models of Systems of Casual Groups"Author(s): James ColemanSource: Sociometry, Vol. 25, No. 2 (Jun., 1962), pp. 172-176Published by: American Sociological AssociationStable URL: http://www.jstor.org/stable/2785948 .

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172 SOCIOMETRY

group on the number of isolates in the system of groups. They rely solely on the shape of the equilibrium size distribution to determine the validity of their model, and it is ironic that the dependence of arrival rate on number of isolates does not affect this shape at all. The difficulties in conceptualization and correct operational specification of parameters in stochastic models of empirical reality are as great as the mathematical difficulties in obtaining solutions. It takes courage to seek affirmative results in this area and it is to be hoped that more elaborate empirical investigations will be undertaken to permit further progress.

Comment on Harrison White, "Chance Models of Systems of Casual Groups"

JAMES COLEMAN, Johns Hopkins University

Because Professor White raises some interesting new points, and because he attacks vigorously some of the aspects of our model, it is useful to consider some of these points, in what I feel to be their order of importance.

1. Single Group vs. System of Groups. White makes a very important point in the discussion surrounding equations (1)-(11), and in the intro- ductory paragraph on closed systems, where he distinguishes a stochastic model for the system of groups from a stochastic model for a single group in a system of groups. In many stochastic processes, a stochastic model for a single element is identical to the model for the mean value of the system of elements. If this were the case in the models under consideration, then our treatment, and his subsequent treatment, would give an exact distribu- tion for the expected number of groups of size i. However, in systems exemplified by these, it is not permissible to do as we did, and set up a stochastic model for a single element's behavior, in order to obtain an expected value. Only if the number of persons approaches infinity would this procedure be permissible. Consequently the model we developed, and those White has developed, are only approximations, as he points out. Alternatively, one can think of our treatment as a deterministic one at the level of the system of groups. When it is so interpreted, then the point made by White is that the deterministic model does not give the same result as the expected value of the stochastic model.

What are the kinds of processes that do not allow the simplifying trick which we attempted? It is interesting to pose this question, for this dif- ference distinguishes those processes which are sociological in a strong sense from those which are not. In general, the distinguishing feature of such systems is this: they are systems in which the transition probabilities

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COMMENT ON "CHANCE MODELS" 173

for one element are contingent upon the state of other elements. In stochastic epidemic models, the probability of an individual becoming infected is con- tingent upon the exact number (not the expected number) of infectious individuals. In the group-size models, the probability of moving to a group of size i is contingent upon the exact number (not the expected number) of groups of size i.

What then does one do for models of this "sociological" sort? The stochastic models are often insoluble by present methods; the deterministic models are only approximations. I suggest the most reasonable strategy is that taken unwittingly by us, and consciously by White: to get an approximate solution rather than none, to develop the deterministic models as far as possible.

2. Open and Closed Systems. White, developing his models from the imagery of queing theory, discusses two kinds of systems, open and closed, the former with persons entering and leaving, and the latter with a fixed set of persons. This is a valuable addition to our treatment, which considered only closed systems, though we never made this clear.

As White points out, we omitted the equation for dp1/dt for our model, which led to confusion about whether the model referred to a closed system or an open one. The equation is given below:

I _= anj > pi-2anjpj+,8:4ipj+4,8p2 dt Ad i=, This equation shows that Pi (the proportion of groups in state 1) is decreased through all the joinings that occur in dt, and increased by all the breakaways that occur in dt. There is no gain from or loss to the outside.

3. Interpretation of na/fl. As White points out, the parameter for which we solve, na/fl, includes a dependent quantity, n (the total number of groups), and not merely the parameters of the system, a (the acquisition rate of groups), 8 (the loss-rate per person of groups), and N (the total number of persons). We overlooked this, leading to an incorrect statement, as White points out in footnote 15.

Because of the nature of the process postulated by our model, it is impos- sible to write the equation for pi (the proportion of groups of size i) explicitly in terms of N, a, and f8. What is possible is to write the equation in terms of k,

as in our eq. ( 15), Pi= ( ki 1) and then write the parameter -a in terms

of k. Using the fact that N =un, /A= k and - - ek- 1 we get

Na = kek. In our eqs. (17) through (22), and in solving for the empirical

examples in our paper, we should have solved for Aa rather than na

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174 SOCIOMETRY

It is unfortunate that the indirect strategy in which Pi must be written

in terms of k, and then k related to N, is necessary, but it is forced by

the nature of the model, in which the density of groups plays an explicit role. White's criticism has forced us to think through more carefully just how this occurs, and how the parameters are to be interpreted. Given two

Na differing values of the parameter Nha, just how are they to be interpreted? The interpretation is this: first, it is necessary to standardize the sample of observations in time or space, so that N refers to the number of indi- viduals per unit of time or space.' If the area covered in one community

sweep is twice that of a second, then Na for the first must be divided by

two to give N the same dimensions in the two cases: the number of persons in a standard unit area. The empirical hypothesis here is that the basic parameter of the system is a/f,, and that as the density increases or decreases, but the situation and the affinities of the persons remain the same, the coalescence into groups will increase or decrease in such a way that a/,3 remains constant.

An example will indicate how values of a/,8 are compared for different situations. In Seoul, Korea, two sets of observations were carried out, resulting in the distribution of Table 1. The first covered 1.5 hours, persons

TABLE 1

Pedestrian Groups Observed in Seoul, Korea

Semi-residential Area Business District Group Size Actual Calculated Actual Calculated

1 818 813.0 897 899.0 2 194 205.0 252 245.0 3 38 34.4 38 44.6 4 6 4.3 7 6.1 5 1 0.4 1 0.7

Total 1057 1195

1 Some of James's observations ("community sweeps") are observations over the total area at a fixed time point. These must be standardized by area covered. Others (e.g., sidewalk observations) sampled at multiple points in time a fixed area through which groups were passing. In these cases, the total area (area of observation x number of observations, or if observations are continuous in time, area x time, assuming that the speed of movement is the same) must be standardized.

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COMMENT ON "CHANCE MODELS" 175

moving past a designated line, in one direction, in a semi-residential area. There were 1349 persons, or 900 per hour. The second covered 1 hour, the same length line, but persons moving in two directions, in a business district. The number of persons was 1548, or 774 per hour in one direction.

In the first case, k is .506, resulting in an Na of .84. The ratio a//3 is

.84/900, or .00093. In the second case, k is .549, resulting in Na 7T

of .95. The ratio a/,( is .00123. Thus the conclusion of this comparison is that the ratio of affiliation to de-affiliation is less in the semi-residential area than in the business district, in the proportion .00093:.00123.

Most of White's models, though they appear similar to ours, are fundamentally different in their assumptions about behavior. His a-model for closed groups, introduced in eq. (27), is identical to ours (as he has pointed out in personal communication), but he later rejects this model as implausible; his principal focus of attention is on the c-model, the a-model, and the y-model. Whereas our rate at which isolates join groups is propor- tional to the product of the number of isolates and the number of groups in a given area (see eq. (2) in the original paper), his rate at which isolates join groups is proportional to either the number of isolates alone (the c- model) or the number of groups alone (the a-model). Such an assumption allows White to escape the inconvenience of solving for his parameters in two steps, as we must do. But this escape, we propose, is bought too dearly, for it loses what we see as a fundamental property of the system: that the rate of isolates joining groups depends on the product of the number of isolates and the number of groups. In his c-model, as many isolates would join per unit time if there were n groups as if there were 10 n groups; in his a-model, as many isolates would join per unit time if there were n1 isolates as if there were 10 n1 isolates.

Which assumption is more nearly correct? Our assumption (of constant a/,8) implies that the average group size increases as the population density in the observed area increases while White's assumption of constant y implies that it remains the same. Probably some systems of freely-forming groups conform more to his model, while others conform better to ours. In crowded cocktail parties, the average group size is probably larger than in uncrowded ones. But on crowded sidewalks, the average group size may be no greater than on uncrowded ones. The point is that the choice one makes between models is not merely a matter of convenience or "efficiency." In a given class of situations, which assumption is most correct? Although the data collected by James do not distinguish between the models, other data would do so.

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176 SOCIOMETRY

4. Contagion vs. Non-contagion. White shows (eq. 39-43) that a con- tagious model (i.e., larger groups attract more members) will give rise to the same distribution, apparently contradicting our statement that no con- tagion is evident in these distributions. He is correct, but trivially so; a contagious process can always be postulated to give rise to the same distribu- tion as a non-contagious one. But parsimony usually leads one to rule out such complications unless the data will fit only a model involving con- tagion or heterogeneity of the parameters. In this case, White must counter- balance the contagious joining process by a conflict-produced leaving process. To be sure, such contagion-and-conflict might occur; but when more parsi- monious assumptions fit the data, it seems reasonable to favor them.

5. Use of X2. White suggests that the use of the X2 test is not valid here. We claim it is valid in a goodness of fit test as to whether the truncated Poisson fits the data. From the data, one estimates one parameter to give a set of pi's, thus using up one degree of freedom. Then with these pi's as expected proportions for cells, a goodness of fit test, with r- 2 degrees of freedom (where r is the number of cells), is clearly valid.2 We erred in using r-1 degrees of freedom rather than r- 2, but not in using a X2 distribution to test the fit of the data to a truncated Poisson distribution. To be sure, this does not test what process generated the distribution; but we can never expect a statistical test to do so.

6. In Conclusion. White has made explicit a wide range of processes, all giving rise to the same equilibrium distribution as our initial process, for systems of freely-forming groups. His paper also clarifies and corrects a number of confusions and inaccuracies in our paper. We believe the process we postulated more nearly conforms to what goes on in many systems of freely-forming groups than do the alternative processes he proposes. But our view is a biased one, and clearly it is necessary to carry out some detailed observations, to learn just what processes are in fact operative. By direct estimation of joining and leaving rates for different-sized groups and in different population densities, it would be possible to learn what kinds of parameters do in fact govern systems such as these.

2 See, for example, A. Hald, Statistical Theory with Engineering Applications, New York: Wiley, 1952, p. 742.

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