Linking Individuals and Societies

Abstract

How do individuals shape societies? How do societies shape individuals? This article develops a framework for studying the connections between micro and macro phenomena. The framework builds on two ingredients widely used in social science—population and variable. Starting with the simplest case of one population and one variable, additional variables and additional populations are systematically introduced. This approach enables simple and natural introduction and exposition of such operations as pooling, matching, regression, hierarchical and multilevel modeling, calculating summary measures, finding the distribution of a function of random variables, and choosing between two or more distributions. To illustrate the procedures we draw on problems from a variety of topical domains in social science, including an extended illustration focused on residential racial segregation. Three useful features of the framework are: First, similarities in the mathematical structure underlying distinct substantive questions, spanning different levels of aggregation and different substantive domains, become apparent. Second, links between distinct methodological procedures and operations become apparent. Third, the framework has a potential for growth, as new models and operations become incorporated into the framework.

Keywords:

• inequality
• micro–macro link
• probability distributions
• segregation
• sociobehavioral forces

ACKNOWLEDGMENTS

This research was supported in part by the U.S. National Science Foundation under Grant No. SBR-9321019 and partly carried out while the author was a Fellow at the Center for Advanced Study in the Behavioral Sciences in 1999–2000. Early versions of portions of this article were presented at the annual meeting of the American Sociological Association, San Francisco, CA, 1989, the Seventh Annual Group Processes Conference, Los Angeles, CA, 1994, and the annual meeting of the Mathematical Association of America, Washington, DC, January 2009. I am grateful to participants at those meetings and especially to John Angle, Joseph Berger, Barbara Meeker, Eugene Johnsen, Samuel Kotz, Geoffrey Tootell, Murray Webster, the anonymous referees, and the Editor for many valuable comments and suggestions. I also gratefully acknowledge the intellectual and financial support of New York University.

Notes

This usage is not universal. For example, in economics the term “micro” refers to decision-making units, which need not be persons, but may be large aggregates such as corporations, and the term “macro” refers to aggregates of decision-making units.

The distinction between censoring and truncation is by now standard (Gibbons, 1988, p. 355; Kotz, Johnson, and Read, 1982a, p. 396). The term “censoring” refers to selection of the units in a subdistribution by their ranks or percentage (or probability) points in the parent distribution; truncation refer to selection of the units in a subdistribution by values of the variate. Thus, the truncation point is the value x separating the subdistributions; the censoring point is the percentage point α separating the subdistributions. For example, the subpopulations with incomes less than $35,000 or greater than $90,000 each form a truncated subdistribution; the top 5% and the bottom 10% of the population each form a censored subdistribution.

The general problem is known as the problem of finding the distribution of a function of a random variable. One of the methods that is used in solving this problem is called the change-of-variable technique. Because the phrase “change-of-variable” is compact and immediately signals the associated operation, in this article we refer to the general model involving the distribution of the function of a random variable as the change-of-variable model.

For comprehensive introduction to probability distributions, see Johnson, Kotz, and Balakrishnan (1994)Johnson, Kotz, and Balakrishnan (1995) and Stuart and Ord (1987). Mathematically specified distributions have associated with them a variety of functions. The most basic is the cumulative distribution function (CDF), which is defined as the probability α that the variate X assumes a value less than or equal to x and is usually denoted F(x). Probably the best known of the associated functions is the probability density function (PDF), denoted f(x), which in continuous distributions is the first derivative of the distribution function with respect to x (and which in discrete distributions is sometimes called the probability mass function). One of the most useful is the quantile function (QF), which among other things provides the foundation for whole-distribution measures of inequality such as Pen's Parade. The quantile function, variously denoted G(α) or Q(α) or F ⁻¹(α), is the inverse of the distribution function, providing a mapping from the probability α to the quantile x.

The shape constant operates as the general inequality parameter specified by Jasso and Kotz (2008).

In this problem, when the distribution of the actual reward is lognormal, the distribution of justice evaluations is normal, and when the distribution of the actual reward is either rectangular or power-function, the distribution of justice evaluations is positive exponential.

In a recent similar case, theoretical analysis of status led to a new variate which did not resemble any known variate and which was termed “Unnamed” (Jasso, 2001, p. 122). Subsequently, Jasso and Kotz (2007) named it the ring(2)-exponential and generalized it to two new families of probability distributions, which they named the mirror-exponential and the ring-exponential.

For exposition of Jensen's inequality, see Stuart and Ord (1987).

Perfect positive association denotes the case in which each individual has the same relative rank on both A and C. Perfect negative association denotes the case in which the rank ordering in A is exactly the reverse of the rank ordering in C; thus one ranking is the conjugate ranking of the other (Kotz, Johnson, and Read, 1982b, p. 145).

Note: J denotes the justice evaluation, A the actual reward, and C the just reward.

Figure 3 also reports, in the lower-lefthand panel, the graph of the probability density function of the distribution of justice evaluations when the actual reward and the just reward are identically and independently distributed exponentials; as shown, in this case the justice evaluation distribution is logistic.

For fuller discussion of the general inequality parameter in continuous univariate two-parameter distributions, see Jasso and Kotz (2008).

St. Anselm's (1033–1109) idea that the created will has two inclinations is worked out first in his famous thought experiment on the devil, reported in the dialogue De Casu Diaboli [The Fall of Satan] which dates from about 1085–90, and again in De Concordia Praescientiae et Praedestinationis et Gratiae Dei [The Harmony of God's Foreknowledge, Grace, and Predestination], written in about 1107–08.

Notes: In all cells, each micro variable generates macro variables and a subpopulation structure; additionally, each quantitative micro variable generates new rank variables. The problem of finding the distribution of a function of several random variables occurs in many specifications of matching problems, but, for simplicity, is omitted from the six cells involving matching. Moreover, all operations that can be performed on a single population can be performed on the superpopulation generated by pooling or matching.

Mathematically, this is an application of the classic Newtonian idea that mass accumulates at a point.

Notes: The proportions in each row sum to 1.

Notes: The proportions within each racial subgroup sum to 1. Within each racial subgroup, the proportion integrationist is always less than half. The proportion integrationist in the Black subgroup always exceeds that in the White subgroup.

Notes: The proportions segregationist Blacks, segregationist Whites, and integrationist total in each row sum to 1. Within the integrationists, the proportions Black and White in each row sum to 1.