Detecting Discrimination: Analyzing Racial Disparities in Public Contracting

InCity of Richmond v. J. A. Croson Co.(1989), the Supreme Court established strict scrutiny as the standard applicable to affirmative action programs which set aside quotas of public contracts for minority-owned businesses, andAderand v. Pena(1995) extended the strict scrutiny standard to federal programs. Although the requirements of these decisions clearly require multivariate statistical analysis, most “disparity studies” have used a univariate comparison between the expected and the observed shares of contracts going to minority-owned firms. We examine four statistical methods—ordinary least square multiple regression, logit and tobit models, and a multivariate procedure for comparing expected and observed outcomes. Because no data are presently available at the level of specificity required byCroson,we constructed synthetic data sets to represent typical variations among large U.S. cities. Applying the statistical methods to each data set allows evaluation of the extent to which each method is able to both remove spurious and detect valid estimates of racial disparity when relevant control variables are added. Findings: (a) All four models removed apparent disparities which, although significant in univariate analysis, were known to be spurious. (b) Tobit and logit models, whose underlying assumptions better fit the nature of public contracting data, provided more accurate and more sensitive estimates than OLS regression. (c) Comparison of expected and observed outcomes within categories of control variables yielded results very similar to logit and tobit models and, because of the nature of the comparison specified inCroson,produced slightly more sensitive probability estimates.     

A note on block variables in multiequation models

This paper seeks to further refine the efforts of Coleman (Social Science Research 5, 1–20, 1976) and Igra (Social Science Research 8, 253–264, 1979) to develop summary regressionlike coefficients for causal models involving blocks of conceptually related variables, using Heise's (Sociological Methods and Research 1, 147–173, 1972) strategy of explicitly representing block concepts as unobserved intervening variables in a multiequation model. Such models can be estimated using the maximum-likelihood techniques of Jöreskog and Sörbom (LISREL IV: Analysis of Linear Structural Relationships by the Method of Maximum Likelihood, International Educational Services, Chicago, 1978). The procedure proposed leads to (a) an unambiguous definition of block variables in terms of their constituent variables; (b) a clear definition of effects of conceptual variables on one another, and on single response variables, and a reduced model formulated in terms of conceptual variables only; and (c) a clear decomposition of correlations between conceptual variables into direct effects, indirect effects, and unanalyzed components. Numerical illustration is provided.     

The analysis of ordinal dispersion: A comparison of Leik's “D” with a variance formula

The frequency of ordinal data and the importance of attitudinal consensus in social research motivated Leik [Leik, R. L. (1966), Pacific Sociological Review 9, 85–90] to formulate a simple, effective measure of ordinal dispersion, but he did not provide a method for its statistical analysis, and its operational interpretation for nonextremal values is unclear. Leik's D is an equal-interval, “folded” scoring vector inner dot-product, scalar transformation of the vector of ordered relative frequencies. The folding point is the median category. The metric analog of D is the difference between the random variable's expected values above and below the median. An equal-interval, pure variance measure of ordinal dispersion is posited and compared to D. They correlate closely but not perfectly, and D has a serious problem when an extremal category has more than half the total cases. Both formulas are easily generalized to non-equal-interval scorings. A general linear model for categorical data permits easy statistical comparisons of D, but not the variance. D tends to underestimate the degree of significant dispersion in comparison to the variance formula, it is suggested.     

A multivariate analysis of response structure: Race attitudes, 1972–1977

A particular kind of latent class model is used to characterize the unobservable variable measured by six discrete indicators of racial stance in 1972 and 1977. Methods recently introduced by Clogg and Goodman, 1982, Clogg and Goodman, 1983) for the simultaneous latent structure analysis of two multidimensional contingency tables are employed in across-year homogeneity tests on the latent class proportions. Trends in multivariate response patterns over the 5-year interval are then examined by cross-classifying the predicted latent variable with selected demographic characteristics of respondents in each year. The results indicate that (1) no significant changes in the distribution of the latent variable occurred over the 5-year time period, and (2) the status of certain demographic variables as predictors of racial stance fluctuated between the two survey years.     

The measurement of diversity: Ethnic and socioeconomic mixing in residential areas

Units of observation such as census tracts continue to be analyzed according to various modal characteristics while the variation or diversity existent in such units is often ignored. The qualitative or nominal-level indicators of diversity are examined which (1) are operative in the polytomous situation, and (2) measure within-unit diversity rather than divergences among units. Six qualitative indicators are explained and compared both theoretically and by example, with the Index of Qualitative Variation suggested as the most appropriate measure of diversity when variables representing a nominal scale are used. Quantitative or interval-level diversity also was examined with six measures analyzed, representing three operational situations. Because of the susceptibilities of five of the quantitative measures to skewness and variable sample sizes, the coefficient of variation was recommended for interval-level variables to evaluate within-unit diversity.     

Components of change in the number of female family heads ages 15 to 44, an update and reanalysis: United States, 1940 to 1983

We review the literature dealing with the various components of change in the number of female family heads and conclude that P. Cutright's (1974, Journal of Marriage and the Family 36, 714–721) four components of change in the number of ever-married female family heads can be usefully applied to a reanalysis of decade-by-decade changes in this quantity among white and nonwhite women ages 15–44, from 1940 through 1980. However, we substantially revise and update Cutright's (1974) original data. Our findings include the fact that while the relative importance of the various components of increases in ever-married female family headship vary from decade to decade, overall trends are fairly similar for white and nonwhite women. Moreover, the rapid rise between 1970 and 1980 in the number of white and nonwhite female family heads among the never-married (single) continued through the 1980 to 1983 period. A decade ago, Cutright (1974) predicted that a change in the number of female family heads from 1970 on would reflect little more than changes in the numbers of ever- and never-married women ages 15–44. We discuss this inaccurate prediction and conclude that wishful thinking is a poor guide in forecasting the future.     

The proportional odds with partial proportionality constraints model for ordinal response variables

The proportional odds assumption in ordered logit models is a restrictive assumption that is often violated in practice. A violation of the assumption indicates that the effects of one or more independent variables significantly vary across cutpoint equations in the model. In order to relax this assumption for the cumulative odds model, researchers may use either a “partial” model that relaxes the assumption for a subset of variables or the “generalized” model that relaxes the assumption for every independent variable. In this paper, we propose a relatively new and under-utilized third alternative, the proportional odds with partial proportionality constraints (POPPC) model, which allows the effects of a subset of variables to vary across cutpoint equations by a common factor. We improve upon an earlier formulation of the POPPC model by offering an additional conceptual justification for the model and an estimation method that does not require the use of person-threshold data. We illustrate the POPPC model with two examples from the 2008 General Social Survey.     

The making of family values: Developmental idealism in Gansu,China

This paper examines the role of developmental thinking in the making of family values. We analyze survey data collected from Gansu Province in China with regular and multilevel logit models. The results show that individuals’ endorsement of neolocal residence, self-choice marriage, gender egalitarianism, late marriage for women, and low fertility depends on the conjunction of preference for development and beliefs in its association with those family attributes, which we term developmental idealism associational evaluation. Furthermore, such impact of developmental thinking on family values holds robust in the presence of indigenous ideational forces, in this case Islamic religion. Although Islam influences family values in the opposite direction than developmental ideas do, the effect of Developmental Idealism associational evaluation does not differ significantly between Muslims and non-Muslims.     

Inequality of educational opportunity in France and the United States

Unequality of educational opportunity is related to school structure. Comparison of the attainment process in France and the United States reveals the consequences of structure for individuals' attainments. Turner's (1960) typology of educational systems is applied. The French educational system is shown to be closer to Turner's “sponsorship system,” and that of the U. S. closer to the “contest” type. Inequality of educational opportunity, as indicated by the effect of father's occupational status on years of schooling completed, is greater in France than in the U. S. The difference is accounted for by school structural variables, indicating the importance of structure for inequality of opportunity. Other differences concerning the causes of educational attainment in France and the U. S. are also discussed.     

Intergenerational and career mobility in Britain: An integrated analysis

Mobility analysis has traditionally concentrated on the task of charting flows in the two-way contingency table which relates occupational origin to occupational destination. It has not yet undertaken a unified analysis of intergenerational and intragenerational aspects of mobility. And only rarely does it incorporate terms which represent ordered concepts such as inequality or vertical movement. In contrast, path analysis has typically imposed strong orderings on the data, and it has been employed to tease out the relations among several occupational variables. Its main defect is that it entirely abstracts from shifts in mean occupational position, and also from change in the shape of the occupational distribution. Yet both of these are important sources of mobility. The present paper shows how two innovations in mobility analysis may be generalized to constitute a technique which unites the strengths of path analysis with those of log-linear analysis. The two innovations are (a) the conceptual disaggregation of perfect and exchange mobility, and (b) the imposition of an ordered component within each (K. Hope, 1981, Sociology 15, 19–55). The outcome of this generalization is a simple yet powerful account of mobility in Britain.     

Recursive models for categorical data

Standard methods for recursive models with continuous endogenous variables are extended to models with categorical endogenous variables. The concept of a reduced-form equation is generalized in a natural way to cover nonlinear regression functions and, in particular, models with categorical endogenous variables. Maximum-likelihood estimation and asymptotic chi-square tests are described. Two numerical examples are presented: a linear recursive two-equation model for all-categorical data, and a combined linear and logit three-equation recursive model with both categorical and continuous endogenous variables. Limitations of the present work and directions for further extension are noted.     

Occupational prestige from preference orientations via the Markov Chain: An alternative to respondent ranking techniques

This paper suggests a method of determining occupational prestige structures through simulation of transfer of credit processes involving the Markov Chain. The method required recording present and preferred occupations of a representative sample of employed persons. Following the example of Blau (1956) and Empey (1956) the concept “preferred occupation” is suggested as clearly distinct from occupational choice which has been treated by other researchers (Musgrave 1967; Carol and Parry 1968; Kuvlesky and Bealer 1966; Haller and Sewell 1967; Solcum 1956; and Philband and Gregory 1956). A series of Markov Chain interactions transforms the present-preferred occupation matrix, “controls” for structure limitations on ambitions and social distance, and generates a hierarchical model of occupational prestige structures. This paper examines the methodological and theoretical issues implicit in the measurement of prestige and the establishment of prestige hierarchies, and describes the procedures of the application of the Markov Chain to occupational preference responses. It reports the results of the application of this technique to a pilot sample and compares the resulting hierarchy with those derived from NORC measures and the Duncan socioeconomic index.     

Coalitional affiliation as a missing link between ethnic polarization and well-being: An empirical test from the European Social Survey

Many studies converge in suggesting (a) that ethnic and racial minorities fare worse than host populations in reported well-being and objective measures of health and (b) that ethnic/racial diversity has a negative impact on various measures of social trust and well-being, including in the host or majority population. However, there is much uncertainty about the processes that connect diversity variables with personal outcomes. In this paper, we are particularly interested in different levels of coalitional affiliation, which refers to people’s social allegiances that guide their expectations of social support, in-group strength and cohesion. We operationalize coalitional affiliation as the extent to which people rely on a homogeneous social network, and we measure it with indicators of friendships across ethnic boundaries and frequency of contact with friends. Using multi-level models and data from the European Social Survey (Round 1, 2002–2003) for 19 countries, we demonstrate that coalitional affiliation provides an empirically reliable, as well as theoretically coherent, explanation for various effects of ethnic/racial diversity.     

Are there sex differences in the utilization of educational capital among college-educated workers?

This paper introduces the concept of educational utilization as an overlooked part of the education-to-work transition and a potential mechanism by which occupational sex segregation is generated among the college-educated labor force. The paper begins with a critical discussion of the operationalization approaches that have been used in prior research that implicitly measures educational utilization. Multiple empirical measure of the concept are then developed using data from the O*NET and the National Surveys of College Graduates. The explanatory power of each measure is assessed using conditional logit models of occupational attainment. A combined measure is then used to assess sex differences in educational utilization using data from the 1993 and 2003 National Surveys of College Graduates for 2 cohorts of college graduates—those who earned their baccalaureate or post-baccalaureate degrees and entered the labor market in the years 1985–1993 and 1995–2003. The analysis identifies sex differences in educational utilization that vary across field, degree level and cohort and concludes with an examination of the implications of sex differences in educational utilization for occupational segregation.     

The social standing of jobs: Scores for all Australian occupations

This paper presents the method used to develop the ANU II status scores for all Australian occupations. For some purposes it supersedes ANU I, a 16-point scale (Broom, Jones, and Zubrzycki, 1965, Australian and New Zealand Journal of Sociology (Suppl.) 1; Broom and Jones, 1969, American Sociological Review 34, 650–658). The two scales correlate at .83. The new scale was calculated using a broad range of data applied to a prestige score criterion to estimate an equation used to predict status scores for all occupations. These scores are available in Broom, Duncan-Jones, Jones, and McDonnell (1977, Investigating Social Mobility) along with further details of the method and results. The new scale is currently being used in the analysis of the ANU 1973 survey of social mobility in Australia, and in the reanalysis of data from the ANU 1965 survey. Preliminary results (also presented in Broom et al., 1977) indicate that the scale will be comparable both across time in Australia and across space when comparisons with the “second generation” of mobility studies is undertaken.     

Intransitivity in the structure of small closed groups

Observations of the structure of closed groups are presented which give rise to four observable tendencies within either a sociogram or the cat_ij matrix as developed by Bernard and Killworth (1973). These observations are used to form theories of social structure of varying degrees of complexity; all these theories, however, are somewhat unusual in that intransitivity of interpersonal sentiment, far from being an artifact, is proposed as an essential part of the structure of human groups, whereas recent papers have stressed the occurrence of transitivity within a group. A statistical theory including the earlier theories shows that at least three subgroups must exist within a group, even if only two are forced by external organization. One of these subgroups is formed of mutually unknown individuals who serve as connecting links between other subgroups formed by strong interpersonal sentiment.     

The Mapping of family-composition preferences: A scaling analysis

When subjects express preference for one type of family composition over another, their judgments have the useful property that they can be referred to the ‘objective’ characteristics defining the stimuli. Pair comparison preference data from the Bollen-Delbeke study are “internally” analyzed according to the Carroll-Chang “Points of View” vector model and Coombs' Unfolding (distance) model. The distance model was found to give a better fit. Secondly, the models are compared within the framework of Carroll's Preference Mapping hierarchy of models, referring the data to the ‘objective’ structure underlying the stimuli. The simple distance model is found to give a much better fit.The methodological and substantive implications of this finding are discussed.¹     

Determinants of fertility levels and change among developed countries: 1958–1978

This article provides a multivariate cross-national test of the hypothesis that national population/family planning policies have effected levels, and changes in fertility in developed nations over the past two decades. Variation is assessed in the total fertility rates (TFR) in 1978, and in the change in these rates between 1958 and 1978, among thirty developed countries. Measures include socioeconomic development, divorce, percent in consensual unions, female labor force participation, abortion policy, and level of contraceptive use by married couples and, government population/family planning policy. Seventy percent of the variation in 1978 TFR is related to the percent contracepting, female labor force, and the population/family planning policy measures. These are the only measures with significant direct effects. A longitudinal analysis of 1958 to 1978 change in TFRs is also conducted. This model increases R² to 75%, and the three independent variables remain significant. Implications of these findings for policy makers interested in increasing or decreasing fertility rates are noted.     

A test for a vanishing tetrad: The second canonical correlation equals zero

Many four variable path models have the overidentifying restriction of a vanishing tetrad: ?₁₃?₂₄ ? ?₁₄?₂₃ = 0. The literature offers but a 1924 test (Spearman and Holzinger) of a vanishing tetrad. We suggest that a canonical analysis can be set up to provide a test of a tetrad difference. Testing the hypothesis that the second canonical correlation of set X₁ and X₂ with set X₃ and X₄ is zero is identical to testing the hypothesis that ?₁₃?₂₄ ? ?₁₄?₂₃ = 0.