Three Dimensions of Change in School Segregation: a Grade-Period-Cohort Analysis

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Online Resource 1

Three Dimensions of Change in School Segregation: A Grade-Period-Cohort Analysis

Jeremy E. Fiel and Yongjun Zhang

Here we provide supplementary materials omitted from the main text for parsimony. Additional details for findings discussed but not presented here are available on request from the authors in a more comprehensive supplement.

Black-Hispanic Segregation

First we discuss our analyses of black-Hispanic segregation. Based on conventional analyses, black-Hispanic dissimilarity declined 3.5 points (-11%) between 1999 and 2013. The APC estimates are included in Table S1. Figure S1 illustrates the results. The grade-adjusted trend reveals a decline practically identical to that under the conventional analysis. It appears that the grade distribution was stable enough over this period to prevent the substantial grade effects from distorting the temporal trend. The decline is driven by period effects, which decreased a net 4.1 points between 1999 and 2013. This dominated a 1.2 point increase across cohorts. The grade effects are substantial, larger than for black-white or Hispanic-white segregation, and concentrated at the transitions to middle and high school. There is a 6.9 point decline in dissimilarity by seventh grade and a net 7.3 point decline by the end of high school.

Figure S2 shows the results separately for metropolitan areas and nonmetropolitan counties. The decline over time was roughly twice as pronounced in metropolitan areas, and these trends weredriven by period effects. Cohort effects increased only slightly in nonmetropolitan areas and were stable in metropolitan areas. And grade effects were comparable in both types of locale. Figure S3 shows the results by region. The decline was most pronounced in the Midwest, followed by the South and Northeast; it was least pronounced in the West. The latter is due to substantial increases across cohorts in the West, which also occurred for black-white segregation in the West. Black-Hispanic segregation declined across periods in all regions, but declined most in the Midwest. Grade effects are most pronounced in the Midwest, followed by the Northeast and West, and are weakest in the South.

School System Fragmentation

To more directly assess the role of school system fragmentation, we replicated our primary analyses for areas with different numbers of school districts. These results are available from the authors on request. The results provide mixed support for our predictions. For black-white segregation, period effects are most positive in areas with one district, as expected given that desegregation policies and hence their reversals were likely more influential in these areas. Cohort effects are more negative in areas with fewer districts, which is not expected, because we assumed declining residential segregation would be more influential in areas with more districts. Grade effects are weakest in areas with the highest number of districts, as expected, but also in areas with the fewest districts, which is surprising. The patterns differ for Hispanic-white segregation, which further complicates our interpretation. We leave a more thorough assessment to future research.

Additional Analyses

We also describe additional analyses testing various assumptions and data limitations. Figure S4 illustrates average public school and private school enrollments across grades during our period of study. Public school enrollment is fairly stable through eighth grade, increases sharply at ninth grade, and declines subsequently. The increase in ninth grade is likely due in part to a drop in private school enrollment at ninth grade, also shown in the graph. It is also likely due to grade retention among high schoolers who do not attain enough credit hours to advance. Declining enrollment throughout high school is likely due to dropout as well.

Figures S5-S7 compare our primary analyses to parallel analyses that add controls for each focal group’s public school enrollment and for total public and private school enrollment in each grade-year. Table S1 provides the estimates and standard errors for these models. In all cases, the grade effects do not change much, suggesting that grade-specific selection into or out of our datais not problematic for our analyses. In the black-white analysis, the coefficients are positive for total public enrollment (p<.01), negative for white enrollment (p<.01), and negative but not statistically significant for black enrollment and private school enrollment. These controls eliminate the increase across cohorts, likely because total enrollments increased while white enrollment shares declined across cohorts. For Hispanic-white and black-Hispanic segregation, private school enrollment is negatively associated with segregation (p<.05), suggesting that declining private school enrollments shift more segregation to the public sector; the other coefficients are not significant. This exacerbates the increases across cohorts and the declines across periods. It is possible thatprivate school enrollment declined across periods but increased across cohorts.

Overall, the estimates are consistent with the idea that higher private school enrollments reduce public school segregation by shifting segregation to the private sector and the public-private divide. They are also consistent with racial competition theories in that black-white segregation increases when whites experience declining enrollment shares. It is possible that these coefficients are also capturing the influence of systematic selection into, out of, or across cohorts due to grade retention and dropout. Nonetheless, the direction of the trends across periods, cohorts, and grades is robust to the inclusion of these controls.

Alternative Constraints

Figures S8-S10 summarize the results under the four grade-based equality constraints, and Figure S11 illustrates the findings when using the intrinsic estimator. The patterns are similar under all of these identification strategies with one exception: the constraint assuming equal first and second grade effects. This constraint always produces stronger negative grade effects, more negative cohort effects, and more positive period effects. Even under this constraint, however, the grade-adjusted trend over time is practically identical to that under the other constraints. As discussed in the manuscript, we suspect this constraint is problematic due to the high retention rates in first grade, which may distort cohort composition in ways relevant to racial imbalance.

Yang and colleagues (2008) propose an asymptotic t-test that compares the solutions from the coefficient-constrained models to those from the IE analyses. Because the IE solution is an estimable function, which means it is a linear combination of the data generating parameters, this comparison assesses whether the constrained solution qualifies as an estimable function. It relies on a t-test of the scalar value s that describes the deviation of the constrained solution from the IE solution (s can also be viewed as a measure of how sensitive the solution is to the shape of the data matrix). Table S2 summarizes the results of this test for our four grade-based constraints as well as all other possible constraints. None of these tests are statistically significant because of the large standard errors, but the magnitude of the s values does distinguish constraints that may be problematic. Our preferred constraint (G3=G2) has a very small s value, suggesting it is an estimable function. The G4=G3 and G8=G7 constraints also fare well with relatively small s values. G2=G1 is more problematic, and the constraints that span common transition grades are most problematic (G6=G5, G7=G6, G9=G8). These tests are generally consistent with logic underlying our constraints.

Measurement Issues

The remainder of this supplement summarizes analyses that probe a variety of measurement issues. We do not present results here, but can provide a more comprehensive supplement with details on request. We replicated our primary analyses using different samples. The first spans all school years from 1998-2013, but suffers from lower coverage of schools due to more missing data. The black-white sample covers roughly 60% of all public school students and 70% of black public school students; the Hispanic-white sample covers about 50% of all public school students and 70% of Hispanic students. The second is our primary sample from 1999-2013 with greater but still incomplete coverage, as described in the main text. The third has almost complete coverage but only spans the 2009-2013 school years. The black-white sample includes 90% of all students and 99% of black students; the Hispanic-white sample includes 84% of all students and 98% of Hispanic students. The grade-adjusted trends are practically identical across samples, and all three samples yield consistent grade, period, and cohort effects for Hispanic-White segregation. The only minor discrepancies are for black-white and black-Hispanic segregation, for which the 2009-2013 sample yields slightly more negative grade effects, less positive cohort effects, and less negativeperiod effects.

We also compare our APC analyses when using different imbalance measures. Our primary analyses use an unweighted average of dissimilarity (D) across metropolitan areas and nonmetropolitan counties; here we also examine average D weighted by the total enrollment of both focal groups, the average Information Theory Index (H) weighted conventionally by area diversity and student population, and an unweighted average of H. Compared to D, H is more sensitive to schools with extreme imbalances, whereas D is equally sensitive to any degree of imbalance across schools where a group is over- and under-represented (Zoloth 1976). A disadvantage of H is that its weights make it sensitive to compositional changes across markets (Reardon and Firebaugh 2002); unweighted D is not, although weighted D is sensitive to changing population sizes. See Reardon and Firebaugh (2002)for details on the calculation and decomposition of H.

We always obtain higher segregation levels for D than H, and for the weighted than unweighted versions. Nonetheless, the trends in the grade, period, and cohort effects are similar across measures and reinforce our main conclusions. One minor difference is that cohort effects are more pronounced with the weighted measures, likely because the weighted measures are sensitive to compositional change, which primarily occurs across cohorts. Another is that most trends are slightly more muted for the H measures, likely due to its lower values. We assessed additional weighting schemes for dissimilarity as well: unweighted averages of metropolitan area/county dissimilarity as in the main text, averages weighted by the total enrollment of both focal groups, and averages weighted separately by group. The latter compares the trends experienced by the typical white, black, and Hispanic students, which may differ given their uneven distributions across areas. Thegeneral patterns are similar regardless of the weights used.

Finally, we analyze white-nonwhite dissimilarity under different treatments of multiracial students, for whom data were reported inconsistently over time. Some schools in some states began to report multiracial data in 2008-2009; reporting was rare early on but increased over time. First, we ignore multiracial students, as in our primary analyses. Second, we assume they are all white. Third, we assume they are all nonwhite. Findings are fairly consistent across these treatments. Minor differences emerge as period effects. This may present a more significant challenge in the future as the reporting of multiracial students increases.

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Table S1. APC Estimates: Black-White Segregation Analysis
Black-White Dissimilarity / Hispanic-White Dissimilarity / Black-Hispanic Dissimilarity
No Controls / Controls / No Controls / Controls / No Controls / Controls
Coeff. / SE / Coeff. / SE / Coeff. / SE / Coeff. / SE / Coeff. / SE / Coeff. / SE
G1 / -- / -- / -- / -- / -- / -- / -- / -- / -- / -- / -- / --
G2 / 0.005** / (0.001) / 0.004* / (0.002) / 0.002 / (0.002) / -0.001 / (0.002) / 0.002 / (0.002) / -0.001 / (0.002)
G3 / 0.005** / (0.001) / 0.004* / (0.002) / 0.002 / (0.002) / -0.001 / (0.002) / 0.002 / (0.002) / -0.001 / (0.002)
G4 / 0.004 / (0.003) / 0.005 / (0.003) / 0.000 / (0.003) / -0.003 / (0.003) / 0.002 / (0.004) / -0.002 / (0.004)
G5 / 0.002 / (0.004) / 0.002 / (0.004) / -0.004 / (0.004) / -0.006 / (0.004) / -0.005 / (0.006) / -0.008 / (0.006)
G6 / -0.035*** / (0.004) / -0.033*** / (0.004) / -0.037*** / (0.005) / -0.035*** / (0.005) / -0.052*** / (0.007) / -0.050*** / (0.007)
G7 / -0.050*** / (0.005) / -0.047*** / (0.005) / -0.051*** / (0.006) / -0.049*** / (0.006) / -0.069*** / (0.008) / -0.065*** / (0.008)
G8 / -0.050*** / (0.006) / -0.046*** / (0.006) / -0.051*** / (0.007) / -0.047*** / (0.008) / -0.068*** / (0.010) / -0.062*** / (0.010)
G9 / -0.071*** / (0.007) / -0.074*** / (0.007) / -0.067*** / (0.008) / -0.061*** / (0.009) / -0.090*** / (0.011) / -0.073*** / (0.012)
G10 / -0.068*** / (0.008) / -0.067*** / (0.008) / -0.066*** / (0.009) / -0.064*** / (0.009) / -0.084*** / (0.012) / -0.079*** / (0.013)
G11 / -0.066*** / (0.009) / -0.062*** / (0.009) / -0.063*** / (0.010) / -0.064*** / (0.011) / -0.078*** / (0.014) / -0.081*** / (0.014)
G12 / -0.064*** / (0.009) / -0.059*** / (0.010) / -0.059*** / (0.011) / -0.062*** / (0.012) / -0.073*** / (0.015) / -0.078*** / (0.016)
Y99 / -- / -- / -- / -- / -- / -- / -- / -- / -- / -- / -- / --
Y00 / -0.001 / (0.001) / -0.004** / (0.002) / -0.005** / (0.002) / -0.007*** / (0.002) / -0.005* / (0.002) / -0.006** / (0.002)
Y01 / -0.003 / (0.002) / -0.009** / (0.003) / -0.008*** / (0.002) / -0.012*** / (0.003) / -0.010** / (0.003) / -0.013** / (0.004)
Y02 / -0.005 / (0.003) / -0.015** / (0.006) / -0.011** / (0.003) / -0.023*** / (0.006) / -0.013** / (0.004) / -0.024** / (0.008)
Y03 / -0.008* / (0.004) / -0.020* / (0.008) / -0.015*** / (0.004) / -0.035*** / (0.009) / -0.017** / (0.006) / -0.036** / (0.012)
Y04 / -0.009* / (0.004) / -0.024* / (0.010) / -0.020*** / (0.005) / -0.044*** / (0.011) / -0.021** / (0.007) / -0.044** / (0.014)
Y05 / -0.011* / (0.005) / -0.029* / (0.011) / -0.024*** / (0.006) / -0.052*** / (0.013) / -0.027** / (0.008) / -0.054** / (0.017)
Y06 / -0.015* / (0.006) / -0.034** / (0.013) / -0.029*** / (0.007) / -0.061*** / (0.014) / -0.030** / (0.010) / -0.061** / (0.019)
Y07 / -0.015* / (0.007) / -0.037* / (0.014) / -0.032*** / (0.008) / -0.066*** / (0.016) / -0.034** / (0.011) / -0.068** / (0.021)
Y08 / -0.018* / (0.008) / -0.043** / (0.016) / -0.035*** / (0.009) / -0.071*** / (0.017) / -0.039** / (0.012) / -0.076** / (0.023)
Y09 / -0.018* / (0.009) / -0.046** / (0.017) / -0.039*** / (0.010) / -0.077*** / (0.019) / -0.036** / (0.014) / -0.075** / (0.024)
Y10 / -0.015 / (0.009) / -0.055** / (0.020) / -0.042*** / (0.011) / -0.086*** / (0.022) / -0.035* / (0.015) / -0.077** / (0.027)
Y11 / -0.018 / (0.010) / -0.061** / (0.022) / -0.048*** / (0.012) / -0.096*** / (0.024) / -0.038* / (0.016) / -0.084** / (0.030)
Y12 / -0.021 / (0.011) / -0.066** / (0.024) / -0.051*** / (0.013) / -0.104*** / (0.026) / -0.042* / (0.018) / -0.092** / (0.033)
Y13 / -0.022 / (0.012) / -0.070** / (0.026) / -0.051*** / (0.014) / -0.107*** / (0.028) / -0.041* / (0.019) / -0.095** / (0.035)
C88 / -- / -- / -- / -- / -- / -- / -- / -- / -- / -- / -- / --
C89 / 0.009** / (0.003) / 0.011** / (0.003) / 0.004 / (0.004) / 0.010* / (0.004) / 0.002 / (0.005) / 0.010 / (0.005)
C90 / 0.006 / (0.003) / 0.010 / (0.005) / 0.007 / (0.004) / 0.019*** / (0.006) / 0.004 / (0.005) / 0.019* / (0.008)
C91 / 0.007 / (0.004) / 0.014 / (0.007) / 0.011* / (0.004) / 0.031*** / (0.008) / 0.000 / (0.006) / 0.026* / (0.011)
C92 / 0.014** / (0.004) / 0.021* / (0.009) / 0.014** / (0.005) / 0.040*** / (0.011) / 0.004 / (0.007) / 0.039** / (0.014)
C93 / 0.014** / (0.005) / 0.017 / (0.011) / 0.014* / (0.006) / 0.045*** / (0.013) / 0.008 / (0.008) / 0.053** / (0.018)
C94 / 0.020*** / (0.006) / 0.021 / (0.014) / 0.016* / (0.007) / 0.054*** / (0.016) / 0.008 / (0.009) / 0.062** / (0.021)
C95 / 0.022** / (0.007) / 0.022 / (0.016) / 0.020** / (0.008) / 0.065*** / (0.018) / 0.014 / (0.010) / 0.078** / (0.025)
C96 / 0.018* / (0.007) / 0.017 / (0.018) / 0.025** / (0.009) / 0.074*** / (0.021) / 0.008 / (0.012) / 0.083** / (0.028)
C97 / 0.022** / (0.008) / 0.015 / (0.020) / 0.026** / (0.010) / 0.079** / (0.024) / 0.014 / (0.013) / 0.097** / (0.032)
C98 / 0.024** / (0.009) / 0.013 / (0.022) / 0.025* / (0.010) / 0.079** / (0.026) / 0.013 / (0.014) / 0.102** / (0.035)
C99 / 0.024* / (0.010) / 0.007 / (0.023) / 0.030* / (0.011) / 0.085** / (0.029) / 0.008 / (0.015) / 0.103** / (0.038)
C00 / 0.023* / (0.011) / 0.002 / (0.025) / 0.030* / (0.012) / 0.086** / (0.031) / 0.013 / (0.017) / 0.113** / (0.041)
C01 / 0.024* / (0.011) / -0.000 / (0.026) / 0.036** / (0.013) / 0.093** / (0.033) / 0.018 / (0.018) / 0.122** / (0.043)
C02 / 0.025* / (0.012) / -0.002 / (0.028) / 0.033* / (0.014) / 0.090** / (0.034) / 0.012 / (0.019) / 0.119** / (0.045)
C03 / 0.027* / (0.013) / -0.002 / (0.029) / 0.033* / (0.015) / 0.090* / (0.036) / 0.016 / (0.021) / 0.126** / (0.048)
C04 / 0.027 / (0.014) / -0.005 / (0.030) / 0.036* / (0.016) / 0.094* / (0.038) / 0.018 / (0.022) / 0.131** / (0.050)
C05 / 0.029 / (0.015) / -0.004 / (0.031) / 0.032 / (0.017) / 0.092* / (0.039) / 0.011 / (0.023) / 0.129* / (0.052)
C06 / 0.033* / (0.016) / -0.004 / (0.033) / 0.034 / (0.018) / 0.095* / (0.041) / 0.011 / (0.025) / 0.134* / (0.054)
C07 / 0.033* / (0.017) / -0.007 / (0.034) / 0.033 / (0.019) / 0.095* / (0.043) / 0.013 / (0.026) / 0.139* / (0.057)
C08 / 0.036* / (0.017) / -0.008 / (0.035) / 0.033 / (0.020) / 0.096* / (0.044) / 0.013 / (0.027) / 0.142* / (0.059)
C09 / 0.043* / (0.018) / -0.003 / (0.037) / 0.031 / (0.021) / 0.095* / (0.046) / 0.009 / (0.029) / 0.142* / (0.061)
C10 / 0.039* / (0.019) / -0.010 / (0.038) / 0.033 / (0.022) / 0.097* / (0.047) / 0.013 / (0.030) / 0.150* / (0.063)
C11 / 0.046* / (0.020) / -0.008 / (0.040) / 0.031 / (0.023) / 0.096 / (0.050) / 0.014 / (0.031) / 0.156* / (0.066)
C12 / 0.045* / (0.021) / -0.014 / (0.041) / 0.026 / (0.024) / 0.092 / (0.051) / 0.013 / (0.032) / 0.159* / (0.068)
C13 / 0.044* / (0.022) / -0.017 / (0.043) / 0.028 / (0.025) / 0.094 / (0.053) / 0.012 / (0.034) / 0.159* / (0.069)
Enrollment controls (100,000s)
Black / -- / -- / -0.011 / (0.008) / -- / -- / -- / -- / -- / -- / -0.003 / (0.013)
White / -- / -- / -0.027** / (0.008) / -- / -- / -0.001 / (0.010) / -- / -- / -- / --
Hispanic / -- / -- / -- / -- / -- / -- / 0.004 / (0.012) / -- / -- / -0.017 / (0.015)
Private school / -- / -- / -0.005 / (0.012) / -- / -- / -0.040* / (0.016) / -- / -- / -0.047* / (0.022)
Total Public / -- / -- / 0.014** / (0.005) / -- / -- / -0.003 / (0.007) / -- / -- / 0.001 / (0.007)
Intercept / 0.469*** / (0.040) / 0.448*** / (0.045) / 0.468*** / (0.064)
Note: ***p<.001, **p<.01, *p<.05. Estimates based on constrained OLS regressions with equality constraint G2=G3.

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Table S2. Tests of Estimability of Constrained Model Estimates
Black-White / Hispanic-White / Black-Hispanic
Constraint / s / se(s) / t / s / se(s) / t / s / se(s) / t
G2=G1* / -0.137 / 3.299 / -0.042 / -0.033 / 2.827 / -0.012 / -0.079 / 2.093 / -0.038
G3=G2* / 0.027 / 3.153 / 0.009 / 0.024 / 2.702 / 0.009 / -0.001 / 2.000 / -0.001
G4=G3* / 0.042 / 3.153 / 0.013 / 0.068 / 2.702 / 0.025 / 0.000 / 2.000 / 0.000
G5=G4 / 0.124 / 3.153 / 0.039 / 0.171 / 2.702 / 0.063 / 0.250 / 2.000 / 0.125
G6=G5 / 1.272 / 3.153 / 0.404 / 1.149 / 2.702 / 0.425 / 1.603 / 2.000 / 0.801
G7=G6 / 0.538 / 3.153 / 0.171 / 0.523 / 2.702 / 0.193 / 0.567 / 2.000 / 0.284
G8=G7* / 0.026 / 3.153 / 0.008 / 0.014 / 2.702 / 0.005 / -0.028 / 2.000 / -0.014
G9=G8 / 0.741 / 3.153 / 0.235 / 0.557 / 2.702 / 0.206 / 0.743 / 2.000 / 0.371
G10=G9 / -0.051 / 3.153 / -0.016 / -0.001 / 2.702 / 0.000 / -0.180 / 2.000 / -0.090
G11=G10 / -0.045 / 3.153 / -0.014 / -0.090 / 2.702 / -0.033 / -0.224 / 2.000 / -0.112
G12=G11 / -0.046 / 3.124 / -0.015 / -0.093 / 2.677 / -0.035 / -0.182 / 1.982 / -0.092
Note: s is a scalar measure of the distance of the constrained solution from the IE solution; deviations from 0 indicate that the constrained solution may not be an estimable function of the data generating parameters. *Denotes the four primary constraints in this analysis.

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Figure S1. APC Analysis of Black-Hispanic Dissimilarity. Each graph plots national-level average dissimilarity within metropolitan areas/nonmetropolitan counties. The conventional trend uses school-level data (aggregating all grades) to calculate segregation. The grade-adjusted trend plots yearly segregation after controlling for grade effects. Period effects are based on period-specific predictions from the APC model, with cohorts and grades held constant; cohort and grade effects are calculated similarly.

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Figure S2. APC Analysis of Black-Hispanic Dissimilarity, by Metropolitan Status.

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Figure S3. APC Analysis of Black-Hispanic Dissimilarity, by Region.

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Figure S4. Changes in Enrollment by Grade.Average enrollment in public (left axis) and private (right axis) schools by grade level, 1999-2013.

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Figure S5. Sensitivity to Controls,Black-White Dissimilarity.“Controls” indicates APC estimates after controlling for black andwhite public school enrollment and overall public and private school enrollment in each grade-year.

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Figure S6. Sensitivity to Controls,Hispanic-White Dissimilarity.“Controls” indicates APC estimates after controlling for Hispanic andwhite public school enrollment and overall public and private school enrollment in each grade-year.

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Figure S7. Sensitivity to Controls,Black-Hispanic Dissimilarity.“Controls” indicates APC estimates after controlling for Hispanic andblack public school enrollment and overall public and private school enrollment in each grade-year.

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Figure S8. Sensitivity to Constraints,Black-White Dissimilarity.Lines for all four equality constraints are overlaid.

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Figure S9. Sensitivity to Constraints,Hispanic-White Dissimilarity.

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Figure S10. Sensitivity to Constraints,Black-Hispanic Dissimilarity.

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Figure S11. APC Analysis with Intrinsic Estimator. Coefficients based on analyses using intrinsic estimator.

REFERENCES

Reardon, S. F., & Firebaugh, G. (2002). Measures of Multigroup Segregation. Sociological Methodology, 32, 33-67.

Zoloth, B. S. (1976). Alternative Measures of School Segregation. Land Economics, 52(3), 278-298.