## Abstract

Several recent studies have concluded that residential segregation by income in the United States has increased in the decades since 1970, including a significant increase after 2000. Income segregation measures, however, are biased upward when based on sample data. This is a potential concern because the sampling rate of the American Community Survey (ACS)—from which post-2000 income segregation estimates are constructed—was lower than that of the earlier decennial censuses. Thus, the apparent increase in income segregation post-2000 may simply reflect larger upward bias in the estimates from the ACS, and the estimated trend may therefore be inaccurate. In this study, we first derive formulas describing the approximate sampling bias in two measures of segregation. Next, using Monte Carlo simulations, we show that the bias-corrected estimators eliminate virtually all of the bias in segregation estimates in most cases of practical interest, although the correction fails to eliminate bias in some cases when the population is unevenly distributed among geographic units and the average within-unit samples are very small. We then use the bias-corrected estimators to produce unbiased estimates of the trends in income segregation over the last four decades in large U.S. metropolitan areas. Using these corrected estimates, we replicate the central analyses in four prior studies on income segregation. We find that the primary conclusions from these studies remain unchanged, although the true increase in income segregation among families after 2000 was only half as large as that reported in earlier work. Despite this revision, our replications confirm that income segregation has increased sharply in recent decades among families with children and that income inequality is a strong and consistent predictor of income segregation.

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## Notes

- 1.
Of the approximately 130 million housing unit addresses in the United States in 2005, this plan aimed to survey 15 million housing units over five years (National Research Council 2007).

- 2.
The census follow-up rate for nonresponse and unmailable addresses varies by the tract characteristics (U.S. Census Bureau n.d.).

- 3.
To see this in a simple (extreme) case, suppose that each neighborhood in a city were 50 % poor and 50 % rich and that we estimated income segregation by drawing a random sample of one person from each neighborhood. Our sample would have no within-neighborhood variation, but considerable variation would exist in the population as a whole, so we would (very wrongly) conclude that the city was completely segregated by income.

- 4.
Logan et al. (2018) described the approach used in these studies; the sampling procedure used is not well documented in the published papers.

- 5.
We have replicated this Logan et al. (2018) finding in our own analyses (not shown). The Reardon and Bischoff (2011) approach does not eliminate differential bias in segregation estimates. The resampling is done from the estimated tract income distributions, not the actual income distributions. This differential bias due to the estimated tract distributions is then carried into the (equally sized) samples drawn in the resampling process.

- 6.
Moreover, although it is straightforward to show that sample-based estimates of other measures of segregation, such as the dissimilarity index and the Gini index, will also be biased upward (see the online appendix, section A9), we do not have a tractable expression for the magnitude of the bias for these indices, because of the presence of the absolute value function in their formulas. For these reasons, we focus on the

*H*and*R*indices. See Napierala and Denton (2017) for some discussion of sampling bias in the dissimilarity index. - 7.
Strictly speaking,

*I*and*E*must also be estimated in Eqs. (2) and (3), and these estimates will be biased downward. However, because*I*and*E*are estimated from the pooled sample over all units, rather than separately within each unit, the sampling bias in \( \widehat{I} \) and \( \widehat{E} \) is small compared with the bias in the \( {\widehat{I}}_j \)s and \( {\widehat{E}}_j \)s. As a result, in general, \( \widehat{R} \) and \( \widehat{H} \) will be biased upward. - 8.
- 9.
The corrections in Eq. (15) rely on first correcting the binary measures and then constructing a rank-order measure from these estimates. An alternate approach would be to use the uncorrected binary segregation measures to construct a (biased) rank-order segregation estimate and then to correct the rank-order measure, using the following formulas:

\( {\displaystyle \begin{array}{c}{\widehat{R}}^{R\ast }=\frac{{\widehat{R}}^R-B}{1-B}\\ {}{\widehat{H}}^{R\ast }={\widehat{H}}^R-B.\end{array}} \)

These formulas will yield estimates of \( {\widehat{R}}^{R\ast } \) identical to those produced by Eq. (15) but will typically yield very slightly different estimates of \( {\widehat{H}}^{R\ast } \). We prefer the approach described by Eq. (15) both because it yields rank-order estimates that are consistent with the binary estimates used to construct them and because in simulations we conducted (not shown), Eq. (15) generally produced very slightly better results (in terms of bias elimination) than this alternate approach.

- 10.
The data and code used in these simulations are available online (https://cepa.stanford.edu/wp18-02).

- 11.
Tract-level unweighted sample sizes of persons and housing units are publicly available from the Census Bureau for each decennial census and ACS five-year aggregate estimate. We downloaded them via Social Explorer. We estimate tract-level sampling rates as the ratio of the unweighted count of housing units to the population estimate of housing units reported by the Census Bureau. We assume that the sampling rates are the same across subpopulations (by race/ethnicity and household type).

The population estimates are subject to margins of error, so the sampling rates we compute are subject to a small amount of error. This will tend to lead to very slight underestimates of the harmonic mean of sampling rates (

*r*̃), which may in turn lead to very slight overcorrections of sampling bias in segregation estimates. However, any such overcorrection will generally be extremely small. If tract-level sampling rates are not available, the overall sampling rate in the larger geographic unit of interest (here, the metropolitan area) could be substituted; as long as the sampling rates do not vary substantially among tracts, the overall sampling rate will be a reasonable approximation of the harmonic mean of tract sampling rates. - 12.
We do the same for white and Hispanic families, but we focus here on the simulation results for black families because they represent the strictest test of the formulas. Failures of the bias-correction formulas are most likely to appear in the black family income segregation estimates because the black population is both smaller and more segregated than the Hispanic or white population in most metropolitan areas.

- 13.
The census categorizes households as either family or nonfamily households. Family households consist of people related by marriage or parenthood; nonfamily households include single people living alone and nonrelated people living together. Data on income by the presence of children prior to 1990 are not publicly available.

- 14.
Cape Coral is excluded from these estimates because of missing data in 1970.

- 15.
As described earlier, our bias-correction method performs best when the average tract population in a metropolitan area is at least 200. The average tract population of both families with children and households without children is greater than 200 in all metropolitan areas in our analysis sample.

- 16.
For comparison with prior research (Bischoff and Reardon 2014), we focus on estimates that include all white and black families regardless of Hispanic ethnicity. Trends for non-Hispanic white families are similar to those for white families, though levels of segregation are 6 % to 14 % lower among non-Hispanic white families. Income data on non-Hispanic black families are not publicly available.

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## Acknowledgments

The research described in this article was supported by a grant from the UPS Foundation Endowment Fund at Stanford University and by fellowships to Bischoff and Owens from the National Academy of Education/Spencer Postdoctoral Fellowship Program. The opinions expressed are ours and do not represent the views of UPS, Stanford University, the National Academy of Education, or the Spencer Foundation.

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Reardon, S.F., Bischoff, K., Owens, A. *et al.* Has Income Segregation Really Increased? Bias and Bias Correction in Sample-Based Segregation Estimates.
*Demography* **55, **2129–2160 (2018). https://doi.org/10.1007/s13524-018-0721-4

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### Keywords

- Income segregation
- Residential segregation
- Segregation methodology
- Sampling bias
- Spatial inequality