## 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.

## Introduction

Several recent studies have documented an increase in residential segregation by income in the United States over the last four decades (Bischoff and Reardon 2014; Jargowsky 1996; Owens 2016; Reardon and Bischoff 2011; Watson 2009). In particular, income segregation in the United States appears to have grown sharply in the 1980s and since 2000, indicating that American society is becoming more spatially polarized. Owens (2016) showed that the increase in income segregation was driven largely by the growing segregation of families with children. Given the importance of neighborhood socioeconomic conditions for young children’s development and opportunities for economic mobility (Brooks-Gunn et al. 1997; Chetty and Hendren 2015; Chetty et al. 2016), the increasing economic segregation of children is of particular concern.

There is reason, however, to doubt the reported increase in economic segregation. The estimates of income segregation reported in the aforementioned studies are based on household or family income data reported on the long form of the U.S. decennial census from 1970 to 2000 and on the American Community Survey (ACS) from 2005 onward. Only a sample of the U.S. population was asked to fill out the decennial census long form from 1970 to 2000 (approximately one in six households), and an even smaller sample is asked to fill out the ACS in any five-year window (approximately one in 12 households). As we describe later, segregation estimates based on random samples are generally biased upward relative to the values that would be measured if the full population were observed. Moreover, because the upward bias is inversely related to the sampling rate, the upward bias is larger for estimates based on the ACS than for estimates based on the census. As a result, we would expect to see an increase in estimated segregation between 2000 and later years—a period in which sample rates declined from approximately 17 % to 8 %—even if there were no true change in levels of income segregation. Without a method of accounting for (or eliminating) the bias in segregation estimates, we cannot make valid comparisons between estimates based on different sampling rates or sample sizes.

We are not the first to raise this concern. Logan et al. (2018) questioned the reported increase in income segregation after 2000, noting that the difference in sampling rates between the census long form and the ACS bias estimates of the post-2000 trend in income segregation upward. Napierala and Denton (2017) noted that sampling variation leads to imprecision in segregation estimates and upward bias when samples are small. Several earlier studies also found upward bias in segregation measures as a result of stochastic processes (Winship 1977) and small sample sizes (Reardon and Bischoff 2011). The authors of some of these papers proposed strategies for constructing unbiased sample-based estimates of segregation (Logan et al. 2018; Reardon and Bischoff 2011). However, the Reardon and Bischoff (2011) approach fails to remedy the problem (see Logan et al. 2018), and the methods Logan et al. (2018) proposed have not been validated across a wide range of data-generating conditions and in some cases require access to restricted microdata. Microdata, as it relates to this research, consist of individual-level census records with exact income values, as opposed to aggregated counts of households with incomes falling in a set of ordered income bins. Our goals in this article are to develop and validate a method of eliminating the bias in sample-based segregation measures that does not rely on access to microdata and to use this method to produce unbiased estimates of segregation trends over the last few decades.

We first derive formulas describing the approximate sampling bias in two measures of segregation. Our formulas allow us to quantify the bias in both binary measures of segregation between two mutually exclusive groups and in measures of rank-order segregation, such as the measures widely used to study income segregation. These formulas describe the approximate bias in segregation estimates as a function of the average unit (e.g., census tract) population and the harmonic mean of the sampling rate across units. If both unit population sizes and sampling rates are known, we show that they can be used to construct bias-corrected segregation estimates without relying on access to sample microdata. Using Monte Carlo simulations, we test these bias-corrected estimators across a wide range of realistic residential patterns and characterize the range of conditions under which they provide approximately unbiased estimates of segregation. 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.

Second, we 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 (Bischoff and Reardon 2014; Owens 2016; Reardon and Bischoff 2011, 2016) to examine whether their reported trends and patterns were an artifact of the biased estimators they relied on. We find that the primary results of these studies hold up, although the true increase in income segregation among families after 2000 was only half as large as that reported by Bischoff and Reardon (2014).

## Sampling in the Decennial Census and American Community Survey

The census is a decennial, housing unit–based survey that collects limited information on the full U.S. population, including housing tenure status and an enumeration of the age, sex, and race/ethnicity of each household member. These data are available for all geographic levels down to the census block, the basic census sampling unit that encompasses approximately one city block (although blocks may be larger in suburban and rural areas). All other sociodemographic information tabulated by the census, including the household and family income data used to estimate income segregation, is collected from a sample of Americans from what was formerly called the census “long form,” also collected every 10 years. These data are generally publicly available and tabulated in slightly larger geographic units, such as block groups (an aggregation of several contiguous blocks) and tracts (an aggregation of several contiguous block groups). Tracts typically contain several thousand residents and have often been used in sociological research to approximate residential neighborhoods.

In censuses up to and including 2000, the long-form data were collected from samples of approximately one in six households, or about 17 % of the U.S. population. This amounted to approximately 18 million households contacted, and 16.4 million usable questionnaires (National Research Council 2007). Using unweighted sample counts and population estimates available from the Census Bureau, we estimate that on average in the 1970 through 2000 censuses, 250–300 households were sampled in each tract, and the household sampling rate ranged from 17 % to 20 %.

The ACS replaced the census long form after 2000 with the promise of providing more frequent (annual) estimates of sociodemographic characteristics of the U.S. population. The cost of these annual estimates is a reduction in sample size. In addition, it is necessary to use aggregate ACS data across five-year windows to accumulate what the Census Bureau deems to be sufficient sample sizes for the smaller geographic units used to compute segregation estimates, such as tracts. In 2005, the first year that the ACS was fully implemented, the Census Bureau aimed to achieve a sampling rate of approximately 12 % over a five-year period by sampling about 3 million unique addresses per year.^{1} The ACS sample, however, is then reduced substantially by subsampling for in-person interviews.^{2} In reality, the ACS sampled approximately 14.5 million housing units in the 2005–2009 period and conducted 9.7 million final interviews to be included in the usable data (approximately 7.5 % of total housing unit addresses). The Census Bureau increased the sample size after 2011, resulting in an original sample of approximately 16.8 million housing units in the 2010–2014 period and a final sample of nearly 11 million usable questionnaires. Using available unweighted sample counts and population estimates, we estimate that in the ACS five-year aggregate data from 2005–2009 to 2012–2016, the average tract sample size is 130–160 households, and the average tract sampling rate ranges from 8 % to 10 %.

Clearly, sample sizes and sampling rates declined with the inception of the ACS. Knowing that small sample sizes would be problematic for researchers, particularly for those interested in neighborhood analyses, the Census Bureau began publishing confidence intervals in ACS data tables to highlight the imprecision in the estimates. The Census Bureau was less transparent about sampling error in census long-form data, although small-geography data from 2000 and earlier also suffered from substantial imprecision (although to a lesser extent than data in the ACS) (National Research Council 2007:65–74).

## Prior Research on Sampling Bias and Corrections in Income Segregation Estimates

Segregation measures are generally based on a decomposition of the population variation in income (or race or any other characteristic) into between- and within-neighborhood components (Reardon 2011; Reardon and Firebaugh 2002). Commonly used measures differ in how they quantify variation: for example, the Neighborhood Sorting Index (NSI) uses the variance of income (Jargowsky 1996); the rank-order information theory index (*H*^{R}) uses a measure of the entropy of income ranks; and the rank-order variance ratio index (*R*^{R}) uses the variance of income ranks (Reardon 2011; Reardon and Bischoff 2011). The bias in sample-based segregation measures arises because the observed variation in a finite sample is generally a downwardly biased estimator of the variation in the full population, and the magnitude of the downward bias is inversely related to sample size. Thus, sample-based segregation estimators underestimate the true extent of within-neighborhood variation much more than they underestimate population variation (because the population variation is estimated from a much larger sample than each neighborhood’s variation). As a result, sample-based segregation estimates assign too much of the variation to the between-unit component of the decomposition—that is, they overestimate segregation.^{3}

Several authors recently proposed methods for correcting the sampling bias in income segregation estimates. Reardon and Bischoff (2011) noted that comparisons of segregation estimates across populations (across metropolitan areas, years, or racial/ethnic groups) are biased if the average within-tract sample sizes differ among populations. To address this, they randomly sampled 10,000 families from the census-reported population in each metropolitan area–year–racial/ethnic group cell, reasoning that by equalizing sample sizes, they would equalize the amount of bias in each estimate. More recently, several researchers used a modified version of this method, sampling a number of families equal to 50 times the number of census tracts in a metropolitan area, reasoning that it was preferable to hold the average sample size per tract constant rather than the total sample size (Bischoff and Reardon 2014; Owens 2016; Reardon and Bischoff 2016).^{4}

Reardon and Bischoff (2011) argued that although this approach would not yield unbiased estimates in any given population, the expected bias would be equal in each estimate, allowing for unbiased estimation of changes over time and differences between racial/ethnic groups. Subsequent analyses of this method, however, indicate that subsampling from populations based on census or ACS estimates does not yield comparable bias across populations (Logan et al. 2018).^{5} The segregation trends and patterns that Reardon and Bischoff and Owens reported are therefore potentially confounded by differential bias across time, place, and/or racial/ethnic group.

Logan et al. (2018) proposed several approaches for constructing unbiased sample-based estimates of segregation. First, they suggested an approach—which they termed “sparse-sampling variance decomposition” (SSVD)—that uses census or ACS microdata and a finite sample correction to obtain an unbiased estimate of the average within-tract variance. This method is useful when one is using a variance-based segregation measure, such as Jargowsky’s (1996) NSI or Reardon’s (2011) rank-order variance ratio index (*R*^{R}), and when one has access to restricted-access microdata through a Federal Statistical Research Data Center (FSRDC). Second, they derived formulas describing the approximate bias in sample-based estimates of the rank-order information theory index (*H*^{R}) and the information theory index measures of segregation of poverty (*H*10) and affluence (*H*90) (Reardon and Bischoff 2011). They proposed estimating *H*^{R}, *H*10, and *H*90 from microdata and then subtracting the corresponding bias terms from these estimates. Logan et al. (2018) showed that both the SSVD approach and the bias-formula correction method yielded approximately—but not perfectly—unbiased segregation estimates in the set of six cities they studied using microdata from the 1940 census.

These methods rely on microdata. When only aggregated data are available (as is the case with publicly available census and ACS data), Logan et al. (2018) suggested a multistep approach: (a) using the binned income data to estimate within-unit income distributions; (b) generating repeated samples from these estimated distributions; (c) applying the microdata-based bias-correction approaches to each of these simulated microdata samples; and (d) averaging the resulting estimates. Using simulations based on individual household-level census data from Chicago in 1940, they showed that this approach to estimating segregation from binned data yielded estimates that were generally less biased than the uncorrected estimates when the sampling rate was low (although the adjusted estimator of the NSI appeared generally worse than the unadjusted estimator, at least for Chicago). The adjusted binned-data estimators for *H*^{R}, *H*10, *H*90, and *R*^{R} appeared to overcorrect in some cases and undercorrect in others, although they were (at least for Chicago) less biased than the unadjusted estimators (Logan et al. 2018). The approaches they recommended for correcting estimates of segregation based on binned data have not been validated across a range of data-generating scenarios, however. The approximations on which they were based may break down when samples are small or when sampling rates and/or sample sizes vary among units.

In this article, we extend the literature in several ways. First, we derive formulas describing the approximate sampling bias in both binary and rank-order measures of segregation, focusing on the information theory index *H* and the variance ratio index *R* because these two measures have the most desirable mathematical properties in an index (James and Taeuber 1985; Reardon 2011; Reardon and Firebaugh 2002). Second, we use these formulas to derive bias-corrected segregation estimators that can be used with binned income data (such as publicly available tract-level data) and do not require access to sample (or simulated) microdata. Third, we use simulations to investigate the performance of the bias-corrected estimators over a wide range of data-generating models; we base these data-generating models on the spatial income distribution patterns found in U.S. metropolitan areas. Fourth, we use the bias-corrected estimators to produce corrected estimates of recent trends and patterns of income segregation in the United States. And fifth, we use the corrected estimates to replicate the key analyses of four prior studies of income segregation.

## A Review of Binary and Rank-Order Segregation Measures

We focus in this article on two binary measures of segregation: (1) the information theory index, denoted *H*; and (2) the variance ratio index, denoted *R*. These indices satisfy a set of important properties, including organizational equivalence, size invariance, organizational decomposability, and the principles of transfers and exchanges (James and Taeuber 1985; Reardon and Firebaugh 2002). The dissimilarity and Gini indices do not satisfy all of these principles and thus are less broadly useful.^{6} In addition, both *H* and *R* can be used to construct measures of rank-order segregation, denoted *H*^{R} and *R*^{R} (Reardon 2011), which can be used to measure segregation along some ordered dimension, such as income (Reardon and Bischoff 2011). We briefly review their formulas here.

*J*units (e.g., census tracts). Let

*p*denote the group proportion in a given unit. For values of

*p*∈ [0,1], define the interaction index (

*I*) and entropy (

*E*):

*I*and

*E*are concave down functions of

*p*, a feature that leads estimates of both to be biased when

*p*is estimated from a sample (see the online appendix, section A1). The binary variance ratio and information theory segregation indices are defined, respectively, as follows:

*I*and

*E*are the values of

*I*and

*E*in the whole population;

*I*

_{j}and

*E*

_{j}are the values of

*I*and

*E*in unit

*j*; and

*t*

_{j}/

*T*is the share of the population in unit

*j*.

*y*is an ordered variable, such as income, the corresponding rank-order income segregation indices are

*I*(

*q*),

*R*(

*q*),

*E*(

*q*), and

*H*(

*q*) are the values of

*I*,

*R*,

*E*, and

*H*when the population is divided into groups defined by whether

*y*is above or below the 100 ×

*q*th percentile of

*y*. For example,

*H*(0.5) is the value of

*H*computed between those with above and below median values of

*y*. The rank-order measures are weighted integrals of the binary indices over values of

*q*∈ (0,1). In practice, when

*y*is available only in coarsened form (such as when income data are binned into 16 income categories in U.S. census or ACS data), we estimate

*H*(

*q*) or

*R*(

*q*) by first computing

*H*or

*R*at the set of finite values of

*q*that correspond to the percentiles of the thresholds used to bin the income data, fitting a polynomial function through the resulting points, and then using the fitted polynomial as an estimate of

*H*(

*q*) or

*R*(

*q*) in Eq. (4) (see Reardon 2011; Reardon and Bischoff 2011).

## Bias in Sample-Based Segregation Estimates

The preceding equations assume that we observe *p*_{j} and *t*_{j} without error in each unit *j*. Instead, here we assume that we know *t* with certainty but must estimate *p* from a sample. As we show in online appendix A, the assumption that *t* is known with certainty is not essential. More specifically, from each unit *j* ∈ {1, . . . , *J*}, we observe a simple random sample of size *n*_{j}, drawn without replacement from the population in the unit, which is of known finite size *t*_{j}. Because *p*_{j} is estimated from a sample, $I^j$ and $E^j$ will be biased downward (given that, as noted earlier, *I* and *E* are concave down functions of *p* on the interval (0,1)). The formulas for *R* and *H* in Eqs. (2) and (3) indicate that downward bias in $I^j$ and $E^j$ will lead $R^$ and $H^$ to be biased upward. This is the source of upward bias in sample-based estimates of segregation.^{7}

In online appendix A, we show that the approximate biases in both $R^$ and $H^$ are functions of a bias term, *B* (defined later), which depends on the arithmetic and harmonic means of the unit populations and the harmonic mean of the sampling rates.

*t*

_{j}denote the population in unit

*j*, and let $t\xaf\u22121$ and $t\u22121~$ denote the arithmetic and harmonic means of

*t*

_{j}– 1, respectively. Finally, let

*C*denote the covariance of

*t*

_{j}and

*I*

_{j}:

*B*, defined as

*z*is a function of the ratio of the arithmetic and harmonic means of

*t*

_{j}– 1:

*z*has a minimum of 1, obtained if $tj=t\xaf$ is constant across units, and grows larger with more variation in the

*t*

_{j}s. Moreover,

*z*≈ 1 unless $t\xaf$ is small and the

*t*

_{j}s are highly variable.

*z*is large. To get a sense of the absolute magnitude of

*B*, note that if

*r*

_{j}=

*r*is constant, then

*B*≈ .92 / 80 = 0.0115. Note also how the bias factor changes as the sampling rate changes: changing the sampling rate by a factor of

*c*changes

*B*by a factor of $1\u2212cr~c\u2212cr~$. So, for example, halving the sampling rate from 0.16 to 0.08 increases

*B*by a factor of 2.19. Halving it again from 0.08 to 0.04 increases

*B*by a factor of 2.09.

*B*. The first component is positive and proportional to 1 –

*R*, so that $R^$ is biased toward 1 by a proportion

*B*. The second component of the bias in Eq. (6) may be positive or negative, depending on the sign of

*C*, the covariance between unit population (

*t*

_{j}) and unit diversity (

*I*

_{j}); it is proportional to

*C*and inversely proportional to $t\xaf\u22121$ and

*I*. This bias term will be small relative to the first bias term unless $t\xaf$ is small and

*I*is small (which occurs if the proportion of individuals in one of the two binary categories is near 0 or 1) and

*C*is large. If

*C*= 0 (which will occur by definition if either $tj=t\xaf$ or

*I*

_{j}=

*I*is constant across units), then

In practice, we assume that the second component of the bias in $R^$ is 0 because in most cases, it will be very small relative to the first term. As we show later, this assumption is reasonable in many of the cases we examine.

The bias in $H^$ described in Eq. (7) has a single component. The bias is positive, proportional to *B*, and inversely proportional to *E*. This bias term will be large when *E* is small (which occurs if the proportion of individuals in one of the two binary categories is near 0 or 1). In the online appendix A, we note that the approximation in Eq. (7) fails substantially when $t\xaf$ is small and/or *E* is near 0.

^{8}

The bias described in Eqs. (12) and (13) results from the fact that income data are based on samples. One might additionally worry that the use of binned income data may lead to additional bias in rank-order segregation estimates because even among sampled households, income is not known exactly. Given that *E*(*q*) and *I*(*q*) in Eq. (4) are known by definition, there will be error in $H^R$ or $R^R$ only to the extent that the estimated functions $H^q$ and $R^q$ are error-prone. Intuitively, the error in the functions $H^q$ and $R^q$ will depend on the number and location of the data points used to estimate them; these are determined by the number and location of the thresholds used to bin the income data. When there are fewer income categories and when these income categories are not relatively evenly spaced across the income distribution, the uncertainty in $H^q$ and $R^q$ will be greater, leading to more uncertainty in the rank-order income segregation estimates. Although binned data may lead to imprecision in income segregation estimates, there is no reason to expect it to lead to systematic bias in rank-order segregation measures or error patterns that differ systematically over time or that are related to sampling rates. Having fewer or differently located income thresholds would not be expected to systematically shift the fitted functions $H^q$ and $R^q$ upward or downward. Moreover, Reardon (2011) showed that choosing different numbers or locations of the income thresholds does not yield systematic differences in estimated segregation levels. Because there is no theoretical reason to expect systematic bias related to the binning of income data, we focus here on the bias that results from sampling.

## Bias-Corrected Segregation Measures

^{9}

## Assessing the Accuracy of the Bias-Corrected Segregation Measures

Equations (14) and (15) will yield unbiased estimates of segregation if the approximations used in deriving Eqs. (6) and (7) are accurate. To assess the validity of these approximations and the resulting formulas, we conduct a series of simulation analyses. In order to ensure that our simulations capture the range of data-generating conditions that arise in practical applications, we use observed tabulations from the 2005–2009 ACS in generating simulated data.^{10}

First, we select all census tracts in a given metropolitan area using the 2003 Office of Management and Budget metropolitan area and division definitions. For each census tract, the ACS provides a tabulation of the estimated family income distribution, with income reported in 16 discrete ordered categories. This tabulation takes the form of a vector $t^1jt^2j\u2026t^16j$, where $t^kj$ is the estimated number of families in income category *k* in tract *j* and where $tj=\u2211kt^kj$ is the total number of families in tract *j*. We also obtain the reported unweighted sample size *n*_{j} in tract *j* and compute the tract-specific sampling rate, *r*_{j} = *n*_{j} / *t*_{j}.^{11}

Second, we construct a simulated population data file with *T* = ∑_{j}*t*_{j} observations, where each observation represents a single family in tract *j* with income in category *k* and where the binned income distribution in each tract is defined by the ACS-reported tabulations. We treat this population as the true population of the metropolitan area; from it we compute the true binary and rank-order segregation indices in this population.

Third, for a given sampling rate *r*, we draw without replacement samples of sizes *r* ⋅ *t*_{j} from each tract *j*. From this sample, we estimate the unadjusted binary and rank-order segregation measures and their bias-corrected analogs. We repeat this step 100 times, producing distributions of uncorrected and corrected segregation estimates.

Finally, we repeat this process for each of the 380 metropolitan areas (excluding Puerto Rico) and for sampling rates ranging from *r* = .02 to *r* = .20. For each metropolitan area and sampling rate (and for both *H* and *R*), we compute the average of the uncorrected estimates; the difference between this average estimate and the true segregation level is the bias of the uncorrected estimate. We do the same with the bias-corrected estimates in order to assess the bias in the corrected estimator. For each metropolitan area and sampling rate, we compute the bias in these estimators of binary segregation at each of 15 income thresholds and for both the corrected and uncorrected versions of each of the two rank-order indices.

## Simulation Results

We first examine the uncorrected and corrected binary segregation measures $R^$ and $H^$ as a function of the proportion of families in the metropolitan area below the income threshold used to define the binary measure. Figure 1 presents the bias in the uncorrected and corrected binary segregation measures at each income threshold in each metropolitan area when the sampling rate is *r* = .08. This results in 380 ⋅ 15 = 5,700 estimates of bias, spanning a wide range of income thresholds and levels of segregation.

Figure 1 shows that at an 8 % sampling rate, the bias in the binary $H^$ is roughly +0.007 when the income threshold is in the middle of the income distribution but is up to three times larger when it is in the tails of the income distribution. The bias in the binary $R^$ is roughly +0.01 at income thresholds across the full income distribution. The bias correction performs well in most cases for both measures of segregation, although it clearly overcorrects binary $H^$ at low and high income percentiles.

For the binary $R^$ measure, both the uncorrected and corrected bias have a slight discernible negative slope with respect to the income percentile. In supplemental analyses, we find that this is driven by the fact that *C* (Eq. 5) is generally not 0 (see online appendix B). Rather, in most metropolitan areas, tract size is often slightly positively correlated with median income. As a result, *C* is generally positive for high income thresholds and negative for low income thresholds. Because we assume that *C* equals 0, the bias-corrected estimator tends to slightly overcorrect the binary $R^$ at high income percentiles and undercorrect at low percentiles. In online appendix B, Figs. B1 and B2 show that when we artificially constrain all tracts in a metropolitan area to have the same population size in our simulations (thereby setting *C* = 0 by construction), the bias-corrected estimator performs equally well at all income thresholds. Nonetheless, the bias correction is generally very good; remaining bias in the corrected measures is generally a very small fraction of original bias of the uncorrected measures.

Figure 2 presents smoothed estimates of the bias in binary uncorrected and corrected $R^$ and $H^$; the dashed lines for the uncorrected and corrected estimates at *r* = .08 are constructed by fitting a smoothed line through the points in Fig. 1. The lines associated with *r* = .04 (solid) and *r* = .16 (dotted) are estimated similarly. For the uncorrected measures, the average bias at any income percentile decreases as the sampling rate increases. The bias corrections, however, appear to perform equally well across a range of sampling rates, except for the bias-corrected $H^\u2217$ measure at income percentiles near the extremes of the income distribution.

We next examine the bias in the rank-order income segregation measures. Figure 3 describes the bias in the uncorrected and corrected rank-order $H^R$ and $R^R$ as a function of sampling rate, where *r* varies from 2 % to 20 %. Bias in the uncorrected measures is clearly decreasing as sampling rate increases, and is roughly twice as large at 8 % as at 16 % for both $H^R$ and $R^R$, suggesting that comparisons of census- and ACS-based rank-order income segregation estimates will be biased. The bars on the figure represent the range in which the bias of 95 % of metropolitan areas fall. For the uncorrected measures of rank-order income segregation, the bias varies considerably among metropolitan areas, particularly at low sampling rates. This is partly because the bias term *B* depends on the average tract population, which varies across metropolitan areas. In the case of $R^R$, the variation also emerges because the bias depends on the true value of *R*^{R} (see Eq. (12)), which varies across metropolitan areas.

The bias in the corrected measures, averaged across all metropolitan areas, is nearly zero at all sampling rates. Moreover, the bias is not only zero on average, but also in every metropolitan area, as seen by the very narrow 95 % range bars (some of which are too narrow to be visible in the figure). At the very lowest sampling rate (2 %), the bias in the bias-corrected rank-order $R^R$ is slightly negative but is still vastly smaller than the bias in the uncorrected estimator. The figure makes clear that the corrected estimators perform very well when used to estimate income segregation among the full population across the range of conditions present in U.S. metropolitan areas.

The derivations of the bias-correction formulas rely on several approximations that are valid when tract populations are relatively large and do not vary much across tracts. These conditions are met when considering the income segregation of all families but may be less true for subpopulations. Consider income segregation among black families, for example. In many metropolitan areas, the average number of black families per tract is much smaller (one-tenth or even smaller) than the total number of families. Moreover, because of residential segregation, the number of black families per tract varies widely across tracts. Both of these conditions might lead to a failure of the simplifying assumptions used to derive the bias and bias-correction formulas.

To assess the performance of the bias-corrected measures under such conditions, we compute uncorrected and bias-corrected measures of income segregation among black families from the presented simulations.^{12} Figure 4 presents the bias in rank-order income segregation among black families by the average number of black families per tract in a metropolitan area. This figure shows two notable findings. First, the bias in estimates of income segregation among black families is very large, particularly in metropolitan areas with few black families. Second, the bias-corrected estimators tend to overcorrect the estimates, producing segregation estimates that are often much too low; this overcorrection is particularly pronounced when the average number of families in a tract is low. For metropolitan areas where the number of black families in the average tract is greater than 200, the bias-corrected estimates are generally less biased (in terms of absolute value of the bias) than the uncorrected estimates, although they are still somewhat negatively biased. Table B1 (online appendix B) quantifies the average extent of overcorrection in $H^R\u2217$ and $R^R\u2217$ at different sampling rates and average tract sizes. At an average tract size of 200, $R^R\u2217$ overcorrects by roughly 30 %, on average, regardless of sampling rate; $H^R\u2217$ overcorrects by even more, particularly at low sampling rates. Below average tract sizes of 200, the bias-corrected estimators do not perform well. This is particularly true for $H^R$, where the cure is worse than the disease when average tract sizes are small.

It is important to note that the failure of the bias-correction formulas in Fig. 4 results from the confluence of three conditions: the tract population sizes are (a) very small on average, (b) highly variable, and (c) correlated with tract median income. Without all three of these conditions present, the bias-correction formulas perform well across a wide range of conditions.

## Bias-Corrected Estimates of Trends in Income Segregation

In this section, we report and compare bias-corrected and uncorrected estimates of income segregation among different populations over the past several decades. We then compare bias-corrected estimates of income segregation with previously published estimates and replicate multivariate regression analyses from several key studies, using the corrected estimates of income segregation. Estimates rely on publicly available counts of families or households in 8 to 25 income categories, depending on the year and population, from the decennial census or ACS.

We begin by estimating rank-order income segregation among four census-defined populations that have been the focus of recent published research: families, households, families with children under the age of 18, and households without children (Bischoff and Reardon 2014; Owens 2016; Reardon and Bischoff 2011, 2016).^{13} Comparisons among estimates for these populations highlight important distinctions among the social contexts and patterns of inequality for children and adults. We present trends in average levels of income segregation within a set of large metropolitan areas used in previous work: those with populations over 500,000 as of 2007 (*N* = 116).^{14} We pay particular attention to trends between the 2000 decennial census and the ACS years because of the change in the sampling rate between these surveys that resulted in reduced tract-level sample sizes.

Table 1 presents uncorrected and bias-corrected estimates of both rank-order *H*^{R} and *R*^{R} using decennial census data since 1970 and ACS data for two nonoverlapping five-year spans, 2005–2009 (labeled 2007, the middle year) and 2012–2016 (labeled 2014). Results for all families (top panel) show that uncorrected estimates are inflated in all years, with larger differences between uncorrected and bias-corrected estimates in the post-2000 ACS years. From 1970 to 2000, estimates of the average uncorrected *H*^{R} and *R*^{R} are 4 % to 6 % higher than the corresponding bias-corrected estimates; in 2007 and 2014, the uncorrected estimates are 8 % to 10 % higher than the bias-corrected estimates. Figure 5 presents trends in uncorrected (dashed line) and bias-corrected (solid line) estimates of *H*^{R} from 1970 to 2014, clearly showing that the uncorrected estimates are more upwardly biased after 2000, when the tract-level sampling rate declined, than in earlier years. Nonetheless, the corrected estimates show that average metropolitan area income segregation increased by roughly 4 % between 2000 and 2014 (*p* < .01; see Table 1).

The lower three panels of Table 1 present the average bias-corrected rank-order income segregation estimates among all households, families with children, and households without children. Similar to the findings for families, bias-corrected estimates for these populations are lower than uncorrected estimates in all years, with greater differences after 2000. Bias is smaller for all households (2 % to 4 % in census years and 6 % to 7 % in ACS years) than for families with children or households without children (5 % to 8 % in census years and 11 % to 15 % in ACS years) because the total household population is larger than the subpopulations by the presence of children.

Figure 6 provides a summary of trends in rank-order income segregation among all four populations, presenting the mean bias-corrected estimates of *H*^{R} found in Table 1. Two key patterns are evident here. First, income segregation is substantially higher among families with children than among households without children. It is slightly higher among family households than among all households after 1980 because the share with children is larger for families than for all households and because segregation is highest among families with children. Second, income segregation among all households did not change substantially after 1990, although this masks divergent trends among families with children and childless households. Income segregation increased by 20 % after 1990 among families with children, with most of the increase occurring between 2000 and 2014 (*p* < .001). Among households without children, income segregation declined by 10 % in the 1990s and thereafter remained stable. The higher segregation levels and divergent trends mean that income segregation among families with children is now more than twice as high as among households without children.^{15} Table 1 shows that these patterns are the same when segregation is measured using the rank-order variation ratio index, *R*^{R}.

In addition to describing average levels of income segregation, binary *H* and *R* can be estimated at any percentile in the income distribution by fitting a polynomial through the binary estimates at each income threshold defined by census or ACS categories and using the fitted polynomial to predict the value of segregation at any given income percentile (for details about this method, see Reardon and Bischoff 2011). Figure 7 and Table C1 (online appendix C) present trends in family income segregation at the 90th (*H*90 and *R*90), 50th (*H*50 and *R*50), and 10th (*H*10 and *R*10) percentiles of the income distribution from 1970 to 2014. The difference between uncorrected and bias-corrected estimates are again larger after 2000.

Figure 7 shows that segregation of affluent families (*H*90) declined in the 1970s, rose in the 1980s, and declined modestly in the 1990s. After 2000, *H*90 increased modestly and then declined again. The difference from 2000 to 2014 is not statistically significant. Segregation between families in the top and bottom halves of the income distribution (*H*50) increased after 1980, with levels significantly higher in 2014 than in 2000. Segregation of poor families from others (*H*10) rose in the 1970s and 1980s, declined in the 1990s, and remained stable after 2000; the difference between the 2014 and 2000 estimates is not statistically significant. Overall, income segregation increased only very modestly after 2000 at the top and bottom of the income distribution, with a slightly larger increase between the top and bottom halves of the income distribution. Notably, segregation of affluent families is much higher than segregation of poor families in all years: in 2014, segregation of affluent families was approximately 30 % higher than segregation of poor families.

Examining trends in income segregation among different racial/ethnic groups is important for understanding the changing relationship between race/ethnicity, socioeconomic status, and residential attainment. Estimates for specific racial/ethnic groups may be particularly prone to bias because the size of these populations is small relative to the total population and, due to racial segregation, unevenly distributed across tracts in many areas of the United States. As previously noted, our bias-correction method performs best when the average tract population in a metropolitan area is at least 200 families of a particular group (although even then, our method tends to overcorrect the bias modestly). Therefore, we estimate rank-order income segregation among white, black, and Hispanic families in only the metropolitan areas in our sample that meet this criterion (116 metropolitan areas for white families, 22 for black families, and 20 for Hispanic families).^{16} We discuss *R*^{R} for the race-specific results because *R*^{R} provides better bias adjustments than *H*^{R} for small populations, as shown in Fig. 4, although trends in *H*^{R} and *R*^{R} (see Table 2) are consistent with one another.

The estimated trends in average within-race segregation are shown in Table 2 and Fig. C1 in online appendix C. In 2000, income segregation levels were similar among white and black families (0.107 and 0.105, respectively) and slightly lower among Hispanic families (0.095). Income segregation among all three groups increased after 2000, but segregation rose much more among black and Hispanic families than it did among white families (by approximately 25 % and 4 %, respectively). Compared with the distribution of segregation levels in 2000 (which had standard deviations of roughly 0.025 for white and black income segregation, and 0.015 for Hispanic segregation; see Table 2), the increases in black and Hispanic average income segregation were very large—about 1 standard deviation—whereas the increase in white income segregation was less than one-quarter of a standard deviation. These results are consistent with findings presented by Logan et al. (2018), whose bias-adjusted estimates showed that income segregation among black families grew substantially and faster than income segregation among the population as a whole. Given that the bias-correction will tend to overcorrect the black and Hispanic segregation estimates (and will do so more when the sampling rate is lower), the estimated 25 % increase in segregation among black and Hispanic families likely underestimates the true trend modestly. The trends for different racial/ethnic groups are not strictly comparable, however, because they reflect a different set of metropolitan areas. In Table 2, we report estimated average income segregation among white families in the sets of 22 and 20 metropolitan areas for which we estimate black and Hispanic family income segregation, respectively. These analyses yield the same conclusion: average income segregation increased more among black and Hispanic families than among white families from 2000 to 2014.

## Replications of Previously Published Research on Income Segregation

Do our bias-corrected estimates alter the conclusions reached in previously published research on the trends and correlates of income segregation? Table 3 presents published estimates of rank-order income segregation (*H*^{R}) among families (Bischoff and Reardon 2014; Reardon and Bischoff 2016), households, families with children, and households without children (Owens 2016). Recall that the authors of these papers adjusted the income segregation estimates in an attempt to address concerns about small-sample bias. Their adjustment methods generally inflated the estimates but did not eliminate the bias. Estimates of *R*^{R} have not been previously published.

Bischoff and Reardon (2014) concluded that income segregation among families declined modestly in the 1970s, rose sharply in the 1980s, was stable in the 1990s, and increased after 2000. As presented in the top panel of Table 3, bias-corrected estimates of *H*^{R} for the same years and metropolitan area sample support these conclusions. Of particular interest are the trends from 2000 onward, when sampling rates declined. The bias-corrected estimates indicate that rank-order income segregation increased by about 4 % from 2000 to 2012, compared with an 8 % increase reported by Reardon and Bischoff (2016). Therefore, about half of the previously reported increase after 2000 was due to bias induced by the changing sampling rate between the census and ACS. The bias-corrected estimates indicate that from 1970 to 2012, income segregation among families increased by about 25 %, slightly less than had been previously reported.

Bischoff and Reardon (2014) also reported estimates of income segregation by race. We do not compare our results to these previously published estimates because our sample restriction (to metropolitan areas with an average of 200 or more black or Hispanic families per tract) differs from that used in earlier work and because previous work did not publish estimates of *R*^{R}, which we prefer for the race-specific results. Our bias-corrected estimates in Table 2 indicate that the level of income segregation among white families in 2000 was more similar to that of black and Hispanic families than Bischoff and Reardon (2014) reported. Our findings are, however, consistent with Bischoff and Reardon’s conclusion that income segregation increased more quickly among black and Hispanic families than among white families.

The lower panels of Table 3 compare published and bias-corrected estimates of rank-order income segregation among all households and by the presence of children. Owens (2016) showed that from 1990 to 2010, income segregation did not increase substantially among all households but did increase among families with children. The bias-corrected estimates for the same years and metropolitan areas support this conclusion: *H*^{R} changed negligibly from 1990 to 2010 among all households and actually declined by about 9 % among households without children. The previously reported increase in income segregation, *H*^{R}, among families with children from 2000 to 2010 was 17 %; the bias-corrected increase is about 14 %. Therefore, our bias-corrected income segregation estimates generally confirm the trends reported in past research, although the magnitude of changes after 2000 is slightly smaller than previously reported.

Past research has also documented a relationship between income segregation and income inequality, demonstrating that rising income inequality is associated with increasing residential sorting by income (Reardon and Bischoff 2011; Watson 2009). We replicate multivariate results from several published papers using bias-corrected *H*^{R} and *R*^{R} and find no substantial change in the results.

Table 4 presents the key coefficients from models estimating the relationship between income inequality and rank-order income segregation (replications of the full tables are presented in online appendix C, Tables C2–C4). Each study measured income inequality using the Gini coefficient. The top panel of Table 4 replicates Reardon and Bischoff (2011), predicting income segregation among large metropolitan areas from 1970 to 2000; the second panel replicates Bischoff and Reardon (2014), which extends the model to 2009. Models include metropolitan area and year fixed effects and metropolitan area-year covariates. As shown in column 1, the published results indicated that an increase of 1 point on the Gini index corresponded to approximately an increase of one-half point in family income segregation between neighborhoods. Columns 2 and 3 present results from the same model using bias-corrected *H*^{R} and *R*^{R}; the coefficients in these models are similar in magnitude and statistical significance to the corresponding previously published estimates. The magnitude of the differences between published and corrected coefficients ranges from –1 % to +14 %.

The bottom panel of Table 4 presents results from Owens (2016), who focused on differences between income segregation among households with and without children. Replications of regression analyses from that study using bias-corrected *H*^{R} and *R*^{R} produce very similar results to those published. The coefficient for income inequality is positive and of similar magnitude to published results, indicating that changes in income inequality from 1990 to 2010 were positively associated with changes in income segregation among childless households. The positive and significant interaction term between income inequality and families with children across all estimates indicates that the relationship between income inequality and income segregation was more than twice as large among families with children as among childless households. Owens (2016) also investigated whether school district fragmentation (the degree to which a metropolitan area was split up between many school districts) contributed to higher residential segregation among families with children. The results predicting bias-corrected *H*^{R} and *R*^{R} are consistent with the published results: income segregation among families with children is higher in metropolitan areas that are more fragmented, as demonstrated by the significant and positive coefficient for fragmentation × families with children.

In summary, our replications of previously published regression models using bias-corrected measures of income segregation do not alter any of the substantive conclusions of prior research. The relationships between income segregation and both income inequality and school district fragmentation remain positive, large, and statistically significant when the bias-correction methods for income segregation are used.

## Discussion

Our investigations of potential bias in recent income segregation trends demonstrate several important facts. First, we confirm that sample-based segregation measures are biased upward. The bias is often moderately large relative to the magnitude of observed differences and changes in segregation. Ignoring the bias may therefore lead to erroneous inferences. Second, we show that it is possible to compute bias-corrected segregation measures that largely eliminate the small-sample bias using publicly available census and ACS tabulations; the correction does not require access to restricted census microdata, nor does it require repeated simulation of microdata. Third, bias-corrected estimates indicate that roughly one-half the increase in family income segregation between 2000 and later ACS years reported in recent research is due to increased upward bias resulting from the lower sampling rate of the ACS relative to the 2000 census. Nonetheless, the bias-corrected trend indicates that income segregation did rise after 2000, albeit more slowly than has been reported. The increase in family income segregation is largely due to trends among families with children, which we find did increase substantially after 2000, consistent with Owens’s (2016) findings. We also find that income segregation among black and Hispanic families increased much more than it did among white families. Furthermore, replications of multivariate analyses from previously published research confirm that rising income inequality is a primary predictor of increases in residential sorting by income.

The bias in segregation measures that we investigate here is not limited to the study of income segregation. All standard segregation indices (binary, multigroup, ordinal, and rank-order measures) are biased upward when computed from sample data. Sample-based measures of gender segregation within firms, for example, will be biased upward—and will be more biased in firms or divisions in which the organizational units are smaller. Likewise, sample-based measures of racial/ethnic segregation among organizations (such as churches, schools, clubs, or firms) will be subject to upward bias. Moreover, the bias we describe is not limited to the two segregation measures we focus on here; similar bias is present in the dissimilarity and Gini indices of segregation, although we do not derive formulas for their biases. Segregation measures are not subject to small-sample bias, however, when computed from full population data. Racial segregation measures computed from the decennial census, for example, are based on race counts from the full population enumeration and thus are not subject to sample-induced bias. Likewise, school segregation measures computed from population-based racial/ethnic or free lunch–eligible enrollment counts—such as in the Common Core of Data (CCD)—are not subject to sampling bias.

We show that the sampling bias is inversely related to the average unit size and the sampling rate. This has implications not just for the measurement of income segregation trends but also for any comparison involving either different sampling rates or units of different average size. For example, a comparison of uncorrected estimates of income segregation in different countries will be biased if the countries use a different sampling rate or if the geographic units (the equivalents of census tracts in each country) differ in size. A comparison of between-tract to between-block group segregation will be biased because the average tract population is roughly three times larger than the average block group population. Thus, any sample-based decomposition of segregation into within- and between-unit components will overstate the within-unit component. And a comparison of segregation in two different subpopulations—for example, income segregation among individuals over age 65 and among those younger than 65—will be biased if the subpopulations have different average within-tract sizes.

The biases in $R^$ and $R^R$ are also inversely related to the true level of segregation. Thus, for example, if we wish to compare the change in segregation between two metropolitan areas—one highly segregated and one much less segregated—the estimated change will be biased upward more in the less-segregated metropolitan area than in the more-segregated one. Finally, the bias in binary $H^$ is inversely related to the entropy *E*. So an estimate of the segregation between the bottom 10 % of earners and all others will be more upwardly biased than an estimate of the segregation between the bottom and the top half of the income distribution.

The bias-corrected estimators we describe here provide a method of obtaining approximately unbiased estimates in such cases, however. Researchers may use these bias-corrected estimators to make valid comparisons in cases where sample rates, unit populations, or true segregation differ (e.g., between countries with different sampling rates or between different levels of geography). In the substantive cases of interest in this article, the methods we describe allow comparisons of income segregation across years with different sampling rates, metropolitan areas with different overall levels of segregation, and subpopulations of different sizes. These methods do not require access to microdata. Instead, one need only know the total unit populations and the harmonic mean of the unit sampling rates, which can be estimated with publicly available data. With these, it is straightforward to implement the bias-corrected estimators we describe. We have written a set of Stata commands, *seg* and *rankseg*, to perform the bias correction. These are publicly available via the Boston College Statistical Software Components (SSC) archive.

As we show, however, there are some cases in which the bias-corrected estimators fail to provide accurate estimates. When sample sizes are small on average and highly variable, the estimators may fail to provide unbiased estimates. We advise caution in such cases; Fig. 4 and the derivations in online appendix A may provide researchers with some guidance regarding the potential magnitude of the bias and extent to which our estimators eliminate this bias.

Note also that the methods and estimates we describe here provide bias correction due to bias that arises in the case of sampling without replacement in each geographic unit, where the sampling rate in each tract is known. The approach can accommodate variation in sampling rates across units. Simple adjustments to the formulas can accommodate cases where sampling is done with replacement. But our methods do not address several other factors that complicate the estimation of income segregation. Our method does not explicitly address cases in which sampling probabilities vary within a unit (as is implicitly the case when sampling weights are used to account for various forms of nonresponse) or missing income data are imputed, reducing the effective sample size. In such cases, however, if the effective sampling rate in each unit is known, it can be used in place of the simple sampling rate in the bias-correction formulas.

Finally, our concern here is the bias in sample-based estimates of segregation. We have not addressed the issue of sampling variability in segregation estimates. Because the substantive focus of this article is on average trends in segregation among many metropolitan areas, the error in each metropolitan area’s estimated income segregation that arises from sampling variability is a secondary concern. In our estimates of average trends, uncertainty resulting from sampling variability is captured in the standard errors of the regression models. In other contexts, though, one might want to know whether income segregation changed in one particular place. In that case, it is essential to quantify the sampling variance in segregation estimates. We know of no published research providing formulas describing the sampling variability of segregation measures, however (although Reardon (2011) provided formulas for the error arising from model (polynomial order) uncertainty). In part, this is because most methodological research on segregation indices has focused on racial segregation, where full population data have been widely available, obviating concerns about sampling variability. Sample-based segregation estimates, however, may have substantial sampling variance. Future work should therefore aim to construct standard errors for sample-based segregation estimators.

We conclude with some practical guidelines for estimating segregation. First, for data based on samples rather than full populations, segregation estimates will be subject to bias. In general, lower sampling rates and smaller unit populations will yield larger bias. Second, our formulas (particularly Eqs. (7), (11), (12), and (13)) allow one to estimate the degree of bias. If the expected bias is large enough to affect the inferences of interest, the bias-correction methods we propose will be useful. Third, if unit-specific sampling rates are not available, we recommend using the overall or average sampling rate in the larger population; unless sampling rates are highly variable, this will yield bias-corrected estimates that are very close to what would be obtained using unit-specific sampling rate information. Fourth, when binary segregation measures are being estimated, if the group proportions are near 0 or 1, we recommend using *R* instead of *H* (see Fig. 1), given the poor performance of the bias correction of *H* in such cases. Fifth, when rank-order segregation measures are being estimated, bias correction works well for both *R*^{R} and *H*^{R} under most conditions. Finally, we advise caution when estimating (binary or rank-order) segregation in cases when the unit populations are small on average (our rule of thumb is 200), highly variable, and correlated with the interaction or entropy indices, as is the case when estimating within-race income segregation under conditions of substantial racial and economic segregation. With the exception of these conditions, the bias-correction methods appear to satisfactorily address the concern about bias in sample-based segregation estimates.

## 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.

## 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 $I^$ and $E^$ is small compared with the bias in the $I^j$s and $E^j$s. As a result, in general, $R^$ and $H^$ will be biased upward.

^{8}

Logan et al. (2018) used a formula that assumes both sampling with replacement (which is not the case with census data) and a low sampling rate. Our formula in Eq. (13) assumes sampling without replacement and accommodates heterogeneity in sampling rates (and unit sizes).

^{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:

$R^R\u2217=R^R\u2212B1\u2212BH^R\u2217=H^R\u2212B.$

These formulas will yield estimates of $R^R\u2217$ identical to those produced by Eq. (15) but will typically yield very slightly different estimates of $H^R\u2217$. 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.