Spatial scale and measuring segregation: illustrated by the formation of Chicago’s ghetto

Abstract

Few studies of residential segregation in cities have directly addressed the issue of spatial scale, apart from noting that the traditional indices of segregation tend to be larger when calculated for small rather than large spatial units. That observation however ignores Duncan et al.’s (Statistical geography: problems in analyzing areal data. Free Press, Glencoe, 1961) explication that any measure of segregation at a fine-grained scale necessarily incorporates, to an unknown extent, segregation at a larger scale within which the finer-grained units are nested. To avoid that problem, a multi-level modelling perspective is introduced that identifies the intensity of segregation at each scale net of its intensity at any larger scale included in the analysis. It is applied to an analysis of the emergence of Chicago’s Black ghetto over the twentieth century’s first three decades, using data at the ward and ED scales. It shows that across Chicago as a whole segregation was equally as intense at the two scales, with statistically significant increases in that intensity at both scales across the three decades. At the finer scale, however, segregation was much more intense across the EDs within those wards that formed the core of the emerging ghetto than it was in the remainder of the city.

Appendices

Appendix 1: The model specification for three-level binomial model

The form of the multilevel model used in the analysis of EDs and Wards (derived from Browne et al. 2005) is as follows, with each census year analysed separately:

$$ PropBlack_{jk } \sim Binomial\left( {Population_{jk} , \pi_{jk} } \right) $$

$$ \log it\left( {\pi_{jk} } \right) = \beta_{0} x_{0} + v_{0k} + u_{0jk} $$

$$ v_{0k} \sim N\left( {0, \sigma_{vo}^{2} } \right) $$

$$ u_{0jk} \sim N\left( {0, \sigma_{uo}^{2} } \right) $$

$$ Var \left( {PropBlack_{jk} |\pi_{jk} } \right) = \frac{{\pi_{jk} \left( {1 - \pi_{ik} } \right)}}{{Population_{jk} }} $$

where $PropBlack_{ijk}$ is the observed response variable, the proportion of Black people in ED j in Ward k that are Black and $Population$ is the denominator of the Black plus White population. The log of the odds of being Black $\left( {\log it \left( {\pi_{jk} } \right)} \right)$, is modelled as a function of a fixed effect where $x_{0}$ is a constant (a set of 1 s for each and every observation) so that $\beta_{0}$ is the overall city-wide average log-odds of being Black. The random part of the model consists of a differential logit for each ward ($v_{0k}$) and a differential logit for each ED within each ward ($u_{0jk}$). These logit differentials are assumed to come from a Normal distribution and are summarised by a variance term at each level so that $\sigma_{vo}^{2}$ summarises the between Ward differences while $\sigma_{uo}^{2}$ summarises the within-Ward between-ED variation. These two variances are our measures of segregation. At the lowest level there is a single variance term and this is assumed to follow a Binomial distribution. In practice this is fitted as a three-level model, the Wards at level 3, and exactly the same set of units—the EDs—at level 1 and level 2; that is, each level 2 unit has exactly one level 1 unit. This views the aggregate proportions at level 2 as consisting of replicated binary responses for ‘individuals’ at level 1. This use of a pseudo-level is fully explained in Browne et al. (2005) and allows the separation of the variance into exact Binomial at level 1 and over-dispersion at higher level so that the higher-level variances summarize differences between areas over and above those expected from a random variation generated by a varying denominator. The estimated higher-level variances are transformed to MORs by using the formula given in Larsen and Merlo (2005).

Appendix 2: Model specification allowing for additional within-ghetto segregation (in three wards) in the model’s fixed part

The specification of the model is as follows:

$$ PropBlack_{jk } \sim Binomial\left( {Population_{jk} , \pi_{jk} } \right) $$

$$ \log it\left( {\pi_{jk} } \right) = \beta_{0} x_{0} + \beta_{1} x_{1jk} + \beta_{2} x_{2jk} + \beta_{3} x_{3jk} + v_{0k} x_{0} + u_{0jk} x_{4jk} + u_{1jk} x_{5jk} $$

$$ v_{0k} \sim N\left( {0, \sigma_{vo}^{2} } \right) $$

$$ \left[ {\begin{array}{*{20}c} {u_{0jk} } \\ {u_{1jk} } \\ \end{array} } \right]\sim N\left( {0,\left[ {\begin{array}{*{20}c} {\sigma_{uo}^{2} } & {} \\ {} & {\sigma_{u1}^{2} } \\ \end{array} } \right] } \right) $$

$$ Var \left( {PropBlack_{jk} |\pi_{jk} } \right) = \frac{{\pi_{jk} \left( {1 - \pi_{jk} } \right)}}{{Population_{jk} }} $$

The underlying model is similar to before with the response being the logit of being Black and the lowest level variance being an exact Binomial distribution. However in the fixed part of the model three dummies ($x_{1jk}$; $x_{2jk}$ and $x_{3jk}$) identify the three wards that constitute the ghetto area. The $\beta_{0}$ then gives the overall log-odds of an individual being Black outside the ghetto while $\beta_{1}$ is the differential logit of being Black in ward 1, and so on. In the random part $v_{0k}$ gives the differential ward effect which will be zero for wards 1, 2 and 3 (which have their own mean in the fixed part), so that the variance $\sigma_{vo}^{2}$ summarizes the between ward variation in the logit of being Black outside the defined ghetto area. Two separately–coded (Bullen et al. 1997) dummy variables ($x_{4jk}$ and $x_{5jk}$) identify EDs that are in and outside the ghetto and the two variance terms ($\sigma_{uo}^{2}$ and $\sigma_{u1}^{2}$) summarise the variations within wards between EDs inside and outside the ghetto area. There is no covariance in the level 2 random part as an ED cannot be simultaneously in and outside the ghetto. The MLwiN (Rasbash et al. 2009) software is particularly flexible in estimating such models with complex variance functions.