Factor Analysis of Segregation Measures: A Replication

While reading A Wealth and Status-Based Model of Residential Segregation, I noticed its discussion of Massey and Denton (1988), who reviewed 20 segregation indices and applied them to 1980 census data from 60 metropolitan statistical areas (MSAs). Using factor analysis, Massey and Denton classified these indices into five underlying dimensions: evenness, exposure, clustering, centralization, and concentration.

However, they appear to have chosen five factors a priori, based on their review of the literature, rather than using the data to determine how many factors to retain. So, I decided to replicate their analysis using parallel analysis to determine the number of factors empirically.

I also broadened the analysis in two ways. Instead of restricting the sample to 60 MSAs, I used every contemporary metropolitan area with at least 1,000 members of the focal racial group and at least 1,000 non-Hispanic White-alone residents¹. And instead of examining only 1980, I repeated the analysis for 1990, 2000, 2010, and 2020 to see how stable the findings are over four decades. The national-sample result is remarkably consistent: parallel analysis retains three factors in every census year. The fixed panel corresponding to the original 60 metropolitan identities produces a more conservative result, as discussed below.

Massey and Denton’s five dimensions describe the different ways in which residential segregation might occur:

Evenness: How evenly distributed is a group across residential areas?
Exposure: How much potential contact do members of one group have with members of another group?
Concentration: How much physical space does a group occupy?
Centralization: How concentrated is a group in the city center?
Clustering: To what degree does a group disproportionately live in contiguous areas?

How many factors are there?

The national sample consistently yields three factors, not five.

Five-panel parallel-analysis chart for the national metropolitan sample in 1980, 1990, 2000, 2010, and 2020 using a 2-kilometer spatial radius. In every year, the first three observed eigenvalues exceed the 95th-percentile null eigenvalues, so three components are retained.

I’ve shown the findings for when the local-environment radius for the spatial measures is set at 2km, but the same result happens when we use 0.5, 1, or 4 km (see the Appendix).

Does the result depend on the sample?

As a robustness check, I repeated the analysis using a fixed panel corresponding to Massey and Denton’s original 60 metropolitan identities, following the same metro names while using each census year’s contemporary boundaries. The number of retained factors is the same at every spatial radius within a given sample and year:

Sample	1980	1990	2000	2010	2020
National sample	3	3	3	3	3
Original-metro panel	2	2	2	2	3

The fixed panel has a much smaller weighted effective sample size (between 69 and 92 observations, compared with 230 to 472 in the national sample) so parallel analysis is more conservative. Still, we can see that the data does not support five factors.

What do the factors measure?

The three factors are Evenness, Isolation, and Concentration. Expand each factor below for its interpretation and loadings.

Evenness

This is pretty similar to the original Evenness dimension that Massey and Denton found; it measures how evenly distributed the minority and majority groups are.

Factor loadings

Index	1980	1990	2000	2010	2020
Atkinson, $b = 0.5$	1.00	0.99	0.99	1.00	0.99
Atkinson, $b = 0.9$	1.00	0.98	0.99	1.00	0.99
Information theory ($H$)	0.99	0.99	0.97	1.00	0.99
Spatial information theory ($H_s$)	0.98	0.99	0.96	1.00	1.00
Spatial dissimilarity ($D_s$)	0.96	0.97	0.99	0.99	0.99
Gini ($G$)	0.98	0.97	0.98	0.98	0.98
Dissimilarity ($D$)	0.96	0.97	0.99	0.98	0.99
Atkinson, $b = 0.1$	0.88	0.93	0.96	0.96	0.96
Correlation ratio ($\eta^2$)	0.94	0.95	0.90	0.94	0.94
Spatial proximity ($\mathrm{SP}$)	0.82	0.92	0.90	0.93	0.91
Divergence index ($\mathrm{DIV}$)	0.81	0.81	0.77	0.85	0.86
Relative clustering ($\mathrm{RCL}$)	0.60	0.69	0.72	0.59	0.55
Relative centralization ($\mathrm{RCE}$)	0.22	0.46	0.64	0.45	0.46

Isolation

Broadly speaking, these indices measure how much contact a random member of the focal group is expected to have with other members of the same group.

Factor loadings

Index	1980	1990	2000	2010	2020
Distance-decay isolation ($\mathrm{DP}_{xx}$)	0.98	0.97	0.98	0.95	0.96
Spatial isolation (${}_xP_x^{(s)}$)	0.89	0.90	0.94	0.92	0.93
Isolation (${}_xP_x$)	0.85	0.87	0.93	0.91	0.93
Absolute concentration ($\mathrm{ACO}$)	-0.83	-0.87	-0.31	-0.90	-0.92

Concentration

This combines indices that measure how concentrated a group is in the city center with indices that measure how concentrated a group is in general.

Factor loadings

Index	1980	1990	2000	2010	2020
Absolute centralization ($\mathrm{ACE}$)	0.84	0.87	0.62	0.90	0.90
Delta ($\mathrm{DEL}$)	0.84	0.82	0.68	0.83	0.86
Proportion central city ($\mathrm{PCC}$)	0.82	0.83	0.61	0.77	0.77
Relative concentration ($\mathrm{RCO}$)	0.82	0.74	0.92	0.50	0.55
Relative centralization ($\mathrm{RCE}$)	0.69	0.46	0.28	0.40	0.39

Why are there fewer than five factors?

So, are there really only three dimensions of segregation? The national sample says three, while the fixed panel generally says two. But neither supports five. I’d argue that there are three relevant dimensions, even if the smaller fixed panel does not always distinguish all three empirically.

A problem that immediately jumped out at me when reading about the five dimensions was that the Centralization dimension makes an assumption not about how a group is distributed, but about where the group is distributed. This assumption makes me uneasy. I don’t like my mathematical measures to bake in such messy, empirical assumptions. Now, it does make sense that, given the fact that minority groups tend to be located near the center of urban areas, we’d expect centralization to be correlated with concentration. But I’d prefer to use concentration measures and then, once we determine that certain groups tend to end up concentrated, ask where those groups are concentrated. If that turns out to be the city center, then great.

What about the two dimensions that don’t show up in my factor analysis, Exposure and Clustering? Well, conceptually, clustering seems related to evenness: if a group is all clustered together, then it isn’t evenly distributed. As for exposure, that seems related to isolation. If a group is isolated, that is, its members are particularly likely to live near one another, then they’re not very exposed to members of other groups.

So, it doesn’t actually seem like we needed five dimensions in the first place.

Which measures should we use?

Which measures should be used for each factor? For the Evenness factor, there are seven indices with consistently near-perfect loadings, so really, we could choose any of them and we’d be fine. But the dissimilarity index ($D$) seems to be the most popular, so we’ll go with that.

For the Isolation factor, the three highest-loading indices are, well, isolation indices. Of these three, distance-decay isolation and spatial isolation seem the most conceptually appropriate: we should expect neighboring tracts to matter when determining isolation. Of these, distance-decay isolation consistently loads higher, so let’s go with that.

For the Concentration factor, I’d prefer to avoid any measures that explicitly refer to or rely on the idea of a city center, which leaves delta ($\text{DEL}$) as our best option.

So, if you want to measure segregation, I would distinguish three dimensions: evenness, isolation, and concentration. If you’re interested in evenness, use the dissimilarity index. If you’re interested in isolation, use distance-decay isolation. If you’re interested in concentration, use the delta index.

Appendix

Measures of Segregation

The analysis uses 22 nonredundant indices: 18 from the original Massey-Denton battery and four later extensions. Expand any measure below for its interpretation and formula.

Population and area notation:

$x_i$ and $y_i$ are the focal-group and non-Hispanic-White populations of tract $i$.
$t_i=x_i+y_i$ is the tract’s two-group population.
$X$, $Y$, and $T$ are the corresponding metropolitan totals.
$p_i=x_i/t_i$ is the focal-group share of tract $i$, and $P=X/T$ is the focal-group share of the metropolitan population.
$a_i$ is tract land area (in square miles).

For the original proximity measures:

$d_{ij}$ is the distance in miles between the representative points of tracts $i$ and $j$.
$z_{ij}=e^{-d_{ij}}$ is the distance-decay weight.
Within a tract, self-distance is approximated as $d_{ii}=\sqrt{0.6\,a_i}$. This is to avoid treating every resident of a tract as if they’re in the exact same place.

For the added local-environment measures:

$r_{ij}$ is the distance in kilometers between tracts $i$ and $j$.
$h$ is the local-environment radius.

Nearby tracts are weighted using the biweight kernel:

\[w_{ij}(h)= \begin{cases} [1-(r_{ij}/h)^2]^2, & r_{ij}<h,\\ 0, & r_{ij}\ge h. \end{cases}\]

The resulting focal-group share of tract $i$’s local environment is:

\[q_i=\frac{\sum_j w_{ij}(h)x_j}{\sum_j w_{ij}(h)t_j}.\]

Indices of Segregation

Dissimilarity ($D$)