Multi-dimensional and multi-scale approaches to residential segregation

EPID 684
Spatial Epidemiology

Jon Zelner
[email protected]


  • Discuss the qualitative dimensions of residential segregation.

  • Looking at interacting dimensions of risk using the index of concentration at the extremes (🧊).

  • Using a hierarchical approach to characterize multi-level patterns of residential segregation.

  • Smoothing pt. 2 (if ⏰…)

In small groups: How would you characterize the patterning and causes of U.S. segregation ca. 2022?

  • Take a look at the map of county-level segregation here.

  • Read some of the (short) regional segregation stories.

  • Pick an area that you are interested in/feel knowledgable about.

  • Look at patterns of change over time.

  • How does changing the scale impact what you see/learn from the map?

What are the characteristics of segregation in the U.S.?

  • A function of local and regional dynamics.

  • Reflects the independent and interacting effects of racial discrimination and economic disparities (1).

  • Huge regional differences in the nature, causes, and intensity of residential segregation.

  • Obviously these dynamics stretch the ability of very reductive approaches to show us much that is meaningful.

The ICE 🧊 approach

  • Capture the impact of extremes

  • For wealth: \[\text{ICE}_i = \frac{A_i - P_i}{T_i}\]

  • Where, \(P_i\) = number of poor people in area \(i\), \(A_i\) number of affluent people, and \(T_i\) is total number.

  • Ranges from -1 to 1

  • Can be extended to multiple groupings, i.e. number of wealthy individuals in advantaged race/ethnic group - number of poor individuals in disadvantaged race/ethnic group.

  • What does the ICE let us learn in the Feldman (2) piece that we wouldn’t have otherwise?

Lichter and Parisi are trying to figure out how to deal with multi-scale segregation

An example where segregation is primarily explained by within-unit variation. (from Lichter & Parisi (3))

This requires dealing with both micro- and macro-level segregation at the same time

A pattern of segregation dominated by between-unit variation (3)

Can classic metrics of segregation capture hierarchical variation?

  • Dissimilarity & Isolation only characterize variation within the lowest levels observed.

  • Moran’s I includes information on proximity or adjacency but is also only focused on lowest-level relationships.

  • Lichter and Parisi (3) use the Thiel index - an entropy-based measure - to characterize the overall intensity and scale of residential segregation.

What is entropy?

  • Entropy (\(E\)) is a measure of uncertainty

  • Maximum value \(\to\) Maximum Uncertainty

  • Minimum value Minimum Uncertainty

  • For two groups: \[E = p \frac{1}{p} + (1-p)\frac{1}{1-p}\]

  • Not limited to binary comparisons

p <- seq(from=0.01, to=1, by = 0.01)
E <- p*log(1/p) + (1-p)*log(1/(1-p))
df <- data.frame(p=p,E=E)
g <- ggplot(df, aes(x=p,y=E)) + 
geom_point() +
xlab("Proportion in group") + 
ylab("Entropy") + 
geom_vline(xintercept = 0.5)

Entropy as a function of the proportion of individuals in a group, p

Thiel index (\(H\))

  • A weighted average of differences in entropy between different levels.

  • If \(H=0\), the entropy within all lower-level units is equivalent to the population-level entropy.

  • In other words, all variation is within-unit.

  • If \(H=1\), the total amount of population-level entropy is explained by between-unit variation.

\[ H_{B \in M} = \frac{1}{N_M E_M} \sum_{b=1}^{B}N_B(E_M-E_b) \]


  • \(N_M\) is the total population
  • \(E_M\) is the entropy for the whole population
  • \(N_b\) is the population of small area \(b\)
  • \(E_b\) is the entropy of small area \(b\)

What does it mean for \(H\) to be decomposable?

  • \(H\) is a sum over neighborhood-level differences from the population entropy.

  • To see what the contribution of a given level is to the overall H, we just look at the sum over all smallest-level units in the one we care about.

  • This lets us ask what proportion of the total \(H\) in an area can be attributed to variation within and between different types of units within a metro area.

Let’s compare the Theil approach to the others


So, what can we learn from this?

Different groups \(\to\) different patterns

Any concerns about the comparison being made?

Do the hierarchical and ICE approaches help us with the tyrrany of neighborhoods?

Can theses approaches help us pick more relevant scales of analysis? (Figure from (4))

Next Time: Mobility!

Relating mobility to COVID-19 spread in the early days of the pandemic (5)


Bruch EE, Mare RD. Neighborhood Choice and Neighborhood Change. American Journal of Sociology [electronic article]. 2006;112(3):667–709. ( (Accessed April 11, 2022)
Feldman JM, Waterman PD, Coull BA, et al. Spatial social polarisation: Using the Index of Concentration at the Extremes jointly for income and race/ethnicity to analyse risk of hypertension. J Epidemiol Community Health [electronic article]. 2015;69(12):1199–1207. ( (Accessed March 31, 2022)
Lichter DT, Parisi D, Taquino MC. Toward a New Macro-Segregation? Decomposing Segregation within and between Metropolitan Cities and Suburbs. American Sociological Review [electronic article]. 2015;80(4):843–873. ( (Accessed March 13, 2020)
Petrović A, Manley D, van Ham M. Freedom from the tyranny of neighbourhood: Rethinking sociospatial context effects. Progress in Human Geography [electronic article]. 2020;44(6):1103–1123. ( (Accessed February 14, 2021)
Glanz J, Carey B, Holder J, et al. Where America Didn’t Stay Home Even as the Virus Spread. The New York Times [electronic article]. 2020;( (Accessed April 11, 2022)