More reliable inference for the dissimilarity index of segregation

Abstract

SummaryThe most widely used measure of segregation is the so‐called dissimilarity index. It is now well understood that this measure also reflects randomness in the allocation of individuals to units (i.e. it measures deviations from evenness, not deviations from randomness). This leads to potentially large values of the segregation index when unit sizes and/or minority proportions are small, even if there is no underlying systematic segregation. Our response to this is to produce adjustments to the index, based on an underlying statistical model. We specify the assignment problem in a very general way, with differences in conditional assignment probabilities underlying the resulting segregation. From this, we derive a likelihood ratio test for the presence of any systematic segregation, and bias adjustments to the dissimilarity index. We further develop the asymptotic distribution theory for testing hypotheses concerning the magnitude of the segregation index and show that the use of bootstrap methods can improve the size and power properties of test procedures considerably. We illustrate these methods by comparing dissimilarity indices across school districts in England to measure social segregation.

Measuring Group Differences in High‐Dimensional Choices: Method and Application to Congressional Speech

“…A prominent example is the measurement of residential segregation (e.g., Reardon and Firebaugh (2002)), where the groups might be defined by race or ethnicity and the choices might be neighborhoods or schools. The finite-sample bias that we highlight has been noted in that context by Cortese, Falk, and Cohen (1976) and addressed by benchmarking against random allocation (Carrington and Troske (1997)), applying asymptotic or bootstrap bias corrections (Allen, Burgess, Davidson, and Windmeijer (2015)), and estimating mixture models (Rathelot (2012), D'Haultfoeuille and Rathelot (2017)). 4 Recent work has derived axiomatic foundations for segregation measures (Echenique and Fryer (2007), Frankel and Volij (2011)), asking which measures of segregation satisfy certain properties.…”

Measuring Group Differences in High-Dimensional Choices: Method and Application to Congressional Speech

“…A prominent example is the measurement of residential segregation (e.g., Reardon and Firebaugh 2002), where the groups might be defined by race or ethnicity and the choices might be neighborhoods or schools. The finitesample bias that we highlight has been noted in that context by Cortese et al (1976) and addressed by benchmarking against random allocation (Carrington and Troske 1997), applying asymptotic or bootstrap bias corrections (Allen et al 2015), and estimating mixture models (Rathelot 2012;D'Haultfoeuille and Rathelot 2017). 4 Recent work has derived axiomatic foundations for segregation measures (Echenique and Fryer 2007;Frankel and Volij 2011), asking which measures of segregation satisfy certain properties.…”

Student Segregation Across and Within Schools - The Case of the Portuguese Public School System

“…More recently, Allen et al (2015) offer several correction procedures to the dissimilarity index in order to make it a measure of systematic segregation. The corrections aim to, precisely, partial-out the "bias" of strictly positive levels of unevenness that may be coherent with random allocation.…”

“…For each of these levels we look at three segregation dimensions, each stemming from specific students' dichotomous characteristics: economic (low-income vs non-low-income), academic (at least one retention vs no retentions), and immigrant (born abroad or with at least one parent born abroad vs all three not born abroad). Segregation is measured with the density-corrected dissimilarity index (Ddc) recently proposed by Allen, Burgess, Davidson, & Windmeijer (2015) to capture systematic segregation more robustly than the classic dissimilarity index (D).…”