We develop and apply a multilevel modeling approach that is simultaneously capable of assessing multigroup and multiscale segregation in the presence of substantial stochastic variation that accompanies ethnicity rates based on small absolute counts. Bayesian MCMC estimation of a log-normal Poisson model allows the calculation of the variance estimates of the degree of segregation in a single overall model, and credible intervals are obtained to provide a measure of uncertainty around those estimates. The procedure partitions the variance at different levels and implicitly models the dependency (or autocorrelation) at each spatial scale below the topmost one. Substantively, we apply the model to 2011 census data for London, one of the world’s most ethnically diverse cities. We find that the degree of segregation depends both on scale and group.

Most world cities can now be characterized as multiethnic and multicultural in their population composition, and the residential patterning of their major component ethnic groups remains a topic of substantial research interest. Many studies of the degree of residential segregation of ethnic groups recognize that this is multiscalar in its composition, but few have incorporated this major feature into their analyses: Those that do mostly conclude that segregation is greater at the microscale than at the macroscale. This article uses a recently developed alternative procedure for assessing the degree of segregation that differs from all others in that it analyzes the geography of all groups simultaneously, providing a single, synoptic view of their relative segregation; can incorporate data for more than one date and therefore evaluate the statistical significance of the extent of any change over time; operates at several geographical scales, allowing appreciation of the extent of clustering and congregation for the various ethnic groups at different levels of spatial resolution; and-most important-is based on a firm statistical foundation that allows for robust assessments of differences in the levels of segregation for different groups between each other at different scales over time. This modeling procedure is illustrated by a three-scale analysis of ethnic residential segregation in Auckland, New Zealand, as depicted by the country's 2001, 2006, and 2013 censuses.

Most studies of ethnic residential segregation that address the issue of spatial scale make it implicitif not explicitthat segregation is greater at smaller than larger scales. Such studies, however, invariably measure segregation separately at those scales, and take no account of the fact that measures at the smaller scale necessarily incorporate that at any larger scales. The present paper rectifies that situation by, for the first time, modelling ethnic segregation in London at the 2001 and 2011 censuses within a Bayesian statistical framework at three scales, which allows for the statistical significance of any changes to be formally assessedsomething not possible before. It finds that for many of the groups studied segregation was as great, if not greater, at the macro-scale as at the micro-scale, with both measures larger than at the meso-scale, with significant reductions in segregation across the decade, especially at the micro-scale.

A model-based approach to measuring residential segregation is further developed by explicitly including spatial effects at multiple scales. This model distinguishes segregation as unevenness and as spatial clustering in the presence of stochastic variation. An accompanying badness-of fit measure allows the identification of the scale and zonation where the spatial patterns come into focus thereby potentially transcending the modifiable areal unit problem. The model is applied to Indian ethnicity in Leicester UK finding segregation as unevenness and as spatial clustering at multiple scales.

This paper introduces the Multilevel Index of Dissimilarity package, which provides tools and functions to fit a Multilevel Index of Dissimilarity in the open source software, R. It extends the conventional Index of Dissimilarity to measure both the amount and geographic scale of segregation, thereby capturing the two principal dimensions of segregation, unevenness and clustering. The statistical basis for the multilevel approach is discussed, making connections to other work in the field and looking especially at the relationships between the Index of Dissimilarity, variance as a measure of segregation, and the partitioning of the variance to identify scale effects. A brief tutorial for the package is provided followed by a case study of the scales of residential segregation for various ethnic groups in England and Wales. Comparing 2001 with 2011 Census data, we find that patterns of segregation are emerging at less localised geographical scales but the Index of Dissimilarity is falling. This is consistent with a process whereby minority groups have spread out into more ethnically mixed neighbourhoods.