We revisit in this chapter a common issue with popular indices used for measuring residential sorting, i.e. the extent to which a subgroup of the population is spatially distributed (sorted or segregated) differently from the remainder of the population. Specifically, we show that three common measures of residential sorting (viz. theIndex of Segregation, theIndex of Isolation and theEntropy Index of Segregation) are affected by group size, i.e. the expected values of the indices are positive rather than zero under random sorting, and the size of this positive bias is related to group size. This is an important issue because it is common to compare sorting indices across groups of rather different sizes, both cross-sectionally and over time. Using New Zealand data, we demonstrate group-size impact on bias in measures of residential sorting by means of scatter plots and regression in four different ways: (1) investigating the relationship between group size and each residential sorting measure calculated with actual data; (2) randomly allocating individuals across area units, calculating the resulting residential sorting measures, and again investigating the relationship between index values and group size; (3) showing that normalised/systematic indices of sorting are also related to group size; and (4) calculating the bias for each sorting index and investigating the relationship with group size. Our empirical illustration uses microdata on the self-reported ethnicity of individuals (with multiple responses possible) from five New Zealand Censuses of Population and Dwellings (1991–2013) for the Auckland region, selected due to its high ethnic diversity. Our results demonstrate that theEntropy Index of Systematic Segregation measure of residential sorting is the measure that is least affected by group size variation. As a result, we strongly recommend using this index of sorting as a preferred measure.

1. 1.

   We use ‘residential sorting’ as a term that encompasses a range of measures of residential segregation that include dissimilarity, isolation, and concentration (e.g. Massey and Denton1988). Our preferred term is not only broader, but carries none of the negative connotations associated with use of the word ‘segregation’.

2. 2.

   A randomised allocation is obtained when the number of persons of the group allocated to an area is given by a draw from a binomial distribution B (n,p) withn equal to the area’s population andp the fraction of the group in the total population.

3. 3.

   The most recent population census was held on March 6, 2018. At the time of collecting the data for this chapter, the results of that census were not yet available. In any case, due to non-response issues, 2018 Census data are of somewhat lesser quality than previous censuses with respect to variables such as ethnicity. Additionally, caution is needed in comparing results of the 2018 Census with those of previous censuses. See 2018 Census External Data Quality Panel (2020)Final report of the 2018 Census External Data Quality Panel. Retrieved fromwww.stats.govt.nz.

4. 4.

   The sum of these percentages exceeds 100 percent, as people can report more than one ethnicity.

5. 5.

   A meshblock is the smallest geographic unit for which Statistics New Zealand collects statistical data. Meshblocks vary in size from part of a city block to large areas of rural land. The country is divided into about 50,000 meshblocks that are aggregated to about 2000 area units. Our analysis is based on data aggregated to the area unit level. Area units are non-administrative areas that are in between meshblocks andterritorial authorities in size (Statistics New Zealand2013b). In urban areas, area units are approximately the size of individual suburbs, and in our dataset they have an average population of 1530.

6. 6.

   In this chapter, we use 2013 area unit boundaries.

7. 7.

   Counts that are already a multiple of three are left unchanged. Those not a multiple of three are rounded to one of the two nearest multiples. For example, a one will be rounded to either a zero or a three. Each value in a table is rounded independently.

8. 8.

   The sum of Level 2 total responses in Table 1 is greater than the sum of Level 1 total responses because some individuals reported multiple ethnicities at Level 2 for which some or all belonged to the same ethnic group at level 1.

9. 9.

   The ethnicity question in the 1996 Census had a different format from that used in 1991 and 2001. In 1996, there was an answer box for ‘Other European’ with additional drop down answer boxes for ‘English’, ‘Dutch’, ‘Australian’, ‘Scottish’, ‘Irish’, and ‘other’. These were not used in 1991 or 2001. Furthermore, the first two answer boxes for the question were in a different order in 1996 from 1991 and 2001. ‘NZ Māori’ was listed first and ‘NZ European or Pakeha’ was listed second in 1996. The 1991 and 2001 questions also only used the words ‘New Zealand European’ rather than ‘NZ European or Pākehā’ (Pākehā is the Māori word referring to a person of European descent). The 2001 question used the word ‘Māori’ rather than ‘NZ Māori’. The format of the 2006 and 2013 questionnaire was the same as that of 2001 (Statistics New Zealand2017).

10. 10.

    We also ran the analysis with not further defined as a separate category, as well as dropping them completely. The ranking of groups, the trends over time, and our key conclusions are not affected.

11. 11.

    Fossett (2017) has introduced an alternative way of generating sorting measures that will have an expected value of zero under random sorting.

12. 12.

    Appendix Table7.8 reports the average of index values obtained from the 100 simulations. We have multiplied the index values by 100 for easy interpretability.

13. 13.

    It can be easily shown by calculus that for a given spatial distribution of the group across areas, theIndex of Isolation is non-decreasing in total group size. It should also be noted that theIndex of Segregation is scale free in the total size in the group of interest for a given spatial distribution of this group. No simple mathematical result can be established in the case of theEntropy index of Segregation. This is because, even ifE_a is scale-invariant for a given distribution of groupg across areas,\( \overline{E} \) and\( \frac{P_{.a}}{P..} \) depend on how relatively important the groupg is in the population and in each area unit ‘a’, respectively. This group size effect has been investigated previously by Fossett (2017) in empirical terms with US data.

14. 14.

    We have multiplied the index values by 100 for easy interpretability.

15. 15.

    Because regression coefficients are linearly related to the dependent variable, the coefficients in Table 6 can of course also be obtained by subtracting the coefficients in Table 5 from the corresponding columns in Table 4. However, Table 6 also reports theR² (within, between, and overall) and the correct standard errors of the regressions of bias on group size.

16. 16.

    This is the case because theEntropy Index of Systematic Segregation is defined as (E –E_R)/(1 –E_R) and the expected value ofE_R is constant across different realisations of the actual spatial distribution of the group. Hence theEntropy Index of Systematic Segregation is a simple linear transformation of theEntropy Index of Segregation. Since the latter index satisfies the James and Taeuber (1985) criteria, the former does also.

Access to data used in this study was provided by Statistics New Zealand under conditions of security and confidentiality provisions of the Statistics Act 1975. The first author acknowledges the support provided by the University of Waikato in the form of the generous University of Waikato Doctoral Scholarship. The first author also acknowledges the generous funding received from CaDDANZ (Capturing the Diversity Dividend of Aotearoa/New Zealand). We thank Arthur Grimes, an anonymous reviewer of the paper and participants at the New Zealand Association of Economists (NZAE) conference, held in Auckland in June 2018, for their detailed comments and suggestions.

Disclaimer

The results in this chapter are not official statistics. They have been created for research purposes from census unit record data in the Statistics New Zealand Datalab. The opinions, findings, recommendations, and conclusions expressed in this chapter are those of the authors, not Statistics New Zealand. Access to the anonymised data used in this study was provided by Statistics New Zealand under the security and confidentiality provisions of the Statistics Act 1975. Only people authorised by the Statistics Act 1975 are allowed to see data about a particular person, household, business, or organisation and the results in this chapter have been confidentialised to protect these groups from identification and to keep their data safe. Careful consideration has been given to the privacy, security, and confidentiality issues associated with using unit record census data.

Authors and Affiliations

1. School of Accounting, Finance and Economics, University of Waikato, Hamilton, New Zealand

   Mohana Mondal, Michael P. Cameron & Jacques Poot

2. National Institute of Demographic and Economic Analysis (NIDEA), University of Waikato, Hamilton, New Zealand

   Michael P. Cameron & Jacques Poot

3. Department of Spatial Economics, Vrije Universiteit Amsterdam, Amsterdam, The Netherlands

   Jacques Poot

Corresponding author

Correspondence toJacques Poot.

Editors and Affiliations

1. Hokkai-Gakuen University, Sapporo, Japan

   Soushi Suzuki

2. University of Bologna, Bologna, Italy

   Roberto Patuelli

Appendix

Table7.7

Table7.8

Table7.9

Fig.7.3

Scatter plot ofIndex of Isolation values and group sizes, based on randomised data: Auckland region, 1991–2013

Full size image

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