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A decomposition method for measuring ethnic bias in police stopsThe previous section drew a distinction between the observed stop rates (OSR) and the PSR associated with the different ethnic groups. The OSR was denoted d_{k} and the PSR was denoted cr_{k} for the sixteen groups, k = 1, , 16. Since the PSR were computed using the method of “recycled predictions”, discussed earlier, the point was made that differences between the different groups in their PSR could be ascribed entirely to ethnic differences. The OSR and the PSR of the different social groups are compared in Figure 5.1. Figure 5.1 Observed and Predicted Stop Rates by Selected Ethnicity Source: Own calculations from www.ethnicityfactsfigures.service.gov.uk/crimejusticeand thelaw/policing/stopandsearch/latest. Note: Stop rates are defined as number of stops per 1,000 of the ethnic population. Following from these observations, the difference in OSR between two groups  say, Black Caribbeans and British Whites (respectively, In the following discussion, it is assumed that a_{BC} > a_{BW}, that is, the term Z > 0. The term A in equation (5.6) represents the difference in PSR between Black Caribbeans and British Whites, that is, the difference in stop rates which is solely due to ethnic difference. Consequently, this difference represents ethnic bias by the police in deciding who to stop. The term В in equation (5.6) represents the difference for Black Caribbeans in their OSR and PSR. The OSR is based on Black Caribbean coefficients applied to Black Caribbean attributes which are the spatial distribution of Black Caribbeans, while the PSR is based on Black Caribbean coefficients applied to the collective attributes of all the ethnicities. The term В can, therefore, be interpreted as a measure of the “inferiority” of the spatial distribution of Black Caribbeans over the collective distribution of all the ethnicities. Similarly, the term C can be interpreted as a measure of the “inferiority” of the attributes of British Whites over the collective distribution of all the ethnicities.^{16} The difference between the two terms В and C in equation (5.6) is a measure of the relative inferiority of the spatial distribution of Black Caribbeans visavis British Whites. From the preceding discussion, the term 6 = А/ Z represents the proportion of the overall difference in stop rates between an ethnicity and British Whites that is due to ethnic bias. The term Л = (В  C) / Z represents the proportion of the overall difference between that ethnicity and British Whites which is the result of the relative inferiority (in spatial distribution) of that ethnicity visavis British Whites. In this context, there are four possibilities:
riority of the spatial distribution of that group visavis British Whites (A < 0) because the effect of ethnic bias exceeds the effect of spatial superiority («5 > IAI). In this situation, A > Z so that S > 1. 3 Z> 0 when A < 0 and (В  C) > 0. In this situation, Z > 0 despite bias in favour of an ethnicity because its relative inferiority in spatial distribution (A > 0) offsets this bias. In this situation, IAI < IZI so that 1 <6 <0. 4 Z < 0 when A < 0 and (В  C) > 0. In this situation, Z < 0 because bias in favour of an ethnicity is not offset by its relative inferiority in spatial distribution (A > 0). In this situation, IAI > IZI so that S < 1. The terms В and C in equation (5.6) could be positive or negative. If, say, C < 0, then a_{BW} < a_{BW} and the OSR of British Whites is smaller than the stop rate which would result if British Whites were assigned the general spatial distribution. This implies that British Whites have a spatial distribution which is superior to the general distribution. On the other hand, if C > 0, then If В > 0, then Table 5.5 shows the empirical results corresponding to equation (5.6). The gaps in the OSR between Black Africans, Black Caribbeans, Indians,
Source: Own calculations from www.ethnicityfactsfigures.service.gov.uk/crimejusticeand thelaw/policing/stopandsearch/latest. Note: Results are defined visavis British Whites. Bangladeshis, Pakistanis, and Chinese and the comparator group, British Whites, were, respectively, 31, 75, 1, 26, 12, and 5, while the gaps in the PSR between the same groups and British Whites were, respectively, 10, 50, 5, 7, 9, and 7. Since the gaps in the PSR are interpreted as due to ethnic bias, the results in Table 5.5 suggest the following: ^{1}

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