A decomposition method for measuring ethnic bias in police stops

The 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.ethnicity-facts-figures.service.gov.uk/crime-justice-and- the-law/policing/stop-and-search/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, bc and cr_BW) - may be written as follows:

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 vis-a-vis 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 vis-a-vis British Whites. In this context, there are four possibilities:

• 1 A > 0 and (В - C) > 0. In this situation, Z > 0 partly because of bias against an ethnic group (5 > 0) and partly because of the relative inferiority in the spatial distribution of that group vis-a-vis British Whites (A > 0). In this situation, A < Z so that 0 < S < 1.
• 2 A > 0 and (В - C) < 0. In this situation, Z > 0 despite the relative supe

riority of the spatial distribution of that group vis-a-vis 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 bw > a_BW and the OSR of British Whites is larger 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 inferior to the general distribution.

If В > 0, thengc > gcand the OSR of Black Caribbeans is larger than the stop rate which would result if Black Caribbeans were assigned the general level of characteristics. This implies that Black Caribbeans have a spatial distribution which is inferior to the general population. On the other hand, if В < 0, then a_BC < a_BC and the OSR of Black Caribbeans is smaller than the stop rate which would result if Black Caribbeans were assigned the general level of characteristics. This implies that Black Caribbeans have a spatial distribution which is superior to the general population.

Table 5.5 shows the empirical results corresponding to equation (5.6). The gaps in the OSR between Black Africans, Black Caribbeans, Indians,

	Stop Rates per 1,000 Ethnic Population Z = A +(B-C)
Black African	31	10	20	-1
Black Caribbean	75	50	24	-1
Indian	1	-5	5	-1
Bangladeshi	26	7	18	-1
Pakistani	12	9	2	-1
Chinese	-5	-7	1	-1

Source: Own calculations from www.ethnicity-facts-figures.service.gov.uk/crime-justice-and- the-law/policing/stop-and-search/latest.

Note: Results are defined vis-a-vis 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

• 1 For Black Africans, 32% of the difference in the OSR between them and British Whites (10/31) was due to ethnic bias against Black Africans with the remaining 68% of the difference (21/31) due to the unfavourable spatial distribution of Black Africans relative to British Whites. So, in terms of equation (5.6), Z > 0, A > О, В > 0, and C > 0 for Black Africans.
• 2 For Black Caribbeans, 67% of the difference in the OSR between them and British Whites (50/75) was due to ethnic bias against Black Caribbeans with the remaining 33% of the difference (25/75) due to the unfavourable spatial distribution of Black Caribbeans relative to British Whites. So, in terms of equation (5.6), Z > 0, A > 0, В > 0, and C > 0 for Black Caribbeans.
• 3 For Indians, ethnic bias worked in their favour but spatial distribution worked against them: Z > 0, A < 0, and (В - C) > 0. Consequently, with ethnic bias but no spatial handicap, the difference in OSR between Indians and British Whites would have been -5; without ethnic bias but with spatial handicap, the difference in OSR between Indians and British Whites would have been 6. The overall difference was 1.
• 4 For Bangladeshis, 27% of the difference in the OSR between them and British Whites (7/26) was due to ethnic bias against Bangladeshis, with the remaining 73% of the difference (19/26) due to the unfavourable spatial distribution of Bangladeshis relative to British Whites. So, in terms of equation (5.6), Z > 0, A > 0, В > 0, and C > 0 for Bangladeshis.

• 5 For Pakistanis, 75% of the difference in the OSR between them and British Whites (9/12) was due to ethnic bias against Pakistanis, with the remaining 25% of the difference (3/12) due to the unfavourable spatial distribution of Pakistanis relative to British Whites. So, in terms of equation (5.6), Z > 0, A > 0, В > 0, and C > 0 for Pakistanis.
• 6 For Chinese, as for Indians, ethnic bias worked in their favour, but spatial distribution worked against them but, unlike Indians, not by as much to offset the positive bias: Z < 0, A < 0, and (В - C) > 0. Consequently, with ethnic bias but no spatial handicap, the difference in OSR between Chinese and British Whites would have been -7; without ethnic bias but with spatial handicap, the difference in OSR between Chinese and British Whites would have been 2. The overall difference was -5.