Decomposition of measure from symmetry for analyzing collapsed ordinal square contingency tables

Figures & data

Table 1 Frequency of spending evenings with friends and neighbors for low-income respondents in 2012 and 2016 from Davern et al. (2021).

	(a) In 2012
	with friends
with neighbors	(1)	(2)	(3)	(4)	(5)	(6)	(7)	Total
(1)	9	6	7	0	3	1	1	27
(2)	1	10	4	9	2	1	6	33
(3)	2	1	0	3	1	0	3	10
(4)	0	3	5	2	6	3	2	21
(5)	0	0	2	1	4	0	1	8
(6)	0	0	1	1	1	0	2	5
(7)	1	8	8	7	4	5	14	47
Total	13	28	27	23	21	10	29	151
	(b) In 2016
	with friends
with neighbors	(1)	(2)	(3)	(4)	(5)	(6)	(7)	Total
(1)	1	7	8	2	0	2	3	23
(2)	2	11	12	13	2	0	1	41
(3)	0	4	5	7	0	0	0	16
(4)	4	2	2	7	2	2	4	23
(5)	0	0	0	1	2	0	2	5
(6)	0	1	3	6	0	4	2	16
(7)	3	10	7	7	8	4	26	65
Total	10	35	37	43	14	12	38	189

Note: Status is (1) almost every day, (2) once or twice a week, (3) several times a month, (4) about once a month, (5) several times a year, (6) about once a year, and (7) never.

Table 2 Frequency of everyday discrimination of low-income respondents in 2021 from Davern et al. (2021).

	Treated as not smart
harassed	(1)	(2)	(3)	(4)	(5)	(6)	Total
(1)	0	1	0	1	2	1	5
(2)	3	0	1	0	1	0	5
(3)	2	1	6	4	1	0	14
(4)	1	0	1	4	0	3	9
(5)	3	3	5	2	6	3	22
(6)	6	9	10	12	15	50	102
Total	15	14	23	23	25	57	157

Note: Status is (1) almost every day, (2) at least once a week, (3) a few times a month, (4) a few times a year, (5) less than once a year, and (6) never.

Table 3 Estimates of measures ${\Psi }_{\text{CS}},{\Psi }_{\text{CoGS}}$, and ${\Psi }_{\mathrm{S}}$, approximate standard errors, and 95% confidence intervals applied to the data in Tables 1 and 2.

Application	Measure	Estimated measure	Standard error	Confidence interval
Table 1a	${\Psi }_{\text{CS}}$	0.100	0.036	$(0.030,0.171)$
	${\Psi }_{\text{CoGS}}$	0.020	0.009	$(0.002,0.037)$
	${\Psi }_{\mathrm{S}}$	0.120	0.042	$(0.038,0.202)$
Table 1b	${\Psi }_{\text{CS}}$	0.133	0.037	$(0.060,0.207)$
	${\Psi }_{\text{CoGS}}$	0.047	0.020	$(0.007,0.087)$
	${\Psi }_{\mathrm{S}}$	0.180	0.052	$(0.078,0.283)$
Table 2	${\Psi }_{\text{CS}}$	0.037	0.020	$(-0.003,0.077)$
	${\Psi }_{\text{CoGS}}$	0.327	0.091	$(0.149,0.506)$
	${\Psi }_{\mathrm{S}}$	0.364	0.088	$(0.192,0.536)$

Fig. F1 The sampling distribution obtained from the structure of probabilities in Table F1a.

Fig. F2 The sampling distribution obtained from the structure of probabilities in Table F1b.

Fig. F3 The sampling distribution obtained from the structure of probabilities in Table F1c.

Fig. F4 The sampling distribution obtained from the structure of probabilities in Table F1d.

Table F1 Patterns of probability structure in a square contingency table.

(a)	1	2	3	4	5	6
1	0.0278	0.0278	0.0278	0.0278	0.0278	0.0278
2	0.0278	0.0278	0.0278	0.0278	0.0278	0.0278
3	0.0278	0.0278	0.0278	0.0278	0.0278	0.0278
4	0.0278	0.0278	0.0278	0.0278	0.0278	0.0278
5	0.0278	0.0278	0.0278	0.0278	0.0278	0.0278
6	0.0278	0.0278	0.0278	0.0278	0.0278	0.0278
(b)	1	2	3	4	5	6
1	0.0244	0.0244	0.0244	0.0244	0.0488	0.0976
2	0.0244	0.0244	0.0244	0.0244	0.0244	0.0488
3	0.0244	0.0244	0.0244	0.0244	0.0244	0.0244
4	0.0244	0.0244	0.0244	0.0244	0.0244	0.0244
5	0.0244	0.0244	0.0244	0.0244	0.0244	0.0244
6	0.0244	0.0244	0.0244	0.0244	0.0244	0.0244
(c)	1	2	3	4	5	6
1	0.0192	0.0192	0.0192	0.0192	0.0769	0.1154
2	0.0385	0.0192	0.0192	0.0192	0.0192	0.0769
3	0.0192	0.0385	0.0192	0.0192	0.0192	0.0192
4	0.0192	0.0192	0.0385	0.0192	0.0192	0.0192
5	0.0192	0.0192	0.0192	0.0385	0.0192	0.0192
6	0.0192	0.0192	0.0192	0.0192	0.0385	0.0192
(d)	1	2	3	4	5	6
1	0.0179	0.0179	0.0179	0.0179	0.0714	0.1786
2	0.0357	0.0179	0.0179	0.0179	0.0179	0.0714
3	0.0179	0.0357	0.0179	0.0179	0.0179	0.0179
4	0.0179	0.0179	0.0357	0.0179	0.0179	0.0179
5	0.0179	0.0179	0.0179	0.0357	0.0179	0.0179
6	0.0179	0.0179	0.0179	0.0179	0.0357	0.0179

Davern M., Bautista R., Freese J., Morgan S. L., and Smith T. W., Bautista R., Freese J., Morgan S. L., and Smith T. W., 2021. General Social Surveys, 1972-2021 Cross-section [machine-readable data file, 68,846 cases]. Principal Investigator, Davern, M.; Co-Principal Investigators, Bautista, R., Freese, J., Morgan, S.L., and Smith, T.W.; Sponsored by National Science Foundation.-NORC ed.- Chicago: NORC at the University of Chicago [producer and distributor]. Data accessed from the GSS Data Explorer website at gssdataexplorer.norc.org.

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