Contingency Tables

Table 1: Data for applicant entrance for 6 departments

Department	Male.Yes	Male.No	Female.Yes	Female.No
1	512	313	89	19
2	353	207	17	8
3	120	205	202	391
4	138	279	131	244
5	53	138	94	299
6	22	351	24	317

dta$OR = with(dta, (Male.Yes * Female.No) / (Male.No * Female.Yes))
dta$se = with(dta, (sqrt(1/Male.Yes + 1/Female.No + 1/Male.No + 1/Female.Yes)))
dta$Conf.lwr = with(dta, OR - (1.96 * se))
dta$Conf.Upr = with(dta, OR + (1.96 * se))

kable(dta, split.tables = Inf)

Department	Male.Yes	Male.No	Female.Yes	Female.No	OR	se	Conf.lwr	Conf.Upr
1	512	313	89	19	0.3492120	0.2627081	-0.1656958	0.8641199
2	353	207	17	8	0.8025007	0.4375926	-0.0551808	1.6601823
3	120	205	202	391	1.1330596	0.1439424	0.8509325	1.4151868
4	138	279	131	244	0.9212838	0.1502084	0.6268753	1.2156922
5	53	138	94	299	1.2216312	0.2002426	0.8291558	1.6141066
6	22	351	24	317	0.8278727	0.3051635	0.2297522	1.4259933

marginal = colSums(dta[, 2:5])
OR = (marginal[1] * marginal[4]) / (marginal[2] * marginal[3])
se = sqrt(1/marginal[1] + 1/marginal[4] + 1/marginal[2] + 1/marginal[3])

## Confidence Interval for the Marginal OR
OR + c(-1, 1) * 1.96 * se

[1] 1.71585 1.96631

library(DescTools)
library(lawstat)

dta = xtabs(freq ~ .,
            cbind(expand.grid(Gender = c("Male", "Female"),
                              Entrace = c("Yes", "No"),
                              Department = c("1", "2", "3", "4", "5", "6")),
                  freq =  c(512, 89, 313, 19, 353, 17, 207, 8, 120, 202, 205, 391,
                            138, 131, 279, 244, 53, 94, 138, 299, 22, 24, 351, 317)
                  )
            )

## Ho: OR = 1, Ha: OR > 1
BreslowDayTest(dta, OR = 1)

    Breslow-Day test on Homogeneity of Odds Ratios

data:  dta
X-squared = 19.938, df = 5, p-value = 0.001283

## Ho: OR_1 = OR_2 = OR_3 = OR_4 = OR_5 = OR_6, Ha: At least one set of OR are not equal
cmh.test(dta)

    Cochran-Mantel-Haenszel Chi-square Test

data:  dta
CMH statistic = 1.52460, df = 1.00000, p-value = 0.21692, MH
Estimate = 0.90470, Pooled Odd Ratio = 1.84110, Odd Ratio of level
1 = 0.34921, Odd Ratio of level 2 = 0.80250, Odd Ratio of level 3
= 1.13310, Odd Ratio of level 4 = 0.92128, Odd Ratio of level 5 =
1.22160, Odd Ratio of level 6 = 0.82787

Based on the Breslow Day test we reject the null hypothesis that the odds ratios are equal to 1. The CMH test fails to reject that gender and entrance are independent.