Question

Based on murder rates in Kenya, a survey has reported that the probability a new born child of eventually being a murder victim is 0.0263 for urban males, 0.0049 for rural males, 0.0072 for rural females and 0.0023 for white Urban's females.

1) Find the conditional odds ratios between region and whether a murder victim ,given gender. Interpret

2) If half the newborns are of each gender, for each region, find the marginal odds ratio between race and a whether murder victim.

Answer 1

1) Consider 2×2×2 table

$θ_{male} = \frac{0.0263 \times 0.9951}{0.0049 \times 0.9737} = 5.485 \\ θ_{female} = \frac{0.0023 \times 0.9928}{0.0072 \times 0.9977} = 0.317 \\ \frac{θ_{male}}{θ_{female}} = \frac{5.485}{0.317}=17.309$

$θ_{male}$ and $θ_{female}$ are different. These variables are not homogeneous association.

2) The probability of urban victim is

$= 0.0263 \times 0.5 + 0.0023 \times 0.5 \\ = 0.01315 + 0.00115 \\ = 0.0143$

The probability of rural victim is

$= 0.0049 \times 0.5 + 0.0072 \times 0.5 \\ = 0.00245 + 0.0036 \\ = 0.00605$

The marginal odds ratio between race and a whether murder victim is

$= \frac{0.0143 (1-0.00605)}{0.00605 (1-0.0143)} \\ = \frac{0.0142}{0.0059} \\ = 2.406$