This is an example for investigating the moderate effect of gender on the association between exposure and outcome disease.
The p1 to p7 are the population parameters, and the odds ratios in women (ORw) and in men (ORm) are calculated based on them.
If a researcher want to do a case control study, the expectation of simple combined OR [OR(combined), ORc] is calculated based on Equation 2.1-3, and it is controlled by proportion of males in the case group (k1) and in the control group (k2).
The ORλ is calculated based on the Equation 2.1-5 using k1 to replace m.
Under two our setting assumptions: (1) rare disease, and (2) independent, the log(ORc) and log(ORλ) will be very close.
Introduction of assumptions:
(1) rare disease
p1, p2, p3, p4 are very rare. For example, p1, p2, p3 and p4 are 0.0001, 0.0001, 0.0002 and 0.0006 (Default).
(2) independent
There is no association between the factor of interest and the major independent variable. For example, gender and exposure are independent, and p6 is equal to p7 (Default).

The x-axis is the proportion of males in the case group (k1).
The y-axis is the log of odds ratios (including log(ORc) and log(ORλ)).
1. Under above two assumptions, you can find the impact of k2 on ORc is very small.
2. Under above two assumptions, you can control the k1 and understand the difference of trajectories is very small.
3. You can change the p1 to p7 and understand the impact of them on the difference between log(ORc) and log(ORλ).
Note: the best range of odds ratios in women (ORw) and in men (ORm) are between 2 to 6.