25.3 Sampling distribution: Comparing odds
From the numerical summary table (Table 25.2), the odds of a student eating most meals off-campus is:
- \(26\) for students living with their parents.
- \(4.375\) for students not living with their parents.
So the OR of eating most meals off-campus, comparing students living with parents to students not living with parents, is \(26 \div 4.375 = 5.943\). The odds are different in each group, and hence the OR is not one. The OR means that the odds of eating most meals off-campus is 5.943 times larger for students living with their parents.
Of course, every sample of students is likey to be different, so the OR varies from sample to sample, so there is sampling variation. This means that the odds ratio has a sampling distribution and a standard error.
Unfortunately, the sampling distribution of the sample OR is not a normal distribution.8 Fortunately, a simple transformation to the sample OR has a normal distribution. For this reason, we will use software output for finding the CI for the odds ratio, and not discuss the sampling distribution directly. In other words, we will rely on software to find CIs for odds ratios.
For those who want to know (this is optional): The OR is only defined for non-negative values so a normal distribution is inappropriate. However, the logarithm of the OR has an approximate normal distribution under certain conditions.↩︎