25.1 Introduction: Odds ratios
A study (Mann and Blotnicky 2017) examined the eating habits of university students. One issue studied was the relationship between eating on-campus, and where the student lived.
The researchers cross-classified the \(n=183\) students into groups: each student (the unit of analysis) was observed on two qualitative variables:
- Where they lived: With their parents, or not with their parents;
- Whether they ate most meals off-campus, or most meals on-campus.
Since both variables are qualitative, means are not appropriate for summarising the data. However, the students can be classified into a two-way table of counts (Table 25.1), called a contingency table. Both qualitative variables have two levels, so the table is a \(2\times 2\) table.
| Lives with parents | Doesn’t live with parents | Total | |
|---|---|---|---|
| Most meals off-campus | 52 | 105 | 157 |
| Most meals on-campus | 2 | 24 | 26 |
| Total | 54 | 129 | 183 |
The purpose of the research is to study the odds (or proportion) of students who eat most meals off-campus, comparing those who live with their parents and those who do not live with their parents?
The parameter of interest could be the difference between the proportions (or percentages) in each group, a comparison between the odds in each group, or the odds ratio.
For reasons that we can’t delve into, usually the odds ratio (OR) is used as the parameter. One important reason is that software produces output related to the sample OR.
To compare two groups with regard to another qualitative variable, software usually works with odds rather than percentages or proportions.
For this reason, writing the RQ in terms of odds is also most appropriate.Using the OR, the RQ could be written as
Among university students, what is the odds ratio of students eating most meals off-campus, comparing those who do and do not live with their parents?
The parameter is the population OR, comparing the odds of eating most meals off-campus for students living with their parents to students not living with their parents.
Take care in defining the odds ratios in the parameter!
Recall (Sect. 14.2 that software usually compares Row 1 to Row 2, and Columnn 1 to Column 2