31.1 Introduction: Meals on-campus
In Sect. 25.1 a study (Mann and Blotnicky 2017) was introduced to examine the eating habits of university students. Researchers cross-classified? \(n=183\) students into groups according to two qualitative variables:
- Where they live: With their parents, or not with their parents;
- Whether they eat most of their meals off-campus, or most of their meals on-campus.
|Lives with parents||Doesn’t live with parents||Total|
Since both variables observed on each student (the unit of analysis) are qualitative, means are not appropriate. However, the data can be compiled into a two-way table of counts (Table 31.1).
Since both qualitative variables have two levels, the table is a \(2\times 2\) table. A graphical summary is shown in Fig. 25.1, and a numerical summary in Table 31.2. (The details of the computations appear in Sect. 25.1).
|Odds of having most meals off-campus||Percentage having most meals off-campus||Sample size|
|Living with parents||26||96.3||54|
|Not living with parents||4.375||81.4||129|
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.
Understanding how software computes the odds ratio is important for understanding the output. In jamovi and SPSS, the odds ratio can be interpreted in either of these two ways:
The odds are the odds of eating most meals off-campus (Row 1 of Table 31.1). Then, the odds ratio compares these odds for students living with their parents (Column 1 of Table 31.1) to those not living with their parents (Row 2 of Table 31.1). That is, the odds are \(52/2= 26\) (for those living with parents) and \(105/24 = 4.375\) (for those not living with parents), so the OR is then \(26/4.375 = 5.943\), as in the output (jamovi: Fig. 31.1; SPSS: Fig. 31.2).
The odds are the odds of living with parents (Column 1 of Table 31.1). Then, the odds ratio compares these odds for students eating most meals off-campus (Row 1 of Table 31.1) to the odds of students eating most meals on-campus (Row 2 of Table 31.1). That is, the odds of living with parents are \(52/105= 0.49524\) (for those eating most meals off-campus) and \(2/24 = 0.083333\) (for those eating most meals on-campus), so the OR is then \(0.49524/0.083333 = 5.943\), as in the output (jamovi: Fig. 31.1; SPSS: Fig. 31.2).
Unlike the previous decision-making RQs, this RQ does not concerns means. Instead, the RQ can be written in terms of comparing proportions, odds, or odds ratios.
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 testing the OR. Using the OR, the RQ could be written as
Is the population odds ratio of eating most meals off-campus, comparing students who live with their parents to students not living with their parents, equal to one?
Alternatively, but probably easier to understand, is to write the RQ in terms of comparing the odds in the two groups explicitly:
Are the population odds of students eating most meals off-campus the same for students living with their parents and for students not living with their parents?
The RQ can also be worded as comparing the percentages (or proportions) of students eating meals off-campus in each group. This is equivalent to the RQs above, but is not directly related to the software output, which works with odds ratios.
Another alternative, which sounds less direct but is useful for two-way tables larger than \(2\times 2\) (see Sect. 31.10), is worded in terms of relationships or associations between the variables:
Is there a relationship (or association) between where students eat most of their meals and whether or not the student lives with their parents?
All of these are equivalent. Usually, for \(2\times2\) tables, working with odds or odds ratios is best, because most software (including jamovi and SPSS) readily produces CIs for the odds ratio.