## 25.2 Numerical and graphical summaries: Comparing odds

With two qualitative variables, an appropriate numerical summary includes the odds and percentages from each comparison group for the outcome of interest, and the sample sizes.

From these data,
the odds of eating most meals *off-campus* is:

- \(52\div 2 = 26\) for students
*living with their parents*. - \(105\div24 = 4.375\) for students
*not living with their parents*.

So the *odds ratio* (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 numerical summary (Table 25.2) shows the percentage and odds of eating most meals off-campus, comparing students living at home and those not living at home.

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 25.1).

Then, the odds ratio compares these odds for students living with their parents (Column 1 of Table 25.1) to those*not*living with their parents (Column 2 of Table 25.1).

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 (Fig. 25.2).The

*odds*are the odds of living with parents (Column 1 of Table 25.1).

Then, the odds ratio compares these odds for students eating most meals off-campus (Row 1 of Table 25.1) to the odds of students eating most meals on-campus (Row 2 of Table 25.1).

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 (Fig. 25.2).

**first row**or

**first column**.

Odds of having most meals off-campus | Percentage having most meals off-campus | Sample size | |
---|---|---|---|

Living with parents | 0.4952 | 16.6 | 54 |

Not living with parents | 0.0833 | 3.8 | 129 |

Odds ratio | 5.943 |

An appropriate graph
(Fig. 25.1)
is a *side-by-side* bar chart
or a *stacked* bar chart.
For comparing the *odds*,
the side-by-side bar chart is better.
(A *stacked* bar chart is
better for comparing *proportions*, but either is fine.)