corn site) and two columns (gender). Note that we are interested in the distribution of the site where the corn is located on the foot within gender, and so the percentages add up to 100 down each column, rather than across the rows.
Table 3.2 Cross‐tabulation of anatomical site of corn by gender for 201 patients with corns on feet
(Source: data from Farndon et al. 2013).
| Gender | ||||
|---|---|---|---|---|
| Anatomical site of index corn on foot | Male | Female | ||
| n | (%) | n | (%) | |
| Apex (end of toe) | 4 | (5) | 9 | (8) |
| Proximal interphalangeal joint (middle part of toe – on the top) | 8 | (10) | 19 | (16) |
| Interdigital (between the toes) | 5 | (6) | 11 | (9) |
| Metatarsal head (ball of the foot – on the bottom) | 54 | (64) | 65 | (56) |
| Plantar calcaneus (heel) | 3 | (4) | 3 | (3) |
| Other part of foot | 10 | (12) | 10 | (9) |
| Total | 84 | (100) | 117 | (100) |
As an example of the importance of considering relative proportions Furness et al. (2003) reported in Auckland, New Zealand over a one‐year period that 25.6% of road accidents were to white cars. As a consequence, a New Zealander may think twice about buying a white car! White cars were the most prevalent colour on the roads with a proportion of 25.9%. So about a quarter of cars on the road are white and this is the same as the proportion of road accidents that were in white cars; thus white cars are not more dangerous than other colours.
Labelling Binary Outcomes
For binary data it is common to call the outcomes ‘an event’ or ‘a non‐event’. So having a car accident in Auckland, New Zealand may be an ‘event’. We often score an ‘event’ as 1 and a ‘non‐event’ as 0. These may also be referred to as a ‘positive’ or ‘negative’ outcome, or ‘success’ and ‘failure’. It is important to realise that these terms are merely labels and the main outcome of interest might be a success in one context and a failure in another. Thus in a study of a potentially lethal disease the outcome might be death, whereas in a disease that can be cured it might be being alive.
Comparing Outcomes for Binary Data
Many studies involve a comparison of two groups. We may wish to combine simple summary measures to give a summary measure that in some way shows how the groups differ. Given two proportions one can either subtract one from the other, or divide one by the other.
Suppose the results of a clinical trial, with a binary categorical outcome (positive or negative), to compare two treatments (a new test treatment versus a control) are summarised in a two by two contingency table as in Table 3.3. The results of this trial can be summarised in a number of way sas in Table 3.3 below.
Table 3.3 Example of two by two contingency table with a binary outcome and two groups of subjects.
| Treatment group | ||
|---|---|---|
| Outcome | Test | Control |
| Positive | a | b |
| Negative | c | d |
| Total | a + c | b + d |
Summarising Comparative Binary Data – Differences in Proportions
From Table 3.3, the proportion of subjects with a positive outcome under the test treatment is
The difference in proportions is given by
In prospective studies the proportion is also known as the risk and the difference in proportions as the risk difference (RD)
When one ignores the sign, the above quantity is also known as the absolute risk difference (ARD), that is.
where the symbols || mean to take the absolute value.
If we anticipate that the treatment to reduce some bad outcome (such as deaths) then it may be known as the absolute risk reduction (ARR). If we anticipate that the exposure/treatment will increase some bad outcome (such as deaths) then it may be known as the absolute risk excess (ARE).
Example – Summarising Results from a Clinical Trial – Corn Plasters RCT: Differences in Proportions
