Table 8.2 The distribution of alcohol intake and deaths by sex and level of alcohol intake. Reproduced from BMJ, 308, 302–6, courtesy of BMJ Publishing Group

*One beverage contains 9–13 g alcohol.

Exercise 8.4 Table 8.2 is from a cohort study into the influence of sex, age, body mass index and smoking on alcohol intake and mortality in Danish men and women aged between 30 and 79 years (Gronbaek et al. 1994). The table shows the distribution of alcohol intake and deaths by sex and level of alcohol intake. Use the information in the table to construct an appropriate contingency table for: (a) men; (b) women. Calculate the absolute risk of death among those subjects who consume: (i) less than one beverage a week; (ii) more than 69 beverages a week. Interpret your results.

Odds

The odds for a particular outcome from an event is closely related to probability, is perhaps a more difficult concept, but important in medical statistics, and we will meet it again later in the book. As you saw above, the probability (or risk) of a particular outcome from an event is the number of outcomes favourable to the event divided by the total number of outcomes. But:

The odds for an event is equal to the number of outcomes favourable to the event divided by the number of outcomes not favourable to the event.

Notice that:

The value of the odds for an outcome can vary from zero to infinity.

When the odds for an outcome are less than one, the odds are unfavourable to the outcome; the outcome is less likely to happen than it is to happen.

When the odds are equal to one, the outcome is as likely to happen as not.

When the odds are greater than one, the odds are favourable to the outcome; the outcome is more likely to happen than not.

Let’s go back to the dice rolling game. The odds in favour of the outcome ‘an even number’, is the number of outcomes favourable to the event (the number of even numbers, i.e. 2, 4, 6), divided by the number of outcomes not favourable to the event (the number of not even numbers, i.e. 1, 3, 5), which is 3/3 = 1/1 or one to one.

So the odds of getting an even number are the same as the odds of getting an odd number. Nearly all the odds in health statistics are expressed as ‘something’ to one. We call this value of one the reference value.

As a further more relevant example, we can also calculate odds from a table such as that for the exercise and stroke case-control study in Table 6.2. For instance:

Among those patients who’d had a stroke, 55 had exercised (been exposed to the ‘risk’ of exercising) and 70 had not, so the odds that those with a stroke had exercised is 55/70 = 0.7857.

Among those patients who hadn’t had a stroke, 130 had exercised and 68 had not, so the odds that they had exercised is 130/68 = 1.9118.

In other words, among those who’d had a stroke, the odds that they had exercised was less than half the odds (0.7857/1.9118) of those who hadn’t had a stroke. We can conclude on the basis of this sample that exercise when young seems to confer protection against a stroke.

Exercise 8.5 Table 8.3 is from a matched case-control study into maternal smoking during pregnancy and Down syndrome (Chi-Ling et al. 1999). It shows the basic characteristics of mothers giving birth to babies with Down syndrome (cases), and without Down syndrome (controls). Use the information in the table to construct appropriate separate 2 × 2 contingency tables for women: (a) aged under 35; (b) aged 35 and over. Hence calculate the odds that they had smoked during pregnancy among mothers giving birth to: (i) a Down syndrome baby; (ii) a healthy baby. What do you conclude?

Why you can’t calculate risk in a case-control study

For most people the risk of an event, being akin to probability, makes more sense and is easier to interpret than the odds for that same event. That being so, maybe it would be more helpful to express the stroke/exercise result as a risk rather than as odds. Unfortunately we can’t, and here’s why.

To calculate the risk that those with a stroke had exercised, you need to know two things: the total number who’d had a stroke, and the number of these who had been exposed to the risk (of exercise). You then divide the latter by the former. In a cohort study you would select the groups on this basis – whether they had been exposed to the risk (of exercising) or not. So one group would contain individuals exposed to the risk and the other those not exposed.

Table 8.3 Basic characteristics of mothers in a case-control study of maternal smoking and Down syndrome. Reproduced from Amer. J. Epid., 149, 442–6, courtesy of Oxford University Press

Selected characteristics of Down syndrome cases and birth-matched controls. Washington State, 1984–1994

But in a case-control study you don’t select on the basis of whether people have been exposed to the risk or not, but on the basis of whether they have some condition (a stroke) or not. So you have one group composed of individuals who have had a stroke, and one group who haven’t, but both groups will contain individuals who were and were not exposed to the risk (of exercising). Moreover, you can select whatever number of cases and controls you want. You could for example halve the number of cases and double the number of controls. This means the column totals, which you would otherwise need for your risk calculation, are meaningless.

The link between probability and odds

The connection between probability (risk) and odds means that it is possible to derive one from another:

risk or probability = odds/(1 + odds)

odds = probability/(1 – probability)

Exercise 8.6 Following on from Exercise 8.5, what is the probability that a mother chosen at random from those aged ≥ 35, will have smoked during pregnancy if they are:

(a) mothers of Down syndrome babies; (b) mothers of healthy babies?