Probability, risk and odds

Explain why it’s not possible to calculate a risk ratio in a case-control study.

Define number needed to treat, explain its use and calculate NNT in a simple example.

Chance would be a fine thing – the idea of probability

Probability is a measure of the chance of getting some outcome of interest from some event. The event might be rolling a dice and the outcome of interest might be getting a six; or the event might be performing a biopsy with the outcome of interest being evidence of malignancy and so on. Some basic ideas about probability:

The probability of a particular outcome from an event will lie between zero and one.

The probability of an event that is certain to happen is equal to one. For example, the probability that everybody dies eventually.

The probability of an event that is impossible is zero. For example, throwing a seven with a normal dice.

If an event has as much chance of happening as of not happening (like tossing a coin and getting a head), then it has a probability of 1/2 or 0.5.

If the probability of an event happening is p, then the probability of the event not happening is 1 – p.

Table 8.1 Frequency table showing causes of blunt injury to limbs in 75 patients

Calculating probability

You can calculate the probability of a particular outcome from an event with the following expression:

The probability of a particular outcome from an event is equal to the number of outcomes that favour that event, divided by the total number of possible outcomes.

To take a simple example: What is the probability of getting an even number when you roll a dice?

Total number of possible outcomes = 6 (1 or 2 or 3 or 4 or 5 or 6)

Total number of outcomes favouring the event ‘an even number’ = 3 (i.e. 2 or 4 or 6)

So probability of getting an even number = 3/6 = 1/2= 0.5

The above method for determining probability works well with experiments where all of the outcomes have the same probability, e.g. rolling dice, tossing a coin, etc. In the real world you will often have to use what is called the proportional frequency approach, which uses existing frequency data as the basis for probability calculations.

As an example, look at Table 8.1 (which is Table 2.3 reproduced for convenience) which shows the causes of blunt injury to limbs. I have added an extra column showing the proportional frequency (category frequency divided by total frequency). Notice that the proportional frequencies sum to one.

Exercise 8.1 Table 1.6 shows the basic characteristics of the two groups of women receiving a breast lump diagnosis in the stress and breast cancer study. What is the probability that a woman chosen at random: (a) will have had her breast lump diagnosed as (i) benign? (ii) malignant?; (b) will be post-menopausal?; (c) will have had three or more children?

Exercise 8.2 Table 1.7 is from a study of thrombotic risk during pregnancy. What is the probability (under classification 1) that a subject chosen at random will be aged: (a) less than 30?; (b) more than 29?

Now ask the question, ‘What is the probability that if you chose one of these 75 patients at random their injury will have been caused by a fall?’. The answer is the proportional frequency for the ‘fall’ category, i.e. 0.613. In other words, we can interpret proportions as equivalent to probabilities. Probability is a huge subject with many textbooks devoted to it, but for our purposes in this book we don’t really need to know any more.

Probability and the Normal distribution

We know that if data is Normally distributed then about 95 per cent of the values will lie no further than two standard deviations from the mean (see Figure 5.5). In probability terms, we can say that there is a probability of 0.95 that a single value chosen at random will lie no further than two standard deviations from the mean. In the case of the Normally distributed birthweight data, this means that there is a probability of 0.95 that the birthweight of one of these infants chosen at random will be between 2890 g and 4398 g.

Exercise 8.3 Using the information on cord platelet count in Figure 4.6, determine the probability that one infant chosen at random from this sample will have a cord platelet count: (a) between 101 × 109/l and 515 × 109/l; (b) less than 239 × 109/l.

Risk

As I mentioned earlier a risk is the same as a probability, but the former word tends to be favoured in the clinical arena. So the definition of probability given earlier applies equally here to risk. In other words, the risk of any particular outcome from an event is equal to the number of favourable outcomes divided by the total number of outcomes. Risk accordingly can vary between zero and one.

As an example, and also to re-visit the contingency table, look again at the table in Table 6.1 from the cohort study of coronary heart disease (CHD) in adult life and the risk factor ‘weighing 18 lbs or less at one year’. The risk (or probability) that those adults who as infants weighed 18 lbs or less at one year will have CHD, is equal to the number who weighed 18 lbs or less at one year and had CHD, divided by the total number who weighed 18 lbs or less. This is equal to 4/15 = 0.2667.

Similarly, the risk (or probability) for those who weighed more than 18 lbs at one year will have CHD equals the number who weighed more than 18 lbs at one year and had CHD, divided by the total number who weighed more than 18 lbs. This is equal to 38/275 = 0.1382 and thus is only half the risk of those weighing 18 lbs or less.

The risk for a single group, as it is described it above, is also known as the absolute risk, mainly to distinguish it from relative risk, which is the risk for one group compared to the risk for some other group (which we’ll come to shortly).