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# Chapter 18 Relative risk, relative and absolute risk reduction, number needed to treat and confidence intervals

## Relative and absolute risks

How do you interpret the results of a randomised controlled trial. A common measure of a treatment is to look at the frequency of bad outcomes of a disease in the group being treated compared with those who were not treated. For instance, supposing that a well-designed randomised controlled trial in children with a particular disease found that 20 per cent of the control group developed bad outcomes, compared with only 12 per cent of those receiving treatment. Should you agree to give this treatment to your child? Without knowing more about the adverse effects of the therapy, it appears to reduce some of the bad outcomes of the disease. But is its effect meaningful?

This is where you need to consider the risk of treatment versus no treatment. In healthcare, risk refers to the probability of a bad outcome in people with the disease.

Absolute risk reduction (ARR) – also called risk difference (RD) – is the most useful way of presenting research results to help your decision-making. In this example, the ARR is 8 per cent (20 per cent - 12 per cent = 8 per cent). This means that, if 100 children were treated, 8 would be prevented from developing bad outcomes. Another way of expressing this is the number needed to treat (NNT). If 8 children out of 100 benefit from treatment, the NNT for one child to benefit is about 13 (100 ÷ 8 = 12.5).

For technical reasons, some other measures are often used. The relative risk (RR) of a bad outcome in a group given intervention is a proportional measure estimating the size of the effect of a treatment compared with other interventions or no treatment at all. It is the proportion of bad outcomes in the intervention group divided by the proportion of bad outcomes in the control group. In this hypothetical case, the RR is 0.6 (12 per cent ÷ 20 per cent = 0.6).

When a treatment has an RR greater than 1, the risk of a bad outcome is increased by the treatment; when the RR is less than 1, the risk of a bad outcome is decreased, meaning that the treatment is likely to do good. For example, when the RR is 2.0 the chance of a bad outcome is twice as likely to occur with the treatment as without it, whereas an RR of 0.5 means that the chance of a bad outcome is twice as likely to occur without the intervention. When the RR is exactly 1, the risk is unchanged. For example, a report may state ‘The relative risk of blindness in people given drug T was 1.5’. This shows that the drug increased the risk of blindness. Another measure that is used is the odds ratio. For practical purposes, assume that the odds ratio is the same as the relative risk. Sometimes the outcome is a good one and the interpretation of relative risk is the opposite of what we have just outlined.

Relative risk reduction (RRR) tells you by how much the treatment reduced the risk of bad outcomes relative to the control group who did not have the treatment. In the previous example, the relative risk reduction of fever and rash in the group of the children on the intervention was 40 per cent (1 – 0.6 = 0.4 or 40

per cent).

The RR (and therefore the RRR ) is often the same in people irrespective of their level of risk. which means that the ARR will be greatest in those at greatest risk, as shown in Table 18.1. The greater your risk, the more you stand to gain from the intervention .

Percentage with poor outcomes.

## Confidence intervals

Confidence intervals (CIs) aim to give you an idea of how confident you can be about a study’s estimate of a treatment’s effects. Even when a study is of impeccable quality, the results may have happened by chance. Statisticians deal with this uncertainly by doing some nifty calculations to determine how confident one can be about the results, which give us the confidence interval. The narrower the range, the more precise the study’s estimates, and the more confident you can be that it is a ‘real’ finding and not due to chance.

This is usually expressed in terms of a 95 per cent confidence interval (95%CI), which represents the range of results within which we can be 95 per cent certain that the true answer lies.

As an illustration of how confidence intervals can help, imagine that you are doing a study investigating whether there is gender bias in the method used by a university to choose its students. If there were no such bias you would expect 50 per cent of its students to be men and 50 per cent to be women. Supposing that you check a small sample, say 10 students, and found 4 of them were men. How sure can you be that this is a true reflection of the student population? Statistical calculations show that you can be 95 per cent certain that the true quota of men in the entire university population is somewhere between 12 and 74 per cent. This is an unhelpfully wide range.

But supposing you randomly sample 100 students and find that 40 are men. Statistical calculations show that you can be 95 per cent certain that the true quota of men in the entire university population is somewhere between 30 and 50 per cent – a narrower range.

Imagine also that you randomly examined a large sample of 1000 students, of whom 400 were men. The 95%CI would be from 37 per cent to 43 per cent – a much narrower range showing a very high level of confidence that this represents a true reflection of the gender ratio in the university.

In the sample of 10 students, finding four men is compatible with our expected value for society at large – 50 per cent males and 50 per cent females. In the group of 1000 students finding that only 40 per cent are men is not expected. The result from this large sample is statistically significant, which means that the disparity between the observed 40 per cent and the expected 50 per cent is real – that is, it is very unlikely to have arisen by chance. In the sample of 100 students, the upper end of the confidence interval is just on the expected value of 50 per cent and therefore just statistically significant.

The same principle applies to studies investigating treatments, except that we might be looking at the relative risks of a poor outcome in the group receiving the intervention compared with the control group.

Source: www.ncbi.nlm.nih.gov
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