To assess the importance of the risk factor, it is necessary to compare the risk for developing a disease in the exposed group with the risk in the nonexposed group. In the study by Rivers and coworkers [6] the risk for death on the early goal-directed therapy is 32.5%, whereas on the standard therapy it is 49.6%. A comparison between the two risks can be made by examining either their ratio or the difference between them.

### Risk ratio

The risk ratio measures the increased risk for developing a disease when having been exposed to a risk factor compared with not having been exposed to the risk factor. It is given by RR = risk for the exposed/risk for the unexposed, and it is often referred to as the relative risk. The interpretation of a relative risk is described in Statistics review 6 [7]. For the Rivers study the relative risk = 0.325/0.496 = 0.66, which indicates that a patient on the early goal-directed therapy is 34% less likely to die than a patient on the standard therapy.

The calculation of the 95% confidence interval for the relative risk [8] will be covered in a future review, but it can usefully be interpreted here. For the Rivers study the 95% confidence interval for the population relative risk is 0.48 to 0.90. Because the interval does not contain 1.0 and the upper end is below, it indicates that patients on the early goal-directed therapy have a significantly decreased risk for dying as compared with those on the standard therapy.

### Odds ratio

When quantifying the risk for developing a disease, the ratio of the odds can also be used as a measurement of comparison between those exposed and not exposed to a risk factor. It is given by OR = odds for the exposed/odds for the unexposed, and is referred to as the odds ratio. The interpretation of odds ratio is described in Statistics review 3 [4]. For the Rivers study the odds ratio = 0.48/0.98 = 0.49, again indicating that those on the early goal-directed therapy have a reduced risk for dying as compared with those on the standard therapy. This will be covered fully in a future review.

The calculation of the 95% confidence interval for the odds ratio [2] will also be covered in a future review but, as with relative risk, it can usefully be interpreted here. For the Rivers example the 95% confidence interval for the odds ratio is 0.29 to 0.83. This can be interpreted in the same way as the 95% confidence interval for the relative risk, indicating that those receiving early goal-directed therapy have a reduced risk for dying.

### Difference between two proportions

#### Confidence interval

For the Rivers study, instead of examining the ratio of the risks (the relative risk) we can obtain a confidence interval and carry out a significance test of the difference between the risks. The proportion of those on early goal-directed therapy who died is p_{1} = 38/117 = 0.325 and the proportion of those on standard therapy who died is p_{2} = 59/119 = 0.496. A confidence interval for the difference between the true population proportions is given by:

(p_{1} - p_{2}) - 1.96 × se(p_{1} - p_{2}) to (p_{1} - p_{2}) + 1.96 × se(p_{1} - p_{2})

Where se(p_{1} - p_{2}) is the standard error of p_{1} - p_{2} and is calculated as:

Thus, the required confidence interval is -0.171 - 1.96 × 0.063 to -0.171 + 1.96 × 0.063; that is -0.295 to -0.047. Therefore, the difference between the true proportions is likely to be between -0.295 and -0.047, and the risk for those on early goal-directed therapy is less than the risk for those on standard therapy.

#### Hypothesis test

We can also carry out a hypothesis test of the null hypothesis that the difference between the proportions is 0. This follows similar lines to the calculation of the confidence interval, but under the null hypothesis the standard error of the difference in proportions is given by:

where p is a pooled estimate of the proportion obtained from both samples [5]:

So:

The test statistic is then:

Comparing this value with a standard Normal distribution gives p = 0.007, again suggesting that there is a difference between the two population proportions. In fact, the test described is equivalent to the χ^{2}test of association on the two by two table. The χ^{2} test gives a test statistic of 7.31, which is equal to (-2.71)^{2} and has the same *P* value of 0.007. Again, this suggests that there is a difference between the risks for those receiving early goal-directed therapy and those receiving standard therapy.