NOTE: The following sections are OPTIONAL and are only provided as a future resource.

# Rate Ratios

Rate ratios are closely related to risk ratios, but they are computed as the ratio of the incidence rate in an exposed group divided by the incidence rate in an unexposed (or less exposed) comparison group.

Rate Ratio = (IR_{e}) / (IR_{u}

Consider an example from The Nurses' Health Study. This prospective cohort study was used to investigate the effects of hormone replacement therapy (HRT) on coronary artery disease in post-menopausal women. The investigators calculated the incidence rate of coronary artery disease in post-menopausal women who had been taking HRT and compared it to the incidence rate in post-menopausal women who had not taken HRT. The findings are summarized in this table:

Post-Menopausal Hormone Use |
Number With Coronary Artery Disease |
Person-Years of Disease-Free Follow-up |
---|---|---|

Yes |
30 |
54,308.7 |

No |
60 |
51,477.5 |

- The rate in those using hormones was 30 / 54,308.7 = 55.2 per 100,000 person-years
- The rate in those NOT using hormones was 60 / 51,477.5 = 116.6 per 100,000 person-years.

So, the rate ratio was 55.2 / 116.6 = 0.47.

Interpretation: Women who used postmenopausal hormones had 0.47 __times the rate__ of coronary artery disease compared to women who did not use postmenopausal hormones.

(Rate ratios are often interpreted as if they were risk ratios, e.g., post-menopausal women using HRT had 0.47 times the risk of CAD compared to women not using HRT, but it is more precise to refer to the ratio of rates rather than risk.)

## Confidence Interval for a Rate Ratio

We can also calculate a 95% confidence interval for the rate ratio to give us an idea of the range of plausible values for the measure based on our sample. Like the risk ratio, the rate ratio is not normally distributed, but the natural log of the rate ratio, log(IRR), where IRR stands for incidence rate ratio, is normally distributed. So, again, we have to work on the log-scale to satisfy the normality requirement, and then take the antilogarithm of the lower and upper confidence limits at the end to compute the confidence limits on the rate ratio scale. The formula for the 95% Confidence Interval for the risk ratio is as follows:

As with the risk ratio, lets walk through this formula step-by-step, using the hormone replacement therapy and coronary artery disease data as an example.

- Step 1: Calculate the natural log of the rate ratio

- Step 2: Calculate the standard error of the log(IRR)

- Step 3: Calculate the lower and upper confidence bounds on the natural log scale

- The lower bound of the 95% confidence interval of the log rate ratio is:

- The upper bound of the 95% confidence interval of the log rate ratio is:

- Step 4: Take the antilogarithm to obtain the upper and lower bounds of the 95% confidence interval for the rate ratio

- Step 5: Report and interpret the estimate and the confidence interval:

In this study, those on hormone replacement therapy had 0.47 times the rate of developing coronary artery disease compared to those who did not take hormone replacement therapy. Based on this sample, we are 95% confident that the true rate ratio lies between 0.303 and 0.729.

As a rule of thumb, you should use the following wording to interpret the 95% confidence interval for the rate ratio: "Based on this sample, we are 95% confident that the true rate ratio lies between [lower bound] and [upper bound]."

## Hypothesis Test for the Rate Ratio

Hypothesis testing for the rate ratio can also be done, and its null and alternative hypotheses are similar to those for the risk ratio. For the rate ratio, the hypotheses are as follows:

H_{0}: IRR = 1 vs. H_{A}: IRR ≠ 1

Or, in words

H_{0}: There is no association between the exposure and the rate of disease

vs.

H_{1}: There is an association between the exposure and the rate of disease

Again, both forms of the null and alternative are equivalent since, if there is no association between the exposure and the rate of disease, the rate of disease among the exposed will be similar to that among the unexposed, which would make the rate ratio close to the null value of 1. In the example of hormone replacement therapy and coronary artery disease, the null and alternative hypotheses would be

- H
_{0}: There is no association between hormone replacement therapy and rate of coronary artery disease (IRR = 1). - H
_{1}: There is an association between hormone replacement therapy and rate of coronary artery disease (IRR ≠ 1).

A statistical test can be performed to evaluate these hypotheses. While the hand calculations of the test statistic are beyond the scope of this class, in later pages of this module we will see how to perform a statistical test for the rate ratio using the R statistical software.