Confidence Interval for the Risk Ratio


When comparing frequencies, the chi-squared test is a useful way of assessing the strength of evidence for rejecting the null hypothesis, and the magnitude of association can be summarized by computing a risk ratio, odds ratio, risk difference, etc. depending on the study design and what we are interested in evaluating. However, these measures of association are also estimates of populations parameters, and our interpretations of the findings are greatly enhanced by computing a confidence interval for these estimates as well.

The samples obtained for a cohort study or a randomized clinical trial provide us with estimates of the cumulative incidence in comparison groups, and the differences can be summarized by computing a risk ratio as an estimate of the magnitude of association. The confidence interval for a risk ratio provides us with a range of plausible values.

An earlier module noted that the general form of a 95% confidence interval is:

 

where SE(estimate) is the standard error of the estimate..

One of the assumptions in building a confidence interval is that the possible values are normally distributed. However, risk ratios and odds ratios are not normally distributed; the distribution of possible values is skewed toward higher values. One can get around this problem by taking the natural logarithm (log) of the risk ratio, because log(RR) will be normally distributed.

The distribution of risk ratio values is skewed to the left.

If one takes the natural log of the risk ratio, the distribution will be normally distributed.

So, to compute a confidence interval for the risk ratio, we have to work on the log-scale and then take the antilogarithm of the lower and upper confidence limits to compute the confidence limits on the risk ratio scale.

The formula for the 95% Confidence Interval for the risk ratio is as follows:

The steps are:

  1. Convert from RR to ln(RR)
  2. Find CI for ln(RR)
  3. Convert CI from ln(RR) to RR

Consider the following results from a cohort study in which the investigators compared the risk of developing cardiovascular disease (CVD) in subjects with hypertension (HTN) compared to subjects without hypertension.

Table - Association Between Hypertension (HTN) and Cardiovascular Disease (CVD)

CVD

No CVD

Total

HTN

992 (0.305)

2260

3252

No HTN

165 (0.140)

1017

1182

Total

1157

3277

4434

RR=.305/.140 = 2.18      

Step 1: Take the natural log of the RR (using R):

> log(2.18)

[1] 0.7793249

Step 2: We compute the 95% confidence inteval for log(RR). We will use a Z=1.96 and the following equation:

Step 3: We convert the log limits back to a normal scale for risk ratios by taking the antilog using R.

> exp(0.628)

[1] 1.873859

> exp(0.930)

[1] 2.534509

 

Therefore, the 95% CI for RR: (1.87, 2.53)

Note: The confidence interval is not symmetric about RR   

Interpretation: In this study subject with hypertension had 2.18 times the risk of developing cardiovascular disease compared to subjects without hypertension. With 95% confidence, the true risk rato lies in the range of 1.87-2.53.