In statistics and epidemiology, the relative risk or risk ratio (RR) is the ratio of probability of an event occurring (as for example, developing a disease) in an exposed group to the probability of the event occurring in a comparison

That is, RR = $\frac{Probability\;of\;event\;under\;exposed}{Probability\;of\;event\;under\;non-exposed}$

Suppose we label the disease outcome by 'D' and the risk factor by 'E' which takes the values 'exposed' or 'not exposed' $\bar E$. Then the relative risk for an outcome D associated with binary risk factor E (i.e. E & $\bar E$ is defined as:

RR = $\frac{P(D।E)}{P(D।\bar E)}$

RR = $\frac{a/a+c}{b/b+d}$

That is, RR = $\frac{Probability\;of\;event\;under\;exposed}{Probability\;of\;event\;under\;non-exposed}$

Suppose we label the disease outcome by 'D' and the risk factor by 'E' which takes the values 'exposed' or 'not exposed' $\bar E$. Then the relative risk for an outcome D associated with binary risk factor E (i.e. E & $\bar E$ is defined as:

RR = $\frac{P(D।E)}{P(D।\bar E)}$

**Consider the table:**RR = $\frac{a/a+c}{b/b+d}$

**Description**:

$\star$As RR is defined by division of probabilities, RR must be a non-negative number, i.e RR $\ge$ 0.

$\star$ RR =1 implies P(D।E) = P(D।$\bar E)$ i.e D and E are independent.

Note: RR = 1 is a null value being kept in null hypothesis

RR > 1 implies there is a greater risk or probability of D when exposed (E) than when unexposed ($\bar E$).

[i.e P(D।E) > P(D।$\bar E)$

RR <1 implies the reverse i.e the risk is greater among non-exposed.

$\star$ RR provides risk of disease for an exposed individual by multiplying RR by baseline risk.

That is, P(D।E) = RR × baseline risk.

$\star$Its disadvantage is the restricted lower limit i.e RR $\ge$ 0, and implicit upper bound i.e. RR$\le \frac{1}{P(D।\bar E)}$ (always) as P(D।E) cannot be larger than 1.

$\star$ RR is not symmetric i.e. $\frac{P(D।E)}{P(D।\bar E)} \ne \frac{P(D।\bar E)}{P(D।E)}$

**Consider an example:**

So RR for infant mortality associated with a mother being unmarried at the time of birth is:

$\frac{16712/1213854}{18784/2897205}$ = 2.12

which implies that the risk of an infant death with an unmarried mother is a little more than double the risk when the mother is married.

**Another example of Relative Risk:**

Suppose, the probability of developing lung cancer among smokers was 20% and among non-smokers was 1%.

RR = $\frac{0.2}{0.01}$ = 20

**Interpretation:**

Smokers will be 20 times as likely as nonsmokers to develop lung cancer.

Statistical use of RR and its meaning:

RR is used frequently in the statistical analysis of binary outcomes where the outcome of interest has relatively low probability.

It is often suited to clinical trial data where it is used to compare the risk of developing a disease in people not receiving the raw material treatment versus who are recieving an established treatment.

**Confidence Interval (CI):**

The log of the relative risk has a sampling distribution that is approximately normal with variance that can be estimated by a formula involving the no. of subjects in each group and the event rates in each group (Delta Method). According to this,

CI = log (RR) ± S.E × Z$_\alpha$

where Z$_\alpha$ is the standard score for the chosen level of significance and S.E is the standard error.

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