Biostatistics / Epidemiology

Risk Ratio, Risk Difference, and Odds Ratio with Examples

When you compare the risk of a disease between groups with different exposures, you can choose from a few core methods. Each method gives a different perspective on how exposure relates to disease. If you understand these, you can interpret studies more clearly and decide which measure makes the most sense for your situation.

Two Main Ways to Compare Risks

You will often see two primary approaches: risk ratio and risk difference.

1. Risk Ratio (Relative Risk)

The risk ratio (often called relative risk, abbreviated as RR) is the ratio of the probability of disease in the exposed group to that in the unexposed group.

The formula is:

RR = \frac{\text{Risk in exposed}}{\text{Risk in unexposed}}

In plain words, it shows how many times more likely someone is to get the disease if they move from the unexposed category to the exposed category. It is a multiplicative model — meaning the increase is expressed as “times” or “fold.”

  • If RR = 1, there is no difference in risk between groups
  • If RR > 1, the exposure increases risk
  • If RR < 1, the exposure reduces risk

Example:
If smokers have a lung cancer risk of 0.08 (8 percent) and nonsmokers have a risk of 0.02 (2 percent):

RR = \frac{0.08}{0.02} = 4

This means smokers are four times more likely to develop lung cancer.

2. Risk Difference

The risk difference (RD) measures the absolute change in risk between the exposed and unexposed groups.

The formula is:

RD = \text{Risk in exposed} - \text{Risk in unexposed}

This is an additive model, the difference is measured in percentage points, not multiples.

Example using the same lung cancer data:

RD = 0.08 - 0.02 = 0.06

This means smoking increases the absolute risk of lung cancer by 6 percentage points.

Which Is More Important?

  • RR is useful for identifying potential causes. For example, an RR above 10 is often seen as strong evidence of causation.
  • RD is more relevant for public health decisions. Even if RR is high, if the disease is rare, the actual burden may be small. RD shows the direct additional cases that would occur in the exposed group.

Dose–Response Relationship

One important clue for causation is the dose–response relationship. If higher exposure leads to higher risk in a consistent pattern, this strengthens the case for a causal link.

Example:
If you measure lung cancer risk in smokers by years smoked:

  • 5 years: RR = 2
  • 10 years: RR = 4
  • 20 years: RR = 8

The steady increase in RR as exposure increases supports the idea that smoking is a cause.

Comparing Rates Instead of Risks

Sometimes studies use rates instead of risks, especially when the observation period varies for different individuals. This often happens in cohort studies where we measure person-time at risk.

Rate Ratio (rr) Using Person-Time Data

If you have new cases and the total person-time for each group, you can calculate the rate ratio (rr):

rr = \frac{r_1}{r_0} = \frac{a / PT_1}{b / PT_0}

Where:

  • a = number of new cases in the exposed group
  • b = number of new cases in the unexposed group
  • PT₁ = total person-time for the exposed group
  • PT₀ = total person-time for the unexposed group

Example:
If the exposed group has 50 cases over 10,000 person-years, and the unexposed group has 30 cases over 15,000 person-years:

rr = \frac{50 / 10000}{30 / 15000} = \frac{0.005}{0.002} = 2.5

The exposed group’s rate is 2.5 times higher.

Comparing Risks with Binomial Data

If the data is collected as number of people at risk and number of new cases, you can directly compute RR.

Risk Ratio (RR) with Binomial Data

RR = \frac{a / N_1}{b / N_0}

Where:

  • a = cases in exposed group
  • b = cases in unexposed group
  • N₁ = people at risk in exposed group
  • N₀ = people at risk in unexposed group

Example:
If 40 out of 2,000 exposed people develop the disease, and 20 out of 2,500 unexposed people develop it:

RR = \frac{40 / 2000}{20 / 2500} = \frac{0.02}{0.008} = 2.5

The exposed group has 2.5 times the risk.

Disease Odds Ratio (DOR)

The disease odds ratio (DOR) compares the odds of disease in the exposed group to the odds in the unexposed group.

The formula is:

DOR = \frac{\frac{a}{N_1 - a}}{\frac{b}{N_0 - b}}

Where:

  • Odds in exposed = cases / non-cases in exposed group
  • Odds in unexposed = cases / non-cases in unexposed group

Example:
Using the previous numbers:

  • Odds in exposed = 40 / (2000 – 40) = 40 / 1960 ≈ 0.02041
  • Odds in unexposed = 20 / (2500 – 20) = 20 / 2480 ≈ 0.00806

DOR = \frac{0.02041}{0.00806} ≈ 2.53

The odds of disease in the exposed group are about 2.53 times those in the unexposed group.

Why Use Odds Ratios?

  • RR is more intuitive but can only be calculated in cohort studies where the risk is directly measurable.
  • DOR can be calculated in case-control studies where you do not know the actual population risk, only the odds.

Summary Table of Measures

Measure Formula Model Type Main Use
Risk Ratio (RR) \frac{p_{exposed}}{p_{unexposed}} Multiplicative Causation
Risk Difference (RD) p_{exposed} - p_{unexposed} Additive Public health burden
Rate Ratio (rr) \frac{a / PT_1}{b / PT_0} Multiplicative Cohort person-time data
Disease Odds Ratio (DOR) \frac{a / (N_1 - a)}{b / (N_0 - b)} Multiplicative Case-control studies

Related Articles