Risk Difference, Risk Ratio, and Odds Ratio

Why Compare More Than One Risk Measure?

A genotype-specific risk describes the probability of a binary outcome for a genotype group over a specified time period. Comparing two such risks can answer different questions. A risk difference describes the absolute change in probability; a risk ratio describes a multiplicative change in risk; and an odds ratio compares the odds in the two groups. These measures are related, but their numerical values are not interchangeable.

The baseline risk matters. If a genotype is associated with the same risk ratio in two populations, the absolute increase in risk can still differ because the reference-group risks differ. The odds ratio can also differ. Distinguishing the measures is important when describing genotype effects, especially when translating a relative contrast into an absolute one.

Definition: Let \(G_0\) denote a reference genotype group and \(G_1\) a comparison genotype group. For the same binary outcome and specified time period, let \(p_0=P(Y=1\mid G_0)\) and \(p_1=P(Y=1\mid G_1)\) be their risks. The risk difference is \(RD=p_1-p_0\). If \(p_0>0\), the risk ratio is \(RR=p_1/p_0\). If both risks are strictly between 0 and 1, the finite odds ratio is \(OR=\{p_1/(1-p_1)\}/\{p_0/(1-p_0)\}\).

The order of subtraction and division matters: these definitions compare \(G_1\) with \(G_0\). Reversing the groups changes the sign of the risk difference and reciprocates the risk ratio and odds ratio, when those ratios are defined. A risk is a probability, so \(0\leq p_0,p_1\leq1\). The risk difference is defined across that full range. The risk ratio requires a positive reference risk, and the finite odds ratio requires both risks to be strictly interior.

Calculating the Three Measures

The risk difference is measured on the probability scale. For instance, \(RD=0.08\) means that the comparison group has an 8-percentage-point higher risk than the reference group. It is not an 8% relative increase. The risk ratio is unitless: \(RR=1.5\) means the comparison-group risk is 1.5 times the reference-group risk, or 50% higher. An odds ratio compares \(p/(1-p)\), the odds corresponding to a risk \(p\); \(OR=1\) indicates equal odds in the two groups.

Measure	Formula	Value indicating no contrast	Scale
Risk difference	\(p_1-p_0\)	0	Probability points
Risk ratio	\(p_1/p_0\)	1	Multiplicative risk scale
Odds ratio	\(\{p_1/(1-p_1)\}/\{p_0/(1-p_0)\}\)	1	Multiplicative odds scale

All three measures describe a comparison of risks in the stated genotype groups. By themselves, they do not establish that genotype causes the outcome. A causal interpretation would require a design and assumptions that support it. Here, the task is to calculate and interpret contrasts between the specified risks.

Worked Example: Calculating Contrasts From Hypothetical Genotype Counts

Suppose, hypothetically, that a study observes 1,000 people in each of two genotype groups over the same follow-up period. The outcome occurs in 100 people with \(G_0\) and 200 people with \(G_1\). Calculate the two risks and their three contrasts.

The observed risks are the number with the outcome divided by the number in each group:

\hat p_0=\frac{100}{1000}=0.10,\qquad \hat p_1=\frac{200}{1000}=0.20.

The estimated risk difference is \(0.20-0.10=0.10\), or 10 percentage points. The estimated risk ratio is \(0.20/0.10=2.00\): the observed risk in \(G_1\) is twice that in \(G_0\). The estimated odds in \(G_0\) are \(0.10/(1-0.10)=0.10/0.90=0.1111\); in \(G_1\), they are \(0.20/0.80=0.2500\). Therefore:

\widehat{OR}=\frac{0.20/0.80}{0.10/0.90} =\frac{0.2500}{0.1111} =2.25.

Thus, the hypothetical comparison has an estimated 10-percentage-point increase in risk, a risk ratio of 2.00, and an odds ratio of 2.25. The odds ratio exceeds the risk ratio here; they quantify distinct contrasts.

How Baseline Risk Changes the Contrasts

Suppose the risk ratio is fixed at a value \(r\), so that \(p_1=r p_0\). The risk difference then follows directly:

$$ RD=p_1-p_0=rp_0-p_0=p_0(r-1). $$

For a fixed \(r\), the risk difference is proportional to the reference risk \(p_0\). If \(r>1\), a higher baseline risk produces a larger absolute increase. If \(0<r<1\), a higher baseline risk produces a more negative risk difference, meaning a larger absolute reduction. Thus, the same relative contrast need not correspond to the same number of additional or fewer outcomes.

The odds ratio also depends on baseline risk when the risk ratio is fixed. Substituting \(p_1=RR\,p_0\) into the odds ratio definition gives:

OR =\frac{RR\,p_0/(1-RR\,p_0)}{p_0/(1-p_0)} =\frac{RR(1-p_0)}{1-RR\,p_0}, \qquad 0<p_0<1,\quad 0<RR\,p_0<1.

The conditions are essential: \(0<p_0<1\) makes the reference risk interior, and \(0<RR\,p_0<1\) makes the comparison risk \(p_1=RR\,p_0\) strictly between 0 and 1. These conditions ensure that both odds, and therefore the finite odds ratio, are defined. A risk ratio of zero, for example, would produce \(p_1=0\) and would not satisfy the conditions for this finite odds ratio.

Worked Example: The Same Risk Ratio at Different Baselines

Consider three hypothetical pairs of genotype-specific risks, all with \(RR=2\). For each pair, calculate the risk difference and odds ratio, then compare the results.

At baseline risk \(p_0=0.05\), the comparison risk is \(p_1=2(0.05)=0.10\), which is a valid probability. The risk difference is \(0.10-0.05=0.05\), or 5 percentage points. The odds ratio is:

OR=\frac{2(1-0.05)}{1-2(0.05)} =\frac{1.90}{0.90} =2.111.

At baseline risk \(p_0=0.10\), the comparison risk is \(p_1=2(0.10)=0.20\). The risk difference is \(0.20-0.10=0.10\), or 10 percentage points, and:

OR=\frac{2(1-0.10)}{1-2(0.10)} =\frac{1.80}{0.80} =2.25.

At baseline risk \(p_0=0.30\), the comparison risk is \(p_1=2(0.30)=0.60\). The risk difference is \(0.60-0.30=0.30\), or 30 percentage points, and:

OR=\frac{2(1-0.30)}{1-2(0.30)} =\frac{1.40}{0.40} =3.50.

In all three pairs, the risk ratio is 2.00. Yet the risk differences are 5, 10, and 30 percentage points, and the odds ratios are 2.111, 2.25, and 3.50. All comparison risks are strictly between 0 and 1, so the odds ratios are defined. The example shows why a risk ratio alone does not specify the absolute risk difference or the odds ratio without the baseline risk.

Why the Odds Ratio and Risk Ratio Differ

Risk is \(p\), whereas odds are \(p/(1-p)\). When a risk is small, \(1-p\) is close to 1, so the odds are close to the risk. This is why an odds ratio may be close to a risk ratio when both risks are low. It is an approximation, not an identity. As risks increase, the denominator \(1-p\) has a larger effect and the measures can differ substantially.

The relationship can be checked algebraically. For \(RR>0\) and risks satisfying the interior conditions above:

\frac{OR}{RR} =\frac{1-p_0}{1-RR\,p_0}.

If \(RR>1\), then \(RR\,p_0>p_0\), so \(1-RR\,p_0<1-p_0\). Both denominators are positive under the stated conditions, and therefore \(OR/RR>1\): the odds ratio is larger than the risk ratio. If \(0<RR<1\), then \(RR\,p_0<p_0\), so \(1-RR\,p_0>1-p_0\), giving \(OR/RR<1\): the odds ratio is smaller than the risk ratio. If \(RR=1\), then \(p_1=p_0\) and both ratios equal 1. This establishes why the two ratios generally differ even though both compare groups multiplicatively.

Worked Example: A Protective Risk Contrast

Suppose, hypothetically, that the risk of an outcome is \(p_0=0.40\) in the reference genotype group and \(p_1=0.20\) in the comparison group. Calculate the three measures and interpret their directions.

The risk difference is \(0.20-0.40=-0.20\), or a 20-percentage-point lower risk in \(G_1\). The risk ratio is \(0.20/0.40=0.50\), so the comparison-group risk is half the reference-group risk. Both risks are strictly between 0 and 1, so the odds ratio is defined. The odds are \(0.20/0.80=0.25\) in \(G_1\) and \(0.40/0.60=0.6667\) in \(G_0\). Therefore:

OR=\frac{0.20/0.80}{0.40/0.60} =\frac{0.25}{0.6667} =0.375.

The odds ratio is below the risk ratio, as expected for \(RR<1\). It means the odds in \(G_1\) are 0.375 times the odds in \(G_0\); it does not mean the risk is 0.375 times as large. The risk ratio, which is 0.50, directly gives the relative comparison of risks.

Common Mistakes and Interpretation Checks

A frequent mistake is to report an odds ratio as though it were a risk ratio. For example, interpreting \(OR=2.25\) as “the risk is 2.25 times as high” is incorrect. The odds are 2.25 times as high; the risk ratio must be calculated from the two risks. Another error is describing \(RD=0.10\) as a 10% relative increase. It is a 10-percentage-point difference. In the example with risks 0.10 and 0.20, the risk ratio is 2.00, corresponding to a 100% relative increase, while the risk difference is 10 percentage points.

Check the following before reporting a contrast:

Name the reference and comparison groups so the direction of the contrast is clear.
State the risks and the outcome time period; do not compare risks defined over different periods.
Use percentage points for a risk difference and “times as high” or “times as low” for a ratio.
Confirm that \(p_0>0\) before calculating a risk ratio, and that both risks are strictly between 0 and 1 before reporting a finite odds ratio.
Do not infer that a genotype causes a difference merely because the groups have different observed risks.

These measures serve complementary purposes. The risk difference communicates absolute impact, the risk ratio expresses relative risk, and the odds ratio expresses relative odds. Reporting which measure is being used—and its value in context—prevents a numerical contrast from being mistaken for another one.

Key takeaway: For reference risk \(p_0\) and comparison risk \(p_1\), calculate \(RD=p_1-p_0\), \(RR=p_1/p_0\), and \(OR=\{p_1/(1-p_1)\}/\{p_0/(1-p_0)\}\) when their respective conditions are met. A fixed risk ratio does not fix the risk difference or odds ratio: both also depend on baseline risk.

Check Your Understanding

For each question, identify the measure and explain the result in context.

If \(p_0=0.08\) and \(p_1=0.12\), calculate the risk difference and risk ratio. State the risk difference in percentage points.
Two genotype comparisons both have \(RR=1.5\), but their reference risks are 0.10 and 0.40. Which comparison has the larger risk difference? Explain using \(RD=p_0(RR-1)\).
Why is a finite odds ratio not defined by the stated definition when either risk is 0 or 1?
A study reports an odds ratio of 1.8. What does this say about the odds, and why does it not by itself give the risk ratio?
For \(p_0=0.20\) and \(RR=2\), verify that \(p_1\) is a valid risk and calculate the odds ratio. Check that both risks satisfy the odds-ratio conditions.