Comparing Odds With the Odds Ratio

From Proportions to Odds

A proportion answers, “What fraction of this group experienced the event?” Odds answer a related but different question: “How many event outcomes are there for each non-event outcome?” Both can compare two groups, but they use different denominators. This distinction matters when describing a group difference: a ratio of odds is not automatically a ratio of event probabilities.

Suppose a group has an event proportion \(p\), where the event is clearly defined and every individual is classified as either having experienced it or not. The non-event proportion is \(1-p\). The odds of the event are the event proportion divided by the non-event proportion.

Definition: For an event with probability \(p\), where \(0<p<1\), the odds of the event are \(p/(1-p)\). In a sample with \(a\) events and \(b\) non-events, the sample odds are \(a/b\), provided \(b>0\).

For example, if 20% of a group experiences an event, the odds are \(0.20/0.80=0.25\), or 1 to 4. This does not mean that the event probability is 25%; it is 20%. Odds can be any positive number, including numbers larger than 1. A group with 70% experiencing an event has odds \(0.70/0.30=2.333\), meaning seven event outcomes for every three non-event outcomes.

Defining and Calculating the Odds Ratio

To compare two independent groups, designate one as group 1 and the other as group 2. The event must mean the same thing in both groups. If \(p_1\) and \(p_2\) are their event proportions, respectively, the population odds ratio compares the odds in group 1 with the odds in group 2. Its sample counterpart is calculated from the observed proportions.

Definition: The odds ratio for group 1 relative to group 2 is the odds of the event in group 1 divided by the odds of the event in group 2. For sample proportions \(\hat p_1\) and \(\hat p_2\), the sample odds ratio is \(\widehat{\mathrm{OR}}=[\hat p_1/(1-\hat p_1)]/[\hat p_2/(1-\hat p_2)]\), when both groups have at least one event and one non-event.

A two-by-two table makes the calculation especially direct. Let group 1 have \(a\) events and \(b\) non-events, and group 2 have \(c\) events and \(d\) non-events. Their sample event proportions are \(a/(a+b)\) and \(c/(c+d)\). The odds ratio compares \(a/b\) with \(c/d\), so it can also be computed by multiplying the diagonal counts and dividing by the other diagonal product.

\widehat{\mathrm{OR}} = \frac{a/b}{c/d} = \frac{ad}{bc}

This cross-product form is not a separate definition; it follows from the odds in the two groups. It is useful for checking calculations, but the odds-based form helps preserve the meaning: group 1’s event-to-non-event ratio divided by group 2’s event-to-non-event ratio.

Worked Calculations

Worked Example: Comparing Event Odds in Two Groups

In a hypothetical comparison of two independent groups, 24 of 120 people in group 1 experience a specified event, while 12 of 120 people in group 2 experience it. Calculate and interpret the sample odds ratio for group 1 relative to group 2.

Check the setup. Each person is counted in one group and classified as either experiencing the event or not. The groups do not overlap, and the event definition is the same for both. The non-event counts are \(120-24=96\) in group 1 and \(120-12=108\) in group 2.

Calculate each group’s odds. Group 1’s odds are \(24/96=0.25\). Group 2’s odds are \(12/108=1/9\).

Divide the odds. Using the exact count fractions, \(\widehat{\mathrm{OR}}=(24/96)/(12/108)=(24\times108)/(96\times12)=2592/1152=2.25\).

In this hypothetical sample, the event odds in group 1 are 2.25 times the event odds in group 2. This is a comparison of odds, not a statement that the event probability in group 1 is 2.25 times the event probability in group 2.

Worked Example: An Odds Ratio Below One

Suppose, hypothetically, that 18 of 60 people in group A and 30 of 75 people in group B experience the same event. Find the odds ratio for group A relative to group B and explain its direction.

Check the setup. The two groups consist of different people, and the same binary event is recorded in each. Group A has \(60-18=42\) non-events; group B has \(75-30=45\) non-events. Both event and non-event counts are positive in each group.

The odds in group A are \(18/42\), and the odds in group B are \(30/45\). Therefore, \(\widehat{\mathrm{OR}}_{A:B}=(18/42)/(30/45)=(18\times45)/(42\times30)=810/1260=0.6429\), rounded to four significant figures.

Because the odds ratio is below 1, the event odds are lower in group A than in group B. Equivalently, reversing the comparison gives \(\widehat{\mathrm{OR}}_{B:A}=1/0.642857\ldots=1.5556\), so the event odds in group B are about 1.556 times those in group A. Reversing the group order reverses the ratio; it does not change the underlying data.

Worked Example: Odds Ratio Versus Risk Ratio

In a hypothetical study with two independent groups of 40 people each, 8 people in group 1 and 2 people in group 2 experience an event. Calculate the odds ratio and compare it with the ratio of event proportions.

Check the setup. Each person contributes to only one group, the event is defined consistently, and the remaining counts are non-events. Group 1 has \(40-8=32\) non-events; group 2 has \(40-2=38\).

The odds are \(8/32=0.25\) for group 1 and \(2/38=1/19\) for group 2. Thus \(\widehat{\mathrm{OR}}=(8/32)/(2/38)=(8\times38)/(32\times2)=304/64=4.75\).

The event proportions, sometimes called risks in this setting, are \(8/40=0.20\) and \(2/40=0.05\). Their ratio is \(0.20/0.05=4\), not 4.75. The odds ratio and the ratio of event proportions answer different questions because odds divide events by non-events, whereas a proportion divides events by the group total. The odds ratio here says the event odds in group 1 are 4.75 times those in group 2; it does not say the event probability is 4.75 times as large.

Why the Cross-Product Formula Works

The cross-product calculation follows directly by expressing each group’s odds in terms of its counts. This short derivation is also a useful way to verify which cell belongs in each position. For group 1, the event proportion is \(a/(a+b)\), so its non-event proportion is \(b/(a+b)\). Their ratio is \(a/b\). Applying the same reasoning to group 2 gives odds \(c/d\). Dividing the first odds by the second yields:

\widehat{\mathrm{OR}} = \frac{\frac{a}{a+b}}{\frac{b}{a+b}} \div \frac{\frac{c}{c+d}}{\frac{d}{c+d}} = \frac{a}{b}\div\frac{c}{d} = \frac{ad}{bc}

The group totals cancel because each odds calculation compares events with non-events within its own group. For example, if the table’s rows are groups and its columns are event status, label the four counts before applying the formula; otherwise it is easy to accidentally compare the wrong odds.

Interpreting Direction and Magnitude

The value 1 is the reference point for an odds ratio. An odds ratio of 1 means that the two groups have equal odds of the event. A value above 1 means higher event odds in the numerator group; a value between 0 and 1 means lower event odds in that group. These statements depend on specifying which group is in the numerator.

Odds ratio for group 1 relative to group 2	Interpretation
Greater than 1	Event odds are higher in group 1 than in group 2.
Equal to 1	Event odds are equal in the two groups.
Between 0 and 1	Event odds are lower in group 1 than in group 2.

For a value below 1, its reciprocal states the comparison in the opposite direction. For example, an odds ratio of 0.5 for group 1 relative to group 2 means group 1’s odds are half of group 2’s; equivalently, group 2’s odds are twice group 1’s. The odds ratio has no units. A clear report names the event, the two groups, the comparison direction, and the fact that the quantity is odds.

An odds ratio describes a contrast, not an absolute event frequency. A ratio alone does not tell a reader whether either group’s event probability is small or large. Reporting the group proportions alongside the odds ratio can make the comparison easier to understand. Also, unless the study design supports a causal conclusion, higher odds in one group should not be described as an effect caused by belonging to that group.

Common Mistakes

Using the event proportion as the odds. A proportion of 0.20 is not odds of 0.20; the odds are \(0.20/(1-0.20)=0.25\). Divide the event count by the non-event count, not by the group total.
Calling the odds ratio a risk ratio. The odds ratio compares event-to-non-event ratios; a risk ratio compares event proportions. As the worked example shows, the two numerical values need not agree.
Reporting direction without naming the reference group. “The odds ratio is 2” is incomplete. State that the event odds in group 1 are twice those in group 2, or specify the reverse comparison.
Interpreting an odds ratio as an absolute change. An odds ratio does not say that the event probability increased by a fixed number of percentage points.
Ignoring zero counts. If a group has no non-events, its sample odds are not finite; if it has no events, its sample odds are zero. The ordinary finite, positive sample odds ratio requires positive event and non-event counts in both groups.

Key takeaway: Convert each group’s event proportion to odds by dividing by the non-event proportion, then divide group 1’s odds by group 2’s. Interpret the result as a relative comparison of odds, with the group order stated explicitly.

Check Your Understanding

For each question, keep the event definition and the comparison direction explicit.

A group has 15 events and 45 non-events. What are the event proportion and the event odds?
Group A has 10 events and 30 non-events; group B has 5 events and 35 non-events. Calculate the odds ratio for A relative to B using both the odds and cross-product forms.
An odds ratio for group 1 relative to group 2 is 0.40. State its direction in context and give the reciprocal comparison.
Why is an odds ratio of 3 not generally equivalent to saying the event probability is three times as large?
What happens to the ordinary sample odds ratio if one group has no observed non-events? Explain using the odds definition.