Risks and odds

Author: Lauren Bishop

Comparisons between what is observed and what is expected are very important when describing and analyzing epidemiological data.

Risks and odds are two measures of association that quantify the risk of occurrence of a health event.

Measures of association in epidemiology are often reported as ratios so that we can compare the risk or odds between two groups. Two such ratios are discussed in more detail below.

Risk ratio

A risk ratio, or the relative risk, compares the risk of occurrence of a health event during a specific time interval for one group with the risk of occurrence of a health event during that same time interval for a second group.

This should sound familiar because a risk ratio is calculated by dividing the incidence proportion for one group by the incidence proportion for the second group!

If we can calculate an incidence proportion, why do we need the risk ratio? For example, maybe we think the incidence of bubonic plague in neighborhood A seems quite high. Is this observed incidence proportion higher than what we expect, or higher than it is in other neighborhoods?

We could use a risk ratio to compare the observed proportion in neighborhood A with neighborhood B, which represents the expected level of plague.

The two groups in a risk ratio are commonly differentiated in epidemiology according to a specific risk factor (i.e., whether they were exposed or unexposed); however, the two groups may also differ according to a demographic factor (e.g., born male versus female).

To calculate a risk ratio to compare the risk of occurrence of a health event between two groups, we need a two-by-two table.

Example


	Cholera	No Cholera	Total
Southwark	28,200	139,800	168,000
Lambeth	600	18,600	19,200
Total	28,800	158,400	187,200

Let us revisit the European past. This time, we will go to 19^th century London to apply a classic example to calculate a risk ratio.

In a cholera outbreak in London in 1853, 28,200 of the 168,000 individuals served by the Southwark water supply company developed cholera, compared to 600 of the 19,200 individuals served by the Lambeth water supply company who developed cholera.

To compare the risk between these two groups, we will summarize them in a two-by-two table, with two rows for the exposure (water supply company) and two columns for the outcome (cholera incidence).

First, we will calculate the proportion of cholera incidence for each of the water companies:

Southwark: 28,200/139,800 = 0.202 = 20.2%

Lambeth: 600/18,600 = 0.032 = 3.2%

Then, we calculate the ratio of these two proportions:

Risk ratio = 20.2/3.2 = 6.3 From these data, we could conclude that individuals served by the Southwark water supply company were 6.3 times more likely to develop cholera than those served by the Lambeth water supply company.

Note
A risk ratio > 1.0 indicates that the exposed group (numerator) has an increased risk for the health outcome, as in the above example. A risk ratio < 1.0 indicates that the exposed group has a decreased risk for the health outcome. In other words, the exposure might be protective against the occurrence of the health outcome. A risk ratio = 1.0 indicates that the risk for the health outcome is the same for both groups.

Odds ratio

An odds ratio also quantifies the risk of occurrence of a health event, but it compares two categories: those who were exposed who have or do not have the health outcome, and those who were unexposed who have or do not have the health outcome.

An odds ratio is also calculated using a two-by-two table, but instead of comparing across rows, we compare the numerator in one column to the denominator in the other.

That is why the odds ratio is sometimes called the cross-product ratio: the calculation creates an X in the two-by-two table.

Example


	Salmonella	No Salmonella	Total
Exposed	a=132	b=1,900	2,032
Unexposed	c=10	d=2,100	2,110
Total	142	4,000	4,142

The Swedish public health agency is investigating an outbreak of food-borne infections potentially caused by Salmonella bacteria. They trace the outbreak to eggs served in a popular café in Södermalm.

The public health agency obtains data for everyone who ate in the café during the weeks immediately before and after the oubreak began, including whether they consumed food made with eggs (exposure) or presented with a Salmonella infection (outcome).

Our two-by-two table is the same as the one we used for the risk ratio. Here, we have added letters to help you visualize the cross-product calculations.

To calculate the odds ratio, we first multiply a x d and b x c to get the cross products, and then divide the product of a x d by the product of b x c:

(132 x 2,100)/(1,900 x 10) = 14.6 Those who ate food containing eggs at this particular café during this two-week period were 14.6 times more likely to have contracted a Salmonella infection compared to those who did not eat eggs.

Note
If the health outcome in question is rare, the odds ratio will be quite similar to the risk ratio. This is especially convenient for case-control studies, where the odds of exposure among cases are compared to the odds of exposure among controls.
For case-control studies, we often do not know the size of the population from which the cases were drawn. This means that risks and rates (and therefore risk and rate ratios) cannot reliably be calculated, and so we instead calculate odds ratios.