# Odds And Risk Ratios

Before we move into our example study we need to spend a few minutes on the difference between risk and odds. It can be difficult to understand, but it is important for interpretation of studies.

When research presents risk, what is being presented is the number of people with the selected characteristic divided by the total number of people.

When research presents odds, what is being presented is the number of people with the selected characteristic divided by the number of people without the characteristic.

Visualize this difference by thinking of a set of dice. With six sides, each having a different number between 1-6, your risk of rolling a 1 is 1 divided by six which is 0.167. But your odds of rolling a 1 is 1 divided by 5 (the total number of non-one options) which is 0.20.

## A Worked Example

So, if you have 1000 pregnant women, and 45 of them needed either vacuum or forceps assistance during delivery, you get the following numbers:

Risk of assisted delivery: 0.045 or 4.5%

Odds of assisted delivery: 0.047 or 4.7%

But look what happens when you have a more common event. What if you have 1000 pregnant women, and 900 of them used pharmacologic pain relief during labor?

Risk of pharmacologic pain relief use: 0.90 or 90%

Odds of use of pharmacologic pain relief use: 9 or 900%

Do you see how not understanding the difference could cause misinterpretation of the research results?

## A Literature Example

Often, the results present the ratio of the risk or odds between two groups. How do these differ? Let’s look at this study, http://www.sciencedirect.com/science/article/pii/S0266613803000524 A prospective randomised trial on the effect of position in the passive second stage of labour on birth outcome in nulliparous women using epidural analgesia, to find out.

This study had 107 participants, and they compared those who used a lateral position to those who used a supported sitting position.

The risk of instrumental birth (the rate) was 33% for lateral and 52% for sitting. To determine the risk ratio, we divide .33 by .52 and arrive at a risk ratio of 0.64. This is interpreted as a 36% lower risk for instrumental birth with a lateral positon. 1-0.64 = 36.

We could present the risk ratio as .52 divided by .33 and find 1.58, or a 58% increased risk for instrumental birth with a sitting positon. So why do they report an odds ratio of 2.3 for instrumental delivery? There are two reasons.

First, this odds ratio was calculated using logistical regression, not a chi-square. That means it was able to account for the risk caused by other things such as the baby’s position. When you control for other factors, the risk caused by the factor of interest may increase or decrease.

Secondly, odds is not the same as risk.

In the full paper, we see 49 participants used the lateral position, and 16 of those had an instrumental delivery. The risk is 16 divided by 49 or 0.33. But the odds is 16 dived by 49-16 (33) which is 0.48.

We also see 58 participants used the sitting position, and 30 of those had an instrumental delivery. The risk is 30 dived by 58 or 0.52. The odds is 30 dived by 58-30, or 28 which is 1.07.

Notice that while risk is easily translated into a percent because it is a rate and the numerator will never exceed the denominator. But odds is not a rate, it is odds. The numerator can exceed the denominator (as it does in this case), but this does not mean 107% of the women experienced an instrumental delivery – that wouldn’t make sense.

Think for a minute about a coin flip. A coin has two sides – heads and tails. Your “risk” of flipping heads is 1 divided by 2 or 50%. But your Odds of flipping heads is 1 divided by 2-1, or 1. You have even odds of flipping either a heads or a tails.

So in our example, an odds of 1.07 means you have slightly better odds of having an instrumental delivery than not having an instrumental delivery.

## Interpreting the Ratios

But what happens when we look at the ratios – comparing one group to the other.

The Risk Ratio (sometimes called the relative risk) was calculated as the difference between risk of instrumental delivery in lateral position and the risk of instrumental delivery in sitting position, which was .33 divided by .55 or 0.64. This can be interpreted as a 36% lower risk of instrumental delivery in a lateral position than a sitting position.

The Odds ratio is calculated as the difference between the odds of instrumental delivery in lateral position and odds of instrumental delivery in sitting position, which is .48 divided by 1.07 or 0.45. How do we interpret this? The odds ratio helps us determine if the woman’s positon during labor is associated with instrumental delivery. Remember when the odds ratio is 1, the exposure does not affect the odds of an outcome. When the odds ratio is higher than 1, the exposure is associated with higher odds of an outcome. When the odds ratio is lower than 1, the exposure is associated with lower odds of an outcome. An odds ratio of 0.45 means there is 0.45 lower odds of instrumental delivery in the lateral position than in a sitting position.

If we flip the question, and want to look at the risk ratio and odds ratio from the perspective of the sitting position, we get:

Risk Ratio: .55/.33 = 1.67

Odds Ratio: 1.07/.48 = 2.23

The risk of an instrumental delivery is 67% higher in a sitting position than in a lateral position, however the odds of instrumental delivery are 2.23 higher.

You may see odds ratio reported as an estimate of the risk ratio for rare events. However in general you cannot interpret an odds ratio as a relative risk, and to report the risk of instrumental delivery as 123% higher would be an alarming over-estimate of the risk.