According to a study done in 2001, “women in the highest quartile (of number blocks walked per week) remained less likely than women in the lowest quartile to develop cognitive decline (for blocks walked: odds ratio, 0.66 [95% confidence interval, 0.54-0.82]”

If you’ve just read the previous blog about Risk Ratios you may jump in and interpret this as meaning that women with the highest number of blocks walked per week are 34% less likely to develop cognitive decline than women who walk the least number of blocks or 0.66 times less likely.

Unfortunately, you wouldn’t be exactly correct.

In this post, you’ll learn about odds ratios: (OR) how to calculate them, how to interpret them, and the difference between them and risk ratios. In a future post we’ll look at the overestimation of relative risk from odds ratios, and constructing the 95% CI of odds ratios.

Like risk ratios, odds ratios are used to compare disease rates of different groups (or any other outcome) and are also a very common method of analysis in clinical studies.

Odds Ratio Calculation: what are the odds?

If you’re familiar with betting, speaking in terms of odds should be pretty natural. I’m not, so it’s still rather unintuitive to me. Odds ratios look similar to risk ratios but are not exactly the same.

To calculate the odds ratio, we first need to calculate the odds and understand what odds are. It’s a relatively simple calculation and looking at the calculation is a good first step in understanding odds.

Let’s look at a really simple example: a coin toss. Assume the coin is fair.

What is the probability it will be heads?
50% or 1/2 or 0.5

What are the odds it will be heads?
No, not 50%, 1/2, nor 0.5.

The odds are simply the probability that one outcome will occur divided by the probability that the other outcome will occur.
The probability of heads is 0.5, therefore the probability of tails (not heads) is also 0.5. The numbers must add up to 1 or 100%.

So, the odds of heads is just 0.5/0.5 which is the same as 1/1. You may often see this written as 1:1, read as 1 to 1.

This is equivalent to the equation:

which is the same as:

In short hand, we write this as (p = probability):

Note that this is not the same as the initial calculation for the risk ratio, which was the incidence rate or probability of event occurrence. To remind you of the difference, here are the two equations side by side.

In words, the incidence rate is the probability that an event will occur (p).
Odds are the probability that the event will occur (p) divided by the probability that it will not occur (1-p).

But What Does That Mean?

Perhaps some example would help. These are some common examples of odds.

• The odds of getting an ace of hearts in a standard deck of cards is 1:51.
• The odds of any day being Monday is 1:6 (Monday vs all other days).
• The odds of a fair die coming up 5 or 6 is 2:4 or 1:2.

Notice a pattern? Imagine we flipped a fair coin 100 times. Theoretically, 50 times it would be heads and 50 times it would be tails. That’s 50/50 or 1:1, but the probability of it being heads is 1/2. I hope it’s becoming clearer. Can you calculate the probabilities for the above examples? Try it.

The probabilities are:

• probability of getting an ace of hearts: 1/52
• probability of any day being Monday: 1/7
• probability of a fair die coming up 5 or 6: 2/6 or 1/3

See the difference? I hope so!

Also note that when talking about odds, we usually don’t  say the odds of a fair coin coming up heads is 1. We include the denominator of the fraction. So we say 1 to 1, or 1:1, implying 1/1.

Odds Ratios

So that’s how to calculate the odds, the first step in calculating the odds ratio. For the odds ratio we have two groups to compare. It’s just the odds of group one divided by the odds of group two.

We can substitute the other equations from above:

where p1 and p2 are the probabilities from group 1 and 2.

Notice how this is different from risk ratios. Here are the two equations side by side.

So the difference in the two is that odds ratios compare two odds, and risk ratios compare to incidence rates or probabilities.

An Alternative Calculation

Often in clinical studies, a table is displayed which allows you to calculate the odds ratio in a slightly different, but equivalent way.  Here’s what the table could look like:

	Group 1	Group 2
Event happens	242	348
Event doesn’t happen	1214	1102

We have two groups, 1 and 2. And we have the number of subjects in each group where the event happens and where the event doesn’t happen. In group 1, 242 subjects have the event occur but in group 2, 348 have the event occur. We could calculate the odds ratio using the above formulas, but remember you’d need to calculate the total for each group first, so that you can get the probability of event occurrence in each group.

Here’s a quicker way:

This is the 0.63 odds ratio from the study with women and the number of blocks walked that we started this post with. Those are the real numbers from the study.

The Generalization

If we imagine that the table looks like this:

	Exposure	Non-exposure
Event happens	a	b
Event doesn’t happen	c	d

Then the general equation becomes:

This is the same as:

Lets Check Our Work

To calculate the 0.63 odds ratio the long way, we need the total number of women in each group, which allows us to calculate the probability for each group. Here’s the table again, with the total for each group.

	Group walking most	Group walking least
Cognitive decline	242	348
No Cognitive decline	1214	1102
Total	1456	1450

Here are the steps to calculate the odds ratio:

Probability of cognitive decline for group walking most: 242/1456 = 0.166

Probability of cognitive decline for group walking least: 348/1450 = 0.240

Odds of cognitive decline for group walking most: 0.166 / (1 – 0.166) = 0.199

Odds of cognitive decline for group walking least: 0.240 / (1 – 0.240) = 0.316

Odds Ratio: 0.199 / 0.316 = 0.63

Looks good!

Remember that we could invert the odds ratio.

Odds Ratio: 0.316 / 0.199 = 1.58

And then we would just have to invert our interpretation.

Interpretation of Odds Ratio

Now we know how to calculate the odds ratio, but what does it mean?

To understand an odds ratio, first you must understand what the odds means and how to interpret it. I talked about the odds and how to calculate it above, but lets reiterate:

Remember that the odds is not the probability that an event will occur. It is the probability that the event will occur or p divided by the probability that the event will not occur 1-p.

The odds ratio is comparing 2 odds.

The risk ratio is comparing 2 incidence rates or probabilities.

If you understand what odds are, you should be able to understand what odds ratios are, likewise with risk ratios. The two are easily confused and are often misinterpreted, even in the scientific literature.

Note well: We usually cannot interpret the odds ratio in the same way as a risk ratio. An odds ratio of 3 does NOT mean that you are 3 times more likely to have the event occur if you have an exposure to whatever factor is being studied. And an odds ratio of 0.6 does not mean that you would have a 40% lower risk of having the event occur if you have exposure to whatever factor is being studied. Your risk is lower, but not 40%!

Here’s the proper way to interpret the odds ratio, using our 0.63 odds ratio above.

The odds ratio of 0.63 indicates that the odds of developing cognitive decline for women in the highest quartile of number of blocks walked are 0.63 times the odds of women who are in the lowest quartile of number of blocks walked.

This is not the same as saying that women in the highest quartile of number of blocks walked are 0.63 times as likely to develop cognitive decline than women in the lowest quartile. And still to me the odds ratio is not very intuitive. I really have to think about what it means.

Why the Confusion?

The reason these two ratios get confused is that they do have some similar properties. If the ratio = 1, then exposure to the factor being studied does not affect the outcome (the event occurring). If the ratio is greater than 1, then exposure is positively associated with the outcome. If the ratio is less than 1, then then exposure is negatively associated with the outcome.

And if the outcome prevalence is less than about 10%, the odds ratio actually is approximate to the risk ratio and can be interpreted in the same way.

But remember, the “odds ratio always overestimates the risk ratio, and this overestimation becomes larger with increasing incidence of the outcome.”

It is generally agreed that whenever possible you should use the risk ratio instead of the odds ratio because is is much more intuitive and the results are easier to interpret.

By now you may be wondering:

Why would anyone use odds ratios?!

Good question. And if you search the scientific literature you will find a lot of articles discussing this issue. The answer to this could get very long so I will just comment briefly without going into much detail.

1. Study Designs – there are some which calculating the risk ratio would be meaningless. In these, the odds ratio must be used.
   • OK, there are actually ways to correct the RR for these situations…but traditionally it has not been done
2. Logistic Regression – this statistical analysis can be used to adjust for confounders (co-variables), and the output of it is an estimate of the odds ratio,  e.g. when you see in a study “odds ratio adjusted for age and economic status.”
   • OK, there are some procedures to do this with RR as well, but not traditionally…
3. Tradition / It’s what’s taught / The software does it
   • Not good reasons, but it takes time to change
4. Some people (I suppose sports fanatics, gamblers, bettors and bookies) are used to odds ratios and interpret them easily.

So for now, we must understand how to use and interpret both odds ratios and risk ratios.

Summary

• Odds ratios are a measure of association between an exposure and an outcome.
• Odds ratios compare two odds, usually that of the exposed and non-exposed group.
• Odds ratios and risk ratios are similar but different, and when the incidence rate is less than ~10%, can be interpreted in the same way.
• An odds ratio of 1 means that there is no association, >1 is a positive association, <1 is a negative association.
• An odds ratio of 4 does not mean that exposure is associated with a 4-fold increase in event occurrence, whatever the scientific article says. One review noted that 26% of the odds ratios reported in a certain scientific journal for one year were “interpreted as a risk ratio without explicit justification.”

I will finish with a quote from a great textbook, “Categorical Data Analysis” by Alan Agresti. If you take nothing else away from this post except an understanding of the following concept, then I’ll be pleased.

when [the odds ratio] = 4, the odds of success in row 1 (the numerator), are four times the odds in row 2. This does not mean that the probability [p₁] = 4*[p₂]; that is the interpretation of relative risk.

In other words, an odds ratio of 4 does NOT mean that you are 4 times more likely to have the event occur if you have an exposure to whatever factor is being studied. It means the odds of having the event occur are 4 times higher.

Remember that and you’ll be ahead of the game.

And keep on walking.