Do you know how to calculate odds? Well, we’re here to show you just that in part 4 of our
ChalkTalk series on statistics.
We’ll start by explaining the concept of odds and look at the differences between risk
and odds. At this stage, it would be very useful to have watched our previous ChalkTalk
episode on risk and relative risk.
Let’s begin with an example. Imagine playing a dice game and you need to roll a six to
win. The probability of winning is 1 out of 6 because there are six ways the dice can
land. But, you can also look at this from a different perspective. You could also say:
There’s only one number on the dice needed to win, a six, and five numbers that’ll
make you lose. So, the odds of rolling a 6 with the dice is 1 to 5. It’s calculated
by dividing the probability of an event occurring by the probability of the event not occurring.
Generally, odds are a second means of showing how often an event occurs. However, it’s
important to make a clear distinction between these two parameters. In both cases, the occurrence
of an event is related to different measurements: Probability refers to the total number of
events, whereas odds relates to the number of alternative events. This can be seen in
our example: The probability of winning is 1 out of 6, or 0.17, and the odds value is
1 to 5, or 0.2.
Let’s go back to our fictional study on lung cancer and take a look at the group of
smokers. Of the 100 smokers, 20 developed lung cancer, whereas the remaining 80 smokers
didn’t.
The risk of smokers developing lung cancer is calculated as the proportion of 20 smokers
with lung cancer from the total number of 100 smokers, that is, 20 divided by 100, which
is 0.2.
The odds of smokers developing lung cancer is calculated by dividing 20 smokers WITH
lung cancer by 80 smokers WITHOUT lung cancer. So, 20 divided by 80 is 0.25. Therefore, the
difference between risk and odds lies only in the comparative value, that is, the denominator
of the fraction.
The difference between risk and odds can also be memorized using a pie chart. To calculate
the risk, a slice of cake is divided by the entire cake. To calculate the odds value,
a slice of cake is divided by the rest of the cake that remains on the cake platter.
This is also reflected in the wording. The risk of smokers can be described as 20 out
of 100. In contrast, the odds represent a chance of 20 to 80.
The odds of developing the disease is calculated by dividing the number of exposed individuals
with the disease by the number of exposed individuals without the disease.
Now if you compare the odds and risk values in our fictional study, you’ll see that
the results 0.2 and 0.25 are relatively close to one another. But that’s not usually the
case. Let’s assume in our example that not 20 of the 100 smokers developed lung cancer
but 99. Now, only one exposed individual didn’t develop lung cancer. If you’d like, pause
the episode to calculate the new odds and risk values of the smokers…... So, what
values did you come up with? The risk of smokers developing lung cancer is 99 divided by 100,
which is 0.99. In contrast, the odds value is calculated by dividing 99 by 1, which is
99.
In this example, you can see that the risk and odds can also differ greatly. Therefore,
an important aspect to keep in mind is that for rare diseases, the risk and odds values
are quite similar. In contrast, the more common the disease in the group examined, the larger
the difference between the odds and risk values.
This example gives you an insight into the different range of values for the risk and
odds. The risk is the number of individuals with the disease divided by the total number
of individuals at risk of developing the disease. Since this is a proportion, the risk can’t
exceed the value 1. Therefore, the risk is always a value between 0 and 1.
In contrast, the odds value is calculated by comparing individuals with the disease
to individuals without the disease. So in this case, it isn’t a proportion. If the
number of individuals with the disease is higher than the number of individuals without
the disease, the odds value can exceed 1.
Now, you might be wondering whether both the risk and odds values can be expressed as a
percentage. In our example, the risk of smokers developing lung cancer is 99 percent. The
risk can be expressed as a percentage because it describes the number of patients with the
disease from the total number of individuals in the study. In contrast, odds compares individuals
with and without the disease. It’s not a proportion, so it can’t be expressed as
a percentage. If we would express the odds value in the case shown here as a percentage,
the answer would be 9,900 percent, which is incorrect. So, the odds value is only expressed
as 99.
Now, we’d like to show you some additional information on risk and odds values that may
come in handy.
Risk and odds values can be mutually converted. The odds of developing a disease is identical
to the risk divided by 1 minus the risk. In our example, this would be 0.2 divided by
1 minus 0.2. So, 0.2 divided by 0.8 equals 0.25. As you can see, you’ll get the same
odds value previously calculated.
Ok, let’s briefly summarize what we’ve learned about odds in this episode.
By using a pie chart, you can see that the risk of developing the disease is calculated
as a proportion of the group examined. In contrast, the odds value is calculated by
dividing the number of individuals with the disease by the number of individuals without
the disease. The risk and odds values are very similar when it comes to rare diseases.
The more common the disease in the group examined, the larger the difference between the odds
and risk values.