11.1 Probability and odds
11.1.1 Reminder of probabilities
Consider the following table:
| Burnout | No burnout | Total | |
|---|---|---|---|
| Musician | 20 | 4 | 24 |
| Non-musician | 10 | 8 | 18 |
| Total | 30 | 12 | 42 |
You may recall that this is a 2x2 contingency table, which we have seen before in a chi-square context. Using this table, we can work out the probability of certain events or outcomes.
If we selected someone randomly from this table, for example, what is the probability that they would be a musician? Well, we can see that there are 24 musicians from the sample of 42, so we could simply say:
\[ P(Musician) = \frac{24}{42} = 0.57 \]
Likewise, what is the probability that someone is burnt out? That would simply be:
\[ P(Burnout) = \frac{30}{42} = 0.71 \]
What about the probability that someone is a musician and and burnt out? We could denote this as follows:
\[ P(Musician \cap Burnout) = \frac{20}{42} = 0.47 \] What about the probability that someone is burnt out, given they are a musician? This would be a conditional probability, where we are finding a probability of something on the condition that the person is burnt out. There are 24 participants who reported burnout, so our calculation would be as follows:
\[ P(Burnout | Musician) = \frac{20}{24} = 0.83 \]
11.1.2 Odds
Now, let’s talk about odds. Odds are simply the likelihood of a particular outcome occuring, and is calculated as the probability that an event will occur, divided by the probability that the event will not occur. In other words, if the probability of an event is denoted as \(A\), the probability of event \(A\) not occuring is \(1-A\). We can then calculate the odds as:
\[ Odds = \frac{A}{1-A} \]
Let’s return to our example above, and print out the table again for ease of reference.
| Burnout | No burnout | Total | |
|---|---|---|---|
| Musician | 20 | 4 | 24 |
| Non-musician | 10 | 8 | 18 |
| Total | 30 | 12 | 42 |
What are the odds of burnout in the musician group? To do this, we need to find the probability of burnout given they are musicians, and divide that by the probability of no burnout given they are musicians. The odds of burnout given that someone is a musician is as we saw above:
\[ P(Burnout | Musician) = \frac{20}{24} \]
And the probability of someone not burning out given that they are a musician must therefore be:
\[ P(No \ burnout | Musician) = \frac{4}{24} \] Now we can divide these two probabilities as follows:
\[ Odds = \frac{20/24}{4/24} = \frac{20}{4} = 5 \]
What this means is that musicians are 5 times as more likely to experience burnout than not experience it.
One more example. What are the odds of burnout in the non-musician group? Using the same principles as above, we can calculate this as follows.
\[ P(Burnout | Nonmusician) = \frac{10}{18} \] \[ P(No \ burnout | Nonmusician) = \frac{8}{18} \]
\[ Odds = \frac{10/18}{8/18} = \frac{10}{8} = 1.25 \]
So even non-musicians are 1.25 times more likely - or, in other words, 25% more likely - to increase burnout than not experience it.
11.1.3 Odds ratios
Now we can take a look at the odds ratio. The odds ratio describes how likely one outcome is given an exposure/group, compared to another exposure/group. The odds ratio is calculated by dividing the odds of event A by the odds of event B. The resulting value gives an indication of how much more likely event A is compared to event B, given differences in exposure.
We have already calculated two sets of odds ratios:
- The odds that a musician experiences burnout; \(Odds = 5\)
- The odds that a non-musician experiences burnout; \(Odds = 1.25\)
We can now calculate an odds ratio for how likely a musician is to experience burnout compared to a non-musician. We simply divide the two sets of odds:
\[ OR = \frac{Odds(A)}{Odds(B)} \]
\[ OR = \frac{5}{1.25} = 4 \]
An odds ratio of 4 indicates that a musician is 4 times as likely to experience burnout compared to a non-musician. Heavens!