Odds-ratio can summarize a value that would otherwise take multiple percentage values to explain the result of an intervention. For example, imagine one group of high school students received the mentoring intervention and the other didn't. The results of on-time high school graduation was:
- Group T: 85% graduated; 15% did not graduate
- Group C: 75% graduated; 25% did not graduate
This is a lot of information to communicate. I could reduce it like this too, but still it takes a lot of words:
- Group T: 85% graduated
- Group C: 75% graduated
Odds ratio can express this with one value.
odds ratio= (P1/(1-P1)) / (P2/(1-P2)
To plug in numbers from the graduation example:
odds ratio= (.85/(1-.85)) / (.75/(1-.75)) = 1.8888
For people who are not used to mathematical notations:
- / means divided by (e.g., 30/3 =10)
- Also notice that algorithms usually use rates rather than percentages (not 85 but .85).
I recommend replicating this result using Excel sheet. Enter these values at the left-top corner of an Excel sheet and confirm that the function (A3/B3) will return 1.888...
| 0.85 | 0.75 |
| =(A1/(1-A1)) | =(B1/(1-B1)) |
| =A3/B3 |
For Excel beginners, A1 means the cell defined by Column A and Row 1 of the Excel sheet.
As you do this replication, try to understand the meaning of a resulting value conceptually. Change the values in Excel from original .85 and .75 to other values to understand how the algorithm works and changes the result. Confirm the following:
- Odds ratio can vary from 0 to infinity (=super big values).
- If the odd ratio is greater than 1, the intervention program made a larger difference.
- If the odd ratio is 1, the program did not make any difference. Try to understand the algorithm by entering the same values to P1 and P2.
- If the odd ratio is small than 1, the program made the situation worse.
Finally, one of the advantages of odds ratio is that when you look at the value, you can immediately tell if the treatment group had more favorable result than the comparison group did. If programmed exactly as above, an odds ratio value greater than 1 means the treatment group performed better. If less than 1, the comparison group did better.