four-field table
| 65.0 |
12.0 |
77.0 |
| 42.0 |
846.0 |
888.0 |
| 911.0 |
54.0 |
965.0 |
\[oddsratio = \frac{ a/b }{ c/d } = \frac{ 65.0 / 12.0 }{ 846.0 / 42.0 }\]
\[SE(lnOR) = \sqrt{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}}\]
\[ \sqrt{\frac{1}{ 65.0 }+\frac{1}{ 12.0 }+\frac{1}{ 846.0 }+\frac{1}{ 42.0 }} = 0.35172\]
\[ln(OR) = ln 0.26891 = -1.31337\]
\[ln(OR) \pm 1.96 \times SE(lnOR)\]
\[ -1.31337 \pm 1.96 \times 0.35172\]
\[ -2.00275 \leq lnOR \leq -0.62399\]
\[ e^{ -2.00275 } \leq OR \leq e^{ -0.62399 }\]
\[ 0.13496 \leq OR \leq 0.5358\]
What is Odds ratio?
Odds ratio is a measure of the Odds of a certain event happening in one group compared to the Odds of the same event happening to another group. A Odds ratio of 1 (or close to 1) means there is no difference
between two groups in terms of their Odds.
A confidence interval is used in Odds ratio to see if the difference between the groups are significant. If the confidence interval for OR (Odds ratio) covers 1 then the difference is not significant.
Since Odds ratio is not normally distributed and a confidence interval requires normally distributed variables you need to take the natural log (ln) for OR and its standard error SE(OR) which is SE(lnOR). Formulas to find the
natural log of the Odds ratio and the standard error easily is provided in the calculation.
Since we are using normally distributed variables (after log transform) we can now use a Z-value to find the confidence interval, Z = 1.96 for a 5% confidence interval.
Example, a study compared people who had suffered a certain type of injury in car accidents with people who were not injured. It was also noted whether they had used seat belts or not.
| Hurt |
Yes |
No |
Sum |
| Yes |
65 |
12 |
77 |
| No |
846 |
42 |
888 |
| Sum |
911 |
54 |
965 |
A = 65
B = 12
C = 846
D = 42
You can use risk ratio for this problem too, but for this example we want to find the odds ratio of not getting injured given you have seatblet.
The confidence interval for lnOR is -2 ≤ lnOR < -0.6239
However you can't interpretet the natural log of odds, in order to get rid of the natural log we take the antilog which is e^(lower interval lnOR) & e^(upper interval lnOR).
The confidence interval after taking the antilog is 0.1349 ≤ OR ≤ 0.535 which means for every person that gets injured without a seatbelt between 0.1349 and 0.535 get injured with a seatbelt. In other words
seatbelts are safe unless you want to get injured. The odds ratio has a statistical significance as the confidence
interval does not cover 1.