Definition of odds
Consider an event with probability \(p\). The odds for this event are defined as:
$$odds=\frac{p}{1-p}$$
That is, the odds indicate how many times is more probable that an event occurs
that it dosen't occur. For instance, if the probability of having a cardiovascular
event at 50 years is 0.3, the odds are 0.3/0.7= 0.43
The odds ratio (OR) is defined as the quotient of the odds of an event in two different siruations.
For instance, if the probability of having a cardiovascular event at 50 years is 0.3, and 0.4 at 60 years,
the OR (60 years vs. 50 years are:
$$OR=\frac{0.4/0.6}{0.3/0.7}=1.56$$
Then, the odds of a cardiovascular disease are 1.56 times greater at 60 years vs. 50 years.
If OR>1 then the putative risk factor is directly correlated with the disease, If OR=1,
then there is no association, whereas of OR<1 then the association is inverse.
In practice, supose you have a group of \(n_1\) cases, and \(x_1\) of them
present the risk In a group of \(n_2\) controls, we observe
\(x_2\) with the risk. Then the OR is computed as:
$$OR=\frac{x_1 \times (n_2 - x_2)}{(n_1 \times x_1)\times x_2 }$$
Computing a CI for OR
In order to obtain a CI \((1-\alpha)\)%, we first calculate:
$$\sigma=\sqrt{\frac{1}{x_1}+\frac{1}{n_1-x_1}+\frac{1}{x_2}+\frac{1}{n_2-x_2}}$$
Then we compute:
$$(a,b) \rightarrow log(OR) \pm z_{1-\alpha/2}\times \sigma$$
Finally, the desired CI for the OR will be:
$$(e^a,e^b)$$
Example
Supose that we have 35 cases and 45 controls. The individulas at risk are 25 and 19 in each
group. Then:
$$x_1=25, n_1=35$$
$$x_2=19, n_2=45$$
$$\sigma=\sqrt{\frac{1}{25}+\frac{1}{10}+\frac{1}{19}+\frac{1}{26}}=0.48$$
$$(a,b) \rightarrow log((26 \times 25)/(19 \times 10)) \pm 1.96 \times 0.48 \rightarrow (0.29,2.17)$$
$$95\% CI \rightarrow (e^{0.29},e^{2.17}) \rightarrow (1.33, 8.78)$$