Measures of Association: Odds Ratio

Public health and measures of association

Establishing association between factors and outcome is one of the most important responsibilities of the epidemiologists and health professionals. We established the association of obesity and cardio-vascular diseases, smoking and lung cancer, alcohol and cirrhosis or liver diseases, junk foods and the obesity, vitamin c and scurvy etc. These are just the handful of examples and many relationships are yet to discover. Here are some of the quantitative measures for establishing association.

1. Relative risk/Risk ratio
2. Odds ratio
3. Risk difference
4. Attributable risk
5. Population attributable risk

What is Odds Ratio?

Here, I would like to explore into odds ratio (OR). Odds ratio is one of the measures of association and is frequently used by public health professionals throughout their professional career. However, we are pretty confused about interpretation of odds ratio sometimes in our professional training or career. Sometimes we interpret it as relative risk, sometimes we would like to simplify it in common words as the probability of happening something, which twist the essence of its meaning in other direction.

Example

Let's delve into calculation of odds ratio through our hypothetical case control study. We would like to assess if hormone replacement therapy affects the occurrence of breast cancer in women. In this study we have two comparison groups: case and control.
Cases and Controls: In this case, Cases would be women with breast cancer and controls would be individually or frequency matched women with no breast cancer.
In order to assess the effect of hormone replacement therapy on breast cancer, we ask their history of past or current hormone replacement therapy depending on the objective of the study. We can visualize the following table with their information on breast cancer and hormone replacement therapy. Here, the hormone replacement therapy is the exposure or the factor of our interest.

Odds ratio is basically the ratio of two odds. In this example, OR is ratio of odds that cases were exposed to hormone replacement therapy and odds that controls were exposed to hormone replacement therapy.

Calculation of odds ratio (OR)

There are two things to understand in calculation of odds ratio. First we need to understand about odds. Odds is the ratio of probability of happening of one event in a group to the probability of not happening of that event in the same group so, odds ratio is simply the ratio of two odds. In odds ratio, we take ratio of those odds in two different groups.

Step 1: At first, we calculate the odds in those comparison groups.

Cases:
$$\text{Odds that a case was exposed}=\frac{\text{Probability of exposures in cases (a)}}{\text{Probability of non-exposures in cases (c)}}$$
$$=\frac{\text{Number of exposed among cases (a)}/\text{total number of cases(a+c)}}{\text{Number of not exposed among cases(c)}/\text{total number of cases (a+c)}}$$
$$=\frac{\text{a}}{/}$$

Controls:
$$\text{Odds that a control was exposed}=\frac{\text{Probability of exposures in controls}}{\text{Probability of non-exposures in controls}}$$
$$=\frac{\text{Number of exposed among controls (b)}/\text{total number of controls(b+d)}}{\text{Number of not exposed among controls(d)}/\text{total number of controls (b+d)}}$$
$$=\frac{\text{b}}{/}$$

Step 2: Finally, we calculate the ratios of these two odds and interpret the differential effect of exposures in those two groups.
$$\text{OR}=\frac{\text{a}/\text{c}}{/}=\frac{\text{ad}}{\text{bc}}$$

Interpretation of Odds Ratio

As we already mentioned elsewhere, Odds ratio is utilized to assess the strengths of association between exposure and the outcome. In our hypothetical illustration, we can see we utilized odds ratio to find the strength of association between exposure as hormone replacement therapy on breast cancer (outcome).
If the odds ratio is more than 1 then the odds of having exposed to hormone replacement therapy is higher in cases than in controls which indirectly implies hormone replacement therapy as a risk factor for breast cancer provided that it fulfills other criteria.



In this study Odds ratio is 9.3 which shows that the odds of having exposed to hormone replecement therapy is 9 times higher for cases compared to controls. Odds ratios are not straight forward like percentage or relative risk but it has close approximation to them.

By Pramila Rai