An odds ratio is a relative measure of effect, allowing the comparison of the intervention group of a study relative to the comparison or placebo group. The odds ratio helps quantify the strength and direction of the relationship between two groups or conditions.
Introduction Odds Ratio
The odds ratio (OR) is a measure of association used in statistics to compare the odds of an event occurring in one group to the odds of it occurring in another group. It is commonly used in case-control studies and logistic regression.
- an OR of 1 indicates no difference between groups,
- an OR greater than 1 suggests higher odds in the first group, and
- an OR less than 1 suggests lower odds in the first group.
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Medical students, students from clinical and psychological sciences, professionals allied to medicine enhancing their understanding and learning of medical literature, and researchers from different fields of life usually encounter Odds Ratio (OR) throughout their careers.
When computing the OR, one would do:
- The numerator is the odds in the intervention arm
- The denominator is the odds in the control or placebo arm= OR
Calculating Odds Ratio
The ratio of the probability of success and failure is known as the odds. If the probability of an event is $P_1$ then the odds are:
\[OR=\frac{p_1}{1-p_1}\]
If the outcome is the same in both groups, the ratio will be 1, implying that there is no difference between the two arms of the study. However, if the $OR>1$, the control group is better than the intervention group while, if the $OR<1$, the intervention group is better than the control group.
The Odds Ratio is the ratio of two odds that can be used to quantify how much a factor is associated with the response factor in a given model. If the probabilities of occurrences of an event are $P_1$ (for the first group) and $P_2$ (for the second group), then the OR is:
\[OR=\frac{\frac{p_1}{1-p_1}}{\frac{p_2}{1-p_2}}\]
If predictors are binary then the OR for $i$th factor is defined as
\[OR_i=e^{\beta}_i\]
Real-Life Examples of Odds Ratio
- Medical Researches
- Consider we are interested in comparing the odds of developing a disease (e.g., lung cancer) in smokers versus non-smokers. Suppose, the OR is 2.5, it means smokers have 2.5 times higher odds of developing lung cancer compared to non-smokers.
- Public Health
- Suppose, we are interested in assessing the effectiveness of a vaccine. For example, comparing the odds of contracting a disease (e.g., COVID-19) in vaccinated versus unvaccinated individuals. An OR less than 1 would indicate the vaccine reduces the odds of infection.
- Social Sciences
- Consider we are interested in studying the odds of students passing an exam based on attendance. For instance, if students who attend extra tutoring have an OR of 3.0 for passing, they have 3 times higher odds of passing compared to those who don’t attend.
- Marketing
- Suppose we need to analyze the odds of customers purchasing a product after seeing an advertisement versus not seeing it. An OR greater than 1 suggests the ad increases the likelihood of purchase.
- Environmental Studies
- Evaluating the odds of developing asthma in people living in high-pollution areas compared to those in low-pollution areas. An OR greater than 1 would indicate higher odds of asthma in high-pollution areas.
The regression coefficient $b_1$ from logistic regression is the estimated increase in the log odds of the dependent variable per unit increase in the value of the independent variable. In other words, the exponential function of the regression coefficients $(e^{b_1})$ in the OR is associated with a one-unit increase in the independent variable.
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