

CURRICULUM IN CARDIOLOGY  STATISTICS 

Year : 2016  Volume
: 2
 Issue : 1  Page : 4951 

The odds ratio: Principles and applications
Aakshi Kalra
Institute of Economic Growth, Delhi University, New Delhi, India
Date of Web Publication  26May2016 
Correspondence Address: Aakshi Kalra Institute of Economic Growth, Delhi University, New Delhi India
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/23955414.182992
The odds ratio (OR) is a simple tool, widely utilized in clinical research. As a simple statistic, it can be hand calculated to determine the odds of a particular event or a disease, and the information provided can be useful for understanding the results of a treatment/intervention. This article discusses the application of OR with examples and shows a simple way of performing the test using an online calculator. Keywords: Hand calculation, odds ratio, online calculators
How to cite this article: Kalra A. The odds ratio: Principles and applications. J Pract Cardiovasc Sci 2016;2:4951 
The odds ratio (OR) is a measurement of association which compares the odds of disease or an event of those exposed to the odds of disease/event in those unexposed. It serves to determine the relation between exposure and outcome.
OR is a numeric expression of the strength of association between cause and an effect when both are expressed as categorical variables. As a rule of thumb, larger the OR, more will be the effect on the outcome.
The odds of an outcome are the ratio of occurrences in the presence of an event to the occurrences in the absence of an event. While odds can be calculated for any event, odds raitio is always with reference to a two different outcomes (like cases and controls).
We wish to answer a question like “what is the likelihood of a person to face an outcome, when a medical condition is present, compared to those where the medical condition is not present.” This is then answered as the ratio of the odds of outcome under the two conditions.
The OR can be used to evaluate whether the odds of an event is same for two groups in a study sample. The OR measures the ratio of the odds that an outcome will occur given a particular exposure to the odds of the outcome occurring in the absence of the exposure.^{[1],[2],[3],[4]}
The OR can also be a measure of the effect size.
Formula
 OR = (odds of disease in exposed)/(odds of disease in the nonexposed)
A contingency table for the OR is created as shown in [Table 1].
The OR is calculated as the odds of being exposed if a case (a:c) divided by the odds of being exposed if a control (b:d) and by transformation, can be depicted as above.
Usually, beginners find it confusing between probability and odds. To simplify, the difference can be explained from the following example:
If the probability of a heart attack in the chronic heavy smoker is 0.80 (80%), then the probability that the event will not occur is 1–0.80 = 0.20, or 20%. However, the odds of an event represent the ratio of the (probability that the event will occur)/(probability that the event will not occur) for a given exposure/risk factor.
Use   
ORs are most commonly used in case–control studies; however, it can be used for other study designs with few modifications and assumptions. In a case–control study, the group of exposed individuals consists of a few cases and few controls, and they are not usually a representative sample of all exposed persons in the population. The same holds true for unexposed group. Here, the OR compares the chances of a case being exposed to the risk factor with the odds of control being exposed. OR can be used to determine whether a particular exposure is a risk factor for the outcome of interest, and to compare the magnitude of various risk factors for that outcome.^{[4]} Analysis of this sort has been commonly employed in largescale studies as well.
Taking an Example   
Supposing twenty children from a school went to a restaurant and eight of them ate a new type of pizza. Of the total school children, seven fell ill. Now, the question arises how to test if it was due to the new pizza they ate or not?
To solve this, OR can be used [Table 2].
What Does Value of Odds Ratio Means?   
 OR of 1: There is no difference between the groups; i.e., there would be no association between the exposure (pizza) and the outcome (being ill)
 OR of >1: Suggests that the odds of exposure are positively associated with the adverse outcome compared to the odds of not being exposed
 OR of <1 Suggests that the odds of exposure are negatively associated with the adverse outcomes compared to the odds of not being exposed.
Interpretation of Odds Ratio   
In the example above, those who ate the pizza (exposure) were 8.3 times more likely (OR = 8.33) to be ill (outcome), compared to those who did not eat the pizza.
Testing the Significance of the Odds Ratio   
Since the OR is calculated based on sample data, we need to test for the significance of the observed value comparing it with a hypothetical value.
Testing the significance of the OR is done as per the calculations below but with available online calculators and software such as STATA, SPSS, and R, all these calculations are now done on computers. The significance can also be tested by a Chisquare test.^{[5],[6],[7]}
The OR standard error and 95% confidence interval (CI) calculations are shown below:
With the standard error of the log OR is:
and 95% CI:
95% CI = Exp (In(OR) −1.96× {In(OR)}) to exp (In(OR) +1.96× {In(OR)})
Basically, the precision of the OR is estimated by the 95% CI. The 95% CI is often used as a proxy for the presence of statistical significance if it does not overlap the null value (e.g., OR = 1).^{[7],[8],[9],[10],[11],[12]}
Calculation of OR can be done easily with the help of an online calculator as discussed below.
A Free Online Calculator Is Available at the Site below   
There are many calculators available online which are free to use. One of them is MedCalc; following is the link for this calculator:
https://www.medcalc.org/calc/odds_ratio.php
For computation, values of a, b, c, and d (as per the formula of OR) are required. Importantly, along with OR, this calculator provides value of 95% CI, Zstatistic, and significance level.
Following are few examples using this calculator:
 An example of change of mortality in coronary artery disease with highdose statins [Figure 1]
 The second example shows testing the results of smoking in a group [Figure 2].
 One of the important parts of understanding scientific research is effective analysis and interpretation, and OR is one such statistical tool. This article covered basics about ORlearning how it is calculated, principles behind, application, and interpretation.
Summary of Some Points   
 The numerator is the odds in the intervention/treatment arm
 The denominator is the odds in the control arm
 If the outcome is the same in both groups, the ratio will be one, which implies there is no difference between the two arms of the study
 If the OR is >1 the control is better than the intervention
 If the OR is <1 the intervention is better than the control (protective effect).
 The 95% CI is often used as a proxy for the presence of statistical significance if it does not overlap the null value (e.g., OR = 1).
Conclusion   
Odds are the ratio of probability of an event occurring in one group to probability of the event not occurring. An OR is the odds of the event in one group, divided by the odds in another group not exposed. The OR is used in case–control studies and logistic regression analysis. This article has introduced the reader to some of the concepts of OR and ways of calculated the OR.
Financial support and sponsorship
Nil.
Conflicts of interest
There are no conflicts of interest.
References   
1.  Altman DG. Practical Statistics for Medical Research. Boca Raton, FL: Chapman & Hall/CRC; 1991. p. 611. 
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3.  Deeks JJ, Higgins JP. Statistical Algorithms in Review Manager 5. Statistical Methods Group of the Cochrane Collaboration; 2010. p. 111. 
4.  Szumilas M. Explaining odds ratios. J Can Acad Child Adolesc Psychiatry 2010;19:2279. 
5.  Parshall MB. Unpacking the 2×2 table. Heart Lung 2013;42:2216 
6.  Sheskin DJ. Handbook of Parametric and Nonparametric Statistical Procedures. 3 ^{rd} ed. Boca Raton: Chapman and Hall/CRC; 2003. 
7.  Cornfield J. A method of estimating comparative rates from clinical data; applications to cancer of the lung, breast, and cervix. J Natl Cancer Inst 1951;11:126975. [ PUBMED] 
8.  Mosteller F. Association and estimation in contingency tables. J Am Stat Assoc 1968;63:1. 
9.  Edwards AW. The measure of association in a 2×2 table. J R Stat Soc Ser Gen 1963;126:109. 
10.  
11.  Morris JA, Gardner MJ. Calculating confidence intervals for relative risks (odds ratios) and standardised ratios and rates. Br Med J (Clin Res Ed) 1988;296:13136. 
12.  Viera AJ. Odds ratios and risk ratios: What's the difference and why does it matter? South Med J 2008;101:7304. 
[Figure 1], [Figure 2]
[Table 1], [Table 2]
