Odds ratio

The odds ratio is a technical term often used in statistics, and especially in medical statistics and epidemiology. The odds ratio is the ratio of the relative incidence (odds) of a target disorder in the experimental group relative to the relative incidence in a control group. It reflects how the risk of having a particular disorder is influenced by the treatment. An odds ratio of 1 means that there is no benefit of treatment compared to the control group.

The odds ratio is a difficult concept for non mathematicians, and recommendations for how to teach its use are available.

For a mathematician, the relative incidence of a disorder would be called the 'odds on having the disorder'. An odds ratio is the ratio between the odds on some event under two circumstances.

If we have A individuals with the disorder and B without in the experimental group, and C individuals with the disorder and D without in the control group, the empirical odds ratio is

$Odds{\text{ }}ratio={\frac {A*D}{B*C}}$

Effect of t'ment on disorder

	Disorder
	Yes	No
Experimental treatment	A	B
Control treatment	C	D

Example

This example is from the Titanic (example from Power):

Fate on the Titanic

	Survival
	Yes	No
Males	142	709
Females	308	154

Male passengers:
142 survived, 709 died

• Odds of survival = 142/709 = 0.20
• Probability (risk or chance) of survival = 142/(142+709) = 17%

Female passengers:
308 survived, 154 died

• Odds of survival = 308/154 = 2.00
• Probability (risk or chance) of survival = 308/(308+154) = 67%

Comparison:

• Odds ratio (OR) for survival = 0.20/2.00 = 0.10
• Relative risk (RR) for survival = 17%/67% = 0.25

The odds on a male passenger surviving the Titanic disaster was 1 to 5 against surviving. The odds on a female passenger surviving the Titanic disaster was 2 to 1 in favour of surviving. The odds ratio for survival between men and women is (1/5) / (2/1) = 1/10

Rather than classifying the passengers on the Titanic as male or female, and comparing their odds on survival, one could classify passengers as survivors or not survivors, and compare the odds on being male or female. The odds ratio for sex (male versus female) between survivors and non-survivors is also 0.10. This follows from the algebraic identity $(AD/BC)=(A/B)/(C/D)=(A/C)/(B/D)$.

The identity between the odds ratios is much used in epidemiology, especially in retrospective case-control studies. Consider for example the relation between lung cancer and exposure to asbestos. Instead of sampling from the two populations of people who are exposed to asbestos and people who have not been exposed to asbestos, and studying the odds ratio for lung cancer between the two populations defined by exposure status, one could just as well sample from the two populations of people who got lung cancer and people who did not get lung cancer, and study the odds ratio for asbestos-exposure between the two populations defined by health status. This is extremely valuable when studying rather illnesses which quite rare, even in the poluation exposed to a risk factor.

Interpretation

The odds ratio is generally used to measure the association between a risk factor and disease. However, using the odds ratio to measure the ability of a risk factor to diagnose disease is problematic. The odds ratio should be at least 16 to have reasonable diagnostic ability.

The odds ratio is similar to the relative risk ratio. The two ratios will be numerically similar if the rates in the two groups being compared are both similar and both less than 20% to 30%.

The odds ratio may be the most stable ratio across different prevalences.

The odds ratio may be used to derive the number needed to treat:

For odds ratios less than 1:

$NNT={\frac {1-CER*(1-OR)}{CER*(1-OR)*(1-CER)}}{\mbox{, where CER is control event rate and OR is odds ratio}}$

For odds ratios greater than 1:

$NNT={\frac {1+CER*(OR-1)}{CER*(OR-1)*(1-CER)}}{\mbox{, where CER is control event rate and OR is odds ratio}}$