# Odds Ratio

# ODDS RATIO

The odds ratio (OR) provides a measure of the strength of relationship between two variables,

*Table 1*

Frequencies in a 2 × 2 Table. | ||

OUTCOME +ve | OUTCOME –ve | |

source: Courtesy of author. | ||

Exposure (outcome positive) | a | b |

Exposure (outcome negative) | c | d |

most commonly an exposure and a dichotomous outcome. It is most commonly used in a case-control study where it is defined as "the ratio of the odds of being exposed in the group with the outcome to the odds of being exposed in the group without the outcome." In the standard 2×2 epidemiological table, this ratio can be expressed as the "cross-product" (ad/bc), as seen in Table 1.

This concept can be extended to a situation with multiple levels of exposure (e.g., low, moderate, or high exposure to an environmental containment). One exposure level is assigned as the "reference" level. For each of the remaining exposure levels, one divides the odds of that exposure level in the outcome positive group (compared with the reference level) by the odds of that exposure level in the outcome negative group.

The OR ranges in value from 0 to infinity. Values close to 1.0 indicate no relationship between the exposure and the outcome. Values less than 1.0 suggest a protective effect, while values greater than 1.0 suggest a causative or adverse effect of exposure.

The OR is closely connected to logistic regression. This analytic method models the natural logarithm of the OR as a linear function of the predictor variables. It is a powerful and very common method for the analysis of epidemiological studies.

The OR is one of the most common measures encountered in observational epidemiology. The value of the OR for case-control research was first

*Table 2*

Frequencies of Erysipelas by Obesity | ||

erysipelas | No erysipelas | |

source: Courtesy of author. | ||

Obese | 68 | 97 |

Non-obese | 61 | 197 |

recognized by Jerome Cornfield in 1951. His work provided the theoretical base for the application of the case-control approach to studying disease etiology. The OR estimates the incidence-density ratio or the cumulative incidence ratio that would have been observed if it had been feasible to perform a cohort study rather than a case-control study. Depending on the method used to obtain control subjects, the OR either is identical to one of the incidence ratios or is close to them if the disease is rare. Some epidemiologists modify the term to reflect the type of study being done (e.g., prevalence odds ratio, exposure odds ratio, or disease odds ratio).

Although mainly used for the analysis of case-control studies, the odds ratio can also be applied in cross-sectional and cohort studies. It also plays a major role in certain approaches to the metaanalysis of randomized clinical trials (e.g., the Peto method).

An example of the use of the odds ratio can be found in a paper published by A. Dupuy et al. This paper studied 129 patients with erysipelas of the leg and a control group of 294 people without erysipelas of the leg. Obesity was considered as a risk factor. Analysis of the data produced the 2×2 table shown in Table 2.

This gives an OR of (68×197)/(61×97) or 2.3. That is, people with erysipelas are 2.3 times more likely to be obese than people without erysipelas. This supports the suggestion that obesity increases the risk of developing erysipelas.

George Wells

(see also: *Case-Control Study; Epidemiology; Statistics for Public Health* )

## Bibliography

Dupuy, A.; Benchikhi, H.; Roujeau, J. C.; Bernard, P.; Vaillant, L.; Chosidow, O.; Sassolas, B.; Guillaume, J. C.; Grob, J. J.; and Bastuji-Garin, S. (1999). "Risk Factors for Erysipelas of the Leg (Cellulitis): Case-Control Study." *British Medical Journal* 318:1591–1594.

# odds ratio

**odds ratio** The odds ratio is the ratio of two odds. If, for example, there is an experimental process (pe) and a standard process (ps) then the odds ratio is defined as pe (1 − ps)/ps (1 − pe). Unlike most measures of association, 1.0 represents a complete absence of relationship, with deviations in either direction representing an increasing relation. In sociology odds ratios have been most widely used in the study of social mobility, where their insensitivity to changes in the marginals of a mobility table is said by class analysts to have enabled the distinction to be made between relative and absolute rates of mobility. The mathematics and characteristics of the odds ratio are perhaps easiest understood in this substantive context.

For example, in the hypothetical situation shown in Table 2 we have simple mobility tables for two societies, in which men can be mobile from working-class or middle-class origins to working-class or middle-class destinations. In society A, three-quarters of men from middle-class origins arrive at middle-class destinations, while the remainder are downwardly mobile. Similarly, three-quarters of those born into working-class homes are to be found in working-class destinations, while one-quarter are upwardly mobile. If we calculate the odds of being mobile using the formula shown—that is, the chances of someone born in the middle class arriving at a middle-class rather than a working-class destination, relative to the chances of someone born into the working class achieving a middle-class position rather than remaining in the working class (or, in other words, the ratio of the former to the latter set of odds)—then simple arithmetic shows that the odds ratio in this particular case is approximately nine. That is to say, in the competition to achieve middle-class and avoid working-class destinations, the chances of someone who starts from middle-class origins are nine times greater than those for someone having a working-class background. This odds ratio is then a measure of the unequal mobility chances of individuals having these different class origins.

Now compare the data for society B. Here too, three-quarters of those born into each of the two classes remain there, while one-quarter are socially mobile. However, it should be noted that the working class is relatively much larger in society B, which has also undergone an intergenerational shift in the class structure that is evident in the marginal totals of the table: the middle class comprises 33 per cent (1,000/3,000) of all class origins but almost 42 per cent (1,250/3,000) of class destinations. If one then considers the inflow rate from working-class origins into the middle class, in comparison with that to be found in society A, it appears that society B is less closed. Some 40 per cent (500/1,250) of the middle class in this society are from working-class origins. In society A this is true of only 25 per cent (250/1,000) of those to be found in middle-class destinations. This straightforward consideration of absolute inflow rates alone suggests that society B is the more open. Furthermore, the working class is twice as large in this society, and greater numbers are upwardly mobile. However, as will be seen from the figure, the chances of mobility for the working class, relative to those of the middle class, are in fact the same in both societies (the odds ratio is approximately 9 in each case). This apparent contradiction is simply a consequence of absolute (inflow) rates failing to take into account structural differences in the sizes of classes in the two societies in question.

Odds ratios, in the context of a social mobility study, therefore allow us to appreciate *comparative* mobility chances—regardless of how class structures may vary across societies (or across time, or between different ethnic groups, or the two sexes) merely because of structural processes which have altered the relative sizes of the classes. In other words, they facilitate a clear distinction between absolute (or total observed) mobility rates, and relative mobility chances (or social fluidity). This is important from the point of view of arguments about equality of opportunity, which is an inherently comparative concept, because it refers to equal opportunities rather than the absolute chance of mobility from any particular class. If, therefore, changes in the class structure create more ‘room at the top’, as for example happens in society B, we are interested in the chance of someone from working-class origins moving there as compared to the chance of someone from middle-class origins staying there. Most researchers argue, therefore, that it is necessary to adopt this comparative approach (and consequently the technique of odds ratios) in order to address issues such as those that are raised by the concept of meritocracy.

Odds ratios form the basis of a family of statistical techniques for multivariate analysis of data comprising categoric variables, including those of loglinear modelling and logistic regression, and are widely used in sociology wherever researchers are interested in modelling relative probabilities or chances—as, for example, in the study of health and illness, labour-market outcomes, and voting behaviour.

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