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 workingclass or middleclass origins to workingclass or middleclass destinations. In society A, threequarters of men from middleclass origins arrive at middleclass destinations, while the remainder are downwardly mobile. Similarly, threequarters of those born into workingclass homes are to be found in workingclass destinations, while onequarter 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 middleclass rather than a workingclass destination, relative to the chances of someone born into the working class achieving a middleclass 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 middleclass and avoid workingclass destinations, the chances of someone who starts from middleclass origins are nine times greater than those for someone having a workingclass 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, threequarters of those born into each of the two classes remain there, while onequarter 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 workingclass 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 workingclass origins. In society A this is true of only 25 per cent (250/1,000) of those to be found in middleclass 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 workingclass origins moving there as compared to the chance of someone from middleclass 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, labourmarket outcomes, and voting behaviour.
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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 casecontrol 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 "crossproduct" (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 casecontrol research was first
Table 2
Frequencies of Erysipelas by Obesity  
erysipelas  No erysipelas  
source: Courtesy of author.  
Obese  68  97 
Nonobese  61  197 
recognized by Jerome Cornfield in 1951. His work provided the theoretical base for the application of the casecontrol approach to studying disease etiology. The OR estimates the incidencedensity ratio or the cumulative incidence ratio that would have been observed if it had been feasible to perform a cohort study rather than a casecontrol 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 casecontrol studies, the odds ratio can also be applied in crosssectional 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: CaseControl 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 BastujiGarin, S. (1999). "Risk Factors for Erysipelas of the Leg (Cellulitis): CaseControl Study." British Medical Journal 318:1591–1594.
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