Review of Kozol’s Essay  Inequalities in American Education
January 10th, 2018
Odds Ratio. Odds ratio (OR) is related to RR, but uses odds rather than probabilities in its evaluation. OR is also a relative measure, that evaluates the association between two variables, by dividing the odds of the group exposed to a risk factor by the odds of the unexposed group (Bland & Altman, 2000). Odds are different from probabilities in that odds express the relationship between two probabilities. For instance, the odds of an event A happening, is the probability of that event happening divided by the probability of the event not happening (Fletcher & Fletcher, 2005, p. 49). Since probabilities add up to one, the probability of the event not occurring is 1 – the probability of the event occurring. OR is thus (Kahn & Sempos, 1989, pp. 5153):

OR can be used both in casecontrol studies and cohort studies. In a cohort study, one can calculate the OR since data is available for the proportion of the subjects exposed to the risk factor who developed the outcome, and the proportion of those not exposed to the risk factor but developed the outcome (Kahn & Sempos, 1989; Grimes & Schulz, 2002). In casecontrol studies, one of the group already exhibits the outcome being evaluated, since the study starts by selecting cases (those with the condition) and controls (those without the condition)(Fletcher & Fletcher, 2005, p. 84). OR in such a study can be calculated as the ratio of the proportion of cases who were exposed to the risk factor, to the proportion of the control who were not exposed (Kahn & Sempos, 1989). Accordingly, Odds ratio would be a better estimate for the relative risk where retrospective study data rather than prospective study data is used (Kahn & Sempos, 1989, pp.5356).
Use of Logistic Regressions in Case control studies. Although logistic regressions can be used for both casecontrol and cohort studies, analysis of the data should not follow the same approach. One of the reasons for such differences arises with timedependent exposure. Failure to account for such timedependent exposure could lead to overestimation of exposure thus resulting into a lower OR as would be revealed by data from a cohort study (Higdon, 2003). Therefore, data from casecontrol studies need to be match controls to the cases based on onset for the case and controls need to be from a random sample of all subjects that were at risk of the disease at the time (Higdon, 2003). Using logistic regression in casecontrol studies as if data were derived from a cohort would thus result in an erroneous OR.
References
Bland, J. M. & Altman, D. G. (2000). Statistics notes: The odds ratio. British Medical Journal, 320, 1468.
Fletcher, R. H. & Fletcher, S. W. (2005). Clinical epidemiology: The essentials (4^{th} ed., illustrated.). Baltimore, MD: Lippincott Williams & Wilkins.
Grimes, D. A. & Schulz, K. F. (2002). Cohort studies: Marching toward outcomes. Lancet, 359, 34145.
Higdon, R. (2003). Time dependent exposure in casecontrol studies [PowerPoint slides]. Retrieved from http://www.alz.washington.edu/NONMEMBER/SANFRAN/Higdon.ppt
Kahn, H. A., & Sempos, C. T. (1989). Statistical methods in epidemiology (illustrated, rev.). New York, NY: Oxford University Press.