Odds ratio, calculation

Fig. 18.1. The evolution of the estimates of treatment effect in the UKTIA-Aspirin Trial (Farrell et at. 1991) of high-dose aspirin versus low-dose aspirin versus placebo in patients with transient ischemic attack or minor stroke. The treatment effect (odds ratio) calculated at each point was based on the outcomes at final follow-up for patients randomized to that point (PM Rothwell, unpublished data). The dashed lines represent the level at which the apparent treatment effect approached statistical significance at the p = 0,05 level.

TABLE 29 Examples of Odds Ratio Calculations for a Firm of 140 Employees... 

An example of the outcome of a Fisher exact test is shown in Figure 4.5. A set of MMoA-annotated compounds were screened for inhibition of TNF-a production in lipopolysaccharide-stimulated THP-1 cells. The TNF-a pathway is well characterized and many known protein nodes, such as IKBKB, were identified in the analysis. Target discovery was consistent with a pathway enrichment analysis and, for each protein, a contigency table was constructed with /i-value and odds ratio calculated. To make sure that the / -values identified were significant when testing many different targets, a Bonferroni correction term - a method used to counteract the problem of multiple comparisons, which occurs when a set of statistical inferences are considered simultaneously - was applied. 

In the case-control design, a group with a disease (cases) is compared with a selected group of nondiscased (control) individuals with respect to exposure. The relative risk in control studies can only be estimated as the incidence rate among exposed individuals and cannot be calculated. The estimator used is the odds ratio, which is the ratio of the odds of exposure among the cases to that among the controls. 

Usually, the main purpose of meta-analysis is quantitative. The goal is to develop better overall estimates of the degree of benefit achieved by specific exposure and dosing techniques, based on the combining (pooling) of estimates found in the existing studies of the interventions. This type of meta-analysis is sometimes called a pooled analysis (Gerbarg and Horwitz, 1988) because the analysts pool the observations of many studies and then calculate parameters such as risk ratios or odds ratios from the pooled data. 

Recent publications on major clinical trials whose implications will involve a recommendation to change clinical practice have included summary statistics that quantify the risk of benefit or harm that may occur if the results of a given trial are strictly applied to an individual patient or to a representative cohort. Four simple calculations will enable the non-statistician to answer the simple question How much better would my chances be (in terms of a particular outcome) if I took this new medicine, than if I did not take it . These calculations are the relative risk reduction, the absolute risk reduction, the number needed to treat, and the odds ratio (see Box 6.3). 

Usually, the odds relating to the test treatment group go on the top when calculating the ratio (the numerator), while the odds for the control group go on the bottom (the denominator). However there is no real convention regarding whether it is the odds in favour of success or the odds in favour of failure that we calculate. Had we chosen to calculate the odds in favour of no SAE, the odds ratio would have been which has the value 0.66(= 1/1.51) so take care that when you see an odds ratio presented you are clear how the calculation has been organised. 

There are also conventions with relative risk. As with the odds ratio we usually put the risk for the test treatment group as the numerator and the risk for the control group as the denominator. But now, because we are calculating risk there should be no confusion with regard to what we view as the event we tend to calculate relative risk and not relative benefit. 

We saw in the previous section methods for calculating confidence intervals for the difference in the SAE rates, or the event rates themselves. We will now look at methods for calculating a confidence interval for the odds ratio. 

Previously when we had calculated a confidence interval, for example for a difference in rates or for a difference in means, then the confidence interval was symmetric around the estimated difference in other words the estimated difference sat squarely in the middle of the interval and the endpoints were obtained by adding and subtracting the same amount (2 x standard error). When we calculate a confidence interval for the odds ratio, that interval is symmetric only on the log scale. Once we convert back to the odds ratio scale by taking anti-logs that symmetry is lost. This is not a problem, but it is something that you will notice. Also, it is a property of all standard confidence intervals calculated for ratios. 

Confidence intervals for the hazard ratio are straightforward to calculate. Like the odds ratio (see Section 4.5.5), this confidence interval is firstly calculated on the log scale and then converted back to the hazard ratio scale by taking anti-logs of the ends of that confidence interval. 

If the treatment effect in each of the individual trials is the difference in the mean responses, then d represents the overall, adjusted mean difference. If the treatment effect in the individual trials is the log odds ratio, then d is the overall, adjusted log odds ratio and so on. In the case of overall estimates on the log scale we generally anti-log this final result to give us a measure back on the original scale, for example as an odds ratio. This is similar to the approach we saw in Section 4.4 when we looked at calculating a confidence interval for an odds ratio. 

Selected characteristics were compared between cases and controls by using test. The analyses of data were performed using the computer software SPSS for Windows version 11.5. Max type 1 error was accept as 0.05. Binary logistic regression was performed to calculate the odds ratios (ORs), and 95% confidence intervals (Cls) to assess the risk of breast cancer. 

Fig. 1. An example of power calculation to detect the indicated odds ratio for a range of risk factor prevalence and event rate with a sample size of 300 patients.

Epidemiological data can be analyzed in various ways to give measures of effect. The data can be represented as an odds ratio, which is the ratio of the risk of disease in an exposed group compared with a control group. Odds ratio can be calculated as... 

As mentioned in Section 26.2, case-control studies do not provide information of disease incidence in the exposed and unexposed population because subjects are selected based on disease status. For these studies, the RR is estimated by calculating the odds ratios of exposure (OR), which is the ratio of the odds of exposure in cases or the diseased group (a/c) compared to the odds of exposure in the controls or nondiseased group (b/d) (see Figure 26.1). For both cohort studies and case-control studies, the equations can be simplified as ad/bc. The OR is equal to the RR when the cases and controls are representative of the population and the disease is rare (thus the number of cases are a negligible part of the population). (See Section 26.2.3 for examples of RRs and ORs from the literature.)... 

In practice, the analysis involves calculating an odds ratio for each trial included in the meta-... 

The equation OR (Formula A)/(Formula B) can also be written in another way, as one can sec by using simple algebra. Odds ratio can also be calculated by... 

Stepping back to view the big pictun , one can see that the values for the risk ratio (3,0) and edds ratio (3.86), as calculated above, are not exactly the same. For both risk ratio and odds ratio, a value of 1.0 corresponds to a situation where no association exists betwHCcn the risk factor and the outcome. Fhe use of the risk ratio or odds ratio for expressing epidemiological data can be a matter of personal taste and custom. However, it should be noted that odds ratio has a valuable property not shared by risk ratio, as illustrated with imaginary data involving cholesterol and diabetes (Feinstein, 1985). 

This concerns whether one should use risk ratio or odds ratio for expressing results when evaluating a retrospective or prospective study. It is possible to calculate risk and risk ratio using data from a case-control study, and to acquire mathematical results, but it is not correct to do so. This is because the denominator (the number in the denominator) in the risk formula does not represent a broad random sampling of the population. The number in the denominator represents two groups of subjects who were screened and recruited into the study, Ksk and risk ratio are used for prospective studies. Odds ratio can be accurately used for both retrospective and prospective studies but is customarily only used for retrospective studies. 

The renal safety of aspirin used as single ingredient is easier to evaluate. From the seven case-control studies, only 3 showed an increased risk. All 3 suffered from the same ingredient bias as previously mentioned for paracetamol. In contrast however, both observational studies reported a robust, slightly decreased, odds ratio for the use of aspirin (Figure 1). In both studies, calculated odds ratio s were based on hundreds of regular users of aspirin [41, 42]. 

ALGEBRA USED FOR CALCULATING RISK RATIO AND ODDS RATIO... 

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