Odds ratio confidence intervals

As a part of logistic regression analysis, odds ratio plots are an excellent way to see how much more likely a condition is to exist based on the presence of another condition. Just by glancing at an odds ratio plot, you can see whether an independent variable is significant to the dependent variable. For instance, if the odds ratio confidence interval does not cross the value of 1, then the independent variable odds ratio is significant. Examine the following graph. 

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. 

In a case-control study of the relation between occupational exposures to various suspected estrogenic chemicals and the occurrence of breast cancer, the breast cancer odds ratio (OR) was not elevated above unity (OR=0.8 95% 01=0.2-3.2) for occupational exposure to endosulfan compared to unexposed controls (Aschengrau et al. 1998) however, the sample sizes were very small (three exposed seven not exposed), and co-exposure to other unreported chemicals also reportedly occurred. Both of these factors may have contributed to the high degree of uncertainty in the OR indicated by the wide confidence interval. 

In this example, we see that Active Therapy vs. Placebo, or drug therapy, has a significant odds ratio because the 95% confidence interval line does not cross 1. It appears that patients on active therapy are almost four times as likely to experience clinical success as those who are not on active therapy, while controlling for the variables White vs. Black, Male vs. Female, and Baseline Pain (continuous). ... 

Fig. 6.6 Effect of beta blockers on post-infarction mortality. Etifference in mortality rates is expressed as a percentage of control rate in six controlled trials of beta blockers (95% confidence intervals based on odds ratio consideration). Narrow confidence intervals are associated with the largest trials that are the major contributors to the pooled result.

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. 

The quantity c is very closely related to the odds ratio in fact c is the log of the OR, adjusted for the covariates. The anti-log of c (given by e"") gives the adjusted OR. Confidence intervals in relation to this OR can be constructed initially by obtaining a confidence interval for c itself and then taking the anti-log of the lower and upper confidence limits for c. 

When dealing with binary data a similar link applies, but now with the confidence interval for the odds ratio and the p-value for the test, with one important... 

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. 

If this confidence interval is on the log scale, for example with both the odds ratio and the hazard ratio, then both the lower and upper confidence limits should be converted by using the anti-log to give a confidence interval on the original odds ratio or hazard ratio scale. 

Note that the confidence intervals in Figure 15.1 are not symmetric around the estimated hazard ratio. This is because confidence intervals for hazard ratios and odds ratio and indeed ratios in general are symmetric only on the log scale (see Section 4.5.5 for further details with regard to the odds ratio). Sometimes we see plots where the x-axis is on the log scale, although it will be calibrated in terms of the ratio itself, and in this case the confidence intervals appear symmetric. 

Pooled 95 per cent confidence interval well away from zero (or unity for odds ratios, or the pre-defined margin for non-inferiority trials)... 

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. 

A recent British trial, UK MRC ALL 97, randomized 1498 children to receive either 6-TG or 6-MP (87). After a median follow-up of 6 years, no differences in event-free survival were detected between the two treatment arms. A large reduction of isolated disease recurrence in the CNS by 6-TG [odds ratio (OR) = 0.53,95% confidence interval... 

RR, relative risk CI, confidence interval SMR, standardized mortality ratio OR, odds ratio SIR, standardized incidence ratio " 90% Cis... 

In the Montreal case-control study carried out by Siemiatycki (1991 see the monograph on dichloromethane in this volume), the investigators estimated the associations between 293 workplace substances and several types of cancer. Isopropanol was one of the substances. About 4% of the study subjects had ever been exposed to isopropanol. Among the main occupations to which isopropanol exposure was attributed in this study were fire fighters, machinists and electricians. For most types of cancer examined (oesophagus, stomach, colon, rectum, pancreas, prostate, bladder, kidney, skin melanoma, lymphoma), there was no indication of an excess risk due to isopropanol. For lung cancer, based on 16 cases exposed at the substantial level, the odds ratio was 1.4 (90% confidence interv al, 0.8-2.7). [The interpretation of the null results has to take into account the small numbers and presumed low levels of exposure.]... 

An association between estimated exposure to diethyl sulfate and risk for brain tumours was suggested in a case-control study of workers at a petrochemical plant in the United States. Seventeen glioma cases and six times as many controls were included and an odds ratio of 2.1 (90% confidence interval [CI], 0.6-7.7) was obtained a parallel study of 21 cases (including the 17 of this other study) and with another set of controls showed no clear increase in risk, however (lARC, 1992a). 

Figure 20.1 Odds ratios and 95% confidence intervals of selected cancers for the highest vs. the lowest levels of vegetable consumption. Italy, 1991-2005.

Table 20.1 Odds Ratios 2 (OR) and 95% Confidence Intervals (Cl) Among 805 Cases of Oral Cavity and Pharyngeal Cancer and 2081 Controls, According to Daily Intake Quintile of Six Classes of Flavonoids and Total Flavonoid. Italy, 1992-2005.

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