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How to calculate confidence intervals for odds ratio (or), relative risk (rr), and their natural logs. It provides formulas for standard errors of log (odds) and illustrates how to use z-critical values to find the confidence intervals. The document also discusses hypothesis tests for association between variables and distinguishes between homogeneity and independence.

Typology: Study notes

2010/2011

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Download Calculating Confidence Intervals for Odds Ratio & Relative Risk: Log Transformation - Prof and more Study notes Data Analysis & Statistical Methods in PDF only on Docsity! 2 November Want confidence intervals for these statistics (RR, Ô, OR) It is hard to calculate the standard error for these three statistics Solution: Find the standard error of their natural logs Notation Natural log is denoted ln in math books It is loge Statisticians use log as the natural log On calculate, log is log10 So whenever I write “log”, I mean the natural log S.E. of log (odds) = √ nx (n−x) x = total successes n = total observations S.E. of log (RR) = √ 1−p1 p1n1 + 1− p2 p2n2 OR √ n1−x1 n1 x1 + n2−x2 n2 x2 S.E. of log (OR) = √ 1x1+ 1 n1−x1 + 1 x2 + 1 n2−x2 So CI for the log of the statistic is log (stat) ± Zcrit (S.E.) Thus CI of log (odds) = log (odds) ± Zcrit √ nx (n−x) CI of log (RR) = log (RR) ± Zcrit √ n1−x1 n1 x1 + n2−x2 n2 x2 CI of log (OR) = log (OR) ± Zcrit √ 1x1+ 1 n1−x1 + 1 x2 + 1 n2−x2 Note that e log e x = x. Thus, to get the CIs for the statistics, we take e to the power of the CI of the logs. So the CIs for the statistics are e¿¿ OR (stat) e± Zcrit (S . E .) We can use the CIs for OR and RR to find the CIs for % change in odds and risk, respectively, by plugging the bounds of their confidence intervals into the respective % change equation. So CI for % change in odds is (OR e ± Zcrit√ 1x1+ 1 n1− x1 + 1 x2 + 1 n2− x2 - 1) 100% Hypothesis Tests for Association Between the Variables 2 different possible hypotheses (2 “tests”) Test of homogeneity Are category proportions the same across all populations?