I let w tau represent the population odds ratio of

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(i) Let W (“tau”) represent the population odds ratio of having lung cancer for those who are regular smokers compared to those who are not regular smokers, so W = S 1 /(1 ± S 1 )/( S 2 /(1 ± S 2 )). State the null and alternative hypotheses in terms of this parameter. (j) Use Fisher’s Exact Test to calculate the p -value. (Note: We get the same p-value no matter which statistic we use, why is that?) But we still need a confidence interval for this new parameter as well.
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Chance/Rossman, 2015 ISCAM III Investigation 3.10 236 Theoretical Result: The sampling distribution of the sample odds ratio also follow a log-normal distribution like the relative risk (for any study design). Thus, we can construct a confidence interval for the population/treatment log-odds ratio using the normal distribution. The standard error of the sample log-odds ratio (using the natural log) is given by the expression: SE ( ln odds ratio ) = D C B A 1 1 1 1 ² ² ² where A , B , C , and D are the four table counts. (k) Calculate this standard error and then use it to find an approximate 95% confidence interval for the log odds ratio. (l) Back-transform the end-points of the interval and (k) and interpret your results. (m) Does your interval contain the value one? Discuss the implications of whether or not the interval contains the value one. (n) Compare your results to the following R output: (o) Summarize (with justification) the conclusions you would draw from this study (using both the p- value and the confidence interval, and addressing both the population you are willing to generalize to and whether or not you are drawing a cause-and-effect conclusion).
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Chance/Rossman, 2015 ISCAM III Investigation 3.10 237 Study Conclusions Because the baseline incidence of lung cancer in the population is so small, the researchers conducted a case-control study to ensure they would have both patients with and without lung cancer in their study (matched by age and economic status). In a case-control study, the odds ratio is a more meaningful statistic to compare the incidence of lung cancer between the two groups. We find that the sample odds of lung cancer are almost ten times larger for the regular smokers compared to the non-regulars in this study. By the invariance of the odds ratio, this also tells us that the odds of being a regular smoker (rather than not) are almost 10 times higher for those with lung cancer. We are 95% confident that in the larger populations represented by these samples, the odds of lung cancer are 5.92 to 15.52 times larger for the regular smokers (F isher’s Exact Test p-value << 0.001). If both success proportions had been small, we could say this is approximately equal to the relative risk and use the words “10 times higher” or “10 times more likely.” The full data set (which broke down the second category further) also shows that the odds of having lung cancer increase with the amount of smoking (light smokers have 2 times the odds, heavy smokers have 11 times the odds, and chain smokers have 29 times the odds!) ± this is called a “dose - response.” We see
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