In this section, just to complete the story, we'll look at doing confidence intervals for incidence rate ratios. Because these are ratios, we'll have to do inferences like we did with relative risks and odds ratios, we'll have to compute the uncertainty on the log scale, create confidence interval for the log ratio, then antilog or exponentiate the results back to the ratio scale. So, upon completion of this lecture section, you will be able to estimate and interpret a 95 percent confidence interval for an incidence ratio comparing time-to-event outcomes between two populations based on the results from two samples. So, let's look at our Pennsylvania lung cancer data. Again, these were the cases or new diagnoses of lung cancer in the US, state of Pennsylvania in the year 2002. Previously, we had computed incidents rates separately for females in the state in that year and males. We saw that the incidence rate for females, the estimated incidence rate, was lesser than that for males, resulting incidence rate ratio for females to males of 0.75. So, interpretation wise, we solve with this incidence rate ratio the risk of getting lung cancer in 2002 in the state of Pennsylvania for females was 0.07 times the risk for males or another way to say it and make it clear, that females had lower risk, was to say the females had 25 percent lower risk of lung cancer in 2002 as compared to males. So, like I said at the preamble is, with other ratios, the 95 percent confidence interval for the incidence rate ratio will need to initially be done on the lateral log scale. So, to start, we'll take the natural log of our observed incidence rate ratio of 0.75. It's a number less than one. So, our natural log value will be negative. It is indeed at negative 0.29. A 95 percent confidence interval for this log incidence rate ratio is computed as taking the log of the incidence rate ratio plus or minus two standard errors of that log incidence rate ratio. In general, we can estimate the standard error for the log incidence rate ratio, for the log of the incidence rate ratio by taking the square root and it's just one over the number of events in the first group and plus one over the number of events in the second group. So, it's simply based on the number of events in each of the two groups. So, if we do this in our Pennsylvania lung cancer case data, if you go back to that summary slide, there were 4,587 diagnoses in 2002 for females and 5,692 among the males. Applying this formula for the standard error of the log incidence rate ratio, is one over that counts for females plus one over the counts for males, taking the square root, gives a standard error for the log incidence rate ratio of 0.02. Then a 95 percent confidence interval for the resulting log incidence rate ratio is that log incidence rate ratio in our study of negative 0.29 plus or minus two times that estimated standard error. This incidence rate ratio interval on the log scale, the log incidence rate ratio, has a confidence interval of negative 0.33 to negative 0.25. Remember, on the log scale, the null value for ratios to zero. When we exponentiate this back to the original ratio scale, we get a confidence interval that goes from 0.72 to 0.78 for the true population level incidence rate ratio. Notice that that clearly does not include one. So, we can say our study estimated that near 2,000 women had a 25 percent lower incidence of lung cancer than males. If we think of this as being some sample from a yearly process or from one state of all US states, the confidence interval for the true reduced risk among women compared to men is 0.72 to 0.8. So, women have anywhere from a 28 percent to a 22 percent lesser risk after accounting for sampling variability. Let's look at our clinical trial from the Mayo Clinic, the randomized trial for subjects with primary biliary cirrhosis. We went to look at the comparison of mortality that was the outcome of interest over time for those who got the treatment DPCA versus those who got the control. We already saw that the estimated incidence rate for those who got the treatment was larger than those who got it in the control. So, at this point, scientifically, we'd be done because we wouldn't matter whether that was statistically significant or not if the group who got treatment had worse outcomes in the study than that who got the placebo. It will be clear that the treatment wasn't a good thing in terms of reducing mortality, but just for the exercise will complete this and put confidence limits on this. So, the incidence rate ratio for these two groups was slightly larger than one for the drug compared to placebo. It's 1.06. So, the risk of drug in the DPCA or a drug group was 1.06 times the risk in the placebo group, or another way to say that is subjects in the drug group had six percent higher risk in the followup period when compared to subjects in the placebo group. We do the standard error computation. There were 65 deaths in the DPCA group, 60 deaths and the placebo. So, by this formula, taking one over the number of deaths in the first group plus one over the number in the second and taking the square root of that sum, standard error for this log incidence rate ratio is 0.18. The log of 1.06 is 0.06. So, this is the log of our observed incidence rate ratio 1.06 plus or minus two times the standard error estimate for the log. We get a confidence interval that goes from negative 0.3 to positive 0.43. So, include zero, which is the null value in the log scale. If we exponentiate those endpoints, the confidence interval goes from 0.74 to 1.52, includes the null value of one. So, even though we saw a slight increase in the risk of mortality in our study, the results are inconclusive and there was no evidence that this drug was either harmful or effective in terms of the mortality outcome in patients with primary biliary cirrhosis. So, we could write this up to say in this study, the a 158 patients with primary biliary cirrhosis randomized to receive the drug DPCA had a slightly elevated risk of death. When compared to a 154 such subjects randomized to the placebo group, the incidence rate ratio was 1.06. After accounting for sampling variability, however, there's little evidence of an association between DPCA and death in the population of patients with primary biliary cirrhosis. A 95 percent confidence interval for the incidence rate ratio of 0.74 to 1.52. One last example, just pulling the results from the study, abstract for this study with extraordinary results on ART or antiretroviral therapy and partner-to-partner HIV infant transmission are the results you may recall was that, there was only one link transmission in the group. We've got the aggressive or early therapy compared to many more in the group that got the standard therapy in the hazard ratio or incidence rate ratio of transmission. For the group who's HIV positive partner got the early treatment compared to the standard, was 0.04, a 96 percent reduction. The confidence interval in this goes from 0.01 to 0.27. So, again, of the 28 linked transmissions, only one occurred in the early therapy group. The incidence rate ratio, what they call a hazard ratio, for the early therapy group to the standard was 0.04 at 96 percent reduction. The incident rate ratio at 95 percent confidence, they're all not going to computable, just take it from the authors because from 0.1 to 0.27 percent, so we can say or something very similar to what they said is the study of 1,763 HIV serodiscordant couples, the risk of partner-to-partner transmission among the 866 randomized to receive early antiretroviral therapy was 96 percent lower than among the 877 randomized to receive standard ART therapy. After counting for sampling variability, the early therapy could reduce the risk of partner transmission from 73 percent on the worst-case scenario and how, still 73 percent reduction to 99 percent in the best-case scenario. So, here we have strong evidence of a highly effective treatment for preventing or reducing significantly both scientifically and statistically the transmission from partner-to-partner. So, in summary, the 95 percent confidence interval for an incidence rate ratio can be computed by creating a 95 percent confidence interval for the log incidence rate ratio and exponentiating or you could call it anti-logging the results. The 95 percent confidence interval for the incidence rate ratio gives a range of plausible values for the true incidence rate ratio for the populations being compared by our two samples. As with the relative risk and odds ratio, the null value for the incidence rate ratio is one and on the log scale, the null value for ratios including the incidence rate ratio is zero.