So let's now look at some additional examples related to the material we've just covered. So first, let's go back and look a trial we looked at before in the additional examples for confidence intervals, the aspirin in cardiovascular disease in woman study that we looked at that was published in the New England journal medicine in 2005. This was a randomized trial done on women, initially a randomized trial had been done on men showed that there was some benefit to taking low dose aspirin on a regular basis. There was some benefit in terms of cardiovascular outcomes. They wanted to see if that replicated in women. They randomly assigned nearly 40,000 women age 45 years or older to receive 100 milligrams of aspirin on alternate days or placebo and then monitored them for about 10 years for the first major cardiovascular event. And during the follow-up, there were 477 major cardiovascular events in the aspirin group as compared to 522 in the placebo group, for a nonsignificant reduction in risk. So let's see where they got that. We've already seen some of the summary results that from a percentage perspective, 2.4% of the women who were randomized to receive aspirin experience cardiovascular events as compared to 2.6% of women randomized to receive the placebo. We already looked at the measures of association in their confidence intervals, and we noted that all of them included their respective null values. So the difference was negligible on the risk difference scale and small reduction on the relative risk and odds ratio scales. And the odds ratios and relative risk were very similar in value. None of the confidence intervals excluded the null value, they all included the null value for their respective measure of association. So we knew the result was not statistically significant at the 5% level. We can either use the two-sample z or chi-square test approach, or Fisher's exact to get a p-value. I ran them both. With the two samples, z or chi-squared approach, we get a p-value 0.151. Fisher's exact is slightly larger at 0.159. So a little bit of a disagreement in terms of the numerical value, but certainly the same conclusion in terms of failing to reject the null hypothesis and not having a statistically significant result. By the way, and we'll again talk about this in the last lecture section. This is an example of a large well-powered study, so the researchers were able to parlay that failure to reject the null hypothesis into a stronger decision. And conclude that there was no ostensible benefit of aspirin supplementation with regards to cardiovascular outcomes in women. Here's another study presented in JAMA back in the 90s, but it was interesting study on intravenous drug users. One of the first studies to look at the risk of tuberculosis outcomes in this population. So a study was performed on a representative sample of 258 intravenous drug users. And of particular interest to the researchers were factors which may influence the risk of contracting tuberculosis among intravenous drug users. And so one of exposures the looked at is whether the user was sharing needles with other users or using only clean needles that they only used. So 97 of the study subjects admitted to sharing needles to shoot drugs. Of these 97, 24 had a positive tuberculin test result. The remaining 161 subjects denied having shared needles. And of those who did not share needles, 28 had a positive tuberculin test result. So here are these data in the two-by-two table format. So about 25% of those who had shared needles had a positive TB result, tuberculin test result or tested positive for tuberculosis, as compared to 17% among those who did not share needles. So a large, a risk difference of 8% in the direction of those who shared needles compared to those who didn't. But there's a lot of uncertainty in this data given the relatively small sample sizes, and so the results are not conclusive. They include the null value of 0 for the risk difference. Again, sizable Increases on the ratio scales but after accounting for sampling variability, again, these are not statistically significant as the confidence intervals include their respective null values for ratios. When you look at p-values the p-values from the two sample z test is 0.14, two sample z test or chi-squared, and for Fisher's exact it's 0.1999. So you see a slightly bigger discrepancy between these values in this smaller sample example. Nevertheless, in this situation, both tests concur in terms of their conclusion about failing to reject the null hypothesis. But because this is a smaller sample study that is not as strong a conclusion as failing to reject was in the previous large powered aspirin study in cardiovascular disease study we looked at. So this left the door open for doing more research to answer the question beyond a reasonable doubt. So an article came out in 2016 in New England Journal of Medicine, a randomized trial to look at the efficacy of treated vaginal rings in terms of HIV prevention in sub-Saharan Africa. So this was done in South Africa and Uganda. The background here, from the abstract, is the incidence of HIV infection remains high among women in sub-Saharan Africa. We evaluate the safety and efficacy of extended use of a vaginal ring containing dapivarine for the prevention of the HIV infection 1959 healthy, sexually active women, 18 to 45 years of age, from seven communities in South Africa and Uganda. So this is a randomized, double-blind, placebo-controlled trial. They randomly assigned participants in a 2 to 1 ration, so not equal numbers in the groups, to receive vaginal rings containing either 25 mg of dapivirine or the placebo or a placebo. So patients inserted the rings themselves every four weeks for up to 24 months. The primary efficacy end point was the rate of HIV type 1 seroconversion. So what they found is a total of 77 participants in the treatment group underwent HIV seroconversion during the 1888 person years of follow up. So approximately an incidence rate of 4.10 conversions per 100 person years. And this was compared with 56 in the placebo group who underwent HIV-1 seroconversion during 917 person years of follow up, 6.1 seroconversions per 100 person-years. The incidence of HIV infection was 31% lower in the dapivirine group than in the placebo group. A hazard ratio which, again, is synonymous with, incidence rate ratio, is 0.69, so that's where they get the 31% reduction, with a 95% confidence interval of 0.49 to 0.99. 0.99, so get's close but does not include 1 and the resulting p-value, and this was from a log rank test, was 0.04. So statistically significant, but certainly from a scientific perspective it's hard to make a definitive conclusion. because on the one end, and the certainly the study results indicated a large reduction in the risk of HIV-1 infection for those women who got the treated vaginal rings. And on the lower end this confidence interval is very impressive. The instance rate ratio has a ratio 0.49, but the upper end it's edging close to 1. So it's hard even though all results are statistical significant, on the lower end they indicate relatively large efficacy of the treated rings versus on the upper end not so much of an impact. Here the Kaplan-Meier curve is showing the cumulative risk, the cumulative proportion of women who contracted HIV over the followup time period in the two groups. So you can see, certainly as we'd expect, we showed that the drug, the vaginal rings with dapivirine, were protective. So we see a lower cumulative proportion contracting HIV over the follow up period. You certainly see that very clearly. And the p-value for this comparison, as we reported before, was 0.04 from the log rank test.