So, let's look at a few more examples to reflect on what we've learned in lecture set four. So, let's look at an example of a large clinical trial looking at the potential impact of aspirin supplementation on women's cardiovascular health. So, the background on this is interesting and said that randomized trials had shown that low-dose aspirin decreases the risk of a first myocardial infarction in men, with little effect on the risk of stroke. However, this has not been replicated or studied up until the point of this research here in women. So, what the researchers did is a randomized clinical trial where they randomized nearly 40,000 initially healthy woman who were 45 years or older, to receive a 100 milligrams of aspirin every other day or placebo and then they were monitored or for up to 10 years after the start of their therapy, whether it was aspirin or placebo, for a major cardiovascular event, which I'll just abbreviate by CVD or cardiovascular disease. I'll just call it generically cardiovascular disease for remedy. During the follow-up period, there were 477 cardiovascular events CVD in the aspirin group, compared with 522 in the placebo group. So, I'm going to show the results of this trial in two-by-two format. So, of to 19,934 women who were randomized to receive aspirin, 477 experienced cardiovascular disease within the first 10 years following the supplementation and that was actually 2.4 percent of the sample, as compared to 522 of the 19,942 women who received the placebo and this was 2.6 of the placebo sample. So, let's quantify the association we have here with aspirin is compared to placebo in the ways that we've done before. So, first, we'll look at the difference in proportions, the risk difference or attributable risk. We take this difference in proportions just as it sounds, the 2.4 percent in the Aspirin group minus the 2.6 percent of the placebo, I've chosen to do it in the direction of aspirin minus placebo, and that's a difference of negative 0.002 or negative 0.2 percent. So, we could say, to interpret this, there's a 0.2 percent absolute reduction in the 10 year risk of cardiovascular disease for women given the low dose aspirin therapy compared to women who were given the placebo and hence were not given low dose therapy. If we apply this reduction to a theoretical population of a 100,000 women, we would expect to see 2.2 percent or 0.002 times a 100,000, 200 fewer woman developing cardiovascular disease if they were given low-dose aspirin compared to not being treated. So, even though the risk difference is small, relatively speaking, it's still could have a noticeable impact on a large group of women. We did this on the relative scale. One option we have is the relative risk. We take those same two inputs we had for the risk difference, 0.024 and 0.026 but instead of taking their difference, we take their ratio. Since we're doing it in the same direction, we get the same overall conclusion, which is that women in the aspirin group had lower risk of cardiovascular events than women with placebo and this ratio is less than one. We can interpret this by saying that 10-year risk of cardiovascular disease for women on a low dose aspirin regimen is 0.92 times the risk for women given placebo and that would work. We could also, to make it clear that this is indicative of a reduction, we could say something like a woman can reduce her personal risk of cardiovascular disease by eight percent, if she takes a low dose of aspirin every other day, compared to not doing anything, ie akin to having the placebo. The odds ratio in this situation, the proportion inputs for the proportion of women experiencing the outcome in both groups for lower. If we look at the odds ratio here, if we round the same way we did with the relative risk, we get the same result, a ratio of 0.92. That's, again, not always the case. In many cases, the odd ratio will differ from the relative risk. But where situations where the outcome in both groups is relatively small, we can get that equivalence especially with rounding. So, we will interpret this very similarly as to the relative risk but instead of putting it in terms of risk, we put it in terms of odds. So, the 10-year odds of CVD for a woman on a low-dose aspirin regimen is 0.92 times the odds for woman given the placebo, or a woman can reduce her personal odds of cardiovascular disease by eight percent if she takes a low-dose of aspirin every other day. So, this just gives us another context for looking at relative risks. Risk differences, relative risks and odds ratios, if you're interested, you can compute them in the other direction, the placebo group compared to the aspirin and see how they compare. We want to just walk through a couple of hypothetical examples to show you that we can have very different underlying proportions when they would all give the same estimate one statistic but are very different on the other. So, for example, let's look at some study results comparing proportions for two groups. Such that all we know is the risk difference in the study for the first group compared to the second is 0.05 or five percent. So, if that's all we know, there's a bunch of values for P1 and P2, that could satisfy this, but let's take a look at the corresponding relative risk and odds ratios under these different scenarios. So, one scenario would be that the risk of the outcome in the first group is six percent or 0.06. The second group is one percent or 0.01. You can confirm that the risk difference there is five percent. If I look at the relative risk though, for this comparison, it's 0.06 over 0.01 or six, relative risk of six. If you were to calculate the odds ratio, I don't know if I can calculate it through mathematically here but you can verify this is actually 6.3. So, here, we have a situation where the odds ratio estimate is larger in magnitude than the relative risk estimate. We've seen this before, that the different value, if we misinterpreted the odds ratio is the relative risk, we could overstate the increase in relative risk, for the first group compared the second, although with a relative risk of six, it's pretty dramatic anyway. Let's look at the second example that will give us a risk difference of 0.05 or five percent. The underlying proportions we're comparing could be 0.38 or 38 percent for the first group and 0.33 for the second. If I take the relative risk here, and it's equal to roughly 1.15 or 15 percent increase on the relative scale, very different than the 500 percent increase over here despite the fact that in both situations the risk difference is five percent. If you do the odds ratio here, it turns out to be 1.24. So, larger than the estimated relative risk but it's closer in magnitude than this previous example. This third example here, and you can confirm again that the risk difference is five percent, 0.05, the relative risk here comes in chunning very similar numerically to the risk difference, five percent larger risk on the absolute scale that risk difference 0.05 and also five percent larger risk on the relative scale as well. Here are the odds ratio if you calculate it, again comes in very different than the relative risk 6.32 versus 1.05. So, we've seen here three different situations. You can play around with others, if you're interested to see what happens to the relative risk and how it compares to the odds ratio under different choices for p1 and p2 such that the difference is 0.05. Just want to note something about a log-scale in that first example, just and you could do this with any of them but where p1 equals 0.06 and p2 equals 0.01. The direction I did the relative risk scheme of p1 over p2, it was six to one, and so a 500 percent increased risk on the relative scale for group one relative to group two. Have I done this in the other direction, though and that P2 compared to p1, that will be 0.01 over 0.06 which is really equal to one over six, the reciprocal or inverse of the relative risk in the other direction which is equal to roughly 0.16, slightly higher than that but I'll estimate that 0.16. That would be indicative of a 84 percent reduction when the comparison is done in the other direction. So, group one has 500 percent greater risk than group two, on the relative scale, but group two has only 84 percent lower risk than group one. Well, again, the reason for this discrepancy is because the scaling of possible values changes when we change direction. That's what we talked about in lecture set 4D. On the long scale though, I'm here to show you that the log of six equals 1.79 and he guesses as to what the log of 0.16 is, what's essentially the log of 16 and you could go through and parse it in terms of the log of the numerator minus the log of the denominator but this is equal to negative 1.79. So, these effects are comparable in magnitude when we look at the relative effects on the log scale but not on the original ratio scale. So, look at one more set of examples comparing relative risks for as differences and odds ratios. Now, let's suppose we have study results comparing proportions for two groups and all we know is the relative risk for these two proportions is three. So, what are some possibilities for the actual values of p1 hat and p2 hat? Well, the first one looks like this, p1 hat equals 0.003, p2 hat equals 0.001. So, the relative risk is three but the proportion with the outcome in each of the two groups is our low, relatively speaking. So, the risk difference here, minus p2 hat plus 0.0003 minus point 0.0001 is 0.0002 or 0.02 percent. So, it looks very different a lot less impressive, so to speak, numerically, then that relative risk of three. Let's look at this situation here. If you were to do the odds ratio here, it would be almost exactly equal to the relative risk of three because these numbers are very low. Let's look at this situation here, you can confirm that the relative risk here is three as well, but the risk difference, sorry for that typo, the risk difference here, is notably larger than in the previous example. It's 20 percent on the absolute scale. Just if I add the odds ratio here is equal to 3.86. So, again, there are was situation where the odds ratio is larger than the corresponding relative risk. Three in the direction of association but numerically it's different as well. Let's look at this last example, you can confirm that the relative risk here is also three. There's quite a difference on the absolute scale between these two values, the difference is 0.9 minus 0.3 or 60.6 percent. So, you can see across these three examples despite the fact that the relative risk is constant at three, there's a lot of variation and the value of the absolute risk difference. Just have RA this last example if you were to compute the odds ratio, it's equal to 21. So, in this situation, it's much larger in magnitude than that relative risk of three. So again, if you misquote the odds ratio as a comparison directly of risks, you could really overstate the relative comparison however it's already pretty large, at three to begin with. So, anyway, hopefully this was helpful. In the next set of lectures, we'll start bringing in the element of time, when we're looking at statistics in summary measures.