So let's look at a few additional examples highlighting some of the ideas we discussed in lecture set 5. So let's look at an example from Canada here, incidence of new HIV cases among intravenous drug users in Vancouve, Canada. So in the background here from the abstract says in 1997, we found a higher prevalence of HIV among female than among male ejection users in Vancouver. Factors associated with HIV incidence among women in this setting were unknown. In this present study, we sought to compare HIV incidence rates among male and female injection drug users in Vancouver and to compare factors associated with HIV seroconversion. So this analysis was based on data from 939 participants recruited between May of 1986 and December of 2,000 who were seronegative at enrollment with at least one follow-up visit completed and they were studied perspectively until March 2001. And incidence rates were calculated using the Kaplan-Meier method and in fact they calculate incidence rates the way we have been and then they look at the percentage of persons who contracted HIV over time using the Kaplan-Meier. So the authors for this study report incidence rate ratio of contracting HIV of 1.4 for women compared to men in the study sample. So the incidence rate of HIV for females was 1.4 times the incidence rate for males in the sample of injection drug users, in other words females had a 40% greater risk or incidence than males of having contracted HIV during the study follow-up period. In the presented actual Kaplan-Meier curve showing the cumulative incidence so that is the proportion who had the event by a given time and not the proportion who remained HIV free beyond a given time, so this is the complimentary version of the curves we looked at with regards to survival curves. And you can see, and this is helpful in interpreting that incidence rate ratio of 1.4 in terms of comparing the absolute proportion of males and females who had developed HIV by certain times since the start of the study. So at 12 months for example, we can see what this 40% increase in risk translates to an absolute proportions, the proportion of women who had developed HIV within a year was on the order of 10% compared to about six or seven percent in men and we can continue to look at these over time. If we go to 24 months, this difference on the absolute scale is roughly12-13% in women versus roughly 10% men. So this gives some context on the absolute scale to what the incidence rate ratio of 1.4 means in terms of the comparison for women to men, and by the end of the study, we can see that differential grows, but is keeping in the idea that across this entire study, women have had a higher incidence of HIV than men by about 40% as measured by the ratio and here we get the measurements in terms the absolute proportion of men and women who contracted HIV. We can certainly look at some percentiles of each time to HIV by sex so we could look at for example the 10th percentile for men and women. So the point where 10% of these respective samples have had the outcome and the remaining 90% had not. So if we look at this for women, it's on the order of 15 months by eyeballing as compared to men where it's on the order of 24 or 25 months. So a 9 to 10 month difference in the 10th percentile between men and women. Once again also gives context on an absolute measure to what that incidence rate ratio of 1.4 means in terms of the impact on these persons over time. So just to remind you that we could have presented this ratio in the opposite direction and that changes our point of reference and makes things sound different in terms of the scaling, but if we put these things on the log scale, the comparison of the log scale would be the same in magnitude, just opposite in direction. So again the incidence rate ratio of HIV contraction for females to males was 1.4. So females had a 40% greater risk than males having contracted HIV. If we put it in the opposite direction, which is just the reciprocal of 1.4, the incidence rate ratio of HIV for men to women is 0.71. So males had 0.71 times the incidence of females of contracting HIV over the study period where they had a 29% lower risk. So the risk increase and the risk decrease when the direction is changed are not numerically equivalent. We've seen this before where we look at these on the log scale though the difference would be comparable on the log scale and only difference in terms of sine. So you could prove it yourself by taking the log of these two values and see that they're equal in magnitude in absolute value but they differ in sine. Here's another example. Spatial data, these were leukemia cases collected in upstate New York over a five-year period, 1978-1982, and these are cases counts collected from 281 census tracks, and across these 281 census tracts in the state of New York, so not the entire state, just a region in Upstate New York, there are total 517 for new cases in this five-year period among 1,057,673 residents estimated to have lived in upstate New York in this five-year period. So the observed overall incidence rate in these data is 574 cases per and we take the number of persons we had assuming they each lived in this, and we have to make this assumption because we don't have any more detailed information assuming they each lived in this region for the entire five years, we take that 1,057,603 and multiply it by five and when the dust settles, we'd have an incidence rate of 0.00011 cases per person years and of course we could rescale this any way we wanted to but one way that may make sentences to multiply it by 100,000 in which case the rate could be expressed as 11 cases per 100,000 person years. Finally, let's go back to a study that we introduced initially in this lecture and then we can now highlight in these additional problems the maternal vitamin supplementation and infant mortality study conducted in Nepal. And so we had a pregnant woman who were randomized to receive vitamin A, beta-carotene or placebo during pregnancy. Of interest was the incidence rates comparing the incidence of infant mortality in the first six months after birth by maternal vitamin supplementation group. And here are the incidence rates for the three different groups. And you can see here's the incidence rate for the children whose mothers received vitamin A, and it's slightly higher numerically than the respective incidence rates for the children born to mothers who got beta-carotene and the mother's assigned to the placebo group those two rates are numerically equivalent. But if we wanted to express things in terms of incidence rate ratios, we've got three groups here. Standard practice would be to designate one of these three as the reference and compute incidence rate ratios for each of the other two groups compared to that same reference. So it probably makes sense although there's no law that says we have to do this since we're interested in the efficacy of vitamins compared to no treatment, we could designate the placebo group as a reference and we can compute separate incidence rates for the beta-carotene group compared to the placebo group, if we did that the incidence rate ratio would be one, and for the vitamin A group also compared to the placebo group and that incidence rate ratio would be slightly larger than one. So we can see numerically whether you are looking at these three rates side-by-side or by there's ratios that essentially the infant mortality story was about the same regardless of whether the mother got vitamin A, beta-carotene or nothing or a placebo during pregnancy. We wanted to see what the absolute mortality was over the six month follow-up period in these three groups. We could plot the Kaplan-Meier curves on the same graphic. And you can see that they all, I mean this is scaled such that the axis here only runs from 90% to 100 but they're all neck in neck but the vitamin A group has slightly lower survival compared to the other two groups and that was reflected in the slightly higher incidence rate in that group as well. But this is helpful because now we can see for all three groups, the majority of infants who died did so in the first two months after birth and then the rate of death slowed down after six months but there were still subsequent deaths and at the end of the follow-up period, approximately six percent slight differences, less than one percent across the three groups, about six percent of the respective three samples had died and the remaining 94% or so survive beyond six months. So hopefully this was a quick but helpful extra take on some of the concepts we discussed regarding incidence rates, their comparisons, and Kaplan-Meier curves and what they add to the story.