So, in this section, we'll talk about comparing time-to-event data between two or more samples graphically, and we'll extend the idea of the Kaplan-Meier we looked at in the last section to show Kaplan-Meiers for different samples presented on the same graphic for comparison purposes. So, we'll talk about how to visually compare time-to-event data across two or more samples and explain how survival proportions across time can remain relatively high and alternatively the cumulative probability of having the event relatively low even if only a small proportion of the original study is around the end of the study period. So, let's first go back to our Primary Biliary Cirrhosis study and look at the comparison of the incidence rates of death between those who were treated with the drug DPCA and those who got the placebo. As we saw in the last section, the group that got the drug DPCA had a slightly higher incidence of death over the follow-up period such that the incidence rate ratio comparing this drug group to the placebo is 1.06. So, again, just for interpretation purposes, we could say the risk of death in the DPCA group in the study follow-up period is 1.06 times the risk in the placebo group, or that subjects in the DPCA group had six percent higher risk of death in the follow-up period when compared to the subjects in the placebo group. So, let's look at this side-by-side though that gives us a sense of the relative comparison. But, again, having Kaplan-Meier graphs even though we know the story is similar in the two groups because the ratio is close to one that helps us ground this in absolute percentages. So, we can see in this graphic here, the percentage or proportion of people who survived beyond certain times by using their Kaplan-Meier curve estimates for both the placebo and the DPCA group. So, we can compare the percentage who survive beyond eight years, for example, in the two groups. Note that it's higher slightly for the placebo group than it is for the treaty group. That's the case between the sixth and 10th year is that those in the placebo group have slightly higher survival than those in the DPCA group. Then, the curves cross over, weave back and forth before that and after that, and again the end result is an incidence rate ratio that shows the curves ultimately are almost equivalent in terms of the underlying risk of death in the two groups. But this helps grounded in terms of absolute proportions. We can see that after 10 years, slightly over 40 percent of the persons in both groups were still alive. We can also present it in the complimentary version where we instead prevent the proportion who had died by a certain time in both groups, and this just shows taking a look at from the opposite perspective. Let's look at the antiretroviral therapy and partner-to-partner HIV transmission results. We saw that incredible incidence rate ratio for the treated group with the early treatment to the delayed therapy of 0.04. A very large reduction in the risk or incidence of being transmitting from the HIV positive partner to the non-positive partner for those where the HIV positive partner got treatment immediately upon enrollment versus the group where they did not start antiretroviral therapy treatment until their CD4 counts went below a certain threshold. So, this incidence rate ratio was 0.04 as we said before 96 percent reduction in the incidents for those who got the early treatment to the standard or delayed. But let's look at what this means in terms of absolute impact. What I'm showing here is from the paper that shows the presentation that tracks the proportion of people who have the outcome by a certain point in time. So, it starts at zero for both groups, and here the outcome is linked transmission, and they actually there were so few events in the early group that you can't even see anything on this larger graphic. They even reduce the scale and blew up the Kaplan-Meier curve in the inset here. But we still can't see anything for the early group because the overall proportion is extremely low because, remember, only one couple out of the hundreds in this group actually experienced the outcome. But what we can see, for example, is after two years, those who got the standard therapy roughly three percent or so of the couples had experienced partner transmission from the HIV positive partner to the person who was not HIV positive. At the start of the study and after about three and a half years or four years, this percentage was estimated to be roughly 10 percent compared to almost nothing, less than one percent in the early group. So, this really puts a face on that incidence rate ratio of 0.04 in terms of the absolute impact of this advanced or early therapy over time, and what it does to impact the proportion of couples that experienced the outcome. Now remember, persons were enrolled at various points during the accrual period and they were followed for various amounts of time till they either experienced the transmission within the couple, or they were lost to follow-up, or they got to the end of the study, and still had not experienced a transmission from the HIV positive partner to the partner who was not HIV positive. So, what this a lot of times frequently on Kaplan-Meier curves and survival curves at the bottom under certain landmark times on the timescale? They'll tell you the number of persons who were still at risk are still being followed at that point in the study. So, you can see, we started with 893 couples randomized to the early group, and 882 randomized to the delayed, and they all had their time zero when they were randomized. But what you can see over time is that the number at risk, the number being followed drops off substantially especially when we get into the third, and fourth, and fifth years of this study. So, at the end of the study, in the early transmission group there were only 24 people still being followed. So, if you look at this at face value you may say, "Well, there's still only 24 people being followed in the study." That's a very small percentage of the original 893 we started with. If you've forgot about censoring and the fact that people were not always followed for the full five years you might say, "Well, because this is the only 24 left who still haven't the proportion who had the event was rather high because the proportion remain without the event is on the order of perhaps two percent, two to three percent, 24 out of 893". So, you might conclude that nearly all 97 or 98 percent of those who got the early therapy had experienced the transmission by the end of the study. But what you'd forgot about by doing that is that there was a lot of censoring in the study in the sense that not everybody was followed for the full five years. So, this can happen where we have very few people remaining at the end of the study as a proportion of the persons who all started at a given time zero like randomization, but were the percentage of persons who've actually had the outcome is incredibly small. So, if you just took the percentage here of those remaining of the total, forgot about censoring, you will come to a very different conclusion than if you looked at the estimate of the proportion who had the event on this Kaplan-Meier curve. So, with censoring, especially censoring that occurs because people are enrolled at different times and can't be followed for the full length of the study, we will see situations like this where the number at risk by the end of the study is a small fraction of the number who were initially enrolled. But that doesn't mean that all those people in between the enrollment period and the end of the study had the event, in fact, many of them were censored. That will be reflected and picked up in the Kaplan-Meier estimate on the curves. So, in general, plotting Kaplan-Meier survival curve estimates were the cumulative event probability curves version like we just looked at for the antiretroviral therapy comparison for those on early therapy, for those on standard therapy, among serodiscordant couples. Plotting this on for multiple samples in the same graphic gives a nice overall visual comparison to complement than the numerical comparison being made by the incidence rate ratio. Kaplan-Meier curve estimates, again, are sample statistics and, hence, estimate the underlying unknown true survival curves in the populations from which the samples are taken. So, this is a visual summary statistic and that allows us to make visual comparisons of the time-to-event experience for different samples on one graphic.