So, in this section, we'll look at comparing time to event data outcomes between two or more samples as an estimate of the comparison at the population level for the two populations from which the samples were drawn and we'll do this numerically. So, upon completion of this lecture section, you will be able to estimate a numerical comparison of time to event outcomes between two populations using sample rate estimates and interpret the resulting estimate, what we call the incidence rate ratio in words and in a public health or scientific context. The incidence rate ratio is akin or analogous to the risk ratios we saw with binary outcomes, except it involves comparisons that involve time as part of the estimates. So, let's look at the Pennsylvania Lung Cancer Data again from 2002, and we may want to ask, how did the rates of lung cancer diagnoses in the year 2002 compare for men and women in the state of Pennsylvania? So, what we'll need are the incidence rates computed separately for all females and all males in the state in two 2002. So, from these data, I broke it out by sex and among the females there were 6,351,391 females in Pennsylvania in the year 2002. So, we assume they all live there for the entire year, we can't know for sure of course. There were 4,587 lung cancer diagnoses among the females in that year 2002. So, the incidence rate for females is given as 4,587 cases over a total of 6,351,391 person years or as a decimal 0.00072 cases per person year. Now, let's look at what the estimate is for males. Five point nine plus million males in the state of Pennsylvania in 2002, and 5,692 lung cancer diagnoses. So, the incidence rate among the males of lung cancer in 2002 was 5,692 cases over the 5,929,663 men who lived in the state that year for the entire year. So, person years were approximately 0.00096 cases per person year. So, the incidence rate ratio comparing the incidence of lung cancer for males in 2002 compared to females, is simply as it sounds we take the ratio of the two rates. So, if we do it in the direction of females to males, it turns out to be 0.75. So, we can see numerically and then through this ratio that females had a lower incidence of lung cancer diagnoses in 2002 as compared to males. There is as such the ratio was 0.75, so two ways to interpret this would be to say, the risk of getting lung cancer in 2002 for females was 0.75 times the risk for males. Or in order to make it clear that this ratio shows a lower incidence for females the numerator group we could talk about it in terms of the percent lower risk of lung cancer females had and that would be 25 percent lower risk. So, in terms of the reduction in percentage-wise, it's the numerator 0.75 minus one over the nominator of one or negative 0.25 indicating the numerator is 25 percent lower than them either. So, again females had 25 percent lower risk of lung cancer in 2002 when compared to males. Let's look at our Primary Biliary Cirrhosis data that we'd seen in the last section as well. Remember this study was undertaken where patients were randomized, patients with primary biliary cirrhosis to either get a drug DPCA or a placebo. If we wanted to see what the incidence of mortality or death was over the followup period for the DPCA group compared to the placebo we can compute the incidence rates for both and take the ratio. So, in the DPCA group, there were a total of 872.5 person years of follow-up time when we added it up the time that each individual was followed in that group and there were 65 deaths. So, the incidence rate of death over the followup period for the drug group was 65 deaths per 872.5 person years over approximately 0.075 deaths per person year. In the placebo group, there were a total of 842.5 person years of follow up and 60 deaths for an incidence rate of 60 deaths per 842.5 person years or approximately 0.071 deaths per person year. So, you can see the incidence was actually larger in the group that got treatment. So, if we take the ratio of these values comparing the drug group, the DPCA group to the placebo, the ratio is 1.06 because the incidence rate in the DPCA group was slightly larger. So, how can we interpret this? 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 in other words subjects in the DPCA group had 6 percent higher risk of death in the follow-up period when compared to the subjects in the placebo group. Let's look at that study we looked at in the first section on antiretroviral therapy and partner-to-partner HIV transmission. So, this is the one where they randomized the HIV positive partner in a serodiscordant relationship where one partner was HIV positive and the other was not. They randomized them to either start antiretroviral therapy immediately or wait until their CD4 counts had gotten below a certain threshold. What they found is that there were 28 linked transmissions, between primary partners where one was HIV positive and one was not and only one of the 28 occurred in the early therapy group, the group that got treated right away. So, a synonym for incidence rate ratio is hazard ratio, hazard is another word for risk. They're saying that the incidence rate ratio or hazard ratio of linked transmissions for the partners where the HIV positive person got early treatment compared to those where they got the standard or delayed treatment was 0.04. So, what they're saying is that the hazard rate, and I don't actually have the person years of follow-up for the two groups but I'm just putting in a place holder to represent how they computed this. Is they took the incidence rate of partner to partner transmission in the group where the HIV positive person got early treatment and that was one link transmission over the total amount of follow-up time for that group, compare the incidence rate in the group that got the standard therapy or the delayed therapy. That was 27 linked transmissions per however much follow-up time there was. That's the incidence rate ratio of 0.04. So, how can we translate that? That's quite a reduction by the way. So, we could say that HIV discordant of baseline couples in which the HIV positive partner was given early antiretroviral therapy had 0.04 times the risk of within couple transmission as compared to couples in which the HIV positive partner was given standard therapy. Another way to say this, is HIV discordant at baseline couples in which the HIV positive partner was given early antiretroviral therapy had 96 percent lower risk of within couple transmission as compared to couples in which the HIV positive partner was given the standard or delayed therapy. So, just to think about this, what could potentially happen if follow-up time were ignored when we have the individual follow-up times and we just treated the outcome is binary or not? This is just generally speaking. Well, it's possible to see a situation where over a certain amount of follow-up time, let's say we follow people for up to five years and we had two groups of treatment under control and we were looking at the instance that the outcome in the two groups over the five-year period. Maybe at the end of the five-year period both groups had very similar proportions who had the outcome, 41 percent in the treatment group and 40 percent in the control. If we just compared this based on the binary basis or whether it occurred in the followup periods are not for each of our subjects and took the cumulative proportion, this could look very similar. But what if among those who were treated, most of those who had the event had it between the fourth fifth year, and most of those in the control group had it very early in their follow-up period? So, despite the fact that we ended up with similar proportions, the actual time to have an event profile is different in these groups and the incidence rates. Comparison would capture that differential because it would include information about the follow-up time which would be different between these groups. But if we just made it binary, we wouldn't miss that. So, its importantly especially when we have these individual follow-up times and the individual subjects in our study, to recognize and respect that time when summarizing the results. So, in summary the incidence rate ratio sometimes called IRR, and estimated from sample data by IRR hat, is a numerical comparison of event rates between two groups. If we have more than two groups, one can be designated as the reference group and incidence rate ratios can be computed for each of the other group incidence rates, each compared to the same reference group incidence rate. There's an example of this in the additional examples section. The incidence rate ratio can be thought of as a relative risk measure that incorporates differences in subject follow-up times into the comparison when the follow-up times are known at the individual level. Certainly, even if we don't have the individual follow-up times, it still respects the element of time in the computation.