So, now we'll continue our discussion about basic measures of association with talking about the relative risk. So, relative risk is a measure of the ratio of risk of disease. Perhaps the best and most common measure of relative risk is the incidence rate ratio, which is the ratio of disease rates in different populations. That is, the incidence rate ratio is the instance rate and population A divided by the incidence rate at population B. So, this measures the relative rate at which disease occurs in a given population, and one means there's no difference between the populations being compared. So, if we go back to our same example of cases of acute watery diarrhea in one month, we have population A having three out of 25 people develop acute watery diarrhea in that month, and in population B, one out of 25 people develop acute diarrhea in that month. The incident rate ratio is three. That is three times the rate of diarrhea in that group. So, like with the risk difference, we have a classical method for calculating the relative risk. That is, if we observe people from the same amount of time, we can calculate the relative risk based on the cumulative incidence or more properly the cumulative incidence ratio. So, once again, we set up our two-by-two table to have exposed and sick people in cell a, exposed and not sick people in cell b, unexposed and sick people in cell c, and unexposed and not sick people in cell d. The relative risk is then the people in cell a divided by all the people in both cells a and b, and then that whole value divided by the people in cell c, divided by all the people in cell c and d. Applying this to the same data from John Snow's investigation, we find that the relative risk of death among people who got their water from Southwark and Vauxhall is 8.4 times that of people who got their water in Lambeth. That is they have 8.4 times the risk of death. As with the risk difference, we can more precisely calculate an incidence rate ratio or relative risk, based on the time at risk of people observed. So, here we set up our two-by-two table, once again, to have exposed sick people in cell a. The total time observed among exposed people in cell b, unexposed sick people in cell c, and the total time observed among unexposed people in cell d. The incidence rate ratio is then a over b divided by c over d. If we apply this calculation to the same data from the Nurses Health Study, we get an incidence ratio of 0.52, saying that women who used hormones had 0.52 times the annual rate of developing disease compared to women who did not use hormones. So, you might ask, how do I choose which to calculate, should I focus on the risk difference? Or should I focus on the relative risk? Both are valid measures of association but we often think of them as telling us different things. This comes to a contrast between personal risk versus public health risks. So, suppose John Snow compares those who get water from Southwark and Vauxhall versus those who get the water from Lambeth in one year. This is illustrated on the left. We have the people who get their water from the Lambeth Company shown in green on the top, with the one red person developing Cholera, and we have the people who get their water from Southwark and Vauxhall shown in blue on the bottom, with the three red people who develop Cholera there. So, we compare the one out of a 100 Lambeth users who died from cholera with the three out of a 100 Southwark and Vauxhall users who died from cholera. So, this gives us a risk difference of two deaths per year. If we calculate the risk difference, this tells us suggested benefits of everyone using Lambeth as their water supplier. In other words, if we got rid of Southwark and Vauxhall we made it. Everybody use Lambeth, how many deaths would we get rid of each year? Here we say two deaths per year for every 100 people. In contrast, the relative risk tells us the increased risk of death from switching providers to Southwark and Vauxhall. So, if your provider was Lambeth and you switch to solve Southwark and Vauxhall, you're three times more likely to die each year compared to if he'd stayed with Lambeth as a provider. That's because the relative risk is three. Now, let's look at the same study but now in that top Lambeth group, we have 10 people in red dying out of the a 100 users we sample. In that bottom Southwark and Vauxhall group, we have now 30 people dying as opposed to three from that group. So, we're now comparing one out of a 100 of Lambeth users dying per year versus three out of a 100 Southwark and Vauxhall users dying. So, the risk difference which again tells us the suggested benefits of anybody using Lambeth as water supplier now gets much bigger. It's saying we would prevent 20 deaths per year as opposed to two if we kick out Southwark and Vauxhall and made everybody use Lambeth. But the relative risk which is telling us our personal increase in risk from switching from Lambeth to Southwark and Vauxhall is about the same. We're just three times more likely to die each year if we switch from Lambeth to Southwark and Vauxhall even though the overall death rate in our overall chances of dying are much higher. It's this contrast between how many times we increase or decrease our personal risk and the number of different cases we see different per year in a full population that leads us to often call the risk difference, the measure of public health risk score as we often refers the relative risk as a more of a measure of personal risk. So, to go over some key points from this lecture. The risk difference and relative risk are two common measures of association. The risk difference that is significantly different than zero suggests there is association. A relative risk that's significantly different than one suggests there is association. If there is a difference in the amount of time individuals observed, rates must be used. To use these metrics inclusion must be independent of having the outcome. These different metrics have different interpretations at the individual and population level. For an exercise for this section, let's compare the relative risk and risk difference for the data, do you think this is strong evidence of association?
