 As environmental scientists and environmental engineers we are often interested in connecting the exposure to a particular toxin or a pathogen to adverse effects within a population, usually a population of people. However, this can be a little difficult sometimes because the adverse effects that we're looking to study might occur naturally within the population. And it may not necessarily be a simple case of establishing whether or not the exposure had a major effect or not. Generally we would like to try to quantify the relative effects of the exposure itself. And we can do so with measurement called the population attributable fraction, which is basically a number that sort of says how much of the population who is suffering from the adverse effect a symptom or a disease. How much of that population is due to an exposure of some portion of the population to some form of toxin or a pathogen. So let's take a look at sort of a generic numbers for generic event here. Let's say we have some sort of event, maybe it's an event of food poisoning, maybe it's an event of exposure to some sort of toxin. And we've gathered, we're able to gather numbers and we can divide the people into two categories. We're going to keep this simple for our first example. But we can divide people into two categories, whether they've experienced symptoms or not experiencing symptoms, whether or not they're being affected in a certain way. And then whether or not the people have been exposed or not exposed to the toxin or pathogen of interest. So in order to get appropriate data, we need to divide people into four different groups, whether they've experienced symptoms and are exposed, whether they've been exposed but have no symptoms, whether they're not exposed but do have symptoms. And similarly, whether they have neither symptoms nor have been exposed. So let's create some numbers here. Let's say we have a group and we've pulled the group and we find there are 500 people that have been exposed but have symptoms. But the population itself is only a small portion of the group of people. There are 10,000 total people that have been exposed and only 500 of them experienced symptoms. And then we go and look at the rest of the population and we find out of the people who are not exposed, there are still 900 people with symptoms. Well, that seems a little concerning. It does seem that maybe it's actually better to be exposed until you consider how many people may not have the symptoms. In this case, we're going to say that it's 89,100. So now it makes a little bit more sense that even though there are more people with symptoms that are not exposed, we can now start thinking about this in terms of fractions. Would you rather be in this group that is exposed or not exposed based on your sort of chances of actually having the symptoms? So let's finish the total. So there are 90,000 people that are not exposed. There are 1,400 people total with symptoms and there are a total of 98,600 people who don't have symptoms. And the overall total is going to be 100,000 people, a nice round number for us to do our percentages. Now, if we look here, one of the things we could actually measure is something called the risk. If I am in one group or another, if I'm in the exposed group, what are my chances of having symptoms? Well, we can compare that risk by comparing the number of symptoms to the total number of people in that group. And in this case, that's 500 over 10,000, which is going to be 5%. If I actually do the same thing with the not exposed group, you can see in the not exposed group we have 900 over 90,000, which is simply 1%. So I'd rather be in the not exposed group, I have much less risk of actually having the symptoms. Notice if we can sort of combine the two things together, if we compare the overall population, we see that there's 1,400 total people with symptoms out of the 100,000, which is just 1.4%. So again, we're interested in figuring out what fraction of the population, if 1,400 people in the population have symptoms, what portion of those people actually got the symptoms from being exposed. It's not as simple as saying that 500 of those people have that because as we can see, there's people in the population who are not exposed who have the symptoms anyway. So the symptoms might be caused by something else. And so if we really want to attribute the symptoms to the exposure, we have to figure out some fraction of this exposure. And again, that's our population attributable fraction. So let's go ahead and consider this population attributable fraction or PAF. In order to do this calculation, what we're going to do is we're going to just make it a fraction of the number of excess affected divided by the number of total affected. Well, what do we mean by excess affected? Well, that would basically represent all the people who are affected after exposure, but that would be more than we would have if they hadn't been exposed. And so we have to use our non-exposed percentages to calculate that. So in that case, it's going to be, let's see if we can consider the number of excess affected. We're going to take the number of actual affected, the exposed cases of which there are 500 exposed cases. And to get the total affected, well, actually the number exposed cases, and then we have to subtract a number of cases that we would expect would be if not exposed. I'm not sure about that quite yet, so we're going to postpone that for the moment and take a look at that in just a second. And then we want to know the total number of affected here. Well, that was pretty simple. That was the 900 cases not exposed plus the 500 cases that were exposed, the same number as above, exposed cases. So now we need to figure out this would be if not exposed number. How many people would have the symptoms even if they were not exposed to our environmental problem? Okay, well, let's think about that. What we might want to do here is let's consider the ratio 900 not exposed cases divided by the 90,000 people who are not exposed total. That's a percentage that were affected but not exposed. Okay, that's the 1% that we calculated a little while later. And then we're going to multiply that by the number of people who were exposed. And when we do so, we get a value. In this case, there were 100 of cases that would have been, would have experienced symptoms, assuming the same percentages as everybody else in the population, would have experienced those symptoms even if they had not been exposed. I'm going to take that 100 value and plug it in up here. So there's the relationship we're looking for where we're taking the sort of the excess affected, the people that are affected beyond what would normally be expected divided by the total. And that's our population attributable fraction. If I do that calculation, I get a number that looks something like this. We get 400 on the top divided by 1400. In other words, out of the 500 cases, 400 of them were above and beyond the percentages expected in the population. If we compare that to the 1400 total cases, then we get a value. And in this case, our value is 28.6%, our population attributable fraction. What does that mean? Well, that number basically says out of our 1400 cases, 28.6% of the cases in the population were a result of whatever exposure occurred. And that gives us a pretty good quantitative measure of the effectiveness of the exposure. Now, if we wanted to consider some other value or compare this to some other type of exposure and how it affected the population, we could get a relative ranking for the danger of the exposure, even if there were many fewer people that were exposed. For example, this affected roughly 10% of the population in exposure. But if we had something that only affected 1% of the population, we could still sort of compare the numbers, even though we might have significantly different numbers on how many people ended up contracting the symptoms. There's also an interesting number here that we're going to look at. If we look at these risk values, there's a value that we call relative risk. Relative risk. And in that case, a relative risk is the amount of risk you have when you compare an exposed group to the not exposed group. It's a ratio. In the case here, our relative risk of exposure is the 5% divided by the 1%. The 5% of people that were exposed as compared to the 1% that were not. And we get a relative risk value of 5. It means you are 5 times more likely to experience symptoms if you have been exposed than if you have not been exposed. And that relative risk number is pretty common for expressing sort of the dangers of exposure. If we do a little bit of the math, we can actually simplify our PAF. Our population attributable fraction is often written in this format. Where we have values pe rr minus 1 over 1 plus pe rr minus 1. Where rr is this relative risk factor, where rr is this relative risk factor we were just talking about. And pe is the percentage of the population that's been exposed. So if we do studies over time and can establish relative risk values for different behaviors or different exposures, then if we know how much of a population is exposed, we can make an estimate of our population attributable fraction. Applying it to this particular case, we know that our percentage of exposure, if we go up here, we'll notice that the number of people exposed was 10,000 out of a total of 100,000. So we have a pe or percent of exposure of 10%. And we've already established our relative risk for exposure with a value of 5. Notice if I plug those numbers in here, we have 10% times 5 minus 1. And then on the bottom, we have the same thing, 1 plus that same value, 10% over 5 minus 1. In this case, you get a value of 0.4 over 1.4, which you see is again the same ratio for our population attributable fraction that we calculated before, the 28.6%.