 Now, it is often the case that if we are trying to connect exposure to some form of symptom or effect that we actually have multiple possible levels of exposure. For example, that it's not just a binary thing like that you actually were exposed to something, but that you might be exposed to a different level. You might have breathed something in at a higher concentration. You might have been exposed to something for a longer period of time. And you might still be interested in seeing how that exposure, even though it has a variety of possible levels, might sort of attribute to the effect in the population. And we can still use a population attributable fraction for doing that. If we go back and recall sort of our definition for the population attributable fraction earlier, we saw that what we could do is we could take a proportion of the population that's exposed, P.E., and we can multiply it by the relative risk, subtracting a value of one. And then the denominator in that case is one plus that same value, the portion exposed, and this relationship with the relative risk. So again, our values are the percentage of the population exposed, or perhaps I should put the fraction of the population exposed, okay, and the relative risk of the exposure. Now it's assumed if you're doing multiple exposure level that the relative risk changes with different levels of exposure. Otherwise this wouldn't be necessary. Well, to consider this for multiple levels, what we're going to do in this case is simply add for each level, we're going to assume that each level has its own level of exposure, we'll index it with the letter I, and that it has its own relative risk indexed with the letter I. And we'll calculate that product and sum over that index I. And that will give us fraction, notice each of those products are going to give us some value, and then we're going to calculate the relative difference between those two fractions. So let's take a look at an example for something that is a pretty strong relationship between exposure and effect. In this case what we're going to look at is exposure to smoking. So in order to consider smoking, this is the lifetime risk of lung cancer from smoking, we have to do some studying to figure out what the relative risk is. We actually have to compare groups of people who haven't been exposed, in other words non-smokers or people who have never smoked, to groups of people who are either former smokers or in this case current smokers or current heavy smokers. Notice we have three additional levels of risk here above and beyond the non-exposure level of risk. And you can see that people who have never smoked still have some level of possible risk to lung cancer. Point two percent of people who have never smoked still will contract lung cancer. Notice those numbers go up very quickly, I should say point two percent of men, point four percent of women. Notice however that those percentages go up substantially for people who have smoked in the past and or for people who are currently smoking or currently smoking at a very high level. So one of the numbers we're very interested in here is calculating the relative risk. Twenty-seven point five times more likely if you're a former smoker to eventually contract lung cancer if you're a man than if you had never smoked. A little bit smaller for women. But the numbers, you can see what these relative values are. And these are our values. And notice we have three categories, number one, number two and number three, including we'll label this at the index zero for our never smoking category. Okay, so we have each of these different categories to sort of put into place for our, we have each of these categories that we've indexed. Okay, so our RRI would be one of these columns. Now let's consider what happens if we're going to sort of put this into our population attributable fraction. How do we go about doing that? Well, first of all, you're going to also need some values for our relative populations, PEI. How much of each population is going to be in each of these categories. And notice that will might vary depending on what populations you're considering. You might consider populations in a certain geographic area. You might consider populations at a certain time because there was, you know, smoking, you might have been more popular in a region before and might be less popular now, for example. And you can talk about how your attributable fraction might change over time. So we can sort of consider the multiple levels of exposure. We see here we can take our population levels. So if we're considering this for men, we can go ahead and look at each of our categories. The categories for men that have never smoked, notice the relative risk is a value of one here. Okay. And then we take the fraction of our population. But notice this first category, this is our base category. So we actually don't count that in our overall totals. Even though that value would be multiplied one times 0.55, we're not actually going to count that in our totals here because we're looking for the attributable to what proportions are actually attributable to smoking. So if we take our relative risk in each case and multiply it by the fraction of the population that's in that category, we'll end up with a value here. If we sum all those values, we end up with a value here of 31.78. Well, if we plug that into our multiple level of exposure, both into the numerator and to the denominator, we get a value here. Notice because the value is so high and it's so high because we have such high percentages here. That value is so high that we end up with a very high fraction. Notice we're always going to have one more, so this value is always going to be less than one, but we end up with a very high fraction. So 37% of people with lung cancer in the population are going to have basically achieved that lung cancer, are going to have that the lung cancer can be attributed to the fact that they were some level of a smoker. So again, smoking has a very unusual at high PAF. Usually we cannot make such a strong demonstration, but you can see with the demonstration like this, why there is lots of evidence to not start smoking. And even if you are smoking, you can reduce some of your risk, 122% down to 27.5%. For example, you can reduce some of your risk. But again, that's an example of using multiple levels of exposure to compare exposure to a particular effect.