Now let's continue our discussion of risk factors studies, and talking about how to perform such studies in an open population. That is where we can't define a clear population at risk. When we're talking about open populations, we're talking about a situation where we see cases but don't know how many people were exposed or at risk. For instance, if inclusion in the study is based on disease status. If that's the case we cannot calculate the relative risk because the attack rate in the exposed is unknown. That is we might know how many people in the group developed a disease but we don't know how many people are in the group itself. The solution to this is a case-control study. So to conduct a case-control study we first identify cases based on our case definition. Then we identify controls in the same population who are at risk but did not get sick. We calculate the odds of exposure in cases and controls, and we use the odds ratio as a measure of association between exposure, and disease. Now I'm going to walk you through exactly how to do this. So the odds is the percentage of a group that have some exposure divided by the percentage that do not. So the odds of exposure is the number of exposed divided by the total, divided by the number of unexposed, divided by the total, and that just comes out to be the exposed divided by the unexposed. In case-control studies we calculate the odds of individual exposures in cases, and in controls. The odds ratio is the relative odds of having some exposure in cases, and controls. So it's the odds of exposure in cases divided by the odds of exposure in controls. Values above one mean the exposure is associated with disease just like with the relative risk, and values of below one or at one mean there's not an exposure or a negative exposure. The odds ratio only approximates the relative risk if the disease is rare. But even if the disease is not rare, it's still a valid measure of association. So let's revisit our example of the prairie dog exposure and monkeypox, and see how the odds ratio works out. So here to calculate the odds of exposure among sick children, we divide the number of children who were exposed, and got sick two, divided by the number of children who are not exposed, and got sick two, and that gives us an odds of one. We then calculate the odds of exposure among children who did not get sick. So that's four children who were exposed to prairie dogs divided by 10 who were not for an odds of 0.4. The odds ratio of exposure is then, one divided by 0.4 or 2.5. So here we once again see an association between exposure to prairie dogs, and getting sick, just like with the relative risk. So you might be asking why does this work, and I'm just going to do a little bit of math here to walk you through it. So if you recall the odds ratio is the odds of exposure in cases divided by the odds of exposure in control. So if we break that down that's exposed cases divided by unexposed cases, and that whole value divided by exposed controls, divided by unexposed controls. We can rearrange that to make it exposed cases, times unexposed controls, divided by exposed controls, times unexposed cases which we can rearrange again to be exposed cases divided by exposed controls, divided by unexposed cases, divided by unexposed controls. This works out to be the odds of being a case unexposed versus the odds of being a case in the unexposed. So what we're saying here to simplify is that the odds ratio of exposure comparing cases to controls is the same as the odds ratio of being a case comparing exposed to unexposed people, and this is why we can calculate the odds ratio from exposure, and have a valid measure of whether or not that exposure is associated with disease. The key points here, under the case-control studies allows us to measure risk even when we cannot determine how many people are at risk. Controls should be recruited from the same population as cases. The odds ratio measures association by comparing the odds of exposure in cases, and controls. It only approximates the relative risk if the disease is rare but it's still a valid measure of association. Values above one indicate association between exposure, and disease. As an exercise, let's consider our outbreak of plague, and using the table on the right, calculate the relative risk among students at a field trip, and then determine what the likely source of exposure is. So to get a fruit isn't exactly what we want you to do here. There were 48 students who visited the animals of the field trip. 10 of them got sick with plague. Some of the students visited one or more animals from an up-close visit that involved touching. These are shown in the table on the right. So for each of those exposures, we want to calculate the relative risk for being a case for each animal visited compared to the category of those students who didn't visit any animal. So that none in the bottom of the table, and then based on this calculation, you should make a determination of what you think the most likely exposure was for causing the outbreak.