So, in this lecture section, we're going to look at one more measure we can use to compare binary outcomes between two or more populations based on two or more samples, something called the odds ratio. This is appropriate to bring in now, I'll be at the odds ratio is not always the favorite measure of comparison, but I'd like to bring it now for reasons that I'll explain in this lecture section. So, upon completion of this lecture section you will be able to quantify the association between a binary outcome variable and two and more than two groups as a ratio of odds or an odds ratio, interpret the risk difference and relative risk in odds ratios in a substantive public health contexts. You've already done that with risk differences in relative risks. Abber will come in and compare those to the odds ratio and then compare and contrast these three estimates, the risk difference, relative risk and odds ratio. So, let's go back to our first example, we're looking at treatment response to antiretroviral therapy in HIV positive individuals. We want to summarize the difference in response between two CD4 counts where we measured their CD4 counts at the start of therapy and looked at the response among those who had CD4 counts of less than 250 CD4 counts or greater than or equal to 250. We saw already the proportion who responded in the group that had starting CD4 counts of less than 250 is 25 percent versus 16 percent responding in the group with CD4 counts of greater than or equal to 250. So, we've talked about these two measures already, the risk difference when we compare the direction of CD4 counts less than 250 to CD4 counts greater than 250 was positive nine percent and the absolute difference and the relative risk also called the ratio proportions the risk ratio is 1.56 or 56 percent greater relative risk of responding. So, let's look at this third measure, the odds ratio also called the relative odds. So, what is odds? Before we get started, what is the odds of an outcome? Well, the estimated odds of an event for any group is a function of the probability or risk of occurring, p hat, but it's that probability divided by the probability of it non-occurring. In other words, p hat divided by one minus p hat. So, the probability of having the outcome divided by the probability of not having the outcome. As our risk increases, our odds increases. So, let's just look at the odds for different values of p hat. So, far p hat is the lowest it can be, our odds at zero, then our odds would be zero over one or zero. If p hat is 10 percent, 10 percent chance of the outcome and our odds is 0.1 over 0.9 or one and nine if our p hat is 0.3, 30 percent. 30 percent of the sample has the outcome, then the odds of the outcome is 0.3 over 0.7 over 37. So, notice that as the p hat or proportion having the outcome increases, the odds is also increasing. So, they track together. What is the probability of having the outcome as 50 percent? Well, then the odds is actually colloquial known as 50 50 or an odds of one equal probability or proportion who have the outcome and who don't. Then when we get a proportion is greater than 0.5, our odds now will be greater than one, so this would be 0.6 over 0.4 or 1.5. We had a probability of proportion of 0.75 or 75 percent of the sample had the outcome, the odds of the outcome in that group would be 0.75 over 0.25 or four to one. So, you can see this continues that the odds is increasing as the proportion having the event increases. If we go up to the highest value of proportion could be a one, then the odds is actually undefined or infinite because it's one, the chance of having the event of one versus the proportion of not having the event is zero, one over zero is infinite or undefined. But the important thing to note is that I have the two track together, higher proportions being higher odds, higher odds being higher proportions. So, in this example, the odds of response for each of the two CD4 counts are given by the proportion who responded divided by the proportion who didn't. So, in the group with CD4 counts of less than 250, 25 percent responded and 75 percent did not. So the odds here is roughly one in three. In the other group, 16 percent responded and the remaining 84 percent did not, so the odds for this group is 0.16 over 0.84. So, we've defined what the odds is but what is an odds ratio? Well, this is the ratio of the estimated odds for two groups. Generically speaking if we had group one compared to group two, we would take the odds of the outcome for the first group and divide it by the odds of the outcome for the second group. So, in our data here, for example, with CD4 counts groups and response, the relative odds of response for the group with CD4 count less than 250 compared to the group with greater than or equal to 250 is the odds for the first group divided by the odds for the second, which these reduced numerically to 0.33 and 0.19 respectively for ratio of 1.75. So, how do we interpret this? We could say the less than 250 CD4 count group has 1.75 times the odds of responding to therapy as compared to the greater than or equal to 250 CD4 count group. We can also say the less than 250 CD4 count group has 75 percent greater odds of responding to therapy than the greater than or equal to 250 CD4 count group. Again, just to tell you why this is, this ratio is 1.75 to one. So, how much larger is the numerator than the denominator? We take 1.75 numerator subtracting the 200, how much of this is percentage, what percentage is this of the denominator, which is one. Let's put it 75 out of one or a 75 percent greater odds in the numerator. So, recall that the relative risk comparing the proportions responding between the two CD4 count groups is 1.56 and the odds ratio is slightly larger, 1.75. So, in general, the relative risk and odds ratio estimates based on the same data, they will always agree in terms of the direction of magnitude, both of these show increased probability or proportion responding in the first group compared to the second, but they may differ in magnitude. So, let's go back to another example we've looked at several times and I'll just cut to the chase since we've done this a couple of times. This is the maternal/infant HIV transmission study in which 180 infants were born to mothers given AZT during pregnancy and seven percent of those infants were diagnosed with HIV. Compared to 22 percent of the infants born to mothers who were not treated, who were given a placebo during their pregnancy. So, the risk difference comparing these two groups we already saw in the direction of the AZT group to the placebo was negative 0.15 or negative 15 percent. The relative risk was 0.32 and so this was negative 15 percent sorry. The relative risk was 0.32 or we can say a 68 percent reduction in the risk of tracking HIV. The odds ratio if we put these odds, compute the ratio, it's actually 0.27. So slightly lower in value than 0.32, the risk ratio. So, how do we interpret this? Well, the AZT group has 0.27 times the odds of HIV transmission to child of the placebo group or the AZT group perhaps, an easier way for people to understand that this is a substantial reduction odd, s is that the AZT group has 73 percent lower odds of HIV transmission to child than the placebo group. So, comparing these relative risks and odds ratios, in this example, the relative risks and odds ratio are 0.32 and 0.27 respectively. So, let's talk about the relative risks versus the odds ratio generally speaking. Both are measures, relative measures of comparison. Neither imparts information about the absolute magnitude of the risks or odds in the groups being compared if we only have the ratio and not the components of it. The relative risk is a direct head-to-head comparison of the risk of the outcome or proportion of having the outcome in each group. The odds is an indirect comparison because it filters that risk or proportion through the odds function and the odds ratio compares the odds for one group to the other. But both measures do use the same exact information but compute things in different ways and they can give numerically different results. However, both will always agree in terms of the direction of association. So, if the relative risk is greater than one when comparing two samples, then the odds ratio comparing this outcome between the same two samples will be greater than one as well always so long as they're both compared in the same direction. If the relative risk is less than one, then the corresponding odds ratio will be less than one. The relative risk is equal to one then the corresponding odds ratio will equal one. The smaller the proportions of the outcome are in the two groups being compared, the closer these two will be in value. Then you might think John, why do we even bother with an odds ratio? It seems like a less intuitive version of a relative risk. I absolutely agree with you. In most studies, studies where we can estimate risk, I'd much rather compute the risk difference and the relative risk and not even bother with the odds ratio. But there are several reasons to bring it up. First and foremost, in certain type of epidemiological study is the only measure that can be computed properly and those are what are called case control studies. So, in case control studies, we cannot directly assess the risks of interest, but we can still estimate the relative odds of an outcome between two groups and this is a function of the study design. So, without the odds ratio, we wouldn't be able to learn properly from case control studies. The second reason is when we get into the second term of this course, we'll be looking at a method called logistic regression. The first pass we get from computer output, the first measure of association we get based on this method is in the form of odds and odds ratios. Even though we can do some work to convert it to the risks scale and compute risk differences and risk ratios, a lot of times what will be initially presented are odds ratios for different comparisons. Can we just talk about again, comparing more than two groups via an odds ratio, not just be an extension of what we did for risk differences and relative risks. This goes back to the study we looked at where a person's randomized to one of four groups within the health plan to ostensibly the hope is that with more intervention on the part of the health plan, the individual will be more likely to get screened for colon cancer and we saw pretty strong sample results that in the usual care group that got business as usual, a little over a quarter of them ultimately got tested for colon cancer or got screened. Then the group that got some automated emails from the health plan in addition to the usual care that percentage almost doubled and then consistently increased more with the more assistance and information help participants got from the health plan in two more groups. So, we already have done the risk differences and the relative risks, but continuing onto the odds ratio and again, making the usual care group, the reference, we could compute the odds ratios of being screened for each of the subsequent groups to the usual care group. I'm putting the next to the corresponding risk ratios here and you can see in this situation that the odds ratios are larger in magnitude than the risk ratios. Just want to remind you that the interpretations are slightly different though. This relative risk of 1.93 indicates that those who got the automated care we're 93 percent more likely or had to be screened. Ninety three percent greater risk of being screened, this is not a direct comparison of the probability risk of being screened but it's filtered through the odds. So, in this case, it's equivalent to saying that those who got the automated care had 189 percent greater odds of being screened than those who got the usual. So, 93 percent greater risk of being screened or probability, but 189 percent greater odds. Those are two different measures on two different quantifications of the outcome. A lot of times what happens is, people will get lazy and they will treat the odds ratio like a relative risk and incorrectly describe this value to the increase or decrease in risk. So, I just want to be careful that the odds ratio is not a direct comparison of risks but a function of risks called the odds in each of the two groups we're comparing. So, in summary, this odds ratio sometimes represented by OR hat, to indicate that it's being estimated. Or a sample provides an alternative to the relative risk indicated by RR hat, for quantifying the association between the binary outcome between two groups, two or more groups actually. The odds ratio is the ratio of odds between two groups as it's related to risk and probability and proportion, those are all synonymous, but it's not risk itself. Odds is the risk or probability or proportion of the outcome over the risk of not having the outcome. The odds ratio and relative risk both estimate the association between binary outcome between groups at the individual level, but these two measures will agree in terms direction but not always magnitude. The smaller that risk in the groups being compared, the more similar in value the relative risk and odds ratio are in magnitude.