In this section we're going to talk about comparing binary outcomes between two or more populations using the results from two or more samples from those respective populations. We're going to talk about two different measures that use the same inputs but do different computation, something called the risk difference and something called a relative risk. So upon completion of this lecture section, you will be able to compute the risk difference and relative risk comparing binary outcomes between two samples, interpret the risk difference and relative risk in a public health personal health context, understand that the risk difference and relative risk will always agree in terms of direction but can differ greatly in terms of magnitude and understand that neither risk difference alone nor the relative risk alone is sufficient to quantify the association of interest. So let's go back to our data on 1,000 HIV positive patients from a citywide clinical population. When we look at all the patients together 206 of the 1,000 responded. So the overall response to treatment in the sample was estimated by p hat, the proportion responding of 20.6 percent. So if we actually wanted to look further and we classify we want to see whether the differences is going to response depending on the CD4 counts at the beginning of the therapy. So what I did was I classify patients as subjects as to whether they had starting CD4 counts of greater than or equal to 250 or less than 250 at the beginning of therapy. So of the 497 who had CD4 counts of greater than 250, 79 responded to the therapy and out of the 503 who had CD4 counts of less than 250 at the start of therapy 127 responded. So my resulting sample proportions in each of these two groups is that the proportion who responded in the group whose starting CD4 count was less than or equal to 250. So I have subscripted to this p half of that was 127 out of 503 or 25 percent as compared to 79 of the 497 whose starting CD4 counts at the time of therapy was greater than or equal to 250 which gives a proportion of 16 percent. So how can I compare these in a single numerical summary measure, I've got two proportions 25 percent responded in one group 16 percent responded in the other. Well one way I can do this is take what's called the difference in proportions. Also if you've taken epidemiology or if you will sometimes called the risk difference or attributable risk. This is simply taking the difference in these proportions, the order is arbitrary I'm going to order such that I take the difference between the groups with starting CD4 count lower or less than 250 minus the proportion responding in the group with CD4 counts of greater than or equal to 250. When I do this I get a difference of nine percent 0.25 minus 0.16 is 0.09 or nine percent. So one way to interpret this is to say that there is a nine percent greater response to therapy, nine percentage point greater response to therapy in the CD4 count less than 250 group as compared to the CD4 count greater than or equal to 250 group or we could say there's a nine percent greater absolute risk of response to therapy, sounds weird colloquial to say risk of a good thing, but technically we are quantifying the risk of response. So nine percent greater absolute risk of response to therapy in the CD4 less than 250 group as compared to the CD4 greater than or equal to 250 group. So summary measure number two is the relative risk. I used the exact same two numbers to calculate this the p hats in each of the two groups being compared. But instead of taking the difference between the two I take the ratio of the two. So I'm going to do it in the same direction which means my numerator my first group will be the group less than 250 CD4 count and to divide that by the p hat for the group with CD4 count greater than or equal to 250 and this ratio is 1.56. So again it shows that the response in the group with CD4 counts of lower than 250 was higher than the other group as we also showed with the risk difference. But how do we interpret this relative risk? Well one way to say this is that those in the CD4 count less than 250 group have 1.56 times the chances or risk are responding to therapy as compared to those in the CD4 count greater than or equal to 250 group. We could also say that 56 percent greater relative risk of response to therapy in the CD4 less than 250 group as compared to the CD4 count greater than or equal to 250. Those are both operable and proper statements to define this ratio. So what gives here with one measure involving p1 hat and p2 hat we see a nine 9 percent difference. With the other the other measure involving the same two numbers, we see a 56 percent difference. So let's look at another example that we actually looked at in the last section, but let's take it one step farther. This is the maternal-infant HIV transmission study we looked at and we looked at samples of infants who were born to HIV positive mothers, 180 who were born to mothers who were given AZT during pregnancy, and 183 born to mothers who were given PLACEBO. We already computed the sample p hats or proportions and saw that seven percent of the 180 children whose mothers were given AZT seven percent of those children developed HIV and as compared to 22 percent of those whose mothers were given the PLACEBO during pregnancy. So summary number one, measure number one, will be to compute the difference in proportions of risk difference. Again the direction is somewhat arbitrary but since we're really interested in the efficacy of treatment here AZT I'm gonna put that first in the computation. So the proportion of the risk difference is the p hat for the AZT group minus the p hat for the PLACEBO or 0.07 minus 0.22 which is negative 0.15 or 15 percent, negative 15 percent. So how do we interpret this is pretty large difference actually, this is a 15 percent absolute reduction in HIV positive transmission to children born to mothers given AZT as compared to children born to mothers given PLACEBO. Fifteen percent fewer children were diagnosed with HIV among those whose mothers were given AZT. Another way of saying the same thing as to say, there's a 15 percent lower absolute risk of HIV transmission to children born to mothers given AZT. Now of course we can come back and take another comparison which instead of taking the difference of those two numbers will take the ratio. We'll compute the relative risk or risk ratio by taking the seven percent of children who developed HIV in the AZT mother's group divided by the 22 percent in the group of mothers who were given PLACEBO for a relative risk of 0.32. So keep in mind here that this numerator 0.07 is substantially lesser in value than the denominator, and hence this ratio comes in less than one. So that's consistent with the fact that the difference in risks will be solved before was negative because the first number is lower in value than the second in the comparison. So how do we interpret this? Well, we could say the risk of mother/child HIV transmission for mothers given AZT is 0.32 times the chances or risk of mother/child HIV transmission for mothers given PLACEBO. That's a literal interpretation of this ratio, essentially those mothers or children born to mothers given AZT had about a third of a risk of developing HIV. In order to emphasize that this group on top has a reduced risk however that this ratio 0.32 shows a large reduction, we could talk about this as a 68 percent lower relative risk of mother/child HIV transmission for mothers given AZT. So one way to think about this is if we were to take 0.07 minus 0.22 and divide it by 0.22 which is akin, which would be that negative 0.15 over 0.22 that is equal to negative 0.68 that is 68 percent reduction. Another way to think about it is just if we had a ratio that becomes 0.32 over one that is our actual relative risks. So if we look into the percentage decrease that is relative to the denominator one is 0.032 minus one over one is negative 0.68, so 68 percent reduction in relative terms. So what gives here we've got two different numbers and they're both based on the same two p hat just different computations with them. So how do we interpret these or what's the difference in interpretation? Because both measures used exactly the same information but it give seemingly different results. With the risk difference we say there's a 15 percent reduction in HIV transmission, and with the relative risk we say there's a 68 percent reduction. It sounds a lot more when we talk about relative terms. Notice that both agree in terms of the direction of the association, both show that the risk in the first group, the children whose mothers were given AZT, is smaller than the rest in the second group, the group of children whose mothers were given the PLACEBO. So, what can we do here to rectify this differs? Well, let's talk about ways we can interpret these and think about what they mean similarly and differently. So, one way to think about using this number substantively is it can be interpreted as the impact (assuming causation) on a fixed number of persons. In other words, if we had a certain number of HIV positive pregnant women, how many fewer transmissions could we expect if we were to treat these women with AZT versus not treat them? So, if we were working in a city where every year we had about a 1,000 HIV pregnant women, we'd expect to see 15 percent fewer mother/child transmission if the 1,000 women were given AZT during pregnancy and that would result in a 150 fewer transmissions out of the 1,000 women, so that will be a substantial decrease numerically. We worked in a large city or at the country or county level, we had a group of 50,000 HIV positive pregnant woman. We'd expect to see 7,500 fewer mother/child transmissions if the 50,000 women were given AZT during pregnancy compared to if they were not treated. Whereas the relative risk can be interpreted as a way of communicating impact at the "individual level". Both of these can be used at the individual level but talks about the reduction in relative risk for an individual compared to not be treated versus being treated. So, for example, the risk that an HIV positive mother who takes AZT during pregnancy transmit HIV to her child is 0.32 times the risk if she did not take AZT. So far we're casting a woman about the potential benefits of taking AZT during pregnancy. Another way to put this is that you could reduce your personal risk of transmitting HIV to your child by 60 percent. In general, the risk that an HIV positive mother transmits HIV to her child is 68 percent lower if she takes AZT during pregnancy as compared to if she did not take AZT. Something to note about these two measures is they will always agree in terms of the direction of association. If in the comparison we're making, if p_1, the first group has lower risk than the second group, then the difference in the two will be less than zero and the relative risk will be less than one, still positive but less than one. If the first group in the comparison has a greater risk than the second group, then the risk difference will be greater than zero and the relative risk will be greater than one. If the two risks are equal in the two samples being compared, then the risk difference will be equal to zero and the relative risk will be exactly equal to one. So they will always agree in their general conclusions. However, the two quantities can appear different in terms of their magnitudes. Something to think about from a scientific perspective is it's possible to see a "large" effect with one measure and a "small" effect with the other. So, for example, if the underlying risk in the two risks being prepared are generally small, let's say it's 0.001 in the first group and 0.004 in the second group or 0.1 and 0.4 percent respectively, then p_1 minus p_2 is equal to negative 0.003 or negative 0.3 percent, an absolute decrease of 0.3 percent. Which doesn't look that large although if this were something we can enact at a population level and give to a large number of people, there still could be a sizeable reduction in the number of people who have the outcome. However, if we were to compare this in the relative scale and take the point 0.001 divided by the point 0.004, that ratio is equal to 0.25. That's a relative decrease of 75 percent. So certainly on the relative scale this is quite an impact even if it doesn't seem to be on the absolute risk difference scale. So, you can see in some of the situations we've looked at the increase or decrease as measured by the relative risk looks more dramatic than that measured by the risk difference. So you can imagine which one of these tends to appear more often in news articles for example. So you always want to think when you're reading the results of a study whether it be in a journal article or a news publication, you want to think about what are they presenting here in terms of the measure of risk comparison. So something to think about having only one of these numbers or the other, will not give the full story. If we have the relative comparison we don't have a sense of the magnitude of the individual proportions nor do we have that if we have the absolute difference. So we need both together to get a sense of that or at least the two proportions to start. Marilyn Vos Savant, who was a newspaper columnist, who takes questions about all kinds of things and she answers them. Sometimes they're riddles submitted by readers, sometimes they're questions about the meaning of life, sometimes they're real scientific questions and I actually really unimpressed with how she handled the response to this and we'll talk about what she's getting at. So this letter says, "I'm a middle-aged woman on hormone replacement therapy." So back in the early 2000s, there was a large trial on hormone replacement therapy done on post-menopausal women in the US. The trials suspended or cancelled early because of what they saw, an increased risk of heart disease in those who were given the replacement. So what this woman is asking is about this. So she says, "I'm a middle-aged woman on hormone replacement therapy (HRT), and the news about HRT is very confusing. For example, I read that heart disease increased by almost a third as a result of meditation. Yet I also read that the increase was slight. Which is it? What do the numbers really mean?" Well, here are the results of the study from the actual publication that came out. This looks at the incidence of heart disease in the women who were given hormone replacement therapy versus the women who were given a placebo, and all of the women were free of coronary heart disease at the beginning of the trial. So out of the 8,508 women who were randomized to have received hormone replacement therapy, 163 developed coronary heart disease for a percentage of 0.019 or 1.9 percent. Out of the 8,102 women who were given placebo, 122 developed coronary heart disease for a percentage of 0.015 or 1.5 percent. So, here are the risk difference and the relative risk. A risk difference is can we compare the proportion who developed coronary heart disease in the hormone replacement therapy group compared to the placebo? It's 0.019 minus 0.015 or 0.04, a difference of 0.4 percent. A 0.4 percent absolute greater risk of coronary heart disease among the women who got hormone replacement therapy. However, the relative risk looks very different if we take this ratio of 0.019 to 0.014 is 1.27. On the relative scale, this is a 27 percent increase in the risk and so this is what the woman who's letter was getting read that she saw both these numbers reported and was confused and Marylin does a very good job of breaking this down. What I want to note is that actually this is kind of not usual because a lot of times the number that the media is conceived upon would be the one that looks sexy or more dramatic. So it's important to know that a 27 percent increase on the relative scales is substantial but you still want to have some sense of what the absolute study values are to figure out what that means. How do we compare more than two groups with these risk differences and relative risk? Well, here is an example of a study. A randomized intervention through a health insurance plan, the hope was to increase colon cancer screening on part of the participants and the designs of four-group parallel design where patients in the health were randomized to one of four groups. They were either given the usual care or standard care that was used up to that point. They were given electronic health record linked-messages, automatic emails for example. Automated messages plus telephone assistance or automated messages and assisted telephone assistance plus nurse navigation to testing completion or refusal. So they have four different groups where the persons were randomized and what they did was they looked at the proportion in each of those groups who got screening within two years of the start of the study. What they found were the proportions in the respective groups were varied a lot and the group that got the usual care only a little more than a quarter past 26.3 percent. The group that take automated messages on top of that, the percentage almost doubled to 50.8 percent. Those who got the additional staff of assisted care, telephone assistance increased to 57.5 percent. Those who got everything posted as navigation, the completion rate or string rate was 64.37 percent. So, it's pretty impressive the gains that were made especially this jump from usual care to automated. So, for both the risk difference and relative risk comparisons, if you wanted to give them the numerical comparison, we can designate one of the four groups as a reference group and then compute the comparisons for each of the other three groups compared to this reference. So here are the risk differences and the relative risks for each of the other three groups each compared to the same reference group. You can see this is a situation where the gains on both scales are very unable. So in summary, the risk difference and the relative risk are two different estimates of the magnitude and the direction of association when comparing binary outcomes between groups. These two estimates are based on the exact same inputs and will always agree in terms of the direction of association but not necessarily the magnitude. The risk difference helps to quantify the potential impact of a treatment or exposure for a group of individuals and the relative risk helps quantify the potential impact or treatment for an individual as well. Neither estimate alone is sufficient to tell the entire "story". So it's important to have, even if you have only one of these, to have at least one of the proportions in the two groups being compared otherwise you're only getting a piece of the story.
