So, I just want to talk briefly about some properties of ratios to point something out here, and we will also revisit this later in the course and drill down in a little more detail but I just want to raise awareness at this point. So, upon completion of this lecture section, you'll be able to understand that the scaling of ratios is not symmetric around the value one, which would indicate equal values in the numerator and denominator. Consider the implications of this previous point when interpreting the size of an association quantified on the ratio of scale. Understand that on the log scale and we'll use the natural log, frequently represented as LN, the values of log ratios are symmetric about the value of zero. So, let me give you an example what I'm talking about here by showing you the results from computing the association between HIV transmission from mother to child in AZT compared to placebo, in both the direction we've been using AZT to placebo and then in the other direction placebo to AZT and then we'll compare the results. So, you recall we've looked at this several times when we looked at this data in a two-by-two table, seven percent of the children born to mothers who received AZT contracted HIV within 18 months after birth compared to 22 percent born to mothers who had a placebo. So, the summary measures we looked at computing both on the absolute and relative scale included the risk difference, the absolute difference in proportions, and properly signed absolute difference between proportion of children in the AZT group, compared to the proportion of the placebo group who contracted HIV, and that was the seven percent minus the 22 percent or a reduction on the absolute scale of 15 percent. A difference of negative 15 percent. On the relative scale, the relative risk if we computed it was 0.07 over 0.22 or 0.32 indicating a 68 percent reduction in the risk of contracting HIV on the relative scale. The odds ratio turned out to be similar but not exactly the same as the relative risk, it was 0.27 or an estimated 73 percent reduction in odds. In the odds of maternal-child HIV infant transmission. So, here we go, I'm just showing this now, probably easier to read than my handwriting, as I've typed out the interpretations in terms of their respective reductions. Suppose though however, there's no reason I had to compare AZT to placebo, even though that may make most sense given that we wanted to evaluate the impact of AZT. We certainly could have compared placebo in the direction of placebo compared to AZT. So, if we did this, the risk difference looks familiar. Except it takes the opposite signs instead of showing a reduction in the absolute scale of 15 percent. These the placebo group had 15 percent greater cases of a mother to child transmission than in the placebo group. So, it's the same absolute value 0.15 but takes the opposite sign. However, the relative risk and odds ratios look strikingly different. In some sense, so the relative risk if we go in the other direction, notice it's just the inverse of the 0.32 we got comparing AZT to placebo. We take that inverse and we get the relative risk comparing placebo to AZT and it's 3.1 which would indicate that those in the placebo group had 210 percent greater risk of maternal to infant HIV transmission than those in the AZT group, and while our odds ratio, which can be shown to be the inverse of the odds ratio in the other direction, is 3.7, a 270 percent greater odds of maternal infant HIV transmission for children whose mothers were in the placebo group. Here we go with just again typed it out, so that you wouldn't have to suffer through my handwriting. Let's just think about this for a minute. Certainly on the risk difference scale, the decrease and increase when we change directions AZT to placebo then placebo to AZT are comparable. They're the same magnitude, so, when we do placebo compared to AZT, that's a 15 percent increase and when we did it in the other direction, AZT to placebo, it was a 15 percent decrease. But look at these other measures. The relative risk we said indicated when we did a placebo to AZT, it was a 210 percent increase but when we did it in the other direction, it was a 68 percent decrease. So now that comparable percentage wise decrease as with the increase we did in the direction of placebo to AZT. Similarly, with the odds ratio, we saw when we compare placebo to AZT, there's a 270 percent increase in HIV transmission odds. But when we did AZT to placebo, it was a 73 percent decrease. So, the increase in the one direction is not numerically comparable to the decrease in the other direction on these ratio based measures. So, why is that? Why do they seem to differ in magnitude, if the direction of comparison is reversed? Let's look at the pictorial for a minute. So, if the two proportions were the same, then the relative risk and the odds ratio would both equal to one. But if we do it in the direction where we compare where the first group has lower risk than the second group and we create ratios comparing the first group to the second, whether it be the relative risk or the odds for the first group relative to the second. This ratio will be less than one but it can only go down to ultimately zero. So, the range of possibilities for the estimated ratios, the range of values range from slightly less than one to slightly larger than zero. If I compute them, putting the group with the lower risk on top. But if I reverse this direction, and instead report with the direction of the group with the higher risk in the numerator, whether that be it for the relative risk or the odds, the possible range of values ranges from slightly greater than one all the way up to positive infinity, theoretically. So, when I flip something that ranges between zero and one, when I take the reciprocal of that, I get a much wider range between one and positive infinity. So, the range of possibilities or possible values changes depending on which direction I expressed my ratios in. That's why we get situations where the increase looks a lot larger than the corresponding decrease as 200 per 10 percent increase in one direction versus a 68 percent decrease in the risk and the other when we're looking for example at relative risks. So, it turns out, if I transform the ratios from the ratio scale to the log ratio scale, this is an equalizing transformation. The log of any base, the log of the value one in any base including the natural log which has the base of the natural constant e, the log of one is zero. So, while one indicates equality on the ratio scale, if p1 equals p2, then their ratio is one. On the log scale, it would be zero, the log of one is zero. Values that are between zero and one in the log scale such as ratios comparing p1 to p2 and p1 is less than p2 or odds one to odds two. These will be between zero and one as we saw on the ratio scale, but when I convert that to the log scale, that range get stretched from just below zero down to negative infinity. So, we stretch or rescale the range from the very tight zero to one, to the entire negative number line. Contrast that with when we take values, if we have presented the ratio in the other direction p2 to p1 or the odds for the second group to the first, which was we said range from just above one up to positive infinity when we rescale that, that now ranges from zero to positive infinity. So, by presenting things on the log scale instead of the original ratio scale, we've equalized the range of possibilities for the measure of association regardless of the direction we presented. So, let me just show you what I mean by this and what the equalizing impact does on the log scale. Again, let's look at our relative risks. When I did this for AZT compared to placebo was 0.32 that 68 percent decrease when we did it to placebo compared to AZT, the ratio relative risk was 3.1, 210 percent increase. But let's look at both of these presented on the log scale. The log of this relative risk of 0.32 and since it's a ratio, you can show that it can be deconstructed into the log of the numerator minus the log of the denominator. That's just a property of ratios, that the log of a ratio equals the log of the numerator minus the log of the denominator and that equals the log of 0.07 minus the log of 0.22 or if we were just to put 0.32 directly in your calculator and take its log, you would get negative 1.11. Well, notice that that is just on the log scale that difference in the opposite direction because we switched the numerator and denominator, so when the ratio is expressed as 0.22 over 0.07, we can re-express the log of this ratio as the log of 0.22 minus the log of 0.07 or if you were to just put 3.1 into your calculator and use the LN key, you get the same result, it's equal to 1.11. So, even though these ratios have different magnitudes in terms of increase or reductions on the ratio scales, on the log scale the absolute difference is the same and the only thing that differs is the sign reflecting the direction of comparison. So, to summarize on the ratio scale, the range of possible values is for ratios between zero and one, for negative associations where the group in the numerator has lower risk than the group in the denominator. The range for positive associations goes from one to positive infinity when we flip the direction and the group in the numerator and the p1 is less than p2. We put that larger value in the numerator. We changed the range from a small tight window to a much larger window. But we equalize this on the log scale and the range of possible values in the log scale for the negative associations gets opened up to the entire negative number line, negative infinity to just less than zero. For positive associations, we move the target endpoint from one down to zero and we get the entire positive number lines, so the ranges are equal. All negative values for negative associations on the log scale and all positive values for positive associations on the log scale. So, the properties of ratios and log ratios have potential implications for displaying associations for different group comparisons making them comparable and performing statistically inference on ratio, something we'll get to later in the course, how to create confidence intervals for ratios. So, this is not the last time we'll look at this, and I will bring this back but I just wanted to plant the seeds of this thinking now, while we were talking about ratios and highlight that discrepancy in terms of things, how they're measured on the ratio scale, depending on the direction we put them in and how that's equalized when we put things on the log scale.