So let's revisit our brief note about ratios, and we'll intersperse this throughout the course. So here's another mention of it. So recall from the first brief note about ratios back from lecture five, and some ideas we've talked about in previous sections of this lecture set. Just to remind you again, that the scaling of ratios is not symmetric around the value of one, but of course, the value of one would indicate equal values in the numerator and denominator. So as we've seen several times before, if I have a ratio, I'll just put it in terms of relative risk, but generically speaking if p1 is less than p2, and I express the ratio is p1 over p2, the range of possibilities for values in that direction go from between zero and one. But if I were to express it in the opposite direction with the larger value on top of the smaller, the ratio could it be go for anywhere from one to positive infinity. So a lot greater range of value possibilities when the numerator is larger than the denominator. So what we've seen is when we take the log of the ratio, we use the natural log, the values of the log ratios are symmetric about the value zero. So as we've seen several times, and I've shown in better graphics than I'm doing now, if we map this to the log scale, the log of one becomes zero, and the range of log ratios when the numerator is greater than denominator goes from zero to positive infinity. In the other direction if we go the log of the ratio where the numerator is less than the denominator, and so the ratio is less than one, the log of that goes from zero to negative infinity. So we equalize the range of possibilities for values on the log scale. So this re-scaling to this egalitarian log scale, also means that the confidence interval limits are comparable in the log scale for both positive and negative associations. So let me show you what I mean with this example with regards to maternal HIV infant transmission. Certainly we know, spoil this in a little more detail as well in the additional examples, but certainly we know we've seen this several times now that the proportion of children who develop HIV born to mothers who were given AZT during pregnancy was 7 percent compared to 22 percent in the group of children born to mothers given placebo. We generally computed our associations in the direction of AZT compared to placebo, so when the relative risk scale this was 7 percent divided by 22 percent are relative risk of 0.32, and the confidence interval goes from 0.18 to 0.58. But if we had instead computed this in the opposite direction, and compare the relative risk of transmission to children for mothers who got the placebo compared to AZT, it's the reciprocal one over 0.32 which is 3.1. If we look at the confidence interval for this comparison, it goes from 1.7 to 5.61. So it looks like this is the same estimate just expressed in the opposite direction and the same confidence limits just expressed in the opposite direction, but if we were to look at these and compare them in terms of the interval widths especially, we can see the interval widths differ greatly, and it would look like the estimate is a lot more precise were we to estimate it in the direction of AZT to placebo, than in the other direction. But that doesn't make a lot of sense that one direction will be more precise than the other because it's the same association just expressed in the opposite direction. Again, this disconnect here and the disparate widths of these intervals is because of the different scaling for associations with ratios greater than one versus associations with ratios less than one, the different range of possible values. So if we take things to the log scale, we see not only are the sizes of the associations equivalent on log scale in terms of absolute magnitude they just differ by sine, so even though the magnitudes in terms of percent decrease in the direction of AZT to placebo, and percent increase with regards to placebo to AZT, we're not comparable. When we log that we get estimates of equal absolute value and just opposite signs, and the confidence interval limits are just the opposite signs of each other as well. So we can see that the width of these confidence intervals is exactly the same, and the endpoints are the same in absolute value just opposite signs. So when we look at things on the log scale, the magnitude of the effect or association is the same, the absolute magnitude and it's only the sign that differs depending on the direction of association and the precision of our estimates are the same on the log scale, because we've re-scale things to a scaling in which the range of possible values is the same regardless of the direction of comparison. So that's the equalizing factor of a log scale and that's why we do our computations on the log scale as well as we've talked about before. But this has implications in terms of also presenting the results, and making ratios, and their precision comparable in the same presentation. So I'm going to show you an example here where they report several relative risks, or in this case, incidence rate ratios, and they want to show them on the same comparable scaling regardless of whether the association is positive or negative. So let me just tell you about the study. Here this was a study done on patients undergoing dialysis, the title of the article that was published in Jama's association of race and age with survival among patients undergoing dialysis. So the context for this is that many studies have reported that black individuals undergoing dialysis survive longer than those who are white. The observation is paradoxical given racial disparities in access to quality to and quality of care, and is inconsistent with observed lower survival among black patients with chronic kidney disease. So we hypothesize that age and the competing risk of transportation modifies survival differences by race. So what they ultimately did was they looked at death as the outcome in a timed event framework comparing black and white patients, but they look at it separately by different age groups in the study. What they present here, incidence rate ratios for mortality of black patients versus white, but presented separately across age groupings and adjusted for other differences between black and white patients, but this is presented on the log scale so let's just take a look at this. So in the group of patients who were 18 to 30 years old undergoing dialysis this dot here is the estimated incidence rate ratio mortality for black patients to white patients, and then these corresponding bars represent the confidence interval around that estimate. So we can see in the youngest age group black patients have a notably higher, somewhere between 1.752 and incidence rate ratio mortality compared to white patients, and this is statistically significance since the confidence interval for the incidence rate ratio does not cross the null value of one. As the age increases of the patients, we see in the group 31 to 40-year-olds, black patients still have a higher risk of mortality but it's lower than that comparison for 18 to 30-year-olds, and then in the 41 to 50-year-olds, black patients still have a higher risk of mortality than white patients, but again it's lower than the previous two age groups. After 50, the direction of association changes. We start getting into the ages 51 to 60, the incidence rate ratio for black patients to white is less than one, and it stays that way in older patients. They're presenting estimates relative incidence rate ratios here on they're presenting them numerically as ratios and the confidence interval, and they've labeled as ratios, but this scaling here if you look at this graphic here, the scaling here is not typical arithmetic scale. The scaling there showing this on is the log scale. The reason is if we do that then the interval widths here and the size of the association magnitude are comparable whether the association is positive or negative. So all these confidence interval widths are comparable. So we can say that the precision of these incidence rate ratios tends to get better, and the width of the confidence interval decreases as age increases. That is not because of the different scaling on the ratio scale for associations less than one, versus associated with greater than one, it's not because of that because this now is graphically been presented on the log scale. So how would you know this is the log scale? Well, it's a little hard to tell in this example we're allowed the estimates are relatively close to one,. But if you look at the distance here between one and 0.75, if you were to compare that to the distance between one and 0.25, so the distance on the ratio scale is the same when you use a 0.25, but if you look carefully, the distances between this to one and 0.75, and 1 and 1.25 or not equivalent. That's because these distances are presented in log units and not actual arithmetic units, and so the scaling here is of the log variety, and not of the arithmetic variety. So it's a little hard to see here because the value ranges isn't that great going from 0.75 up to 2.25, but they are ostensibly presenting this on the log scale that so that we can compare the magnitude of the estimates and the precision, but they're labeling with their ratio counterparts numerically so that we don't have to take the values here and exponentiate them. So it's a nice way to present things, and make things comparable visually, and still retain the essence of the ratio values we're looking at. So in summary again because the inequity and ranges for negative and positive associations measured via ratio, the size was association, it's precision can appear very different depending on the direction of the association. However, on the log scale for ratios, the size of the association and the uncertainty are the same regardless of the direction of comparison. This is why the 95 percent confidence intervals for ratios are first computed on log scale, then exponentiated to present on the ratio scale. This is again why ratios and their confidence intervals are sometimes presented on the log scale in a graphical comparison like we saw in that previous example.
