Hazard Ratios and Survival Curves

all right welcome we're going to talk now about hazard ratios and survival curves and probably the best way to get into this it's a look at an example first so here's a table from one of the papers that we're going to be looking at and you can see here hazard ratios let's break this term down so a hazard typically refers to a bad event that's going to happen and in the case of this paper its death we have all kinds of deaths we have all death from all causes cardiovascular death and non cardiovascular death but I should mention that the event that happens doesn't have to be death it could be a time to having a heart attack or it could be something good actually like even time to being cured from a disease but in this paper we're looking at death so it's a bad thing now in the denominator we have the word ratio edit that out and the next half of this term is the word ratio and what that implies is we're going to be comparing two things by dividing we're going to compare one thing to another thing and in this paper we're looking at vitamin D levels and we're looking at a bunch of groups right here we have five different groups at different levels we've got groups with levels less than 25 25 to 50 50 to 75 75 to 100 and greater than 100 so we will be looking at the death in rate in each one of these groups and comparing them by the by way of a ratio and our denominator in these groups is going to be this group here the one that is marked as referent so that's going to be the one that has a vitamin D level that is greater than 100 so let's take a look at this group right here and so we're going to be looking at the group in the numerator that has a vitamin D level less than 25 comparing it to the group that has a vitamin D level greater than 100 and we're saying that the ratio of the outcome which is death is 2.5 times greater in the group that has low vitamin D than the group that has high vitamin D so it would seem here it's not a good thing to have a low vitamin D level there are some subtleties to saying that it's point five times greater really isn't it we'll go into that in a little bit but I think if you think of think of it that way in your head you pretty much I'm going to understand what this paper is trying to say so 2.5 times greater this one here is one point two four times greater to have a level that is 50 to 75 compared to having a vitamin D level that is greater than 100 however can't we can not forget to look at the confidence intervals right so here this is the confidence interval for this one and it says that our point estimate is that it is one point two four but the value could just as easily be zero point nine five or one point six two and what that means is it could actually be worse to have a vitamin D level that is 50 to 75 or it could actually be better we don't know or it could actually be the same if the number is if the ratio was one now if you look at this confidence interval it does not include the point of no difference in a ratio which is one and so it's saying even at our worst estimate at one point six four you're still better off having a vitamin D level of greater than 100 you're still more likely to die with a vitamin D level less than 25 compared to the group that has a vitamin d level of greater than hundred and so what they did for us here is they put in boldface any of these numbers that has a statistically significant confidence interval that means the confidence interval does not include the number one in this case point seven it's lower than one but the high end of the comfortable is much less than one as well in this case two point oh two it's higher than one and the low end of the confidence interval is greater than one now the Mexica we should look at our survival curves and here's an example of one here and what you'll see on one axis you have the survival probability that probability goes from zero percent survival needs a pretty much toast to 100% survival whoo-hoo you live right and then you got time on the other axis and here it's called follow-up in months of zero fifty 150 and so you can see with each one of these curves your chance of survival is decreasing your probability of survival is decreasing and again we're comparing groups and you'll see here the I've color-coded each one of these groups that we can compare them now and the group that has a high vitamin D level is dying off but it's dying off at less of a rate than the group that has a vitamin D level of less than 25 it's dying off and let's separate them then all of these as well and so these are called survivor survival curves or kaplan-meier curves named after the people who came up with it and this is just a way to visually understand what's going on now you might be asking yourself and quite legitimately what's the difference between hazard ratios and risk ratios they actually sound very similar but they're really not the same thing and let me explain why so let's say that we're doing a study over the course of two times and over the course of time and we're looking at two groups one who is exposed to something and one who is not exposed then we decide we're going to take a look at the death rates of these groups that measure that over time first for the that exposed group looks like most of them die pretty early and the exposed group seems to live a lot longer they die more along the end so when you calculate a risk ratio you are looking at this entire time interval you're looking at the risk of death in the exposed group versus the risk of death in the not exposed group and we would calculate something like this the number of dead over the total in the exposed number of death or the total and they're not exposed and divide them and we got a risk ratio now let's look at how we calculate the hazard ratios it's just a little bit different and that we're going to slice this time and travel up into different intervals and we could calculate the hazard rate the hazard rate for each one of these intervals so for this not exposed we know that four of these people died of the 21 so about 19% and the hazard rate for the exposed group only one out of fifteen died and so that's about a 6% rate and if you're the calculator for the next interval got one two three four five six out of 17 that were you know alive at after this point 35 percent that event right here and we had one person die in the exposed group out of a remaining 14 that were left so about a 7% hazard rate and you could calculate the ratios of these in each one of these intervals as well and the thought is that the ratio between these two will stay about the same that is the hazard rates are proportional so let's talk now about survival probability if we look at one interval let's say this interval right here the survival probability depends on two things it depends depends on the product the probability of surviving this interval times the probability of even making it this far can the survival probability and the hazard probability are related by this horribly ugly formula but don't worry a computer will do that for you don't have to memorize that we're just going to think about the concepts here and so that there is a relationship between dying of this but you know or surviving and it's a kind of an inverse relationship and it's complicated because it's using intervals and remember if you made these intervals infinitesimally small you would have an instantaneous probability and now we're going into the realm of calculus which is why you see funny things like this here but we're not going to think about any of that we're just saying that the hazard ratio is the ratio of one group surviving compared to another group within a specific time interval and you can also think of it as a survival probability which is the probability of surviving to this interval times the probability of surviving inside that interval and if you graph all of these survival probabilities that's when we get our kaplan-meier curves so if we look now at this survival curve that I've drawn here of the exposed versus a not exposed group one thing that you'll see is that this line and this line are are diverging they're going farther and farther apart because a survival in this group is getting worse than this five on that remember we talked about that it usually is going to be the hazard ratio is going to be a constant rate it's going to be proportional and that means this is going to be a grow at a normal rate now this doesn't have to be the case though you could have one survival curve that first starts out getting worse and then it actually starts getting better so if you look at this part of the curve over here the exposed group here has a less of a event rate it's got a better survival rate than this part but if you go down over here the survival rate is worse for the exposed group and better for the not exposed group and so this one does not have that proportional hazard rate I only mention that because you're probably going to at some point say something where they talk about Cox models and proportional Cox proportional models and those models allow you to do things with confounding using hazard rates and survival curves but they really really assume that the rate is proportional and so it doesn't have to be the case but it usually is proportional and then you're able to use those models to deal with the confounding so this is in a nutshell what hazard rates are review the points again remember hazard ratios are not the same as risk ratios but they're about the same thing right we're going to look at the hazard ratio and we're going to as a number and if one group is 2.5 if the hazard ratio is 2.5 it means that in a particular given interval one group has a 2.5% 2.5 times more likely chance of experiencing the event in this case death then the other group does it doesn't mean that the group in total is 2.5 times more likely remember because each one of these looks at particular intervals but you can think of it like that like this looking at the whole time and all that one group is more likely than the other group to be experiencing that hazard that element of the time interval is important with the hazard ratios and remember the survival curves it's these little jaggedy things like this the kaplan-meier curves that shows you who's dying off more quickly than than the other group you can you can compare groups that way and that's it for hazard ratios if you have any questions please put them in the comments below thank you for watching

Channel: Rahul Patwari

Views: 135,182

Rating: 4.9488273 out of 5

Keywords:

Id: p1wa8W11JLI

Length: 10min 59sec (659 seconds)

Published: Mon Dec 28 2015