Survival Analysis Part 2 | Survival Function, Hazard, & Hazard Ratio

[Music] we previously introduced the idea of what censoring is in survival analysis what we're gonna do now is just kind of a brief overview to some important things that we want to know through survival analysis so what kind of lay out a foundation what is the survival function what is the hazard and hazard ratio of those sorts of things we'll look at some plots of what is a survival function look like and after we do some some preamble and some intro eventually we get into looking at different survival models the kaplan-meier survival model the exponential or viable survival model as well as Cox proportional hazards model but before we do that I want to lay some foundation some of these concepts will be a little bit abstract until we encounter them in these exact models but the goal is to lay out some conceptual foundation that's a bit abstract then to start to look at the details and these will make more sense and we'll will revisit these concepts as we progress through all the ideas so the first thing of interest is the survival function and often this is written as big s of T and this is the probability that Big T is greater than little T or the probability that once a survival time is beyond time T it's been words this is looking at the probability of surviving beyond time T if we're looking at survival in years the probability that C is greater than 5 is the probability of surviving beyond 5 years then we also have that EMP hazard and I'm just going to abbreviate this as haz as we progress through these ideas in notation this is the probability that T is less than T plus Delta given that T is greater than little T it in words this is we can think of it as the probability of dying in the next few seconds given you're alive now okay again given that use your survival time is greater than T what's the probability of survival time is less than T plus Delta given that you're alive at the five-year mark what's the probability you die just after the five-year mark passes and again should we mentioned the hazard on its own doesn't have that much of a meaningful interpretation in a moment we'll look at survival plots plot of survival function we'll talk about what the survival function is and what the hazard is for the survival function so we'll give this a bit more meaning in a moment David is note it doesn't really have much kind of intuitive meaning say what's the probability I die within the next few seconds given that I'm alive now but where this does become meaningful is when we look at the idea of a hazard ratio a hazard ratio and we're gonna read it abbreviated HR is the hazard save for x equals 1 divided by the hazard for x equals 0 and what I mean by that is when we look at relative hazard what's the hazard for someone who's exposed relative to someone who's not exposed okay so for the sake of discussion suppose that the hazard doesn't have that meaningful interpretation on its own but suppose that the hazard ratio came out to be 2 what this is telling us is at a given instant in time someone who is exposed to the risk factor their risk of dying is double someone who's not exposed okay so has it become more meaningful to interpret we look at hazard ratios here the ratio of two members that's good to get the idea of looking at the survival function what is a survival function what's the hazard in there and so on thinking of these survival function here is time tt-they here's the survival big essence which is the probability the survival time is greater than little T so time starts at time zero let's just say there's time 1 2 3 4 5 so on and you survival goes from 0 up to then you just point it up here it's a little bit messy 1 or 100% at time 0 everyone is alive now the first survival function Network and look at or a survival model is gonna be the camp in minor model you may have seen this before if you haven't we're going to introduce it in a following video as if you haven't seen it before but often people have encountered this so they tend to look something like this this here is the survival function we can use it say what's the probability of surviving beyond time to that time to the probability of surviving down there that's ballpark that about 80% the probability of surviving beyond time 5 is about 50% the problem is surviving beyond time let's just suppose this is here probably surviving beyond time 10 it's about 25% okay so this allows us to make statements about survival of the probability of surviving beyond a certain time okay so this looks like well this is what gets called the survival function and the one I've drawn in here oh that's right KN this is the kaplan-meier survival model we'll get to talk about all the details of that in following videos so this is kind of a nonparametric they don't write that yeah this is a non parametric model and there's no parameters that describe its shape okay it's just a step function that steps down we can imagine trying to have a nice smooth curve that fits through this and the one we're gonna fit we'll talk about this again after the kaplan-meier model a little bit we can imagine say fitting a exponential a negative exponential shape curve to that so this here what I've gone in is the exponential survival this one is a parametric and what we're going to use there is a negative exponential curve to represent the survival so again you can see this can be used in the same way it's the survival function right the probability of surviving beyond time for is about let's say 60% the probability of surviving beyond time 10 it looks like it's about 20% okay so these describe these survival or the probability of surviving beyond time T now the second thing of interest is the hazard they said the hazard we can think of it conceptually as the probability of dying in the next few seconds given that you're alive what's the probability of failing and given that it's working right now what that is in terms of the survival function the hazard is actually the rate of decrease so in the case of the exponential this is a nice smooth function the rate of decrease is going to be the parameter that defines the shape resist one more we're going to later I'm just going to mention it now for the sake of trend mention the three survival models will look at Cox proportional hazards model this is called a semi parametric semi-parametric sort of meaning it's sort of a combination of these two here and as we start to talk about these three in more detail well expand on exactly what semi-parametric means and how it tries to be a combination of these two here but that's the general idea what we do next is in kind of a big-picture overview for these three different models that were in explore in detail we've progressed to this what are the pros and cons of using each of these and again the pros and cons are going to be a little bit abstract at the start and as we progress through and looking at the details of these the pros and cons are going to become a bit more meaningful so what I'm going to do is talk about the pros and cons well we haven't really talked much about the details of these models yet then we're look at the details of them and then after that we'll come back to the pros and cons again and remind ourselves what the pros and cons were once we've gone through the details pick around guys there's more to see at least a seat [Music]

Channel: MarinStatsLectures-R Programming & Statistics

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Id: MdmWdIV5k-I

Length: 10min 11sec (611 seconds)

Published: Tue Mar 24 2020