Enter the fourth week of the course today to build risk assessment models using linear and tree-based models
risk score
This week, you will learn strategies for building and evaluating survival prediction models that will allow you to compare individual patient risks. You will learn about two such models: Cox proportional hazards and survival trees. Finally, you'll learn about Harrell's c-index, which will allow you to evaluate the predictive performance of the survival model you'll build.
This will be a very exciting week as it will bring together the concepts you have learned in this class. In this lesson, you will learn about two other tools that go hand in hand with survival functions: the hazard function and the cumulative hazard function. You'll see that the survival, hazard, and cumulative hazard functions are all interrelated and can help answer different questions.
risk
In this lesson, let's talk about danger. So far we have looked at survival models, where we had the question, what is the probability of surviving after an arbitrary time t?
To answer this question, we see the survival function which tells us the probability of a certain event occurring after time t, we can represent the survival function with the x-axis, let's call it t, and the y-axis represents the probability of survival, which starts at 1 and tends to 0 over time.
But suppose we are now interested in another question, not what is the probability of survival after time t, but what is the patient's direct risk of death. We might want to know this to understand whether a patient is more at risk in the first year or ten years, and we can use this information to inform treatment when the patient is most at risk.
So, how do we express the immediate risk of death for a patient if they live to time t?
This is called the risk, and risk is denoted by the Greek letter "λ". The risk at t is the probability that an event will occur at time t
Like survival functions, hazard functions can also be represented graphically.
What does it tell us, over time, what is the risk at a given moment? Notice the shape, and we can tell what the patient's instantaneous risk of death is at any point in time. We can see that it is highest at 0, then quickly drops to 4, and then increases again to 10, with a similar level of danger as at 0.
So that allows us to see that the patient's risk is high in the left zone, not so high in the middle zone, and then increases. This is called the bathtub curve
At time zero, the risk of death is very high, then falls, and then increases over time. It can happen that with some treatments, such as surgery, the risk of complications can be high, but over time, they decrease. However, patients are at increased risk over time.
survive to risk
Now, the hazard function can actually be used to create a survival function. They are relatives and we won't discuss their relationship too much. But it's important to understand that there is a formula that allows us to derive survival from danger.
Therefore, we can use this formula to get the corresponding survival curve from the hazard curve.
This is the survival curve versus the hazard curve. We see, what is the probability of survival after any time t, easily. For example, if we want to know the probability of survival after t=4, then we can read from the plot that it is slightly above 0.2. At the same time, we can answer questions like, what is the patient's immediate risk of death if they live to time t, by looking at the risk at t=4 and realizing that the risk at time 4 is less than the risk at time 8.
This relationship works both ways, so we can actually infer risk from survival curves as well. Again, we don't have to worry too much about giving the formula for their relationship
but you can interpret the formula as saying that if age is t, then the hazard is the mortality rate. Therefore, using this survival curve, we can generate a corresponding hazard curve.
Here we can see that the risk for this patient population is a constant function which means that the risk of immediate death is constant at any time t; which is 1. Here is the corresponding survival curve. We have now considered survival and hazard as two functions, one can be used to derive the other and vice versa.
cumulative risk
So far, we have looked at the probability of survival after time t, and the risk of dying immediately when a patient arrives at time t, known as the hazard, through the survival function.
Now, we will explore another question, that is, what is the cumulative risk or cumulative risk (Cumulative Hazard) of the patient before time t? We will use the Greek letter "Λ" (uppercase Lambda) to represent this concept, remembering that the lowercase letter "λ" means hazard, and "Λ" means the cumulative hazard before time t, which is "Λ"(t).
Of course, the cumulative risk of patients before time t is closely related to their risk at time t, and the relationship between them can be expressed in the following way
This means that for any time point t, we are looking at all time points before it and summing them. This is the expression when time can take discrete values, such as 0, 1, 2, 3. We also saw the case when time can take on a continuous number from 0 to 3 or whatever we want.
Therefore, we write the continuous expansion of the cumulative hazard as the integral of the lambda from 0 to t. If you have never been exposed to integrals before, don’t worry too much about how this formula works. You just need to understand that the cumulative risk rate will become a summation, and this formula is used to represent the situation when time is a continuous variable.
We have seen risk curves previously, with instantaneous risk on the y-axis and time on the x-axis. We can use this hazard curve to tell us what the corresponding cumulative hazard curve will look like, using the formula we saw earlier. Again, if you haven't touched integrals before, don't worry too much about what this formula does, it's just a way of summing hazard rates along the time axis.
So this is a corresponding cumulative hazard ratio that tells us how the patient's cumulative risk changes over time.
This is a constant hazard curve, so we can see that the hazard has not changed over time, and its corresponding cumulative hazard curve is shown below; you can see that, straight up, we are summing over the time dimension.
Previously we saw that survival models can be used to output survival functions, but now we know that it can be used to output not only survival functions, but also hazard functions or cumulative hazard functions. We can use any of these functions to formulate the other function in order to answer any question we want to answer.
This class introduces some new concepts, without detailed derivation of their formulas. It may lie in its complexity, and interested friends can learn from other materials.
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I'm Tina, see you in the next blog~
Working during the day and writing at night, working hard
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