Nonlinear Risk Assessment Model Using Survival Numbers
In this class, you will learn about survival trees. You will learn how to take patient variables into account and compare risks across different patients.
You'll see that survival trees are similar to binary decision trees, allowing you to build models that capture non-linear relationships in patient data.
In this class, we're going to talk a little bit about survival trees. Previously, we studied survival models for population populations as well as individual hazard functions. We now see a key feature of these individual risk functions:
它们在风险上是成比例的
we can express one risk factor as a multiple of another. In the formula, we see that a patient's risk is the base risk rate multiplied by a factor that is determined by their variables.
Here we see the variables age and smoking with weights associated with these variables. Let's see what this expression does. This means that the risk increases with age. Because remember, this is a positive number with an exponent greater than 1.
But suppose we have a situation where young people are more dangerous than middle-aged people; older people are more dangerous than middle-aged people. We were unable to capture this relationship as a linear function of age. Because a linear function of age can only model a rise or fall in risk. You can't model it as going down and then up. So this is one of the disadvantages of using linear functions and Cox proportional hazards models.
Another disadvantage of the Cox proportional hazards model is that the hazard functions of the two patients are always proportional. This means we assume that the shape of risk looks the same for similar patients. However, patients may have very different risk profiles over different time periods.
For example, suppose we observe patients undergoing treatment such as chemotherapy. For chemotherapy, let's say we can have high-dose and low-dose chemotherapy. If we do high-dose chemotherapy, then we can expect that the post-treatment risk is high, but the long-term risk is low. Low doses are less harmful initially, with lower initial risks but higher long-term risks.
Therefore, we cannot represent these situations with a Cox proportional hazards model. Because in a Cox proportional hazards model, the difference in risk factors for these two curves should always be the same. Whereas here, we can clearly see that they cross, so the difference in risk factors is not fixed. In fact, they change over time.
So our goal is to be able to solve this problem, if we have a population of patients, we should have different risk functions for different types of people in that population. When referring to hazard functions, keep in mind that hazard functions can be used to obtain cumulative hazard functions. This can be used to obtain survival functions and vice versa. So our goal is to be able to build survival trees based on patient data, capturing the non-linear relationships in patient data.
number of survivors
Let's say we're looking at a group of patients and we're looking at variables, blood pressure on the y-axis and age on the x-axis. So in the data we got, we can clearly see three groups of people.
We have a group of older patients. We have a group of young hypertensive patients. We have a group of patients with hypotension and young age. Note that each point representing a patient is marked with their survival time.
White dots are censored data, blue indicates events (diseases). Therefore, we can try to manually classify, assuming that the age threshold is set to 60, and the blood pressure is 160, we can get three different groups.
We have seen decision trees in previous modules. Let's see what they look like.
Remember earlier we had a decision tree where we were trying to classify whether a person was at risk of heart attack or death. We see that we can build these lines, we can classify patients, they are in the appropriate risk category, we have low risk, and high risk category, which we propose as part of our decision tree.
There are a few key differences here.
The first is the time-to-event model we are dealing with now. So in our time-to-event model, we're not just saying whether something is low risk or high risk, we're actually saying what the risk is at each point in time. Remember, we can denote the risk at each point in time by (t), or we can denote the cumulative risk, or the survival function, since they are interchangeable, and in this particular example we are concerned with the cumulative risk.
The second difference is that we are now dealing with survival data. So these patients are not just red or blue like before to indicate whether they died or survived 10 years, but survival time. There is also censored data.
When we had a new patient come in, the patient was 52 years old with a blood pressure of 132. Cumulative risk estimate for group b
Nelson AalenAssessment
Let's talk about how to actually estimate cumulative risk for groups.
Here we have six patients, and for each patient we write the outcome, either event time or censored
time.
Now, once we have such a table, we've seen how to use the Kaplan-Meier method for survival estimation to derive a survival function using these data.
But Kaplan-Meier estimation is not the only estimation function we can use.
Another called Nelson-Aalen估计器allows us to estimate the cumulative risk of the population.
We will not deduce this in depth. We will illustrate with the example of generating a cumulative risk estimate for a population.
Let's take a closer look at this expression. We have the summation of di/ni from i equals 0 to t. The definitions of di and ni are the same as before. di is the number of deaths at time i. ni is the number of people alive at time i ( T i >= i T_i >= iTi>=i）。
Let's take age 33 as an example and try to use this Nelson-Aalen to estimate cumulative risk at age 33. So we have the summation of di/ni from i equals 0 to 33. This gives a sum of 0.75. 0.75 is our estimated cumulative risk at age 33. Of course, our time can be any number between 0 and infinity, and we can get cumulative risk estimates for any time.
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I'm Tina, see you in the next blog~
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