Survival Analysis: find marginal CDF from marginal hazard rates?

• NewNameStat
In summary, the conversation discusses a formula given by a professor involving cause-specific hazard rates and marginal hazard rates for competing risks. The attempt at a solution involves using independent competing risks and a mixture of treatments, but there is uncertainty about its accuracy due to the value of F1 not equalling 1. The conversation also mentions potential issues with unequal numbers of patients receiving treatments.
NewNameStat

Homework Equations

My professor gives us the following formula: [ tex ] F_j(t) = \int\limits_0^t exp\Big\{-\sum\limits_{j=1}^J \int\limits_0^u \lambda_j^{\#}(v)dv \Big\} \lambda_j^{\#}(u)du[ / tex ] where [ tex ]\lambda_j^{\#}(t) [ / tex ] are the cause-specific hazard rates for cause $j$.

The Attempt at a Solution

Assuming we have independent competing risks (we have not yet talked about dependent competing risks so I think it's safe to assume this), $\tilde{F}_j(t) = \int\limits_0^t exp\Big\{-\sum\limits_{j=1}^J \int\limits_0^u \tilde{\lambda}_j(v)dv \Big\} \tilde{\lambda}_j(u)du$ where $\tilde{\lambda}_j(u)$ is the marginal hazard rate for cause $j$.

So, $\tilde{F}_1(t) = \int\limits_0^t exp\Big\{- \int\limits_0^u 1 dv - \int\limits_0^u .5\sqrt{v} \; dv \Big\} .5\sqrt{u} \; du = .5\int\limits_0^t exp\Big\{- u - \frac{1}{3}u^{3/2}\; dv \Big\} \sqrt{u}\;du$.

However, I do not think this is right because $\tilde{F}_1(\infty)\neq 1$ (it equals something around $0.25$).

NewNameStat said:

Homework Equations

My professor gives us the following formula: [ tex ] F_j(t) = \int\limits_0^t exp\Big\{-\sum\limits_{j=1}^J \int\limits_0^u \lambda_j^{\#}(v)dv \Big\} \lambda_j^{\#}(u)du[ / tex ] where [ tex ]\lambda_j^{\#}(t) [ / tex ] are the cause-specific hazard rates for cause $j$.

The Attempt at a Solution

Assuming we have independent competing risks (we have not yet talked about dependent competing risks so I think it's safe to assume this), $\tilde{F}_j(t) = \int\limits_0^t exp\Big\{-\sum\limits_{j=1}^J \int\limits_0^u \tilde{\lambda}_j(v)dv \Big\} \tilde{\lambda}_j(u)du$ where $\tilde{\lambda}_j(u)$ is the marginal hazard rate for cause $j$.

So, $\tilde{F}_1(t) = \int\limits_0^t exp\Big\{- \int\limits_0^u 1 dv - \int\limits_0^u .5\sqrt{v} \; dv \Big\} .5\sqrt{u} \; du = .5\int\limits_0^t exp\Big\{- u - \frac{1}{3}u^{3/2}\; dv \Big\} \sqrt{u}\;du$.

However, I do not think this is right because $\tilde{F}_1(\infty)\neq 1$ (it equals something around $0.25$).

I have re-formatted your tex commands, to make them work properly. You should:
(1) remove the spaces in "[ tex ]" and " [ / tex ]", writing "[TEX]" and "[/TEX]" (but all in lower-case letters); and
(2) use "\exp" instead of "exp".
Also: removing the superscript # from the lambdas would make it look better; those superscripts serve no useful purpose. Just writing ##\lambda_j## is good enough. However, I did not do that, since what you wrote is not incorrect---just unnecessary.

$$F_j(t) = \int\limits_0^t \exp\Big\{-\sum\limits_{j=1}^J \int\limits_0^u \lambda_j^{\#}(v)dv \Big\} \lambda_j^{\#}(u)du$$

Finally: I do not understand the above formula. If, say, there were 1000 patients involved and each treatment involved 500 patients, the probability that a randomly-chosen patient falls in class 1 or class 2 is 1/2, giving F1 as a mixture:
$$F_1(t) = \frac{1}{2} F_{11}(t)+\frac{1}{2} F_{12}(t).$$
However, if unequal numbers got treatments 1 and 2, such as 400 receiving treatment 1 and 600 receiving treatment 2, we should have
$$F_1(t) = \frac{4}{10} F_{11}(t) + \frac{6}{10} F_{12}(t).$$
Here, ##F_{11}## uses ##\lambda_1 = \sqrt{t}## and ##F_{12}## uses ##\lambda_2 = 0.5\sqrt{t}##.

I did not even attempt to read your analysis, since the tex commands are not working; please edit and re-submit.

1. What is Survival Analysis?

Survival Analysis is a statistical method used to analyze the time until an event occurs. It is commonly used in medical and social science research to analyze the survival rates of patients or participants. It takes into account both censored data (where the event of interest has not yet occurred) and uncensored data (where the event has occurred).

2. What are marginal CDF and marginal hazard rates?

Marginal CDF (Cumulative Distribution Function) and marginal hazard rates are two important concepts in Survival Analysis. Marginal CDF is a function that gives the probability that an event occurs at or before a certain time. Marginal hazard rates, on the other hand, are a measure of the instantaneous risk of an event occurring at a particular time, given that it has not occurred yet.

3. Why is it important to find marginal CDF and marginal hazard rates?

Finding marginal CDF and marginal hazard rates is important because they provide valuable information about the probability of an event occurring and the risk associated with that event. This information can help researchers make predictions and identify factors that may affect the event of interest.

4. How do you find marginal CDF from marginal hazard rates?

To find the marginal CDF from marginal hazard rates, you can use a mathematical formula known as the Nelson-Aalen estimator. This estimator calculates the cumulative hazard rate at each time point and then converts it to the cumulative probability using the exponential function.

5. What are some real-world applications of Survival Analysis?

Survival Analysis has many real-world applications, including medical research (such as studying the survival rates of cancer patients), social science research (such as analyzing the time until a person finds a job), and business analytics (such as predicting the survival time of a product in the market). It is also used in engineering, economics, and many other fields.

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