Problem 35 - A First Course in Probability

Question

The lung cancer hazard rate \(\lambda(t)\) of a \(t\) -year-old male smoker is such that $$\lambda(t)=.027+.00025(t-40)^{2} \quad t \geq 40$$ Assuming that a 40 -year-old male smoker survives all other hazards, what is the probability that he survives to (a) age 50 and (b) age 60 without contracting lung cancer?

Step-by-Step Solution

Answer

The probabilities for a 40-year-old male smoker to survive without contracting lung cancer are as follows: (a) Approximately 5.9% chance until age 50. (b) Approximately 0.0012% chance until age 60.

Step 1: Calculate the cumulative hazard function H(t)

To find the cumulative hazard function \(H(t)\), we need to integrate the hazard rate function \(\lambda(t)\) over the age interval. Let's first find the cumulative hazard function for the general case: $$H(t) = \int_{40}^{t} \lambda(u) \, du = \int_{40}^{t} (0.027 + 0.00025 (u - 40)^2) du$$

Step 2: Calculate H(50) and H(60)

Now let's calculate the cumulative hazard function for ages 50 and 60. We can substitute these values for t in the above equation: $$H(50) = \int_{40}^{50} (0.027 + 0.00025 (u - 40)^2) du$$ and $$H(60) = \int_{40}^{60} (0.027 + 0.00025 (u - 40)^2) du$$

Step 3: Solve the integrals

To find the probabilities, we need to solve the above integrals. Using a calculator or software to solve them, we get: $$H(50) = 2.833$$ and $$H(60) = 11.333$$

Step 4: Calculate the survival probabilities S(t)

Now we can find the survival probabilities using the survival function \(S(t) = e^{-H(t)}\). Calculate the survival probabilities for ages 50 and 60: $$S(50) = e^{-H(50)} = e^{-2.833}$$ $$S(60) = e^{-H(60)} = e^{-11.333}$$

Step 5: Find the numerical probabilities

Finally, we can calculate the numerical values for the survival probabilities at ages 50 and 60: (a) The probability that a 40-year-old male smoker survives to age 50 without contracting lung cancer: $$S(50) = e^{-2.833} \approx 0.059$$ So, there is approximately a 5.9% chance of the smoker surviving until age 50 without contracting lung cancer. (b) The probability that a 40-year-old male smoker survives to age 60 without contracting lung cancer: $$S(60) = e^{-11.333} \approx 0.000012$$ So, there is approximately a 0.0012% chance of the smoker surviving until age 60 without contracting lung cancer.

Key Concepts

Hazard Rate

The hazard rate is an essential concept in understanding the likelihood of an event, such as contracting a disease. It's often used in survival analysis to represent the instantaneous risk of failure or occurrence of an event at a given time.

• In this scenario, the hazard rate function for lung cancer is given by \( \lambda(t) = 0.027 + 0.00025 (t-40)^2 \) for a 40-year-old male smoker.
• The function represents how the risk of developing lung cancer changes as a smoker gets older.

The hazard rate tells us that the baseline rate is 0.027, and it increases quadratically based on age. This rate helps determine the probability of survival against the onset of the cancer risk as age progresses.

Cumulative Hazard Function

The cumulative hazard function, represented by \( H(t) \), is calculated by integrating the hazard rate over time. This action accumulates the risk over an interval, providing insight into the total risk experienced over time.

• The formula for finding \( H(t) \) involves calculating \( H(t) = \int_{40}^{t} \lambda(u) \, du \), which accumulates risk from age 40 up to age \( t \).
• For instance, \( H(50) \) reflects the cumulative risk up to age 50, based on the integral calculation.
• Similarly, \( H(60) \) represents the cumulative risk up to age 60.

The cumulative hazard function thus gives us the total risk burden over a certain age interval. Understanding \( H(t) \) is crucial for determining survival probabilities.

Survival Probability

Survival probability quantifies the likelihood of a subject surviving without experiencing the event of interest up until a specific time. It's a crucial part of survival analysis.

• The survival probability is calculated using the exponential function of the cumulative hazard function, \( S(t) = e^{-H(t)} \).
• In this context, it represents the probability a smoker survives to ages 50 and 60 without lung cancer.

Calculating \( S(50) \) gives us a probability of about 5.9%, meaning only a small fraction of 40-year-old smokers make it to 50 years without lung cancer.
Similarly, the survival probability \( S(60) \) is incredibly low at roughly 0.0012%, highlighting a steep decrease in survival probability as the risk period extends.

Integration

Integration is a mathematical process of finding the accumulated value, representing the sum of continuous quantities. In this exercise, integration allows us to sum up all the small hazards over time to get a cumulative measure.

• For the hazard rate function \( \lambda(t) \), the integral \( \int_{40}^{t} \lambda(u) \, du \) is computed to find \( H(t) \).
• Integration helps transition from a rate, \( \lambda(t) \), to a cumulative value, \( H(t) \).
  

Integrating the hazard function is crucial for calculating the cumulative hazard, which then informs the survival probability. It's a step that blends theoretical math with practical interpretation in probability analysis.

Exponential Function

The exponential function in this context is applied to transform the cumulative hazard into a survival probability. It’s a key tool in probability, particularly in handling time-to-event data.

• The function \( S(t) = e^{-H(t)} \) expresses survival probability directly from the cumulative hazard.
• This function is exponential because the risk's effect is compounded over time.

The exponential nature underlies many natural processes, which is why it's fitting in survival analysis.
This model effectively captures how survival probability diminishes in response to rising cumulative hazards over time.