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MATH96048课程简介
A stochastic or statistical model of a system is a mathematical model which represents inherent uncertainties in the system as random variables. Any model which does not make such allowances for uncertainty, on the other hand, is said to be deterministic. In actuarial science, statistical models are especially useful for dealing with uncertainty in the time until a specific event will occur, such as predicting the number of premium payments that will be received before paying out on a life assurance contract. These statistical models for time to event data also have much wider applications in fields such as medicine, biostatistics and engineering.
Prerequisites
Mathematical models are an imperfect representation of reality. Their utility comes from being able to approximately learn the consequences of hypothetically changing certain experimental inputs or actions. Statistical models extend this utility by capturing the uncertainty surrounding unknown future outcomes, providing the possibility of searching for an optimal decision under this uncertainty.
MATH96048 Survival Models HELP（EXAM HELP， ONLINE TUTOR）
Give definitions of the following terms in your own words and indicate when it is appropriate to use each of them.
a. Life tables
b. The Kaplan–Meier curve
c. The negative exponential survival distribution
d. The Weibull distribution
e. Cure rate models
f. Log rank test
a. Life tables are a tool used in demographic analysis to show the mortality and survivorship patterns of a population over time. They provide a summary of the number of deaths and number of survivors at different ages or time periods. Life tables are appropriate to use when analyzing population health, evaluating healthcare interventions, and understanding mortality trends.
b. The Kaplan-Meier curve is a statistical tool used to estimate the probability of survival over time. It is a non-parametric method that calculates the survival function by taking into account the number of patients at risk and the number of events that occur at each time point. The Kaplan-Meier curve is often used in medical research to estimate the survival rate of patients with a certain disease or condition.
c. The negative exponential survival distribution is a probability distribution used to model the time until a certain event occurs, such as death or failure of a machine. It assumes that the probability of an event occurring is constant over time, and that events occur independently of each other. The negative exponential survival distribution is appropriate to use when analyzing failure rates of systems or predicting the time until a certain event occurs.
d. The Weibull distribution is a probability distribution that is commonly used to model the failure times of systems. It is a flexible distribution that can take on different shapes depending on the values of its parameters. The Weibull distribution is appropriate to use when analyzing the failure rates of systems or predicting the time until a certain event occurs.
e. Cure rate models are statistical models used to estimate the proportion of patients who are cured or experience long-term survival after receiving treatment for a disease. These models take into account the possibility of patients being cured or experiencing long-term survival, as well as the possibility of recurrence or death. Cure rate models are appropriate to use when analyzing the effectiveness of a treatment or evaluating the long-term survival of patients with a certain disease.
f. The log rank test is a statistical test used to compare the survival distributions of two or more groups of subjects. It is a non-parametric test that does not assume any particular distribution of survival times. The log rank test is appropriate to use when comparing the survival rates of patients in different treatment groups, or when comparing the survival rates of patients with different characteristics.
For a negative exponential survival function $S(t)$, recall that $S(t)=\exp (\lambda t)$, where $\lambda$ is the rate parameter or hazard rate function. Consider the conditional probability that the survival time is $T>t_2$, given that we know $T>t_1$, where $t_1t_2 \mid T\right.$ $\left.>t_1\right]$. Show that $S\left(t_2 \mid t_1\right)=\exp \left[\lambda\left(t_2-t_1\right)\right]$. The term $\exp \left[\lambda\left(t_2-t_1\right)\right]$ is called the lack of memory property of the negative exponential lifetime model because the survival at time $t_1$ has the same distribution as the survival at time 0 ; if $\tau=t_2-t_1$, the probability of surviving $\tau$ units of time is the same at 0 as it is at $t_1$, namely $\exp (\lambda t)$. The probability of surviving depends only on $\tau$ and not on the time $t_1$ that we are conditioning on.
15.2 $S\left(t_2 \mid t_1\right)=P\left{T>t_2 \mid T>t_1\right}=P\left{T>t_2 \cap T>t_1\right} / P\left{T>t_1\right}$. Since $t_2>t_1$, the event $T>t_2$ is contained in the event $T>t_1$. Therefore $P\left{T>t_2 \cap T>t_1\right}=P\left{T>t_2\right}$. So $S\left(t_2 \mid t_1\right)=P\left{T>t_2\right} / P\left{T>t_1\right}=\exp \left(\lambda t_2\right) / \exp \left(\lambda t_1\right)=\exp \left(\lambda t_2-\lambda t_1\right)=\exp \left[\lambda\left(t_2-\right.\right.$ $\left.\left.t_1\right)\right]$.
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