Question

what is a survival model

Answer 1

Survival analysis is a branch of statistics for analyzing the expected duration of time until one or more events happen, such as death in biological organisms and failure in mechanical systems. A survival model can be used for different kinds of analysis, for example, to study age at marriage, the duration of marriage, the intervals between successive births to a woman, the duration of stay in a city (or in a job), the length of life etc.

The following terms are commonly used in a survival model analysis:

• Event: Death, disease occurrence, recovery, or other experience of interest.
• Time: The time from the beginning of an observation period (such as surgery or beginning treatment) to (i) an event, or (ii) end of the study, or (iii) loss of contact or withdrawal from the study.
• Censored observation: If a subject does not have an event during the observation time, they are described as censored. A censored subject may or may not have an event after the end of observation time.
• Survival function S(t): The probability that a subject survives longer than time t.

The following three functions are essential in designing a basic survival model:

• Survival Function - S(t)
• Hazard Function- λ(t)
• Likelihood Function - L

1. Survival Function

Let T be a non-negative random variable representing the waiting time until the occurrence of an event. For simplicity we will adopt the terminology of survival analysis, referring to the event of interest as `death' and to the
waiting time as `survival' time.

We will assume for now that T is a continuous random variable with probability density function (p.d.f.): f(t) and cumulative distribution function (c.d.f.): F(t) = Pr {T < t}. We can now define the Survival function S(t) which gives the probability of being alive just before duration t, or more generally, the probability that the event of interest has not occurred by a duration t. Then, we have S(t) as follows:

2. Hazard Function

An alternative characterization of the distribution of T is given by the Hazard function λ(t), or instantaneous rate of occurrence of the event, defined as follows:

The numerator of this expression is the conditional probability that the event will occur in the interval [t; t+dt) given that it has not occurred before, and the denominator is the width of the interval. Dividing one by the other we
obtain a rate of event occurrence per unit of time. Taking the limit as the width of the interval goes down to zero, we obtain an instantaneous rate of occurrence.

3. Likelihood Function

Suppose then that we have n units with lifetimes governed by a survival function S(t) with associated density f(t) and hazard λ(t). Suppose unit i is observed for a time t_i. If the unit died at t_i, its contribution to the likelihood function (L_i) is the density at that duration, which can be written as the product of S(t) and λ(t). Let d_i be a death indicator, taking the value one if unit i died and the value zero otherwise. Then the likelihood function (L) may be written as follows:

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Short Note: Model Fitting

There are essentially three approaches to fitting survival models:

• The first and perhaps the most straightforward approach is the parametric strategy, wherein we assume a specific functional form for the baseline hazard λ₀(t). Examples are models based on the exponential, Weibull, gamma and generalized F distributions.
• A second approach is what might be called a flexible or semi-parametric strategy, wherein we make mild assumptions about the baseline hazard λ₀(t). Specifically, we may subdivide time into reasonably small intervals and assume that the baseline hazard is constant in each interval,
  leading to a piece-wise exponential model.
• The third approach is a non-parametric strategy that focuses on estimation of the regression coefficients leaving the baseline hazard λ₀(t) completely unspecified. This approach relies on a partial likelihood function proposed by Cox (1972) in his original paper.

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Hope this helps!