Editing Survival analysis (section)

==Fitting parameters to data==

Survival models can be usefully viewed as ordinary regression models in which the response variable is time. However, computing the likelihood function (needed for fitting parameters or making other kinds of inferences) is complicated by the censoring. The [[likelihood function]] for a survival model, in the presence of censored data, is formulated as follows. By definition the likelihood function is the [[conditional probability]] of the data given the parameters of the model.
It is customary to assume that the data are independent given the parameters. Then the likelihood function is the product of the likelihood of each datum. It is convenient to partition the data into four categories: uncensored, left censored, right censored, and interval censored. These are denoted "unc.", "l.c.", "r.c.", and "i.c." in the equation below.

<math display="block"> L(\theta) = \prod_{T_i\in unc.} \Pr(T = T_i\mid\theta)
 \prod_{i\in l.c.} \Pr(T < T_i\mid\theta)
 \prod_{i\in r.c.} \Pr(T > T_i\mid\theta)
 \prod_{i\in i.c.} \Pr(T_{i,l} < T < T_{i,r}\mid\theta) .</math>
For uncensored data, with <math>T_i</math> equal to the age at death, we have

<math display="block"> \Pr(T = T_i\mid\theta) = f(T_i\mid\theta) .</math>
For left-censored data, such that the age at death is known to be less than <math>T_i</math>, we have

<math display="block"> \Pr(T < T_i\mid\theta) = F(T_i\mid\theta) = 1 - S(T_i\mid\theta) .</math>
For right-censored data, such that the age at death is known to be greater than <math>T_i</math>, we have

<math display="block"> \Pr(T > T_i\mid\theta) = 1 - F(T_i\mid\theta) = S(T_i\mid\theta) .</math>
For an interval censored datum, such that the age at death is known to be less than <math>T_{i,r}</math> and greater than <math>T_{i,l}</math>, we have

<math display="block"> \Pr(T_{i,l} < T < T_{i,r}\mid\theta)
 = S(T_{i,l}\mid\theta) - S(T_{i,r}\mid\theta) .</math>
An important application where interval-censored data arises is current status data, where an event <math> T_i</math> is known not to have occurred before an observation time and to have occurred before the next observation time.