The object of primary interest in Survival Analysis is the
survival function which is defined as 
where t denotes time. That is: the survival function is the probability
that death occur after a specified moment in time. The survival function
is also called the survivor function or survivorship function in problems
of biological survival, and the reliability function in mechanical
survival problems (usually the reliability function is denoted as
R(t)).
The hazard function is defined as the event
rate at time t conditional on survival until time t or later:
The force of mortality is a synonym for the hazard function which is used
particularly in demographics. The term hazard rate is also used. The
hazard function can alternatively be represented in terms of the
cumulative hazard function: 
Other functions are defined in terms of the survival and hazard
functions. The lifetime distribution function
is defined as the complement of the survival
function:
and the derivative of F, the probability
density function of the lifetime distribution, 
is sometimes called the event density; it is the rate
of death or failure events per a unit of time.
Observation are called censored when the dependent variable represents the time to a terminal event and the duration of the study is limited in time.
Type I censoring describes the situation when a process is terminated at a particular point in time and the remaining individuals are known to have survived up to that moment. In this case the censoring time is fixed and the number of events is a random variable.
Type II censoring concerns cases when the experiment would be continued until a fixed proportion of events has occurred, so the number of events is fixed and the survival time is the random variable.
There are situations in which censoring can occur at different times (multiple censoring), or only at a particular point in time (single censoring).
A distinction can also be made to reflect the "side" of the timescale at which censoring occurs. In most cases the researcher knows when exactly the experiment started, and the censoring always occurs on the right side of the timescale. Sometimes the researcher does not know the starting time (for example, when exactly the symptoms of a disease first occurred) and in this case the censoring occurs on the left side (left censoring).
The censoring mechanism in the Survival Analysis module is assumed to be type I single censoring on the right side of the timescale.
The two nonparametric estimators: Kaplan-Meier and Life-Table are the simplest cases of survival analysis. The distribution of the survival time is estimated for the population as a whole, solely on the basis of the distribution of the dependent variable and the values of the censor attribute. The outcome of the non-parametric survival analysis is an empirical (baseline) survivorship function and an empirical (baseline) hazard function.
Let: 
be the k-th ordered time point (event
time) in the data, 
be the number of alive observations, for
which the event time or censored time is at least 
, 
be the number of deaths (uncensored
events) in the interval 
, 
be the number of censored events in the
interval 
, 
be the number of observations that survived in the interval
and
be the length of the k-th time interval.
In the Kaplan-Meier model we estimate the conditional
probability of an event (death) occurrence in the interval
by
and let the probability of a non-event be equal to
. Then the non-parametric survivorship function estimator can be expressed as
and its standard error is
. The non-parametric hazard function estimator can be expressed as
and its standard error is
. The survival probability density function can be expressed as
and its standard error is
The Life-Table model takes into account the censoring mechanism. Let
be the estimator of the number of observations alive in the
interval
. The conditional probability of an event (death) occurrence in
the interval
can be estimated by
and the opposite probability is
. Then the non-parametric survivorship function estimator can be expressed as
and its standard error is
. The non-parametric hazard function estimator can be expressed as
and its standard error is
. The survival probability density function can be expressed as
and its standard error is
100(1-Confidence Level)% confidence interval
for all of the above estimators is calculated for Confidence
Level= 
specified in the non-parametric
survival algorithm settings. The upper and lower bounds are calculated
as:
where 
is the 
percentile of the standard
normal distribution, and 
is the standard
error of the given estimator.
The Cox model or the proportional hazard model is the most general regression model used for survival time modeling. The Cox model tries to predict the distribution of the survival time for individuals in a given population, investigate the strength of influence of particular variables on the expected survival time, and compare survival time distributions among different subpopulations. The model is not based on any assumptions about the underlying survival distribution. It assumes that hazard is a function of the independent variables (covariates); no assumptions are made about the nature or shape of the hazard function.
In the Cox model the survivorship function for an individual
can be expressed as a combination of a non-parametric baseline survivorship function and a parametric part (in the form analogous to that found in regression models):
where 
is the baseline survivorship
function.
The Cox model provides an estimate of the treatment effect on survival after adjustment to other explanatory variables. It allows us to estimate the hazard (or risk) of death or other event of interest for individuals, given their prognostic variables. The Cox model is a semi-parametric model and incorporates the effect of covariates, uses proportional hazards assumptions and does not require a specification of the parametric survival distribution.
The survival (hazard) function 
of survival time
for each individual 
is assumed to have the
form
where 
is a vector of unknown
parameters, 
is the baseline hazard function and
is a vector of explanatory
variables characterizing the given individual. The baseline hazard
function characterizes how the hazard function changes as a function
of survival time.
It is possible to linearize this model by dividing both sides of
the equation by 
and taking a logarithm of both
sides.
The Cox model has no implicit assumptions about the shape of the underlying hazard function, but the model equations do imply two assumptions:
Proportionality assumption: there is a multiplicative relationship between the underlying hazard function and the log-linear function of the covariates.
There is a log-linear relationship between the independent variables and the underlying hazard function.
The proportionality assumption means that for given two observations with different values of the independent variables the ratio of the hazard functions for those two observations does not depend on time.
To estimate a vector 
a partial likelihood function
that depends only on the parameters of interest has been introduced by
Cox 1972,1975 (see also Cox and Oakes 1984). In the present settings the exact
partial likelihood function is:
In the present settings the Breslow approximation of the log partial likelihood function is:
where 
is the group of dead subjects at the
time 
and 
is a group of all subjects in the risk
set at the time 
, 
is number of events (deaths) in the i-th
time interval, N is the number of observations.
In order to find a solution of the maximization problem for the partial likelihood the module applies the widely used Newton-Raphson algorithm, which tries to zero the first order partial derivatives of the log-likelihood function:
where
Partial likelihood allows to employ time-dependent variables in the survival model (i.e. variables whose value for any given individual can change over time). On one hand, it allows to include and describe more complex factors (such as blood pressure), on the other hand, time-dependent variables can be used to test the validity of the proportional hazards assumption. If hazards among different groups (subpopulations) are proportional then the plots of the log hazard function estimates for each group versus log(time) should be parallel. Automatic or semi-automatic variable selection methods, helping to choose the optimal subset of available variables (the one which describes the target variable most accurately), are analogous to those known from regression models. Among the implemented heuristics are: Forward Selection, Backward Elimination and Stepwise.
The Berndt, Hall, Hall, and Hausman estimator (so-called BHHH estimator: for details see Berndt et al. 1974) is used for the estimation of the model parameters:
The Information matrix is approximated by the outer product of the gradient, calculated as:
where 
and the BHHH estimators uses the inverse matrix of
The Survival model quality measures are, to some extent, similar to those of non-linear regression and are based on residual error analysis (for details see the descriptions of Lift and ROC). Among the most frequently used residuals are: Schoenfeld, score, and martingale. The residual is defined as the difference between the actual and predicted survival time. The mean value of the residual is also assessed to check if the model fit is not "skewed". The mean values of the Schoenfeld residuals and the Scaled Schoenfeld residuals are calculated. Under the assumption of proportional hazards model the Schoenfeld residuals have the sample path of a random walk.
For testing the global significance of the estimated parameters
(null hypothesis that 
) three statistics are
calculated:
All three statistics have an asymptotic chi-squared distribution
with the number of degrees of freedom equal to the dimension of the
vector 
.
For testing a linear hypothesis about the estimated parameters
(a null hypothesis that 
, where 
is the matrix of linear
coefficients for the null hypothesis), the Wald statistic for parameter
estimator is calculated. Under the null hypothesis the Wald statistic
has an asymptotic chi-squared distribution with the number of degrees
of freedom equal to the rank of 
.
100(1-Confidence Level)% confidence interval
for the parameters estimates is calculated for Confidence
Level= 
specified in the current algorithm
settings. The upper and lower bounds are calculated as:
where 
is the 
percentile of the standard
normal distribution, and 
is the i-th diagonal element of the estimated covariance
matrix.