Survival analysis analyses and models the time it takes for events to occur. The basic concept of survival analysis is the hazard of an event in a time period, which can be estimated using the number of events and the number of people at risk. This hazard distribution function is then used to calculate the survival function, which is an estimation of the chance of experiencing the event before a specified time.
A popular method of analysing the relation between predictors (e.g. age or treatment) and outcome (e.g. death or relapse) is Cox survival analysis. The relation between predictors and outcome is modelled as a linear regression model of the log hazard. The Cox model is a semi parametric model because it leaves the baseline function (h_{0}(t)) unspecified.
Formula 1: The Cox model
h_{i}(t) = h_{0}(t) exp ( β_{1}x_{i1} β_{2}x_{i2} … β_{k}x_{ik })

A special property of this model is that the hazard ratio of two individuals is constant over time; the Cox model is a socalled proportional hazard model. There are several statistical software packages that can calculate Cox survival analysis, for example SPSS, SPlus and R.
Example of a Cox analysis
Cox analysis results in a model and the values for the fitted parameters for the terms in the model. The result can also have one or more baseline survival functions. The interpretation of the parameter values is the contribution in risk when this variable increases with one. For example the table below shows the output of a Cox analysis in SPSS. The first row in this model give values of B = 0.20 and Exp(B) = .980 for the term PMNX50. This means that an increase of one in the PMNX50 value will give a decrease in hazard by a factor of .980. There is also an interaction term in this model with PMNX50. So there is also a decrease in hazard of (0.9996* Age20) for an increase of one in the PMNX50 count. This effect can only be calculated if the age20 value is known.

B

SE

Wald

df

Sig.

Exp(B)

PMNX50

.020

.003

52.061

1

.000

.980

AGE20

.013

.007

3.863

1

.049

.987

YEAR6

.068

.008

80.408

1

.000

.934

TMXP50

.015

.005

9.900

1

.002

1.015

TMXA20

.024

.008

8.630

1

.003

1.025

P50XA20

3.8E04

.000

3.472

1

.062

0.9996

Table 1: Example of SPSS Cox Analysis
With the table of parameter values and the Cox model we can fill in the model. This formula calculates the hazard ratio of an individual compared to an individual for which all variables are zero. For example the SPSS model gives the following formula.
Formula 2: Example of a formula for a Cox model
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HR = exp ( 0.020 * (Pmnx50)
0.013 * (Age20)
0.068 * (Year6)
0.015 * (Treatm * (Pmnx50))
0.024 * (Treatm * (Age50))
0.00038 * ((Pmnx50) * (Age20))
)

This formula completely describes the effect of the parameters on the hazard for the event, but it is very hard to understand how this formula relates to the survival for a patient and it is difficult to get practical conclusions on the combination effects of the variables.
Chart 1: Example of the baseline output from SPSS
The statistical software can also calculate a baseline survival. This is the survival function for an individual where all variables are zero. Because of the proportionalhazard property the survival line of an individual can be calculated by
<div v:shape="_x0000_s1028">
S = S_{0}^{HR}

Formula 3: Calculation of a Survival function
The survival function can be plotted in a chart and by drawing several survival lines in one chart gives an overview of the effect of the variables on the survival. When a model contains interaction effects then the comparison of a single variable is dependent on other variables and this becomes difficult to display in charts.