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Tuesday, September 26, 2017
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Visualisation of survival analysis

The survival visualisation application is a web application and runs from any web browser.

Start a web browser and open the website on: http://www.msbi.nl/sv

Visualize the outcome of Cox survival analysis models with a comprehensive chart interface that provides direct click and plot ability. Simply click on the parameter values to plot the survival prediction.

Background information on COX models and the underlying analysis

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 (h0(t)) unspecified.

 Formula 1: The Cox model

hi(t) = h0(t) exp ( β1xi1 β2xi2 … βkxik )
 
 
A special property of this model is that the hazard ratio of two individuals is constant over time; the Cox model is a so-called proportional hazard model. There are several statistical software packages that can calculate Cox survival analysis, for example SPSS, S-Plus 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
-3.8E-04
.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

<div v:shape="_x0000_s1026">
HR = exp (      -0.020 * (Pmnx-50)
-0.013 * (Age-20)
-0.068 * (Year-6)
0.015 * (Treatm * (Pmnx-50))
0.024 * (Treatm * (Age-50))
-0.00038 * ((Pmnx-50) * (Age-20))
            )
 
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 proportional-hazard property the survival line of an individual can be calculated by

 
<div v:shape="_x0000_s1028">
S = S0HR
 
 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.