MELJUN CORTES IBM SPSS Survival Analysis

7/30/2019 MELJUN CORTES IBM SPSS Survival Analysis

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Survival Analysis

This contains my personal notes only thus, this is not

complete. Most of the contents were taken from the training

manual of IBM SPSS Modeler. Please refer to the training

manual for a complete discussion.


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Survival Analysis

Survival analysis studies the length of time

of an event of interest.

Originally, the technique was used in

medical research to study the amount of

time patients survive following onset of adisease.


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The length of t ime a person subs cr ibes to a newspaper ormagazine

The time employees s pend w ith a company

The time to fai lure of an electr ic or mechanical compon ent

The time to prom otion o f employee with in an org anizat ion

The time to complete a complex transact ion suc h as a loanappl icat ion

Time it takes to c omplete a requ iremen t (e.g., a PhD at aunivers i ty)

It has been used to model:


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Survival analysis can be done with or

without predictors.

Kaplan Meierfor analysis without

predictors

Cox Regressionfor analysis withcategorical and interval predictors


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Censoring vs Missing Data

Missing data observation for which novalid value is recorded

Censored data - contain information but

the final value is unknown. Examples: Medical studies: at the end of the study some

patients are still alive or in remission, somemove or refuse to participate, some die of

other causes

studying time to promotion: some employeeleft the company before being promoted


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Censored data are included in the survival

analysis

Censored data cannot be included in regressionanalysis, t-test, anova.

Aside from the censoring issues, in regression

model, the residuals are unlikely normallydistributed because time-to-event distribution is

likely to be non-normally distributed.


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Concepts

Length of time is the main outcome in survival analysis Mean and median of length of time can be used to summarize

Confidence interval also can be used to summarize.

Means and Medians for Survival Time

groupMeana Median

Estimate Std. Error

95% Confidence Interval

Estimate Std. Error


Lower Bound Upper Bound Lower Bound Upper Bound

Control 72.545 14.839 43.462 101.629 40.000 12.899 14.719 65.281

Treatment (Prednisolone) 125.264 13.402 98.996 151.532 146.000 28.786 89.580 202.420

Overall 98.925 10.812 77.733 120.117 89.000 21.232 47.385 130.615

a. Estimation is limited to the largest survival time if it is censored.


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Cumulative survival plot

Graphically, survival data can be summarizedusing survival function over time.

The survival function shows

the probability of surviving

longer than the time

displayed in the chart.

For example, the probability

that a person in the

treatment group survives

beyond t=96 is .63.

Generally, the survival ratedecreases over time.

The treatment group has a

higher survival rate than the

control group. Is this finding

significant?


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Cumulative Hazard plot

Survival data can be summarized using hazardfunction over time.

The cumulative hazard plot

shows the risk of an event

occurring at a particular

time.



treatment group does not

survive at t=100 is .49.

Generally, the hazard rateincreases over time.


lower hazard rate than the


significant?


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Survival Procedures in SPSS


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The Life Tables procedure is appropriate when the time

to critical event measure is recorded in broad ranges (for

example in six-month periods, or whole years) so that

there are many ties among the data values, or if there is

no interest in differentiating between small timedifferences.


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The Kaplan-Meierprocedure is

appropriate when the time to critical event

measure is precise enough so there are

relatively few ties in the data. Examplesmight be number of months surviving, or

the fractional number of years a retail

space is occupied by a tenant.


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Cox Regression (also called a proportional

hazard model) posits that the hazard rate can be

a function of both categorical and interval scale

predictor variables. It assumes that the hazard functions for different

groups are proportional to each other over time.

This assumption can be examined and a variant

of Cox regression (Cox Regression with timevarying covariates) can be applied when the

assumption doesnt hold.


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Kaplan-Meier Example: Survival time for

patients with chronic active hepatitis

The group variabledivides the group intotreatment (prednisolonetherapy) and control

patients

The time variablerecords time to thecritical event (death) ortime when censoringoccurred

The status variableindicates whether thecritical event occurred(1= death) or that thecase was censored (2=censored).

Data set: KM.sav


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Null Hypothesis

There is no significant difference in the

survival rate of the treatment and control

groups.


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ResultsMeans and Medians for Survival Time

group Meana Median

Estimate Std. Error


Estimate Std. Error


Lower Bound Upper Bound Lower Bound Upper Bound

Control 72.545 14.839 43.462 101.629 40.000 12.899 14.719 65.281

Treatment (Prednisolone) 125.264 13.402 98.996 151.532 146.000 28.786 89.580 202.420

Overall 98.925 10.812 77.733 120.117 89.000 21.232 47.385 130.615

a. Estimation is limited to the largest survival time if it is censored.

Overall Comparisons

Chi-Square df Sig.

Log Rank (Mantel-Cox) 4.660 1 .031

Breslow (Generalized Wilcoxon) 6.543 1 .011

Tarone-Ware 6.066 1 .014

Test of equality of survival distributions for the different levels of group.

Reject

Ho


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Survival table for the Control Group

The first event which occurred at time 2 (2 months) has a cumulative survival value of .955 or (1 (1/22)). The estimatedprobability of surviving beyond two months in this group is 95.5%.

Survival Table

group

Time Status

Cumulative Proportion Surviving at the

TimeN of Cumulative

Events

N of Remaining

CasesEstimate Std. Error

Control 1 2.000 Died .955 .044 1 21

2 3.000 Died .909 .061 2 20

3 4.000 Died .864 .073 3 19

4 7.000 Died .818 .082 4 18

5 10.000 Died .773 .089 5 17

6 22.000 Died .727 .095 6 16

7 28.000 Died .682 .099 7 15

8 29.000 Died .636 .103 8 14

9 32.000 Died .591 .105 9 13

10 37.000 Died .545 .106 10 12

11 40.000 Died .500 .107 11 11

12 41.000 Died .455 .106 12 10

13 54.000 Died .409 .105 13 9

14 61.000 Died .364 .103 14 8

15 63.000 Died .318 .099 15 7

16 71.000 Died .273 .095 16 6

17 127.000 Censored . . 16 5

18 140.000 Censored . . 16 4

19 146.000 Censored . . 16 3

20 158.000 Censored . . 16 2

21 167.000 Censored . . 16 1

22 182.000 Censored . . 16 0


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Survival Table for the Treatment Group

Survival Table

group

Time Status

Cumulative Proportion Surviving at the

TimeN of Cumulative

Events

N of Remaining

CasesEstimate Std. Error

Treatment (Prednisolone) 1 2.000 Died .955 .044 1 21

2 6.000 Died .909 .061 2 20

3 12.000 Died .864 .073 3 19

4 54.000 Died .818 .082 4 18

5 56.000 Censored . . 4 17

6 68.000 Died .770 .090 5 16

7 89.000 Died .722 .097 6 15

8 96.000 Died . . 7 14

9 96.000 Died .626 .105 8 13

10 125.000 Censored . . 8 12

11 128.000 Censored . . 8 11

12 131.000 Censored . . 8 10

13 140.000 Censored . . 8 9

14 141.000 Censored . . 8 8

15 143.000 Died .547 .117 9 7

16 145.000 Censored . . 9 6

17 146.000 Died .456 .129 10 5

18 148.000 Censored . . 10 4

19 162.000 Censored . . 10 3

20 168.000 Died .304 .151 11 2

21 173.000 Censored . . 11 1

22 181.000 Censored . . 11 0


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Cumulative Hazard plot

Survival data can be summarized using hazardfunction over time.

The cumulative hazard plot

shows the risk of an event

occurring at a particular

time.



treatment group does not

survive at t=100 is .49.

Generally, the hazard rate

increases over time.


lower hazard rate than the


significant?


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The Cox Regression Model

The Cox Regression Model posits that the

hazard rate can be a function of both

categorical and interval scale predictorvariables.


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The hazard function at time t as a function of

predictors X1, X2,Xp:h(t|X1, X2,,Xp) = h0(t)*e

(B1X1+B2X2+BpXp)

h0(t) = the base hazard function that changes over time andis independent of the predictors

e(B1X1+B2X2+BpXp) = the factor and covariate effects which

are independent of time and adjust the base hazard function

The eBXj is the change in the hazard function associated

with a unit change in the predictor (Xj), controlling for the

other effects in the model.


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Assumption:

The effects of the predictors are constant over time. If this assumptionis not met, then the Cox Regression Model will not provide the best fitto the data.

If such assumption is not met, use the Cox Model with time-dependentcovariates.


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Cox Regression Example 1: Survival time in heroin

addiction treatment program

Outcome measure(survtime) time (in days)spent in a program forheroin addicts

Terminating event (status) departure from quittingthe program. Data werecensored for participants

still in the program whenthe study was completed

Predictors: clinic(there were two

clinics whose programsdiffered)

prison (whether or not theaddict had a prisonrecord),

methadone dose(methdose, measured inmg/day)

Data set: addicts.sav


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Results

Omnibus Tests of Model Coefficientsa

-2 Log Likelihood

Overall (score) Change From Previous Step Change From Previous Block

Chi-square df Sig. Chi-square df Sig. Chi-square df Sig.

1347.345 56.193 3 .000 64.351 3 .000 64.351 3 .000

a. Beginning Block Number 1. Method = Enter

Null Hypotehsis: The effect of one or more of the three predictor

variables are significantly not different from zero in the population.

Reject Ho if chisquare is significant (p


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Variables in the Equation

B SE Wald df Sig. Exp(B)

clinic 1.009 .215 22.087 1 .000 2.743

prison -.315 .167 3.543 1 .060 .730

methdose -.035 .006 30.659 1 .000 .965

The B coefficient estimates relate the change in natural log of the hazard per

one unit change in the predictor. (difficult to understand?)

The Exp(B) column presents the estimated change in risk (hazard)

associated with a one-unit change in a predictor, controlling for the otherpredictors. When the predictor is categorical and indicator coding is used,

Exp(B) represents the change in hazard when changing from the reference

category to another category and is referred to as relative risk. Exp(B) is also

called the hazard ratio, since it represents the ratio of the hazards for two

individuals who differ by one unit in the predictor of interest.


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Variables in the Equation


clinic 1.009 .215 22.087 1 .000 2.743

prison -.315 .167 3.543 1 .060 .730

methdose -.035 .006 30.659 1 .000 .965

For clinic, Exp(B) =2.743: Other things equal, the hazard in clinic 0 is 2.743

times greater than the hazard in clinic 1. Thus patients in clinic 0 exhibit

greater risk and lower survival times. Be careful of the reference category!

For methdose, Exp(B)=.965: A one-unit (one mg/day) increase in dosage isassociated with a decrease (.965) in hazard.

For prison, exp(B)=.730: Non- significant! If significant, it would be

interpreted as: The group with no prison record is at less risk than the group

with a prison record.


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Cumulative Survival Plot


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Cumulative Hazard Plot


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Checking the Proportional Hazard Assumptions(The hazard functions of any two individuals or groups remain in constant proportion over time)

1. Examine the survival or hazard plots (inKaplan-Meier) with the categorical predictor asthe factor

2. Examine the survival or log-minus-log plot in

Cox Regression with the categorical predictorspecified as a strata variable

3. Save partial residuals and plot them againsttime (see Cox Regression case study for an

example)4. Fit a Cox Regression model with a time-varyingcovariate; examine its significance andcontribution

Note: we will use #2.


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The survival plots for the two clinics diverge substantially over time,suggesting that the hazard ratio for the two groups is not constant

over time.


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Log Minus Log (LML) Plot

The proportional hazard model holds (in our example it means that over time, thehazard functions of the clinics differ by a constant proportion), then the natural log ofthe negative of the natural log of the survival functions for different groups over timewill form parallel lines.

Here the lines are not parallel, indicating that the proportional hazards assumptiondoes not hold forclinic.


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Cox with Time-Dependent Covariate

Create T_COV_ a time-dependent

variable to be used in the extended cox

model.

T_COV_ = (T_>365)*clinic

Zero(0) when survival time is 365 or less

One (1) when survival time is >365


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It can be created by SPSS as follows:


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Include T_COV_ as covariate in the Model


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Results

The clinic predictor (which now represents the clinic effect during the first 365 days) isno longer significant (p=.06). It should be retained in the model since it was used indefining the time by clinic interaction.

In the first year (time 365) the hazard in clinic 0 is 6.123 times greater than the hazard in clinic 1. The6.123 value is obtained by multiplying the clinic effect (1.616) by the clinic by timeinteraction (3.789); thus after the first year, the clinic effect is estimated to increase bya factor of 3.789.


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Cox Regression Example 2

Consider the database with 5000 records fromcustomers of a telecommunication firm.

The firm has collected a wide variety of consumerinformation of its customers including gender, age,

education, income, marital status, card tenure, annualfee for primary credit card, # of years held the primarycredit card, then churn (switched providers within lastmonth).

We are interested of studying the length of time

customers retain their primary credit card.In other words, we will model the time for thesecustomers to churn-not renew- their primary credit card.


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Churn rates are initially around 50% in the early

years.

People who have recently obtained card People who have had their card for a long time


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Cox Regression Results


gender .048 .056 .739 1 .390 1.050

age -.080 .003 862.033 1 .000 .923

educati

on.120 .009 169.987 1 .000 1.128

income -.003 .001 18.387 1 .000 .997

marital .385 .057 45.457 1 .000 1.469

cardfee .046 .073 .398 1 .528 1.047

Increasing income and

age lead to reduce

hazard for churn.


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Cox Regression Results


gender .048 .056 .739 1 .390 1.050

age -.080 .003 862.033 1 .000 .923

educati

on.120 .009 169.987 1 .000 1.128

income -.003 .001 18.387 1 .000 .997

marital .385 .057 45.457 1 .000 1.469

cardfee .046 .073 .398 1 .528 1.047

Increasing education

and being unmarried

are associated with

increasing hazard for

churn.


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Survival Table (first ten only)

Time

Baseline

Cum Hazard

At mean of covariates

Survival SE Cum Hazard

0 .019 .997 .000 .003

1 .100 .986 .001 .014

2 .170 .976 .002 .024

3 .256 .964 .002 .037

4 .327 .954 .003 .047

5 .409 .943 .003 .058

6 .489 .933 .004 .070

7 .565 .922 .004 .081

8 .650 .911 .005 .093

9 .749 .899 .005 .107

10 .848 .886 .005 .121

Baseline cum hazard is the

hazard rate for the model when

all predictors are zero.

Both survival and cumulative

hazard were computed at the

mean of all predictors

For our model, survival

retaining the primary credit

card-dropped to 93.3% by the

sixth year.

The probability of retainingthe primary credit card beyond

6 years is 93.3%.


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The differences in

survival gradually

increase over time

between groups

The differences in

hazard gradually

increase over time

between groups


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Cox Node Model Options

The following options are available forentering predictors into the model:

enter

stepwise

backward stepwise


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Cox Node Model Options

Specifying a group field causes the node to

compute separate models for each category

of the field. It can be any categorical field

(Flag or Set) with string or integer storage.


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These options allow you to control the parameters for

model convergence.

When you execute the model, the convergence settingscontrol how many times the different parameters are

repeatedly run through to see how well they fit.

The more often the parameters are tried, the closer the

results will be (that is, the results will converge).


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Maximum iterations.Allows you to specify the

maximum iterations for the model, which controls

how long the procedure will search for a solution.

Log-likelihood convergence. Iterations stop if therelative change in the log-likelihood is less than

this value. The criterion is not used if the value is 0.

Parameter convergence. Iterations stop if the absolute

change or relative change in the parameter

estimates is less than this value. The criterion is not used

if the value is 0.


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These options allow you to

request additional statistics and

plots, including the survival

curve, that will appear in theadvanced output of the

generated model built by the

node.


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You can obtain statistics for your model parameters,

including confidence intervals for exp(B) and correlation

of estimates. You can request these statistics either at

each step or at the last step only.


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Display baseline function.Allows you to

display the baseline hazard function andcumulative survival at the mean of the

covariates.


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Plots can help you to evaluate your estimated

model and interpret the results. You can plot

the survival, hazard, log-minus-log, and one-

minus-survival functions.


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This option is available only for categorical fields.

Value to use for plots Because these functions depend on values of the


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Value to use for plots. Because these functions depend on values of the

predictors, you must use constant values for the predictors to plot the

functions versus time.

The default is to use the mean of each predictor as a constant value, but you

can enter your own values for the plot using the grid.

For categorical inputs, indicator coding is used, so there is a regression

coefficient for each category (except the last).

Thus, a categorical input has a mean value for each indicator contrast, equal

to the proportion of cases in the category corresponding to the indicator

contrast.


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Predict survival at future times. Specify one or more

future times. Survival, that is, whether each case is likely to

survive for at least that length of time (from now) without theterminal event occurring, is predicted for each record at

each time value, one prediction per time value.

Note that survival is the false value of the target field.


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Regular intervals. Survival time values are generated from the

specified Time interval and Number of time periods to score.

For example, if 3 time periods are requested with an interval

of 2 between each time, survival will be predicted for future times

2, 4, 6. Every record is evaluated at the same time values.


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Time fields. Survival times are provided for each

record in the time field chosen (one prediction

field is generated), thus each record can beevaluated at different times.


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Specify the survival time of the record so farfor

example, the tenure of an existing customer as afield. Scoring the likelihood of survival at a future

time will be conditional on past survival time.


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Append all probabilities. Specifies whether probabilities for each

category of the output field are added to each record processed

by the node. If this option is not selected, the probability of only

the predicted category is added. Probabilities are computed foreach future time.


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Calculate cumulative hazard function. Specifies whether the value of

the cumulative hazard is added to each record. The cumulative hazard is

computed for each future time.


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Use customer_dbase.sav for the demo

Inputs: gender, age, ed, income,

marital, cardfee, cardtenure

Output: churn

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MELJUN CORTES IBM SPSS Survival Analysis