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Survival Analysis Survival Analysis Biomedical ApplicationsBiomedical Applications

Halifax SAS User GroupHalifax SAS User Group

April 29/2011April 29/2011

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Why do Survival AnalysisWhy do Survival Analysis

Aims :Aims : How does the risk of event occurrence How does the risk of event occurrence

vary with time?vary with time? How does the distribution across states How does the distribution across states

change with time?change with time? How does the risk of event occurrence How does the risk of event occurrence

depend on explanatory variables?depend on explanatory variables?

• Paul Allison, 2002 Lecture. University of Pennsylvania at Paul Allison, 2002 Lecture. University of Pennsylvania at PhiladelphiaPhiladelphia

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Survival DataSurvival Data

Time from randomization until time of the Time from randomization until time of the event of interestevent of interest

Classified as event time dataClassified as event time data

Generally not symmetrical distributionGenerally not symmetrical distribution

Positive skew (tails to right)Positive skew (tails to right)

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Survival DataSurvival Data

ExampleExample -Time from onset of cancer to death/remission -Time from onset of cancer to death/remission

-Time from implant of pacemaker to lead survival-Time from implant of pacemaker to lead survival

Fixed start pointFixed start point- recruitment in study- recruitment in study

- onset of cancer- onset of cancer

- insertion of Pacemaker- insertion of Pacemaker

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CensoringCensoring

RightRight: occurs to the right of the last : occurs to the right of the last known survival timeknown survival time

LeftLeft: actual survival time is less than : actual survival time is less than observed, common in reoccurrenceobserved, common in reoccurrence

IntervalInterval: : survival time is between two survival time is between two time points, a and btime points, a and b

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Kaplan-MeierKaplan-Meier

Non-parametric methodNon-parametric method

Assumes events depend only on time, Assumes events depend only on time, and censored and non-censored and censored and non-censored subjects behave the samesubjects behave the same

Descriptive method primarily used for Descriptive method primarily used for exploratory analysisexploratory analysis

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Kaplan–Meier EstimateKaplan–Meier Estimate

Based on life-table methodsBased on life-table methods

Arbitrarily small intervals, continuous Arbitrarily small intervals, continuous functionfunction

Expressed as a probabilityExpressed as a probability

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Kaplan-Meier Data Kaplan-Meier Data

Time variableTime variable CensoringCensoring

StrataStrata: Categorical variable that : Categorical variable that represents group effectrepresents group effect

ex. Flu strainex. Flu strain

FactorFactor: Categorical variable that : Categorical variable that represents causal effectrepresents causal effect

ex. Type of treatmentex. Type of treatment

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SAS Proc LifetestSAS Proc Lifetest

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Proc GplotProc Gplot

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proc gplot data=out3;proc gplot data=out3;

title 'Adjusted Survival Curve by Gender';title 'Adjusted Survival Curve by Gender';

axis1 label=('Years') order=(0 to 12 by 1);axis1 label=('Years') order=(0 to 12 by 1);

axis2 label=(angle=90 'Proportion Surviving');* axis2 label=(angle=90 'Proportion Surviving');* order=(0.6 to 1.0 by 0.1);order=(0.6 to 1.0 by 0.1);

plot surv*time=gender / haxis=axis1 vaxis=axis2 plot surv*time=gender / haxis=axis1 vaxis=axis2 legend=legend1;legend=legend1;

run;run;

quit;quit;

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SURVPLOT MACROSURVPLOT MACRO%survplot (DATA=xxx_aug , TIME=time_death , %survplot (DATA=xxx_aug , TIME=time_death , EVENT=death ,CEN_VL=0, CLASS=event_anyshock ,EVENT=death ,CEN_VL=0, CLASS=event_anyshock ,

TESTOP=1, CLASSFT=cchrl , CMARKS=0, PLOTOP=0 , TESTOP=1, CLASSFT=cchrl , CMARKS=0, PLOTOP=0 , PRINTOP= 0,PRINTOP= 0,

POINTS='1 2 5 10' , SCOLOR=black, XDIVISOR=1, LABELS= , POINTS='1 2 5 10' , SCOLOR=black, XDIVISOR=1, LABELS= ,

LABCOL=black, BY= , WHERE= , LEGEND=1 , YAXIS=2, LABCOL=black, BY= , WHERE= , LEGEND=1 , YAXIS=2, XAXIS=1,XAXIS=1, XMAX=15 , LCOL=black red blue, PERCENT=0, XMAX=15 , LCOL=black red blue, PERCENT=0, FONT=SWISS, FONT=SWISS,

F1=3, F2=3, F3=3, F4=3, PLOTNAME= , ANNOTATE= , F1=3, F2=3, F3=3, F4=3, PLOTNAME= , ANNOTATE= , RTFEXCL=0,POPTIONS=);RTFEXCL=0,POPTIONS=);

Survplot Macro: Created by Ryan Lennon. 2009 Mayo Clinic College of Medicine.Survplot Macro: Created by Ryan Lennon. 2009 Mayo Clinic College of Medicine.

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Kaplan–Meier EstimateKaplan–Meier Estimate

Product-Limit Survival Estimates Timelist Survival Failure Error # Failed # Left 365.00 365.00 0.5634 0.4366 0.0206 258 292

Point 95% Confidence Interval Percent Estimate [Lower Upper) 50 1323.00 720.00 1905.00 Mean Standard Error 1398.44 54.91

The estimated probability that a patient will survive for 365 The estimated probability that a patient will survive for 365 days or more is 0.56days or more is 0.56

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Kaplan – Meier EstimateKaplan – Meier Estimate

Test of Equality over Strata

Test Chi-Square DF Pr > Chi-Square

Log-Rank 6.8529 1 0.0088

Wilcoxon 3.9626 1 0.0465

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Modeling Survival DataModeling Survival Data

Model the survival “experience” of the Model the survival “experience” of the patient and the variablespatient and the variables

Focus on the risk or hazard of death at Focus on the risk or hazard of death at anytime after the time origin of the studyanytime after the time origin of the study

How explanatory variables affect “FORM” How explanatory variables affect “FORM” of the hazard functionof the hazard function

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Modeling Survival DataModeling Survival Data

Obtain an estimate of hazard Obtain an estimate of hazard function for individualfunction for individual

Estimate the median survival time Estimate the median survival time for current or future patientsfor current or future patients

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Cox Regression ModelingCox Regression Modeling

Proportional hazard assumptionProportional hazard assumption

Semi- Parametric modelSemi- Parametric model

Coefficient is the log of the ratio of hazard Coefficient is the log of the ratio of hazard of death at time tof death at time t

No assumption about shape but restrained No assumption about shape but restrained to be proportional across covariate levelsto be proportional across covariate levels

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Cox Regression ModelingCox Regression Modeling

CategoricalCategorical

Hazard Ratio: Ratio of estimated hazard Hazard Ratio: Ratio of estimated hazard for those with Diabetes to those without for those with Diabetes to those without (controlling for other variables) = 0.250(controlling for other variables) = 0.250

The hazard of death for those with diabetes is The hazard of death for those with diabetes is 25% of the hazard for those without25% of the hazard for those without

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Hazard Ratio = 2Hazard Ratio = 2

“ The treatment will cause the patient to progress more quickly, and that a treated patient who has not yet progressed by a certain time has twice the chance of having progressed at the next point in time compared with someone in the control group.”

What are hazard ratios?. Duerden, M. What is series by Hayward Group Ltd, 2009.

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Violations of PH AssumptionViolations of PH Assumption

PH assumes effect of each covariate PH assumes effect of each covariate is same at all time pointsis same at all time points

1.1. Time dependant covariates Time dependant covariates

2.2. Stratification Stratification

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Time dependent covariatesTime dependent covariates

Variables whose value change over timeVariables whose value change over time

No longer proportional hazard modelNo longer proportional hazard model

Method to deal with violation of PH assumptionMethod to deal with violation of PH assumption

Positive Coefficient: Effect of covariate increases linearly with Positive Coefficient: Effect of covariate increases linearly with timetime

Example: GVHD in model to look at leukemia relapseExample: GVHD in model to look at leukemia relapse

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Surviving Survival Analysis – An Applied Introduction. NESUG 2008. Williams, Surviving Survival Analysis – An Applied Introduction. NESUG 2008. Williams, C.C.

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Survival PredictionSurvival Prediction

Baseline survivor functionBaseline survivor function

Estimate survivor function for any set of Estimate survivor function for any set of covariates = Mean of covariate methodcovariates = Mean of covariate method

Adjusted survival curve methodAdjusted survival curve method

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Mean of Covariate MethodMean of Covariate Method

Mean value of covariate inserted into survival Mean value of covariate inserted into survival function of PH modelfunction of PH model

Limitations regarding mean value and Limitations regarding mean value and dichotomous variablesdichotomous variables

Calculated for ‘average’ personCalculated for ‘average’ person

Easily generated from SAS using baseline Easily generated from SAS using baseline statementstatement

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Corrected Group Prognosis Corrected Group Prognosis MethodMethod

Survival curve generated for each unique Survival curve generated for each unique combination of covariatescombination of covariates

Actual averaging of survival curvesActual averaging of survival curves

Can be computer intensiveCan be computer intensive

Comparison of 2 Methods for Calculating Adjusted Survival Curves from Proportional Comparison of 2 Methods for Calculating Adjusted Survival Curves from Proportional Hazard models. Ghali, W.A., Quan, H., Brant, R., et al. JAMA. 2001; 286(12):1494-Hazard models. Ghali, W.A., Quan, H., Brant, R., et al. JAMA. 2001; 286(12):1494-1497.1497.

http://people.ucalgary.ca/~hquan/Weight.htmlhttp://people.ucalgary.ca/~hquan/Weight.html

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Direct Adjusted Survival Direct Adjusted Survival CurvesCurves

Average of individual predicted survival curvesAverage of individual predicted survival curves Relative risk of survival between treatment arms Relative risk of survival between treatment arms

adjusted for covariates adjusted for covariates Beneficial in non-randomized studiesBeneficial in non-randomized studies Variance estimation and difference of direct Variance estimation and difference of direct

adjusted survival probabilitiesadjusted survival probabilities SAS MacroSAS Macro %ADJSURV %ADJSURV

http://www.mcw.edu/FileLibrary/Groups/Biostatistics/Software/AdjustedSurvivahttp://www.mcw.edu/FileLibrary/Groups/Biostatistics/Software/AdjustedSurvivalCurves.pdflCurves.pdfA SAS Macro for Estimation of Direct Adjusted Survival Curves Based on A A SAS Macro for Estimation of Direct Adjusted Survival Curves Based on A

Stratified Cox Regression Model. Zhang, X., Loberiza, F.R., Klein, J.P. and Stratified Cox Regression Model. Zhang, X., Loberiza, F.R., Klein, J.P. and Zhang, M-J.Zhang, M-J.

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Competing Risks ModelsCompeting Risks Models

Occurrence of one type of event removes Occurrence of one type of event removes individual from risk of all other typesindividual from risk of all other types

Ignoring other event can lead to bias in Ignoring other event can lead to bias in Kaplan-Meier estimatesKaplan-Meier estimates

Assumption of independence of the Assumption of independence of the distribution of the time to the competing distribution of the time to the competing events does not holdevents does not hold

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Competing Risks ModelsCompeting Risks Models

FunctioningShunt

Shunt FailureInfection

Shunt FailureBlockage

Shunt FailureOther Cause

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Cumulative Incidence Cumulative Incidence AnalysisAnalysis

Create data set with multiple strata per ‘failure’Create data set with multiple strata per ‘failure’ If K competing risks, then K rows of dataIf K competing risks, then K rows of data

%CumInc Macro%CumInc Macro %CumIncV Macro%CumIncV Macro Allows some covariates to have same effect on Allows some covariates to have same effect on

several types of outcome eventseveral types of outcome event

Rosthoj, S., Anderson, P. and Adildstrom, S. SAS macros for estimation of the cumulative incidence Rosthoj, S., Anderson, P. and Adildstrom, S. SAS macros for estimation of the cumulative incidence functions based on a Cox regression model for competing risks survival data. Computer Methods functions based on a Cox regression model for competing risks survival data. Computer Methods and Programs in Biomedicine, (2004) 74,69-75. and Programs in Biomedicine, (2004) 74,69-75.