Survival Analysis using R - Western...

Survival Analysisusing R

Bruce L. Jones

Department of Statistical and Actuarial SciencesThe University of Western Ontario

March 24, 2010

Outline

• What is R?

• Why use R?

• A bit about R

• What is Survival Analysis?

• The survival package in R

• Example

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What is R?

• R is a free software environment for statistical computing and graphics.

• It compiles and runs on a wide variety of UNIX platforms, Windowsand MacOS.

• R is very popular among researchers in statistics.

• R is similar in appearance to S.

• R was initially written by Ross Ihaka and Robert Gentleman

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Why use R?

• It contains advanced statistical routines not yet available in otherpackages.

• It provides an unparalleled platform for programming new statisticalmethods in an easy and straightforward manner.

• It has state-of-the-art graphics capabilities.

• It’s free. Just go to http://www.r-project.org

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Assignment, Vectors and Arrays

> 1+2*3

[1] 7

> x=3

> y<-2

> x+y

[1] 5

> z=c(2,3,4,5)

> z

[1] 2 3 4 5

> 2*z

[1] 4 6 8 10

>

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Assignment, Vectors and Arrays

> 1+2*3

[1] 7

> x=3

> y<-2

> x+y

[1] 5

> z=c(2,3,4,5)

> z

[1] 2 3 4 5

> 2*z

[1] 4 6 8 10

>

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Assignment, Vectors and Arrays

> z=2:5

> z

[1] 2 3 4 5

> z=seq(2,5,1)

> z

[1] 2 3 4 5

> zz=seq(10,300,3)

> zz

[1] 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64[20] 67 70 73 76 79 82 85 88 91 94 97 100 103 106 109 112 115 118 121[39] 124 127 130 133 136 139 142 145 148 151 154 157 160 163 166 169 172 175 178[58] 181 184 187 190 193 196 199 202 205 208 211 214 217 220 223 226 229 232 235[77] 238 241 244 247 250 253 256 259 262 265 268 271 274 277 280 283 286 289 292[96] 295 298

>

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Assignment, Vectors and Arrays

> z=2:5

> z

[1] 2 3 4 5

> z=seq(2,5,1)

> z

[1] 2 3 4 5

> zz=seq(10,300,3)

> zz

[1] 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64[20] 67 70 73 76 79 82 85 88 91 94 97 100 103 106 109 112 115 118 121[39] 124 127 130 133 136 139 142 145 148 151 154 157 160 163 166 169 172 175 178[58] 181 184 187 190 193 196 199 202 205 208 211 214 217 220 223 226 229 232 235[77] 238 241 244 247 250 253 256 259 262 265 268 271 274 277 280 283 286 289 292[96] 295 298

>

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Assignment, Vectors and Arrays

> mat=array(1:12,c(3,4))

> mat

[,1] [,2] [,3] [,4]

[1,] 1 4 7 10

[2,] 2 5 8 11

[3,] 3 6 9 12

> mat=matrix(1:12,3,4)

> mat

[,1] [,2] [,3] [,4]

[1,] 1 4 7 10

[2,] 2 5 8 11

[3,] 3 6 9 12

>

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Assignment, Vectors and Arrays

> mat=array(1:12,c(3,4))

> mat

[,1] [,2] [,3] [,4]

[1,] 1 4 7 10

[2,] 2 5 8 11

[3,] 3 6 9 12

> mat=matrix(1:12,3,4)

> mat

[,1] [,2] [,3] [,4]

[1,] 1 4 7 10

[2,] 2 5 8 11

[3,] 3 6 9 12

>

11

Functions

> plus=function(a,b) a+b> plus(3,4)[1] 7> plus(3)

Error in plus(3) : element 2 is empty;the part of the args list of ’+’ being evaluated was:(a, b)

> plus=function(a,b=0) a+b> plus(3,4)[1] 7> plus(3)[1] 3> plus(1:3,4:5)

[1] 5 7 7Warning message:In a + b : longer object length is not a multiple of shorter object length

>

12

Functions

> plus=function(a,b) a+b> plus(3,4)[1] 7> plus(3)

Error in plus(3) : element 2 is empty;the part of the args list of ’+’ being evaluated was:(a, b)

> plus=function(a,b=0) a+b> plus(3,4)[1] 7> plus(3)[1] 3> plus(1:3,4:5)

[1] 5 7 7Warning message:In a + b : longer object length is not a multiple of shorter object length

>

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What is Survival Analysis?

Survival Analysis is the study of lifetimes and their distributions. It usuallyinvolves one or more of the following objectives:

• to explore the behaviour of the distribution of a lifetime.

• to model the distribution of a lifetime.

• to test for differences between the distributions of two or more lifetimes.

• to model the impact of one or more explanatory variables on a lifetimedistribution.

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The Nature of Lifetime Data

• It’s almost always incomplete.

– It often involves right-censoring.

– It sometimes involves left-truncation.

• The methods of survival analysis allow for this incompleteness.

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The survival Package in R

> install.packages("survival") # first time only

--- Please select a CRAN mirror for use in this session ---trying URL ’http://probability.ca/cran/bin/windows/contrib/2.10/survival_2.35-8.zip’Content type ’application/zip’ length 2445387 bytes (2.3 Mb)opened URLdownloaded 2.3 Mb

package ’survival’ successfully unpacked and MD5 sums checked

The downloaded packages are inC:\Documents and Settings\jones\Local Settings\Temp\RtmpEQ5ZaF\downloaded_packages

> library(survival)

Loading required package: splines

>

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Creating a Survival Object

Example 1. Complete data lifetimes: 26, 42, 71, 85, 92.

> ex1.times=c(26,42,71,85,92)

> ex1.surv=Surv(ex1.times)

> ex1.surv

[1] 26 42 71 85 92

> class(ex1.surv)

[1] "Surv"

> class(ex1.times)

[1] "numeric"

>

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Creating a Survival Object

Example 2. Right-censored lifetimes: 26, 42, 71, 80+, 80+.

> ex2.times=c(26,42,71,80,80)

> ex2.events=c(1,1,1,0,0)

> ex2.surv=Surv(ex2.times,ex2.events)

> ex2.surv

[1] 26 42 71 80+ 80+

>

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Creating a Survival Object

Example 3. Left-truncated and right-censored lifetimes:Left-truncation time is 40 for all individuals;Event/right-censoring times are 42, 71, 80+, 80+.

> ex3.lttimes=rep(40,4)

> ex3.times=c(42,71,80,80)

> ex3.events=c(1,1,0,0)

> ex3.surv=Surv(ex3.lttimes,ex3.times,ex3.events)

> ex3.surv

[1] (40,42 ] (40,71 ] (40,80+] (40,80+]

>

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Real Data Example

Lifetimes: Times until death of 26 psychiatric patients

Number of deaths: 14

Number of censored observations: 12

Covariates: patient age and sex (15 females, 11 males)

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Real Data Example

The Data

patient sex age time death patient sex age time death

1 2 51 1 1 14 2 30 37 02 2 58 1 1 15 2 33 35 03 2 55 2 1 16 1 36 25 14 2 28 22 1 17 1 30 31 05 1 21 30 0 18 1 41 22 16 1 19 28 1 19 2 43 26 17 2 25 32 1 20 2 45 24 18 2 48 11 1 21 2 35 35 09 2 47 14 1 22 1 29 34 010 2 25 36 0 23 1 35 30 011 2 31 31 0 24 1 32 35 112 1 24 33 0 25 2 36 40 113 1 25 33 0 26 1 32 39 0

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Real Data Example

Questions

• Does the lifetime distribution behave the way we expect?

• Are the lifetimes different for females and males?

• Do the lifetimes depend on age?

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Estimating the Survival Function

We can explore the lifetime distribution by examining nonparametricestimates of the survival function.

The R function survfit allow us to do this.

> library(KMsurv) # get the data> data(psych)> attach(psych)> names(psych)

[1] "sex" "age" "time" "death"

> psych.surv=Surv(age,age+time,death) # create a survival object

> psych.fit1=survfit(psych.surv˜1) # obtain the estimates

> plot(psych.fit1,xlim=c(40,80),xlab="age",ylab="probability",+ main="Survival Function Estimates") # plot the estimates>

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Estimating the Survival Function

40 50 60 70 80

0.0

0.2

0.4

0.6

0.8

1.0

Survival Function Estimates

age

prob

abili

ty

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Estimating the Survival Function

Now let’s consider females and males separately.

> psych.fit2=survfit(psych.surv˜sex) # separate by sex

> plot(psych.fit2,xlim=c(40,80),xlab="age",ylab="probability",+ main="Survival Function Estimates for Males (red) and Females",+ col=c("red","blue"))

> plot(psych.fit2,xlim=c(40,80),xlab="age",ylab="probability",+ main="Survival Function Estimates for Males (red) and Females",+ col=c("red","blue"), conf.int=T)>

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Estimating the Survival Function

40 50 60 70 80

0.0

0.2

0.4

0.6

0.8

1.0

Survival Function Estimates for Females (blue) and Males

age

prob

abili

ty

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Estimating the Survival Function

40 50 60 70 80

0.0

0.2

0.4

0.6

0.8

1.0

Survival Function Estimates for Females (blue) and Males

age

prob

abili

ty

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Testing for Differences

The R function survdiff allow us to test for differences between lifetimedistributions.

> survdiff(psych.surv˜sex)

Error in survdiff(psych.surv ˜ sex) : Right censored data only

> psych.surv2=Surv(time,death) # create new survival object> survdiff(psych.surv2˜sex)

Call:survdiff(formula = psych.surv2 ˜ sex)

N Observed Expected (O-E)ˆ2/E (O-E)ˆ2/Vsex=1 11 4 6.24 0.807 1.61sex=2 15 10 7.76 0.650 1.61

Chisq= 1.6 on 1 degrees of freedom, p= 0.205

>

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Testing for Differences

The R function survdiff allow us to test for differences between lifetimedistributions.

> survdiff(psych.surv˜sex)

Error in survdiff(psych.surv ˜ sex) : Right censored data only

> psych.surv2=Surv(time,death) # create new survival object> survdiff(psych.surv2˜sex)

Call:survdiff(formula = psych.surv2 ˜ sex)

N Observed Expected (O-E)ˆ2/E (O-E)ˆ2/Vsex=1 11 4 6.24 0.807 1.61sex=2 15 10 7.76 0.650 1.61

Chisq= 1.6 on 1 degrees of freedom, p= 0.205

>

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Fitting a Proportional Hazards Model

The model: h(t|x1, . . . , xp) = h0(t) exp(β1x1 + · · · + βpxp)

• The PH model is often used when we are interested in the impact ofthe covariates, x1, . . . , xp, but not the lifetime distributions themselves.

• We can estimate and make inferences about β1, . . . , βp without esti-mating h0.

• The R function coxph allows us to do this.

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Fitting a Proportional Hazards Model

> psych.coxph1=coxph(psych.surv˜sex)> summary(psych.coxph1)

Call:coxph(formula = psych.surv ˜ sex)

n= 26

coef exp(coef) se(coef) z Pr(>|z|)sex 0.3900 1.4770 0.6102 0.639 0.523

exp(coef) exp(-coef) lower .95 upper .95sex 1.477 0.677 0.4466 4.884

Rsquare= 0.016 (max possible= 0.926 )Likelihood ratio test= 0.43 on 1 df, p=0.5141Wald test = 0.41 on 1 df, p=0.5227Score (logrank) test = 0.41 on 1 df, p=0.5203

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Fitting a Proportional Hazards Model

Next we use our survival object psych.surv2, which does not involve left-truncation.

> psych.coxph2=coxph(psych.surv2˜sex)> summary(psych.coxph2)

Call:coxph(formula = psych.surv2 ˜ sex)

n= 26

coef exp(coef) se(coef) z Pr(>|z|)sex 0.7511 2.1194 0.6055 1.241 0.215

exp(coef) exp(-coef) lower .95 upper .95sex 2.119 0.4718 0.6469 6.944

Rsquare= 0.062 (max possible= 0.945 )Likelihood ratio test= 1.66 on 1 df, p=0.1981Wald test = 1.54 on 1 df, p=0.2148Score (logrank) test = 1.61 on 1 df, p=0.2046

Note that the last test is exactly that performed using survdiff.

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Fitting a Proportional Hazards Model

Finally, consider

> psych.coxph3=coxph(psych.surv2˜age+sex)> summary(psych.coxph3)

Call:coxph(formula = psych.surv2 ˜ age + sex)

n= 26

coef exp(coef) se(coef) z Pr(>|z|)age 0.20753 1.23063 0.05828 3.561 0.00037 ***sex -0.52374 0.59230 0.73753 -0.710 0.47762---Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

exp(coef) exp(-coef) lower .95 upper .95age 1.2306 0.8126 1.0978 1.380sex 0.5923 1.6883 0.1396 2.514

Rsquare= 0.553 (max possible= 0.945 )Likelihood ratio test= 20.91 on 2 df, p=2.879e-05Wald test = 14.3 on 2 df, p=0.0007866Score (logrank) test = 21.27 on 2 df, p=2.409e-05

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Conclusions about this Example

• There is great uncertainty due to the small number of observations.

• Times until death depend on age at first admission to the hospital.

• We cannot conclude that the lifetimes are different for females andmales.

32

Fitting an Accelerated Failure Time Model

• This is a popular fully parametric model for which the lifetime distrib-ution is the same for different covariate values, except that the timescale is multiplied by a different constant.

• The R function survreg can be used to fit an AFT model.

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Summary

• R is a flexible and free software environment for statistical computingand graphics.

• The survival package contains functions for survival analysis.

– Surv creates a survival object.

– survfit estimates (nonparametrically) the survival function.

– survdiff performs tests for differences in lifetime distributions.

– coxph fits the proportional hazards model.

– survreg fits the accelerated failure time model.

These slides are here:

http://www.stats.uwo.ca/faculty/jones/survival_talk.pdf

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