Although the title says “Applied”, this book has a fair amount of mathematics in it, which helps you understand why certain approaches work well. The sample outputs for this book are reproduced in several different software programs at the UCLA Institute for Digital Research and Education.

David W. Hosmer, Jr. Stanley Lemeshow, Susanne May. Applied Survival Analysis, Second Edition. 2008. Wiley_interscience. ISBN-13: 978-0471754992.

This page shows R code to handle the tricky data sets where the response is known to be inside some interval.

UCLA Institute for Digital Research and Education. Interval Regression | R Data Analysis Examples. Available at https://stats.idre.ucla.edu/r/dae/interval-regression/.

Michael Fay gave a short course on interval censored data at the 2010 useR! meeting. The slides from this short course provide a nice overview of the complexities of this type of data.

Michael P. Fay. Interval Censored Data Analysis. useR! 2010 meeting. Available at https://www.r-project.org/conferences/useR-2010/tutorials/Fay.html.

This is a nice overview of how to use R to adjust survival curves in an observational study. It covers weighting and modeling with covariates and criticizes several approaches.

Terry M. Thernau, Cynthia S. Crawson, Elizabeth J. Atkinson. Adjusted survival curves. 2015 (January). Available at https://cran.r-project.org/web/packages/survival/vignettes/adjcurve.pdf.

I’m teaching an online workshop for The Analysis Factor on survival analysis. It’s not announced yet, and I have a LOT of work to do before it is ready. One thing that will save me time is that I am taking many of my examples from the excellent textbook, Applied Survival Analysis Second Edition. One nice perk of this book is that the helpful folks at UCLA have taken every textbook example, and written up code (with comments!) to reproduce the book’s results. With the exception of a few advanced methods in later chapters, where only one or two software packages have the right capability, the code is written in parallel in R, SAS, SPSS, and Stata. They also have links to the raw data at the publishers website, and datasets stored in SAS format and SPSS format. How nice! Browse around and you’ll find software code for all the examples in other popular statistics textbooks as well.

Warning! The R examples look like they are from the first edition, not the second edition. A small nitpick for an otherwise very nice resource.

UCLA Institute for Digital Research and Education. Textbook Examples Applied Survival Analysis. Second edition by David W. Hosmer, Jr., Stanley Lemeshow, and Susanne May. Available at https://stats.idre.ucla.edu/other/examples/asa2/.

First, here’s how to interpret a Kaplan-Meier curve. All survival curves start in the upper right corner. Curves that end up in the lower left corner represent survival rates that descend rapidly to 0%. If you’re looking at an event like mortality, relapse, re-hospitalization, or another unwanted event, then such a curve is very bad news. Curves that end up in the upper right corner represent survival rates that stay high for a long period of time. For unwanted events, such a curve is very good news.

A few graphs will reverse this scale by looking at the probability of death rather than the probability of survival.  Here, the upper left corner is bad and the lower right corner is good.

Each step represents one or more events (deaths if your outcome is survival.

Most software will place a vertical bar at each censoring time.

There are a couple of other things to look for. The size of the steps is an indication of the sample size. Small steps are good. They tell you that the survival probabilities are precisely estimated because they are based on a large number of patients at risk. Large steps could mean that a bunch of people died all at once, but more likely it means that the survival probabilities are imprecise because they are based on a small number of patients at risk. In many survival curves, the steps are small early in the study, but become much larger later in the study because fewer patients have a long term follow-up time.Does the curve tend to level off

From http://journalotohns.biomedcentral.com/articles/10.1186/1916-0216-42-53.

From http://bmccancer.biomedcentral.com/articles/10.1186/1471-2407-6-194.

From http://bmcpulmmed.biomedcentral.com/articles/10.1186/s12890-015-0025-z.

From http://bsd.biomedcentral.com/articles/10.1186/2042-6410-3-16.

From http://jiasociety.biomedcentral.com/articles/10.1186/1758-2652-12-38.

From http://wjso.biomedcentral.com/articles/10.1186/1477-7819-4-55.

From http://diagnosticpathology.biomedcentral.com/articles/10.1186/1746-1596-5-22.

From http://lipidworld.biomedcentral.com/articles/10.1186/1476-511X-9-100.

From http://molecular-cancer.biomedcentral.com/articles/10.1186/1476-4598-13-241.

From http://ro-journal.biomedcentral.com/articles/10.1186/1748-717X-7-78.

This builds on the work in the previous two blog entries.

This is the documentation of the data file.

# Documentation found at http://lib.stat.cmu.edu/datasets/pbc
# The data set is a study of primary biliary cirrhosis
# Variables:
# V1  case number
# V2  number of days between registration and the earlier of death,
#     transplantion, or study analysis time in July, 1986
# V3  status
#     0=censored, 1=censored due to liver tx, 2=death
# V4  drug
#     1= D-penicillamine, 2=placebo
# V5  age in days
# V6  sex
#     0=male, 1=female
# V7  presence of asictes
#     0=no 1=yes
# V8  presence of hepatomegaly
#     0=no 1=yes
# V9  presence of spiders
#     0=no 1=yes
# V10 presence of edema          
#     0   = no edema and no diuretic therapy for edema;
#     0.5 = edema present without diuretics, or edema resolved by diuretics;
#     1   = edema despite diuretic therapy
# V11 serum bilirubin in mg/dl
# V12 serum cholesterol in mg/dl
# V13 albumin in gm/dl
# V14 urine copper in ug/day#
# V15 alkaline phosphatase in U/liter
# V16 SGOT in U/ml
# V17 triglicerides in mg/dl
# V18 platelets per cubic ml / 1000
# V19 prothrombin time in seconds
# V20 histologic stage of disease

The dataset is already part of the survival package.

library("survival")

## Loading required package: splines

head(pbc)

##   id time status trt      age sex ascites hepato spiders edema bili chol
## 1  1  400      2   1 58.76523   f       1      1       1   1.0 14.5  261
## 2  2 4500      0   1 56.44627   f       0      1       1   0.0  1.1  302
## 3  3 1012      2   1 70.07255   m       0      0       0   0.5  1.4  176
## 4  4 1925      2   1 54.74059   f       0      1       1   0.5  1.8  244
## 5  5 1504      1   2 38.10541   f       0      1       1   0.0  3.4  279
## 6  6 2503      2   2 66.25873   f       0      1       0   0.0  0.8  248
##   albumin copper alk.phos    ast trig platelet protime stage
## 1    2.60    156   1718.0 137.95  172      190    12.2     4
## 2    4.14     54   7394.8 113.52   88      221    10.6     3
## 3    3.48    210    516.0  96.10   55      151    12.0     4
## 4    2.54     64   6121.8  60.63   92      183    10.3     4
## 5    3.53    143    671.0 113.15   72      136    10.9     3
## 6    3.98     50    944.0  93.00   63       NA    11.0     3

tail(pbc)

##      id time status trt      age sex ascites hepato spiders edema bili
## 413 413  989      0  NA 35.00068   f      NA     NA      NA     0  0.7
## 414 414  681      2  NA 67.00068   f      NA     NA      NA     0  1.2
## 415 415 1103      0  NA 39.00068   f      NA     NA      NA     0  0.9
## 416 416 1055      0  NA 56.99932   f      NA     NA      NA     0  1.6
## 417 417  691      0  NA 58.00137   f      NA     NA      NA     0  0.8
## 418 418  976      0  NA 52.99932   f      NA     NA      NA     0  0.7
##     chol albumin copper alk.phos ast trig platelet protime stage
## 413   NA    3.23     NA       NA  NA   NA      312    10.8     3
## 414   NA    2.96     NA       NA  NA   NA      174    10.9     3
## 415   NA    3.83     NA       NA  NA   NA      180    11.2     4
## 416   NA    3.42     NA       NA  NA   NA      143     9.9     3
## 417   NA    3.75     NA       NA  NA   NA      269    10.4     3
## 418   NA    3.29     NA       NA  NA   NA      350    10.6     4

pmod <- pbc[1:312,]
tail(pmod)

##      id time status trt      age sex ascites hepato spiders edema bili
## 307 307 1149      0   2 30.57358   f       0      0       0     0  0.8
## 308 308 1153      0   1 61.18275   f       0      1       0     0  0.4
## 309 309  994      0   2 58.29979   f       0      0       0     0  0.4
## 310 310  939      0   1 62.33265   f       0      0       0     0  1.7
## 311 311  839      0   1 37.99863   f       0      0       0     0  2.0
## 312 312  788      0   2 33.15264   f       0      0       1     0  6.4
##     chol albumin copper alk.phos ast trig platelet protime stage
## 307  273    3.56     52     1282 130   59      344    10.5     2
## 308  246    3.58     24      797  91  113      288    10.4     2
## 309  260    2.75     41     1166  70   82      231    10.8     2
## 310  434    3.35     39     1713 171  100      234    10.2     2
## 311  247    3.16     69     1050 117   88      335    10.5     2
## 312  576    3.79    186     2115 136  149      200    10.8     2

summary(pmod$time)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##      41    1191    1840    2006    2697    4556

table(pmod$status)

## 
##   0   1   2 
## 168  19 125

p.surv <- Surv(pmod$time,pmod$status==2)
print(p.surv)

##   [1]  400  4500+ 1012  1925  1504+ 2503  1832+ 2466  2400    51  3762 
##  [12]  304  3577+ 1217  3584  3672+  769   131  4232+ 1356  3445+  673 
##  [23]  264  4079  4127+ 1444    77   549  4509+  321  3839  4523+ 3170 
##  [34] 3933+ 2847  3611+  223  3244  2297  4467+ 1350  4453+ 4556+ 3428 
##  [45] 4025+ 2256  2576+ 4427+  708  2598  3853  2386  1000  1434  1360 
##  [56] 1847  3282  4459+ 2224  4365+ 4256+ 3090   859  1487  3992+ 4191 
##  [67] 2769  4039+ 1170  3458+ 4196+ 4184+ 4190+ 1827  1191    71   326 
##  [78] 1690  3707+  890  2540  3574  4050+ 4032+ 3358  1657   198  2452+
##  [89] 1741  2689   460   388  3913+  750   130  3850+  611  3823+ 3820+
## [100]  552  3581+ 3099+  110  3086  3092+ 3222  3388+ 2583  2504+ 2105 
## [111] 2350+ 3445   980  3395  3422+ 3336+ 1083  2288   515  2033+  191 
## [122] 3297+  971  3069+ 2468+  824  3255+ 1037  3239+ 1413   850  2944+
## [133] 2796  3149+ 3150+ 3098+ 2990+ 1297  2106+ 3059+ 3050+ 2419   786 
## [144]  943  2976+ 2615+ 2995+ 1427   762  2891+ 2870+ 1152  2863+  140 
## [155] 2666+  853  2835+ 2475+ 1536  2772+ 2797+  186  2055   264  1077 
## [166] 2721+ 1682  2713+ 1212  2692+ 2574+ 2301+ 2657+ 2644+ 2624+ 1492 
## [177] 2609+ 2580+ 2573+ 2563+ 2556+ 2555+ 2241+  974  2527+ 1576   733 
## [188] 2332+ 2456+ 2504+  216  2443+  797  2449+ 2330+ 2363+ 2365+ 2357+
## [199] 1592+ 2318+ 2294+ 2272+ 2221+ 2090  2081  2255+ 2171+  904  2216+
## [210] 2224+ 2195+ 2176+ 2178+ 1786  1080  2168+  790  2170+ 2157+ 1235 
## [221] 2050+  597   334  1945+ 2022+ 1978+  999  1967+  348  1979+ 1165 
## [232] 1951+ 1932+ 1776+ 1882+ 1908+ 1882+ 1874+  694  1831+  837+ 1810+
## [243]  930  1690  1790+ 1435+  732+ 1785+ 1783+ 1769+ 1457+ 1770+ 1765+
## [254]  737+ 1735+ 1701+ 1614+ 1702+ 1615+ 1656+ 1677+ 1666+ 1301+ 1542+
## [265] 1084+ 1614+  179  1191  1363+ 1568+ 1569+ 1525+ 1558+ 1447+ 1349+
## [276] 1481+ 1434+ 1420+ 1433+ 1412+   41  1455+ 1030+ 1418+ 1401+ 1408+
## [287] 1234+ 1067+  799  1363+  901+ 1329+ 1320+ 1302+  877+ 1321+  533+
## [298] 1300+ 1293+  207  1295+ 1271+ 1250+ 1230+ 1216+ 1216+ 1149+ 1153+
## [309]  994+  939+  839+  788+

Run a simple Cox proportional hazards model

cox.mod1 <- coxph(p.surv~sex,data=pmod,na.action=na.omit)
print(cox.mod1)

## Call:
## coxph(formula = p.surv ~ sex, data = pmod, na.action = na.omit)
## 
## 
##        coef exp(coef) se(coef)     z     p
## sexf -0.484     0.616    0.236 -2.05 0.041
## 
## Likelihood ratio test=3.77  on 1 df, p=0.0521  n= 312, number of events= 125

summary(cox.mod1)

## Call:
## coxph(formula = p.surv ~ sex, data = pmod, na.action = na.omit)
## 
##   n= 312, number of events= 125 
## 
##         coef exp(coef) se(coef)      z Pr(>|z|)  
## sexf -0.4839    0.6164   0.2365 -2.046   0.0407 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
##      exp(coef) exp(-coef) lower .95 upper .95
## sexf    0.6164      1.622    0.3877    0.9798
## 
## Concordance= 0.532  (se = 0.015 )
## Rsquare= 0.012   (max possible= 0.983 )
## Likelihood ratio test= 3.77  on 1 df,   p=0.05214
## Wald test            = 4.19  on 1 df,   p=0.04075
## Score (logrank) test = 4.27  on 1 df,   p=0.03887

To plot the data, you need to create predictions at the specific values of interest. With multiple indpendent variables, any variable left out would be predicted at its average value.

pred.mod1 <- survfit(cox.mod1,newdata=data.frame(sex=c("f","m")))
plot(pred.mod1,xlim=c(0,6000))
title("sex")
end.pts <- dim(pred.mod1$surv)[1]
end.x   <- rep(pred.mod1$time[end.pts]+100,1)
end.y   <- pred.mod1$surv[end.pts,]
end.nm  <- c("f","m")
text(end.x,end.y,end.nm,cex=1.5,col="darkred",adj=0)

The Cox model works fairly similarly for continuous independent variables.

cox.mod1 <- coxph(p.surv~sex,data=pmod,na.action=na.omit)
print(cox.mod1)

## Call:
## coxph(formula = p.surv ~ sex, data = pmod, na.action = na.omit)
## 
## 
##        coef exp(coef) se(coef)     z     p
## sexf -0.484     0.616    0.236 -2.05 0.041
## 
## Likelihood ratio test=3.77  on 1 df, p=0.0521  n= 312, number of events= 125

summary(cox.mod1)

## Call:
## coxph(formula = p.surv ~ sex, data = pmod, na.action = na.omit)
## 
##   n= 312, number of events= 125 
## 
##         coef exp(coef) se(coef)      z Pr(>|z|)  
## sexf -0.4839    0.6164   0.2365 -2.046   0.0407 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
##      exp(coef) exp(-coef) lower .95 upper .95
## sexf    0.6164      1.622    0.3877    0.9798
## 
## Concordance= 0.532  (se = 0.015 )
## Rsquare= 0.012   (max possible= 0.983 )
## Likelihood ratio test= 3.77  on 1 df,   p=0.05214
## Wald test            = 4.19  on 1 df,   p=0.04075
## Score (logrank) test = 4.27  on 1 df,   p=0.03887

cox.mod2 <- coxph(p.surv~age,data=pmod,na.action=na.omit)
print(cox.mod2)

## Call:
## coxph(formula = p.surv ~ age, data = pmod, na.action = na.omit)
## 
## 
##     coef exp(coef) se(coef)    z       p
## age 0.04      1.04  0.00881 4.54 5.7e-06
## 
## Likelihood ratio test=20.5  on 1 df, p=5.95e-06  n= 312, number of events= 125

summary(cox.mod2)

## Call:
## coxph(formula = p.surv ~ age, data = pmod, na.action = na.omit)
## 
##   n= 312, number of events= 125 
## 
##         coef exp(coef) se(coef)     z Pr(>|z|)    
## age 0.039995  1.040806 0.008811 4.539 5.65e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
##     exp(coef) exp(-coef) lower .95 upper .95
## age     1.041     0.9608     1.023     1.059
## 
## Concordance= 0.625  (se = 0.029 )
## Rsquare= 0.064   (max possible= 0.983 )
## Likelihood ratio test= 20.51  on 1 df,   p=5.947e-06
## Wald test            = 20.6  on 1 df,   p=5.651e-06
## Score (logrank) test = 20.86  on 1 df,   p=4.942e-06

pred.mod2 <- survfit(cox.mod2,newdata=data.frame(age=c(30,40,50,60)))
plot(pred.mod2,xlim=c(0,6000))
title("age")
end.pts <- dim(pred.mod2$surv)[1]
end.x   <- pred.mod2$time[end.pts]+100
end.y   <- pred.mod2$surv[end.pts,]
end.nm  <- c(30,40,50,60)
text(end.x,end.y,end.nm,cex=1.5,col="darkred",adj=0)

I am building on a previous blog entry and this referred to an earlier webpage. Here is the R code. The comment section at the top is a cut-and-paste from the documentation of the data file.

# Some simple examples of survival analysis
# The data set is a study of primary biliary cirrhosis
# Variables:
# V1  case number
# V2  number of days between registration and the earlier of death,
#     transplantion, or study analysis time in July, 1986
# V3  status
#     0=censored, 1=censored due to liver tx, 2=death
# V4  drug
#     1= D-penicillamine, 2=placebo
# V5  age in days
# V6  sex
#     0=male, 1=female
# V7  presence of asictes
#     0=no 1=yes
# V8  presence of hepatomegaly
#     0=no 1=yes
# V9  presence of spiders
#     0=no 1=yes
# V10 presence of edema          
#     0   = no edema and no diuretic therapy for edema;
#     0.5 = edema present without diuretics, or edema resolved by diuretics;
#     1   = edema despite diuretic therapy
# V11 serum bilirubin in mg/dl
# V12 serum cholesterol in mg/dl
# V13 albumin in gm/dl
# V14 urine copper in ug/day#
# V15 alkaline phosphatase in U/liter
# V16 SGOT in U/ml
# V17 triglicerides in mg/dl
# V18 platelets per cubic ml / 1000
# V19 prothrombin time in seconds
# V20 histologic stage of disease

Read the data in from the website. The first 60 rows are documentation about the file and the actual data starts on line 61.

> fn <- "http://lib.stat.cmu.edu/datasets/pbc"
> pbc <- read.table(fn,header=FALSE,skip=60)
> head(pbc)
  V1   V2 V3 V4    V5 V6 V7 V8 V9 V10  V11 V12  V13 V14    V15    V16 V17 V18  V19 V20
1  1  400  2  1 21464  1  1  1  1 1.0 14.5 261 2.60 156 1718.0 137.95 172 190 12.2   4
2  2 4500  0  1 20617  1  0  1  1 0.0  1.1 302 4.14  54 7394.8 113.52  88 221 10.6   3
3  3 1012  2  1 25594  0  0  0  0 0.5  1.4 176 3.48 210  516.0  96.10  55 151 12.0   4
4  4 1925  2  1 19994  1  0  1  1 0.5  1.8 244 2.54  64 6121.8  60.63  92 183 10.3   4
5  5 1504  1  2 13918  1  0  1  1 0.0  3.4 279 3.53 143  671.0 113.15  72 136 10.9   3
6  6 2503  2  2 24201  1  0  1  0 0.0  0.8 248 3.98  50  944.0  93.00  63   . 11.0   3

The data looks like it read in properly, so let’s load the survival library.

> library("survival")
Loading required package: splines

Attaching package: ‘survival’

The following object is masked _by_ ‘.GlobalEnv’:

    pbc

Now that’s strange. There is an object in the survival library that has the same name (pbc). Let’s investigate.

> rm(pbc)
> head(pbc)
  id time status trt      age sex ascites hepato spiders edema bili chol albumin copper alk.phos
1  1  400      2   1 58.76523   f       1      1       1   1.0 14.5  261    2.60    156   1718.0
2  2 4500      0   1 56.44627   f       0      1       1   0.0  1.1  302    4.14     54   7394.8
3  3 1012      2   1 70.07255   m       0      0       0   0.5  1.4  176    3.48    210    516.0
4  4 1925      2   1 54.74059   f       0      1       1   0.5  1.8  244    2.54     64   6121.8
5  5 1504      1   2 38.10541   f       0      1       1   0.0  3.4  279    3.53    143    671.0
6  6 2503      2   2 66.25873   f       0      1       0   0.0  0.8  248    3.98     50    944.0
     ast trig platelet protime stage
1 137.95  172      190    12.2     4
2 113.52   88      221    10.6     3
3  96.10   55      151    12.0     4
4  60.63   92      183    10.3     4
5 113.15   72      136    10.9     3
6  93.00   63       NA    11.0     3

Wow! The data set I was importing was already part of the survival library. Let’s use their version, since it has better names for the variables than V1-V20.

We have to subset the data because the first 312 observations represent a randomized trial and the remaining observations are an add-on study of baseline values for some non-randomized patients.

> tail(pbc)
     id time status trt      age sex ascites hepato spiders edema bili chol albumin copper
413 413  989      0  NA 35.00068   f      NA     NA      NA     0  0.7   NA    3.23     NA
414 414  681      2  NA 67.00068   f      NA     NA      NA     0  1.2   NA    2.96     NA
415 415 1103      0  NA 39.00068   f      NA     NA      NA     0  0.9   NA    3.83     NA
416 416 1055      0  NA 56.99932   f      NA     NA      NA     0  1.6   NA    3.42     NA
417 417  691      0  NA 58.00137   f      NA     NA      NA     0  0.8   NA    3.75     NA
418 418  976      0  NA 52.99932   f      NA     NA      NA     0  0.7   NA    3.29     NA
    alk.phos ast trig platelet protime stage
413       NA  NA   NA      312    10.8     3
414       NA  NA   NA      174    10.9     3
415       NA  NA   NA      180    11.2     4
416       NA  NA   NA      143     9.9     3
417       NA  NA   NA      269    10.4     3
418       NA  NA   NA      350    10.6     4
> pmod <- pbc[1:312,]
> tail(pmod)
     id time status trt      age sex ascites hepato spiders edema bili chol albumin copper
307 307 1149      0   2 30.57358   f       0      0       0     0  0.8  273    3.56     52
308 308 1153      0   1 61.18275   f       0      1       0     0  0.4  246    3.58     24
309 309  994      0   2 58.29979   f       0      0       0     0  0.4  260    2.75     41
310 310  939      0   1 62.33265   f       0      0       0     0  1.7  434    3.35     39
311 311  839      0   1 37.99863   f       0      0       0     0  2.0  247    3.16     69
312 312  788      0   2 33.15264   f       0      0       1     0  6.4  576    3.79    186
    alk.phos ast trig platelet protime stage
307     1282 130   59      344    10.5     2
308      797  91  113      288    10.4     2
309     1166  70   82      231    10.8     2
310     1713 171  100      234    10.2     2
311     1050 117   88      335    10.5     2
312     2115 136  149      200    10.8     2

Take a look at the two key variables needed to define the survival object. Dividing by 365.25 gives the times in years rather than in days.

> summary(pmod$time)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
     41    1191    1840    2006    2697    4556 
> summary(pmod$time/365.25)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 0.1123  3.2610  5.0360  5.4930  7.3850 12.4700 
> table(pmod$status)

  0   1   2 
168  19 125

The values for status, if you read the documentation, are 0=censored, 1=transplant, and 2=death. You could define an event as either 1 and 2 or 2 by itself. Choose the latter for this example.

> p.surv >- Surv(pmod$time,pmod$status==2)
print(p.surv)
  [1]  400  4500+ 1012  1925  1504+ 2503  1832+ 2466  2400    51  3762   304  3577+ 1217  3584 
 [16] 3672+  769   131  4232+ 1356  3445+  673   264  4079  4127+ 1444    77   549  4509+  321 
 [31] 3839  4523+ 3170  3933+ 2847  3611+  223  3244  2297  4467+ 1350  4453+ 4556+ 3428  4025+
 [46] 2256  2576+ 4427+  708  2598  3853  2386  1000  1434  1360  1847  3282  4459+ 2224  4365+
 [61] 4256+ 3090   859  1487  3992+ 4191  2769  4039+ 1170  3458+ 4196+ 4184+ 4190+ 1827  1191 
 [76]   71   326  1690  3707+  890  2540  3574  4050+ 4032+ 3358  1657   198  2452+ 1741  2689 
 [91]  460   388  3913+  750   130  3850+  611  3823+ 3820+  552  3581+ 3099+  110  3086  3092+
[106] 3222  3388+ 2583  2504+ 2105  2350+ 3445   980  3395  3422+ 3336+ 1083  2288   515  2033+
[121]  191  3297+  971  3069+ 2468+  824  3255+ 1037  3239+ 1413   850  2944+ 2796  3149+ 3150+
[136] 3098+ 2990+ 1297  2106+ 3059+ 3050+ 2419   786   943  2976+ 2615+ 2995+ 1427   762  2891+
[151] 2870+ 1152  2863+  140  2666+  853  2835+ 2475+ 1536  2772+ 2797+  186  2055   264  1077 
[166] 2721+ 1682  2713+ 1212  2692+ 2574+ 2301+ 2657+ 2644+ 2624+ 1492  2609+ 2580+ 2573+ 2563+
[181] 2556+ 2555+ 2241+  974  2527+ 1576   733  2332+ 2456+ 2504+  216  2443+  797  2449+ 2330+
[196] 2363+ 2365+ 2357+ 1592+ 2318+ 2294+ 2272+ 2221+ 2090  2081  2255+ 2171+  904  2216+ 2224+
[211] 2195+ 2176+ 2178+ 1786  1080  2168+  790  2170+ 2157+ 1235  2050+  597   334  1945+ 2022+
[226] 1978+  999  1967+  348  1979+ 1165  1951+ 1932+ 1776+ 1882+ 1908+ 1882+ 1874+  694  1831+
[241]  837+ 1810+  930  1690  1790+ 1435+  732+ 1785+ 1783+ 1769+ 1457+ 1770+ 1765+  737+ 1735+
[256] 1701+ 1614+ 1702+ 1615+ 1656+ 1677+ 1666+ 1301+ 1542+ 1084+ 1614+  179  1191  1363+ 1568+
[271] 1569+ 1525+ 1558+ 1447+ 1349+ 1481+ 1434+ 1420+ 1433+ 1412+   41  1455+ 1030+ 1418+ 1401+
[286] 1408+ 1234+ 1067+  799  1363+  901+ 1329+ 1320+ 1302+  877+ 1321+  533+ 1300+ 1293+  207 
[301] 1295+ 1271+ 1250+ 1230+ 1216+ 1216+ 1149+ 1153+  994+  939+  839+  788+

First, let’s look at an overall survival curve.

> plot(survfit(p.surv~1))

Now let’s define some functions that will calculate and compare Kaplan-Meier curves across all the possible covariates in the model. For categorical variables, draw Kaplan-Meier curves for each category level. For continuous variables, split the data into quartiles and then draw Kaplan-Meier curves for each quartile.

km.cat <- function(vn) {
print(vn)
print(table(pmod[,vn]))
km.all(pmod[,vn])
title(vn)
}
km.con <- function(vn) {
print(vn)
br <- quantile(pmod[,vn],na.rm=TRUE)
br[1] <- 0
quartiles <- cut(pmod[,vn],breaks=br)
print(table(quartiles))
km.all(quartiles)
title(vn)
}
km.all <- function(x) {
tb <- table(x)
mi <- sum(is.na(x))
sf <- survfit(p.surv~x)
print(paste("There are",mi,"missing values."))
plot(sf,xlim=c(0,6000))
end.pts <- cumsum(sf$strata)
end.x <- sf$time[end.pts]+100
end.y <- sf$surv[end.pts]
end.nm <- names(tb)
text(end.x,end.y,end.nm,cex=1.5,col="darkred",adj=0)
}
v.cat <- names(pmod)[c(4,6:10,20)]
v.con <- names(pmod)[c(5,11:19)]
for (v in v.cat) {
km.cat(v)
}
for (v in v.con) {
km.con(v)
}

Here is the output.

v.cat <- names(pmod)[c(4,6:10,20)]
v.con <- names(pmod)[c(5,11:19)]
for (v in v.cat) {
  km.cat(v)
}

## [1] "trt"
## 
##   1   2 
## 158 154 
## [1] "There are 0 missing values."

## [1] "sex"
## 
##   m   f 
##  36 276 
## [1] "There are 0 missing values."

## [1] "ascites"
## 
##   0   1 
## 288  24 
## [1] "There are 0 missing values."

## [1] "hepato"
## 
##   0   1 
## 152 160 
## [1] "There are 0 missing values."

## [1] "spiders"
## 
##   0   1 
## 222  90 
## [1] "There are 0 missing values."

## [1] "edema"
## 
##   0 0.5   1 
## 263  29  20 
## [1] "There are 0 missing values."

## [1] "stage"
## 
##   1   2   3   4 
##  16  67 120 109 
## [1] "There are 0 missing values."

for (v in v.con) {
  km.con(v)
}

## [1] "age"
## quartiles
##    (0,42.2] (42.2,49.8] (49.8,56.7] (56.7,78.4] 
##          78          78          78          78 
## [1] "There are 0 missing values."

## [1] "bili"
## quartiles
##     (0,0.8]  (0.8,1.35] (1.35,3.42]   (3.42,28] 
##          90          66          78          78 
## [1] "There are 0 missing values."

## [1] "chol"
## quartiles
##        (0,250]      (250,310]      (310,400] (400,1.78e+03] 
##             71             71             72             70 
## [1] "There are 28 missing values."

## [1] "albumin"
## quartiles
##    (0,3.31] (3.31,3.55]  (3.55,3.8]  (3.8,4.64] 
##          81          76          81          74 
## [1] "There are 0 missing values."

## [1] "copper"
## quartiles
##  (0,41.2] (41.2,73]  (73,123] (123,588] 
##        78        80        75        77 
## [1] "There are 2 missing values."

## [1] "alk.phos"
## quartiles
##             (0,872]      (872,1.26e+03] (1.26e+03,1.98e+03] 
##                  78                  78                  78 
## (1.98e+03,1.39e+04] 
##                  78 
## [1] "There are 0 missing values."

## [1] "ast"
## quartiles
##   (0,80.6] (80.6,115]  (115,152]  (152,457] 
##         79         78         79         76 
## [1] "There are 0 missing values."

## [1] "trig"
## quartiles
##   (0,84.2] (84.2,108]  (108,151]  (151,598] 
##         71         71         70         70 
## [1] "There are 30 missing values."

## [1] "platelet"
## quartiles
##   (0,200] (200,257] (257,322] (322,563] 
##        77        77        77        77 
## [1] "There are 4 missing values."

## [1] "protime"
## quartiles
##      (0,10]   (10,10.6] (10.6,11.1] (11.1,17.1] 
##          89          85          61          77 
## [1] "There are 0 missing values."

I wrote about the Kaplan-Meier curve in a previous webpage, but that was a generic example. First, you read the data in R and transform it into a survival object.

day <- c(37,40,43,44,45,
 47,49,54,56,58,59,60,61,62,68,
 NA,71,NA,NA,75,
 NA,NA,89,NA,96)
death <- is.finite(day)
tim <- day
tim[is.na(day)] <- 70
library("survival")
fly.surv <- Surv(tim,death)

The Surv function has options for left censored and interval censored observations. Read the help file for details.

There is a print method for survival objects.

print(fly.surv)

The survival object prints with a “+” attached to any censored observation.

[1] 37  40  43  44  45  47  49  54  56  58  59  60  61  62  68  70+ 71  70+ 70+ 75  70+ 70+ 89 
[24] 70+ 96

The survfit function creates a new object that summarizes the data in a survival object using a Kaplan-Meier curve or a Cox regression model. The input for survfit is a formula with a survival object on the left side of the equation. A model with “~1″ fits a single Kaplan-Meier curve to the entire survival object.

fly.fit <- survfit(fly.surv~1)

There is a print method for survfit objects .

print(fly.fit)

which lists some basic statistics

Call: survfit(formula = fly.surv ~ 1)

records   n.max n.start  events  median 0.95LCL 0.95UCL 
     25      25      25      19      61      56      NA

There is also a summary method

summary(fly.fit)

which produces a more detailed set of statistics.

Call: survfit(formula = fly.surv ~ 1)

records   n.max n.start  events  median 0.95LCL 0.95UCL 
     25      25      25      19      61      56      NA 
 time n.risk n.event survival std.err lower 95% CI upper 95% CI
   37     25       1     0.96  0.0392       0.8862        1.000
   40     24       1     0.92  0.0543       0.8196        1.000
   43     23       1     0.88  0.0650       0.7614        1.000
   44     22       1     0.84  0.0733       0.7079        0.997
   45     21       1     0.80  0.0800       0.6576        0.973
   47     20       1     0.76  0.0854       0.6097        0.947
   49     19       1     0.72  0.0898       0.5639        0.919
   54     18       1     0.68  0.0933       0.5197        0.890
   56     17       1     0.64  0.0960       0.4770        0.859
   58     16       1     0.60  0.0980       0.4357        0.826
   59     15       1     0.56  0.0993       0.3956        0.793
   60     14       1     0.52  0.0999       0.3568        0.758
   61     13       1     0.48  0.0999       0.3192        0.722
   62     12       1     0.44  0.0993       0.2827        0.685
   68     11       1     0.40  0.0980       0.2475        0.646
   71      4       1     0.30  0.1136       0.1428        0.630
   75      3       1     0.20  0.1114       0.0672        0.596
   89      2       1     0.10  0.0900       0.0171        0.584
   96      1       1     0.00     NaN           NA           NA

Most importantly, there is a plot method.

plot(fly.fit)

The graph includes the survival curve (either from a Kaplan-Meier estimate or a Cox regression model) and confidence limits. The graph displays a “+” at any censored value.

There’s a lot more on survival models which I hope to cover in another blog entry.