1. The Cox model in R Gardar Sveinbjörnsson, Jongkil Kim, Yongsheng Wang

2. OUTLINE  Recidivism data  Cox PH Model for Time-Independent Variables in R  Model Selection  Model Diagnostics  Cox PH Model for Time-Dependent Variables in R  Summary 2 18,April 2011Department of Mathematics, ETHZ

3. Recidivism data The data is from an experimental study of recidivism of 432 male prisoners, who were observed for a year after being released from prison. Half of the prisoners were randomly given financial aid when they were released. 3 18,April 2011Department of Mathematics, ETHZ The data is from an experimental study of recidivism of 432 male prisoners, who were observed for a year after being released from prison. Half of the prisoners were randomly given financial aid when they were released. Recidivism data

4. Variables in Recidivism Data  week: week of first arrest after release, or censoring time.  arrest: the event indicator, 1 = arrested, 0 = not  fin: 1=received financial aid, 0= not  age: in years at the time of release  race: 1= black, 0= others  wexp: 1= had full-time work experience, 0= not  mar: 1= married, 0= not  paro: 1= released on parole, 0= not  prio: number of prior convictions  educ: codes 2 (grade 6 or less), 3 (grades 6 through 9), 4 (grades 10 and 11), 5 (grade 12), or 6 (some post-secondary).  emp1— emp52: 1= employed in the corresponding week, 0 = not 4 18,April 2011Department of Mathematics, ETHZ

5. Recidivism Data > Rossi <- read.table(’Rossi.txt’, header=T) > Rossi[1:5, 1:10] ＃＃ omitting the variables emp1 — emp52 5 18,April 2011Department of Mathematics, ETHZ week arrest fin age race wexp mar paro prio educ 1 20 1 0 27 1 0 0 1 3 3 2 17 1 0 18 1 0 0 1 8 4 3 25 1 0 19 0 1 0 1 13 3 4 52 0 1 23 1 1 1 1 1 5 5 52 0 0 19 0 1 0 1 3 3

6. Cox PH Model for Time-Independent Variables in R

7. 7  Surv and coxph function in R  Cox Regression  Adjusted survival curve 18,April 2011Department of Mathematics, ETHZ

8. 8 Surv function in R  Surv(time, event)  time: survival or censoring time  event: the status indicator  0=censored  1=observed  Left-truncated and right- censored data  Surv(time, time2, event)  time: left-truncation time  time2: survival or censoring time  event: the status indicator  0= censored  1= observed 18,April 2011Department of Mathematics, ETHZ > Surv(time, time2, event, type=c('right', 'left', 'interval', 'counting'), origin=0) Right-censored data

9. 9 Coxph function in R > coxph(formula, data=, weights, subset, na.action, init, control, method=c("efron","breslow","exact"), singular.ok=TRUE, robust=FALSE, model=FALSE, x=FALSE, y=TRUE,...)  Most of the arguments are similar to lm 18,April 2011Department of Mathematics, ETHZ

10. 10 Coxph function in R  Formula  The right-hand side: the same as a linear model  The left-hand side: a survival object  Method : The method for tie handling. If there are no tied survival times all the methods are equivalent.  Breslow: the default for most Cox PH models  Efron: used as the default and much more accurate than Breslow when dealing with tied survival times  Exact: computes the exact partial likelihood 18,April 2011Department of Mathematics, ETHZ

11. 11 Cox regression > mod.allison <- coxph(Surv(week, arrest) ~ fin + age + race + wexp + mar + paro + prio + as.factor(educ), data=Rossi) > mod.allison 18,April 2011Department of Mathematics, ETHZ

12. 12 Cox regression 18,April 2011Department of Mathematics, ETHZ Call:coxph(formula = Surv(week, arrest) ~ fin + age + race + wexp + mar + paro + prio + as.factor(educ), data = Rossi) coef exp(coef) se(coef) z p fin -0.4027 0.669 0.1930 -2.087 0.0370 age -0.0514 0.950 0.0222 -2.316 0.0210 race 0.3615 1.435 0.3122 1.158 0.2500 wexp -0.1200 0.887 0.2135 -0.562 0.5700 mar -0.4236 0.655 0.3822 -1.108 0.2700 paro -0.0982 0.906 0.1959 -0.501 0.6200 prio 0.0794 1.083 0.0293 2.707 0.0068 as.factor(educ)3 0.5934 1.810 0.5196 1.142 0.2500 as.factor(educ)4 0.3284 1.389 0.5437 0.604 0.5500 as.factor(educ)5 -0.1210 0.886 0.6752 -0.179 0.8600 as.factor(educ)6 -0.4070 0.666 1.1233 -0.362 0.7200 Likelihood ratio test=38.7 on 11 df, p=6.01e-05 n= 432, number of events= 114

13. 13 Adjusted survival curve  > plot(survfit(mod.allison), ylim=c(.7, 1), xlab=’Weeks’, ylab=’Proportion Not Rearrested’) 18,April 2011Department of Mathematics, ETHZ

14. 14 Adjusted survival curve  We may wish to display how estimated survival depends upon the value of a covariate. Because the principal purpose of the recidivism study was to assess the impact of financial aid on rearrest, let us focus on this covariate.  We construct a new data frame with two rows, one for each value of fin; the other covariates are fixed to their median. 18,April 2011Department of Mathematics, ETHZ

15. 15 Adjusted survival curve 18,April 2011Department of Mathematics, ETHZ > Rossi.fin <- data.frame(fin=c(0,1), age=rep(median(age),2), race=rep(median(race),2),wexp=rep(median(wexp),2), mar=rep(median(mar),2), paro=rep(median(paro),2), prio=rep(median(prio),2), educ=as.factor(rep(median(educ),2)) > plot(survfit(mod.allison, newdata=Rossi.fin), conf.int=T, lty=c(1,2), col=c(‘red’, ‘blue’), ylim=c(.5, 1), xlab='Weeks', ylab='Proportion Not Rearrested')

16. Model Selection 16

17. Model Selection  Why variable selection?  Purposeful selection  Stepwise selection  Best Subset Selection of Covariates 18,April 201117Departement of Mathematics, ETHZ

18. Why variable selection?  We generally want to explain the data in the simplest way.  Unnecessary predictors in a model will effect the estimation of other quantities. That is to say, degrees of freedom will be wasted  If model is to be used for prediction, we will save effort, time and/or money if we do not have to collect data for predictors that are redundant. 18 18,April 2011Department of Mathematics, ETHZ

19. Why variable selection?  We must decide on a method to select a subset of variables.  Purposeful selection  Stepwise selection - using P-values - using AIC  Best subset selection 19 18,April 2011Department of Mathematics, ETHZ

20. Purposeful selection 1.We fit a multivariable model containing all variables that were significant in a univariable analysis at the 20-25% level. 2.We use the p-values from the Wald statistic to remove variables from our model. We also confirm the non-significance by a likelihood ratio test. 3.We check whether the removal has produced an “important” change in coefficients of other variables. 4.We check again all the variables that we removed. 5.We check for nonlinearity. 6.We look for interactions. 7.We check assumptions. 20 18,April 2011Department of Mathematics, ETHZ

21. Stepwise selection  Stepwise selection is a mix between forward and backward selection.  We can either start with an empty model or a full model and add/remove predictors according some criteria.  At each step we reconsider terms that were added or removed earlier. → Often applied in practice → Done argument in the step() function in R → In practice often based on AIC/BIC 21 18,April 2011Department of Mathematics, ETHZ

22. Stepwise selection  The AIC is a measure of the relative goodness of fit of a statistical model.  It does not only reward goodness of fit, but also includes a penalty that is an increasing function of the number of parameters.  AIC = 2k – 2max(loglikelihood), where k is the number of parameters in the model.  This means the smaller the better 22 18,April 2011Department of Mathematics, ETHZ

23. Stepwise selection using our data Step: AIC=1327.35 Surv(week, arrest) ~ fin + age + mar + prio Df AIC 1327.3 - mar 1 1327.7 - fin 1 1329.0 - age 1 1335.4 - prio 1 1336.2 23 18,April 2011Department of Mathematics, ETHZ

24. 24 Best Subset Selection  Stepwise only considers a small number of all the possible models  Best subset provides a way to check all the possible models  The same as in linear regression: need a criterion to judge the models Idea: not only based on goodness-of- fit, but also penalizes for the model size. 18,April 2011Department of Mathematics, ETHZ

25. 25 Best Subset Selection  Mallow’s C: C=W+(p-2q) smaller C is better  p: number of variables under consideration  q: number of variables not included in the subset model  W=W(p)-W(p-q), where W(p) is the Wald statistics for the model containing all p variables and W(p-q) denotes the Wald statistics for the subset model 18,April 2011Department of Mathematics, ETHZ

26. Best Subset Selection of Covariates  Check the model 26 18,April 2011Department of Mathematics, ETHZ VariablesMallow’s C fin, age, mar, prio6.56 fin, age, mar, prio, race7.22 fin, age, mar, prio, wexp7.81 fin, age, mar, prio, paro8.47 fin, age, mar, prio, educ5.39 fin, age, mar, prio, race, paro9.09 fin, age, mar, prio, wexp, paro9.75 fin, age, mar, prio, race, wexp8.53 fin, mar, prio11.77 fin, age, prio6.60 fin, age, mar17.34 age, mar, prio8.28

27. Model Diagnostics 27

28. Model Diagnostics  Analyze PH assumption with residuals  Influential observations  Checking nonlinearity 18,April 201128Departement of Mathematics, ETHZ

29. 18,April 2011Department of Mathematics, ETHZ Analyze PH assumption with residuals  We have a strong evidence of non-PH assumption for age  plot with cox.zph shows us plots of scaled Schoenfeld residuals. > cox.zph(mod.allison.4) rho chisq p fin -0.000159 2.99e-06 0.99862 age -0.221020 7.38e+00 0.00659 prio -0.077930 7.32e-01 0.39237 mar 0.131485 2.08e+00 0.14937 GLOBAL NA 8.88e+00 0.06406

30. 18,April 2011Department of Mathematics, ETHZ Analyze PH assumption with residuals > plot(cox.zph(mod.allison.4))

31. 18,April 2011Department of Mathematics, ETHZ Analyze PH assumption with residuals  For the variable age, the plot of residuals changes over time.  There are two possible solutions.  The effect of variable age is different with regards to time intervals: Age is a strata variable  The effect of age is declining over time: Interaction between age and time exists

32. 18,April 2011Department of Mathematics, ETHZ Analyze PH assumption (Strata)  The variable age is “strata” variable  We separate observations into 4 groups by their ages  [ < 19]  [20 ~ 25]  [26 ~ 30]  [31 < ] > library(car) > Rossi$age.cat <- recode(Rossi$age, “lo:19=1;20:25=2;26:30=3;31:hi=4") > table(Rossi$age.cat) 1 2 3 4 66 236 66 64

33. Analyze PH assumption (Strata)  Use this separated age groups as strata variables and check the PH assumption again > mod.allison.6 <- coxph(Surv(week, arrest) ~ fin + prio + strata(age.cat) + mar, data=Rossi) > cox.zph(mod.allison.6) rho chisq p fin -0.0164 0.0315 0.859 prio -0.0721 0.5946 0.441 mar 0.1337 2.0830 0.149 GLOBAL NA 2.7478 0.432 18,April 201133Departement of Mathematics, ETHZ

34. Analyze PH assumption (Interaction)  To analyze the interaction between age and time, we should transform the data.  Ex) 1 st observation, we change (0,20] to (0,1+], (1,2+], (2,3+],..., (19,20] 18,April 201134Departement of Mathematics, ETHZ > Rossi[1,1:10] week arrest fin age race wexp mar paro prio educ 1 20 1 0 27 1 0 0 1 3 3 > Rossi2[1:20,1:10] start stop arrest.time week arrest fin age... prio educ 1.1 0 1 0 20 1 0 27... 3 3... 1.19 18 19 0 20 1 0 27... 3 3 1.20 19 20 1 20 1 0 27... 3 3

35. 18,April 2011Department of Mathematics, ETHZ Analyze PH assumption (Interaction)  The interaction exists between time and age coxph(formula = Surv(start, stop, arrest.time) ~ fin + age + age:stop + prio + mar, data = Rossi.2) coef exp(coef) se(coef) z p fin -0.35971 0.698 0.19042 -1.89 0.05900 age 0.03552 1.036 0.03899 0.91 0.36000 prio 0.09868 1.104 0.02721 3.63 0.00029 mar -0.50512 0.603 0.37302 -1.35 0.18000 age:stop -0.00371 0.996 0.00145 -2.56 0.01100 Likelihood ratio test=38.1 on 5 df, p=3.55e-07 n= 19809  Age has a positive partial effect on the hazard but this effect gets smaller with time, even becoming negative effect about 10 weeks.

36. 36 18,April 2011Department of Mathematics, ETHZ Influential observations  For each covariate we look at how much the regression coefficients change if we remove one observation.  In R the argument type=dfbeta to the residuals() function produces a matrix of estimated changes in the regression coefficients upon deleting each observation in turn.  We then plot these changes. 3

37. 37 18,April 2011Department of Mathematics, ETHZ 4

38. 38 18,April 2011Department of Mathematics, ETHZ Influential observations(Just for fun)  Let see what happens if I change some observations.  I take the first age observation and change it. First to age=60 and then to age=110.  See R 5

39. 39 18,April 2011Department of Mathematics, ETHZ Checking nonlinearity  Nonlinearity is a problem in Cox regression as it is in linear and generalized linear models.  To detect nonlinearity we plot the Martingale residuals against covariates.  We add a smooth produced by local linear regression using the loess function and try to detect deviations from zero. 6

40. Martingale residuals  The Martingale residual for individual i on time t i is  Where δ i is the event indicator  is the cumulative hazard function for individual i.  t i is the time at the end of follow up for individual i. 18,April 201140Departement of Mathematics, ETHz

41. 41 18,April 2011Department of Mathematics, ETHZ 7

42. 42 18,April 2011Department of Mathematics, ETHZ In case of nonlinearity  We try to transform our covariate which is not linear.  We can try several transformations, e.g. log or sqrt.  We can also include higher order terms in our model and compare with the original model using likelihood ratio test. 8

43. Cox PH Model for Time-Dependent Variables in R 43

44. 18,April 2011Department of Mathematics, ETHZ Time-Dependent Variables  Cox PH Model for Time-Dependent Variables  Data Transformation  Model with Time-Dependent Variables  Model with Lagged Time-Dependent Variable

45. 18,April 2011Department of Mathematics, ETHZ Cox PH Model for Time-Dependent Variables  Recall the Cox PH model for Time-Dependent Variables  g i (t) which depend on time t can be 0 (time-independent) t, ln(t), etc... One variable at a time g i (t) = 1 (t = t 0, t 1, t 2,..) = 0 (otherwise) Heavyside function g i (t) = 1 (t ≥ t 0 ) = 0 (t < t 0 )

46. 18,April 2011Department of Mathematics, ETHZ Data Transformation  Now, we want to assess the effect of weekly employment on rearrest harzard.  Weekly employment indicators appear as a single row in 52 columns => Weekly employment indicators in rows > Rossi[1,] week arrest fin age race wexp mar paro prio educ emp1 emp2 1 20 1 0 27 1 0 0 1 3 3 0 0 emp3 emp4 emp5 emp6 emp7 emp8 emp9 emp10 emp11 emp12 emp13 1 0 0 0 0 0 0 0 0 0 0 0 emp14 emp15 emp16 emp17 emp18 emp19 emp20 emp21 emp22 emp23 1 0 0 0 0 0 0 0 NA NA NA emp24 emp25 emp26 emp27 emp28 emp29 emp30 emp31 emp32 emp33 1 NA NA NA NA NA NA NA NA NA NA emp34 emp35 emp36 emp37 emp38 emp39 emp40 emp41 emp42 emp43 1 NA NA NA NA NA NA NA NA NA NA emp44 emp45 emp46 emp47 emp48 emp49 emp50 emp51 emp52 1 NA NA NA NA NA NA NA NA NA

47. 18,April 2011Department of Mathematics, ETHZ Data Transformation  Transformed data (Weekly employment indicators in rows) > Rossi.2[1:50,] start stop arrest.time week arrest fin age race wexp mar paro prio educ employed 1.1 0 1 0 20 1 0 27 1 0 0 1 3 3 0... 1.19 18 19 0 20 1 0 27 1 0 0 1 3 3 0 1.20 19 20 1 20 1 0 27 1 0 0 1 3 3 0 2.1 0 1 0 17 1 0 18 1 0 0 1 8 4 0 2.2 1 2 0 17 1 0 18 1 0 0 1 8 4 0... 2.16 15 16 0 17 1 0 18 1 0 0 1 8 4 0 2.17 16 17 1 17 1 0 18 1 0 0 1 8 4 0... 7.1 0 1 0 23 1 0 19 1 1 1 1 0 4 1 7.2 1 2 0 23 1 0 19 1 1 1 1 0 4 1... 7.22 11 12 0 23 1 0 19 1 1 1 1 0 4 1 7.23 12 13 1 23 1 0 19 1 1 1 1 0 4 0...

48. 18,April 2011Department of Mathematics, ETHZ Model with Time-Dependent Variables  We treat weekly employment as a predictor depended on time to rearrest.  Suggested model:  X employed (t) means whether people are employed at week t (0 or 1)  We estimate coefficient β i, δ employed

49. 18,April 2011Department of Mathematics, ETHZ Model with Time-Dependent Variable  The weekly employment variable has an apparently large effect.  The hazard of rearrest is smaller by a factor of e -1.3289 = 0.265 (declined by 73.5%) when people are on a employed status. coxph(formula = Surv(start, stop, arrest.time) ~ fin + age + mar + prio + employed, data = Rossi.2) n= 19809 coef exp(coef) se(coef) z Pr(>|z|) fin -0.33898 0.71250 0.19037 -1.781 0.07498. age -0.04598 0.95507 0.02059 -2.233 0.02552 * mar -0.36119 0.69684 0.37334 -0.967 0.33331 prio 0.08419 1.08784 0.02775 3.034 0.00241 ** employed -1.32897 0.26475 0.24979 -5.320 1.04e-07 ***

50. 18,April 2011Department of Mathematics, ETHZ Model with Lagged Time-Dependent Variables  Claim: The direction of causality is not clear, because a person cannot work when he is in jail. Weekly Employment at time t Arrest at time t Weekly Employment at time t-1 Arrest At time t Ambiguous causality

51. Model with Lagged Time-Dependent Variables  Use a lagged value of employment from the previous week  Model with lagged time  We apply lagged property to the data and then use the same command and arguments in R arrest.time are shifted by a lagged time 18,April 201151Departement of Mathematics, ETHZ

52. 18,April 2011Department of Mathematics, ETHZ Model with Lagged Time-Dependent Variables  The coefficient for the lagged employment variable is still significant, but the estimated effect is much smaller: e -0.7891 = 0.45 coxph(formula = Surv(start, stop, arrest.time) ~ fin + age + mar + prio + employed, data = Rossi.3) n= 19377 coef exp(coef) se(coef) z Pr(>|z|) fin -0.33471 0.71554 0.19104 -1.752 0.079758. age -0.05014 0.95110 0.02075 -2.416 0.015674 * mar -0.41269 0.66187 0.37385 -1.104 0.269650 prio 0.09217 1.09655 0.02752 3.350 0.000809 *** employed -0.78918 0.45422 0.21704 -3.636 0.000277 ***

53. 53 Summary

54. 54 Final model  After we introduced the weekly employment into our model the marriage variable has become non-significant. We therefore remove it.  We also choose to have age as a strata variable for ease of interpretation because it does not satisfy the PH assumption. 18,April 2011Department of Mathematics, ETHZ

55. 55 Final model coxph(formula = Surv(start, stop, arrest.time) ~ fin + strata(age.cat) + prio + employed, data = Rossi.3) coef exp(coef) se(coef) z Pr(>|z|) fin -0.33454 0.71567 0.19078 -1.754 0.079502. prio 0.08984 1.09400 0.02707 3.319 0.000902 *** employed -0.82758 0.43710 0.21583 -3.834 0.000126 *** --- exp(coef) exp(-coef) lower.95 upper.95 fin 0.7157 1.397 0.4924 1.0402 prio 1.0940 0.914 1.0375 1.1536 employed 0.4371 2.288 0.2863 0.6673 Rsquare= 0.002 (max possible= 0.053 ) Likelihood ratio test= 30.08 on 3 df, p=1.325e-06 Wald test = 30.32 on 3 df, p=1.182e-06 Score (logrank) test = 31.42 on 3 df, p=6.933e-07 18,April 2011Department of Mathematics, ETHZ

56. 56 Final model  Financial aid coef exp(coef) se(coef) z Pr(>|z|) fin -0.33454 0.71567 0.19078 -1.754 0.079502.  The estimated hazard ratio for receiving financial aid is 0.71567.  This means, holding the other covariates constant, the rearrested rate of subjects with financial aid reduces 29%. 18,April 2011Department of Mathematics, ETHZ

57. 57 Final model  Number of prior convictions coef exp(coef) se(coef) z Pr(>|z|) prio 0.08984 1.09400 0.02707 3.319 0.000902 ***  The estimated hazard is 1.094.  Holding the other covariates constant, an additional time of prior convictions increases the weekly hazard of rearrest by 9 percent. 18,April 2011Department of Mathematics, ETHZ

58. 58 Final model  Employment coef exp(coef) se(coef) z Pr(>|z|) employed -0.82758 0.43710 0.21583 -3.834 0.000126 ***  The estimated hazard ratio is 0.43710.  This means that the hazard of rearrest is smaller by a decline of 56 percent during a week in which the former inmate was employed. 18,April 2011Department of Mathematics, ETHZ

59. 59 Summary  Cox PH Model for Time-Independent Variables in R  Surv and coxph function in R  Cox Regression  Adjusted survival curve  Model Selection  Why variable selection?  Purposeful selection  Stepwise selection  Best Subset Selection 18,April 2011Department of Mathematics, ETHZ

60. 60 Summary  Model Diagnostics  Analyze PH assumption with residuals  Influential observations  Checking nonlinearity  Cox PH Model for Time-Dependent Variables in R  Model description  Analysis for the result  Lagged variables  Final model 18,April 2011Department of Mathematics, ETHZ