Liver Resection Ananlysis

We will take a look at the file of liverdata.RData which contains data on liver resection. The goal is to carry out a statistical analysis in order to investigate the survival functions of the individuals in the dataset.

load("F:\\TUD lecture\\SA Lecture\\R-Programming\\liverdata.RData")
liver_data<-as_tibble(liver_data)
colnames(liver_data)

 [1] "overall"         "diseasefree"     "death"           "recurrence"     
 [5] "age"             "height"          "weight"          "bmi"            
 [9] "sex"             "complications"   "resection_vol"   "pringle"        
[13] "overall_yrs"     "diseasefree_yrs"

#Total number of death and censored:
table(liver_data$death)

  0   1 
135 157 

#Death proportions between gender  :
with(liver_data,prop.table(table(death,sex),2)) 

     sex
death      male    female
    0 0.4717949 0.4432990
    1 0.5282051 0.5567010

#Proportions death rate and censored respective to the gender:
with(liver_data,prop.table(table(death,sex),1)) 

     sex
death      male    female
    0 0.6814815 0.3185185
    1 0.6560510 0.3439490

#Total number of death based on SEX:
table(liver_data$death,liver_data$sex) #or

   
    male female
  0   92     43
  1  103     54

with(liver_data,table(death,sex))

     sex
death male female
    0   92     43
    1  103     54

Visualization:

library(ggplot2)
ggplot(liver_data,aes(x=sex,y=overall_yrs))+ 
  geom_boxplot()+
  labs(x = "Sex", y = "Overall years") +
  theme_bw()

Female are less likely to spread than males in the upper quantile

Non-Parametric Estimation:

Kaplan-Meier estimation:

library(survminer)
library(survival)
liver_data<-mutate(liver_data,sex=as.character(sex))
km<- survfit(Surv(overall_yrs,death)~sex,data=liver_data)
km1<-survfit(Surv(diseasefree,recurrence)~1,data=liver_data)
surv_pvalue(km)

summary(km)

Call: survfit(formula = Surv(overall_yrs, death) ~ sex, data = liver_data)

                sex=female 
   time n.risk n.event survival std.err lower 95% CI upper 95% CI
 0.0383     97       1    0.990  0.0103        0.970        1.000
 0.1013     95       1    0.979  0.0145        0.951        1.000
 0.1834     94       1    0.969  0.0177        0.935        1.000
 0.4901     93       1    0.958  0.0203        0.919        0.999
 0.5640     92       1    0.948  0.0226        0.905        0.993
 0.6516     91       1    0.938  0.0247        0.890        0.987
 0.7337     88       1    0.927  0.0266        0.876        0.981
 0.7858     87       1    0.916  0.0283        0.862        0.974
 0.8323     86       1    0.906  0.0299        0.849        0.966
 0.8679     85       1    0.895  0.0314        0.835        0.959
 1.1499     84       1    0.884  0.0328        0.822        0.951
 1.1882     83       1    0.874  0.0341        0.809        0.943
 1.2320     82       1    0.863  0.0353        0.797        0.935
 1.2731     80       1    0.852  0.0365        0.784        0.927
 1.3388     79       1    0.841  0.0376        0.771        0.918
 1.3689     78       1    0.831  0.0386        0.758        0.910
 1.3744     77       1    0.820  0.0396        0.746        0.901
 1.5332     75       1    0.809  0.0405        0.733        0.892
 1.5962     74       1    0.798  0.0414        0.721        0.884
 1.8207     72       1    0.787  0.0423        0.708        0.874
 1.8344     71       1    0.776  0.0432        0.696        0.865
 1.8398     70       1    0.765  0.0439        0.683        0.856
 1.9384     69       1    0.754  0.0447        0.671        0.847
 2.1793     68       1    0.743  0.0454        0.659        0.837
 2.2697     66       1    0.731  0.0461        0.646        0.827
 2.3244     64       1    0.720  0.0467        0.634        0.818
 2.5927     62       1    0.708  0.0474        0.621        0.808
 2.6858     61       1    0.697  0.0480        0.609        0.797
 2.9322     57       1    0.684  0.0487        0.595        0.787
 2.9843     56       1    0.672  0.0494        0.582        0.776
 3.0116     54       1    0.660  0.0500        0.569        0.765
 3.2060     53       1    0.647  0.0506        0.555        0.754
 3.3320     51       1    0.635  0.0511        0.542        0.743
 3.4031     49       1    0.622  0.0517        0.528        0.732
 3.4114     48       1    0.609  0.0522        0.515        0.720
 3.4579     47       1    0.596  0.0527        0.501        0.709
 3.7098     44       1    0.582  0.0532        0.487        0.696
 3.7864     41       1    0.568  0.0538        0.472        0.684
 3.8138     40       1    0.554  0.0543        0.457        0.671
 4.0575     36       1    0.538  0.0549        0.441        0.658
 4.3094     35       1    0.523  0.0555        0.425        0.644
 4.6023     34       1    0.508  0.0559        0.409        0.630
 4.6242     33       1    0.492  0.0563        0.393        0.616
 4.6626     30       1    0.476  0.0568        0.377        0.601
 4.7228     25       1    0.457  0.0576        0.357        0.585
 4.9090     23       1    0.437  0.0584        0.336        0.568
 5.1882     18       1    0.413  0.0600        0.310        0.549
 5.2594     17       1    0.388  0.0612        0.285        0.529
 5.3607     16       1    0.364  0.0620        0.261        0.508
 5.4894     14       1    0.338  0.0628        0.235        0.487
 5.5277     13       1    0.312  0.0631        0.210        0.464
 5.8207     12       1    0.286  0.0630        0.186        0.441
 6.0123     11       1    0.260  0.0624        0.163        0.416
 6.6886      6       1    0.217  0.0653        0.120        0.391

                sex=male 
   time n.risk n.event survival std.err lower 95% CI upper 95% CI
 0.0137    195       2    0.990 0.00722        0.976        1.000
 0.0383    190       1    0.985 0.00886        0.967        1.000
 0.0575    189       1    0.979 0.01023        0.959        1.000
 0.0712    188       1    0.974 0.01143        0.952        0.997
 0.0767    187       1    0.969 0.01250        0.945        0.994
 0.0958    186       1    0.964 0.01347        0.938        0.990
 0.1424    185       1    0.958 0.01437        0.931        0.987
 0.2245    184       1    0.953 0.01521        0.924        0.984
 0.3723    182       1    0.948 0.01600        0.917        0.980
 0.3943    181       1    0.943 0.01675        0.911        0.976
 0.4025    180       1    0.938 0.01745        0.904        0.972
 0.4298    179       1    0.932 0.01813        0.897        0.969
 0.5886    174       1    0.927 0.01880        0.891        0.965
 0.6297    173       1    0.922 0.01944        0.884        0.961
 0.6543    171       1    0.916 0.02006        0.878        0.956
 0.7036    170       1    0.911 0.02065        0.871        0.952
 0.7337    169       1    0.905 0.02122        0.865        0.948
 0.8159    166       2    0.895 0.02232        0.852        0.939
 0.8761    164       1    0.889 0.02284        0.845        0.935
 0.8816    163       1    0.884 0.02334        0.839        0.931
 0.9391    162       1    0.878 0.02383        0.833        0.926
 0.9446    161       1    0.873 0.02430        0.826        0.922
 0.9500    160       1    0.867 0.02475        0.820        0.917
 0.9966    159       1    0.862 0.02519        0.814        0.913
 1.0021    158       1    0.856 0.02561        0.808        0.908
 1.1855    157       1    0.851 0.02602        0.801        0.903
 1.2074    156       1    0.845 0.02642        0.795        0.899
 1.2676    155       1    0.840 0.02681        0.789        0.894
 1.2923    154       1    0.835 0.02718        0.783        0.890
 1.3443    151       1    0.829 0.02756        0.777        0.885
 1.5578    148       1    0.823 0.02794        0.770        0.880
 1.6044    147       1    0.818 0.02830        0.764        0.875
 1.6290    146       1    0.812 0.02866        0.758        0.870
 1.6728    145       1    0.807 0.02900        0.752        0.865
 1.7714    142       1    0.801 0.02935        0.745        0.861
 1.8261    141       1    0.795 0.02969        0.739        0.856
 1.8809    140       1    0.790 0.03001        0.733        0.851
 1.9603    137       1    0.784 0.03034        0.727        0.846
 1.9986    136       1    0.778 0.03066        0.720        0.841
 2.1793    133       1    0.772 0.03098        0.714        0.835
 2.2177    132       1    0.766 0.03130        0.707        0.830
 2.2505    131       1    0.760 0.03160        0.701        0.825
 2.2752    128       1    0.755 0.03191        0.695        0.820
 2.3053    127       1    0.749 0.03220        0.688        0.814
 2.3244    125       1    0.743 0.03250        0.682        0.809
 2.3326    124       1    0.737 0.03278        0.675        0.804
 2.3819    123       2    0.725 0.03333        0.662        0.793
 2.4285    119       1    0.719 0.03360        0.656        0.788
 2.4778    118       1    0.712 0.03386        0.649        0.782
 2.5079    117       1    0.706 0.03411        0.643        0.777
 2.6256    115       1    0.700 0.03437        0.636        0.771
 2.6831    114       1    0.694 0.03461        0.629        0.765
 2.7296    112       1    0.688 0.03485        0.623        0.760
 2.7351    111       1    0.682 0.03508        0.616        0.754
 2.8611    110       1    0.676 0.03531        0.610        0.748
 2.8720    109       1    0.669 0.03552        0.603        0.743
 2.9350    107       1    0.663 0.03574        0.597        0.737
 2.9377    104       1    0.657 0.03596        0.590        0.731
 3.0144    101       1    0.650 0.03619        0.583        0.725
 3.0171    100       1    0.644 0.03640        0.576        0.719
 3.0253     97       1    0.637 0.03663        0.569        0.713
 3.0308     96       1    0.630 0.03684        0.562        0.707
 3.2088     93       1    0.624 0.03706        0.555        0.701
 3.2416     92       1    0.617 0.03728        0.548        0.694
 3.4141     90       1    0.610 0.03749        0.541        0.688
 3.4552     87       1    0.603 0.03771        0.533        0.682
 3.5921     86       1    0.596 0.03791        0.526        0.675
 3.6249     85       1    0.589 0.03811        0.519        0.669
 3.6797     83       1    0.582 0.03831        0.511        0.662
 3.6988     82       2    0.568 0.03866        0.497        0.649
 3.8084     79       1    0.560 0.03884        0.489        0.642
 3.9179     74       1    0.553 0.03904        0.481        0.635
 3.9890     72       1    0.545 0.03925        0.473        0.628
 4.0027     71       1    0.538 0.03944        0.466        0.621
 4.0657     68       1    0.530 0.03964        0.457        0.613
 4.3066     67       1    0.522 0.03983        0.449        0.606
 4.3258     66       1    0.514 0.04001        0.441        0.599
 4.3559     65       1    0.506 0.04016        0.433        0.591
 4.3723     64       1    0.498 0.04031        0.425        0.584
 4.6735     63       1    0.490 0.04044        0.417        0.576
 4.7967     60       1    0.482 0.04058        0.409        0.568
 4.8706     57       1    0.473 0.04074        0.400        0.560
 5.1280     54       1    0.465 0.04092        0.391        0.552
 5.1855     52       1    0.456 0.04109        0.382        0.544
 5.3279     50       1    0.447 0.04127        0.373        0.535
 5.3634     49       1    0.438 0.04142        0.363        0.527
 5.7166     44       1    0.428 0.04166        0.353        0.518
 5.7276     43       1    0.418 0.04186        0.343        0.508
 5.7413     42       1    0.408 0.04203        0.333        0.499
 5.7659     41       1    0.398 0.04216        0.323        0.490
 5.7851     40       1    0.388 0.04226        0.313        0.480
 5.8754     39       1    0.378 0.04233        0.303        0.471
 6.0205     35       1    0.367 0.04248        0.293        0.461
 6.2122     33       1    0.356 0.04262        0.282        0.450
 6.5517     26       1    0.342 0.04313        0.267        0.438
 6.8118     23       1    0.327 0.04374        0.252        0.425
 7.2060     21       1    0.312 0.04435        0.236        0.412
 8.1424     14       1    0.290 0.04644        0.211        0.396
 8.3066     12       1    0.265 0.04843        0.186        0.380

Separating gender data:

male<-filter(liver_data,sex == "male")
female<-filter(liver_data,sex == "female")

male_km<-survfit(Surv(overall_yrs,death)~1,data=male)
female_km<-survfit(Surv(overall_yrs,death)~1,data=female)

Visualization:

library(GGally)
male_viz<-ggsurv(male_km, CI=TRUE,plot.cens=TRUE, surv.col= "gg.def", cens.col="gg.def", lty.est=1, size.est= 2, size.ci=1, back.white= TRUE, xlab="Time", ylab="Survival")+ggtitle("Survival of Males")
female_viz<-ggsurv(female_km,CI=TRUE)+ggtitle("Survival of Females")+theme_bw()

library(patchwork)
male_viz|female_viz

Survival rates of both gender monotonically decreasing and on comparison males have a bit high survival chance than females

Nelson-aalen estimation:

na<-survfit(coxph(Surv(diseasefree,recurrence)~1,data=liver_data),type='aalen')
na_d<-ggsurv(na,CI= TRUE)+ 
  ggtitle("Nelson-AAlen Estimation")+
  theme_bw()+ 
  ylim(c(0,1))
na_d

Lifetable analysis:

library(discSurv)
liver_data<-mutate(liver_data, diseasefree_months=ceiling(diseasefree/30.4),diseasefree_years=ceiling(diseasefree_yrs))
data<-as.data.frame(liver_data)

#Days:
lifetable_liverresection_days<-lifeTable(dataSet= data,timeColumn="diseasefree", censColumn="recurrence")
lifetable_liverresection_days<-lifetable_liverresection_days$Output
#Months:
lifetable_liverresection_months<-lifeTable(dataSet= data,timeColumn="diseasefree_months", censColumn="recurrence")
lifetable_liverresection_months<-lifetable_liverresection_months$Output
#years:
lifetable_liverresection_years<-lifeTable(dataSet= data,timeColumn="diseasefree_years", censColumn="recurrence")
lifetable_liverresection_years<-lifetable_liverresection_years$Output

alpha=0.5
#Days
lifetable_liverresection_days_v<-ggplot(lifetable_liverresection_days,aes(x=1:nrow(lifetable_liverresection_days),y=S))+
  geom_line()+
  geom_ribbon(data=lifetable_liverresection_days, aes(ymin=(S-(qnorm(1-(alpha/2))*seS)),ymax=(S+(qnorm(1-(alpha/2))*seS))),alpha=0.2)+
  lims(y=c(0,1))+
  xlab("Days")+
  ylab("S(t)")+
  ggtitle("Lifetable Estimation")+theme_bw()

#Months
lifetable_liverresection_months_v<-ggplot(lifetable_liverresection_months,aes(x=1:nrow(lifetable_liverresection_months),y=S))+
  geom_line()+
  geom_ribbon(data=lifetable_liverresection_months, aes(ymin=S-qnorm(1-alpha/2)*seS,ymax=S+qnorm(1-alpha/2)*seS),alpha=0.2)+
  lims(y=c(0,1))+
  xlab("Months")+
  ylab("S(t)")+
  ggtitle("Disease Free Survival- Lifetable")+theme_bw()

#Years
lifetable_liverresection_years_v<-ggplot(lifetable_liverresection_years,aes(x=1:nrow(lifetable_liverresection_years),y=S))+
  geom_line()+
  geom_ribbon(data=lifetable_liverresection_years, aes(ymin=S-qnorm(1-alpha/2)*seS,ymax=S+qnorm(1-alpha/2)*seS),alpha=0.2)+
  lims(y=c(0,1))+
  xlab("Years")+
  ylab("S(t)")+
  ggtitle("Disease Free Survival- Lifetable")+theme_bw()

lifetable_liverresection_days_v | lifetable_liverresection_months_v / lifetable_liverresection_years_v

Comparing all Three estimation:

lifetable_liverresection_days_v|km_d|na_d

Filtering all old females aged above 70 and using different Confidence Interval to distinguish the results:

female_old<-filter(liver_data , sex=="female" & age>70)
summary(female_old)

    overall      diseasefree         death          recurrence    
 Min.   :  14   Min.   :  14.0   Min.   :0.0000   Min.   :0.0000  
 1st Qu.: 733   1st Qu.: 278.0   1st Qu.:0.0000   1st Qu.:1.0000  
 Median :1246   Median : 547.0   Median :1.0000   Median :1.0000  
 Mean   :1186   Mean   : 735.4   Mean   :0.5652   Mean   :0.8261  
 3rd Qu.:1570   3rd Qu.:1197.0   3rd Qu.:1.0000   3rd Qu.:1.0000  
 Max.   :2443   Max.   :2443.0   Max.   :1.0000   Max.   :1.0000  
      age           height         weight           bmi       
 Min.   :71.0   Min.   :1.53   Min.   :48.00   Min.   :18.29  
 1st Qu.:72.0   1st Qu.:1.60   1st Qu.:59.50   1st Qu.:22.46  
 Median :74.0   Median :1.62   Median :65.00   Median :25.39  
 Mean   :74.7   Mean   :1.62   Mean   :65.37   Mean   :24.97  
 3rd Qu.:77.0   3rd Qu.:1.65   3rd Qu.:71.00   3rd Qu.:27.79  
 Max.   :82.0   Max.   :1.70   Max.   :83.00   Max.   :32.88  
     sex            complications resection_vol    pringle 
 Length:23          minor:17      Min.   :   0.0   no :19  
 Class :character   major: 6      1st Qu.:  48.5   yes: 4  
 Mode  :character                 Median : 250.0           
                                  Mean   : 401.7           
                                  3rd Qu.: 632.5           
                                  Max.   :1980.0           
  overall_yrs      diseasefree_yrs   diseasefree_months diseasefree_years
 Min.   :0.03833   Min.   :0.03833   Min.   : 1.0       Min.   :1.000    
 1st Qu.:2.00685   1st Qu.:0.76112   1st Qu.:10.0       1st Qu.:1.000    
 Median :3.41136   Median :1.49760   Median :18.0       Median :2.000    
 Mean   :3.24614   Mean   :2.01351   Mean   :24.7       Mean   :2.565    
 3rd Qu.:4.29979   3rd Qu.:3.27721   3rd Qu.:40.0       3rd Qu.:4.000    
 Max.   :6.68857   Max.   :6.68857   Max.   :81.0       Max.   :7.000    

Kaplan-Meier

library(broom)
km_f_log<- survfit(Surv(diseasefree,recurrence)~1,data=female_old)
km_f_logit<- survfit(Surv(diseasefree,recurrence)~1,data=female_old, conf.type="logit")
km_f_loglog<- survfit(Surv(diseasefree,recurrence)~1,data=female_old, conf.type="log-log")
km_f_plain<- survfit(Surv(diseasefree,recurrence)~1,data=female_old, conf.type="plain")


t<- function(data){
  data1<- tidy(data)
  return(data1)
}


km_f_loglog<-t(km_f_loglog)
km_f_logit<-t(km_f_logit)
km_f_plain<-t(km_f_plain)


#visulation:
km_f_d_ci<-ggsurv(km_f_log)+
  geom_step(data=km_f_loglog,aes(x=time,y=conf.low),col="blue")+
  geom_step(data=km_f_loglog,aes(x=time,y=conf.high),col="blue")+
  geom_step(data=km_f_logit,aes(x=time,y=conf.low),col="green")+
  geom_step(data=km_f_logit,aes(x=time,y=conf.high),col="green")+
  geom_step(data=km_f_plain,aes(x=time,y=conf.low),col="red")+
  geom_step(data=km_f_plain,aes(x=time,y=conf.high),col="red")+
  labs("Kaplan-Meier")+
  theme_bw()+
  xlim(c(0,2000))

Nelson-Aalen

na_f_log<- survfit(Surv(diseasefree,recurrence)~1,data=female_old,stype = 2)
na_f_logit<- survfit(Surv(diseasefree,recurrence)~1,data=female_old, conf.type="logit",stype = 2)
na_f_loglog<- survfit(Surv(diseasefree,recurrence)~1,data=female_old, conf.type="log-log",stype = 2)
na_f_plain<- survfit(Surv(diseasefree,recurrence)~1,data=female_old, conf.type="plain",stype = 2)



na_f_loglog<-t(na_f_loglog)
na_f_logit<-t(na_f_logit)
na_f_plain<-t(na_f_plain)



#visulation:
na_f_d_ci<-ggsurv(na_f_log)+
  geom_step(data=na_f_loglog,aes(x=time,y=conf.low),col="blue")+
  geom_step(data=na_f_loglog,aes(x=time,y=conf.high),col="blue")+
  geom_step(data=na_f_logit,aes(x=time,y=conf.low),col="green")+
  geom_step(data=na_f_logit,aes(x=time,y=conf.high),col="green")+
  geom_step(data=na_f_plain,aes(x=time,y=conf.low),col="red")+
  geom_step(data=na_f_plain,aes(x=time,y=conf.high),col="red")+
  labs("Nelson-Aalen")+
  theme_bw()+
  xlim(c(0,2000))

Comparing Aged Females using Kaplan-Meier vs Nelson-Aalen:

ci<- km_f_d_ci|na_f_d_ci
ci+ plot_annotation(caption = "CI legend: black: default, green: logit scale, red: plain (Wald),blue: loglog scale")

Testing

#Log-rank test:
survdiff(Surv(overall,death) ~ sex,data = liver_data)

Call:
survdiff(formula = Surv(overall, death) ~ sex, data = liver_data)

             N Observed Expected (O-E)^2/E (O-E)^2/V
sex=female  97       54     50.6     0.222      0.33
sex=male   195      103    106.4     0.106      0.33

 Chisq= 0.3  on 1 degrees of freedom, p= 0.6 

  
#Wilcoxon test:
survdiff(Surv(overall,death) ~ sex,data = liver_data, rho=1)

Call:
survdiff(formula = Surv(overall, death) ~ sex, data = liver_data, 
    rho = 1)

             N Observed Expected (O-E)^2/E (O-E)^2/V
sex=female  97     37.3     36.1    0.0416    0.0825
sex=male   195     71.8     73.0    0.0206    0.0825

 Chisq= 0.1  on 1 degrees of freedom, p= 0.8 

Hazard rate comparing for Non-,Semi- parametric model:

km_hz<- survfit(Surv(diseasefree,recurrence)~1,data=liver_data)
str(km_hz)

List of 16
 $ n        : int 292
 $ time     : num [1:272] 1 5 10 11 12 14 16 21 26 27 ...
 $ n.risk   : num [1:272] 292 291 289 288 287 286 284 283 282 281 ...
 $ n.event  : num [1:272] 1 2 0 0 0 2 0 1 1 1 ...
 $ n.censor : num [1:272] 0 0 1 1 1 0 1 0 0 0 ...
 $ surv     : num [1:272] 0.997 0.99 0.99 0.99 0.99 ...
 $ std.err  : num [1:272] 0.00343 0.00596 0.00596 0.00596 0.00596 ...
 $ cumhaz   : num [1:272] 0.00342 0.0103 0.0103 0.0103 0.0103 ...
 $ std.chaz : num [1:272] 0.00342 0.00595 0.00595 0.00595 0.00595 ...
 $ type     : chr "right"
 $ logse    : logi TRUE
 $ conf.int : num 0.95
 $ conf.type: chr "log"
 $ lower    : num [1:272] 0.99 0.978 0.978 0.978 0.978 ...
 $ upper    : num [1:272] 1 1 1 1 1 ...
 $ call     : language survfit(formula = Surv(diseasefree, recurrence) ~ 1, data = liver_data)
 - attr(*, "class")= chr "survfit"

hz<- with(km_hz,data.frame(
          time=time[n.event!=0],
          hazard=diff(c(0, cumhaz[n.event !=0]))/
          diff(c(0, time[n.event !=0]))
          ))

Non-parametric using Muhaz:

hz_piecewise<- with(liver_data, 
                    muhaz::pehaz(diseasefree,recurrence,max.time = 2500))

max.time= 2500
width= 108.0886
nbins= 24

hz_piecewise_bin_specific<-with(liver_data,
                                muhaz::pehaz(diseasefree,recurrence,max.time = 2500, width=20))

max.time= 2500
width= 20
nbins= 125

hz_piecewise_smooth<-with(liver_data,
                          muhaz::muhaz(diseasefree,recurrence,max.time = 2500))

hz_piecewise<- data.frame(time=hz_piecewise$Cuts[-1],
                          hazard=hz_piecewise$Hazard,
                    CHaz=cumsum(c(0,hz_piecewise$Hazard*hz_piecewise$Width))[-1]
                          )

hz_piecewise_smooth<- data.frame(time=hz_piecewise_smooth$est.grid,
                          hazard=hz_piecewise_smooth$haz.est,
                    CHaz=cumsum(hz_piecewise_smooth$haz.est*diff(c(0,hz_piecewise_smooth$est.grid))))

hz_piecewise_bin_specific<- data.frame(time=hz_piecewise_bin_specific$Cuts[-1],
                          hazard=hz_piecewise_bin_specific$Hazard,
                    CHaz=cumsum(c(0,hz_piecewise_bin_specific$Hazard*hz_piecewise_bin_specific$Width))[-1])

hz$CHaz<-cumsum(c(0,(diff(hz$time)))*hz$hazard)

hz_piecewise<-mutate(hz_piecewise,method=rep("Piecewise"))
hz_piecewise_smooth<-mutate(hz_piecewise_smooth,method=rep("Smooth"))
hz_piecewise_bin_specific<-mutate(hz_piecewise_bin_specific,method=rep("Bin=10"))
hz<-mutate(hz,method=rep("Definition"))

hz_c<-rbind(select(hz,time,hazard,CHaz,method),
            hz_piecewise,
            hz_piecewise_bin_specific,
            hz_piecewise_smooth)

Visualization:

##hazard##
ggplot(hz_c,aes(time,hazard,col=method))+
  geom_step()+
  labs(y=expression(hat(lambda)(t)),x="time")+
  scale_color_viridis_d()+
  theme_bw()+
  theme(legend.position = "top")

##Cumulative hazard##
ggplot(hz_c,aes(time,CHaz,col=method))+
  geom_line()+
  labs(y=expression(hat(Lambda)(t)),x="time")+
  scale_color_viridis_d()+
  theme_bw()+
  theme(legend.position = "top")