Slides

1 Introduction: research question

1.1 Survival after transplantation (M)

1.2 Identify Predictors (M)

1.3 Why Survival Analysis (L)

Motivation: study the distribution of time to event \(T\).

Example: time of death after kidney transplant.

ID	Transplant	Death
1	2000	2005
2	2000	2009
3	2001	2005
4	2003	2004
5	2004	2016

We then calculate time to death for each respondents.

With this, all we need is to fit a linear model t ~ X or log(t) ~ X

ID	Transplant	Death	Time to death
1	2000	2005	5
2	2000	2009	9
3	2001	2005	4
4	2003	2004	1
5	2004	2016	12

What if we do not know when exactly some respondents die?

Scenario 1: the study ends at the year 2008?

ID	Transplant	Death	Time to death
1	2000	2005	5
2	2000	> 2008	> 8
3	2001	2005	4
4	2003	2004	1
5	2004	> 2008	> 4

Right Censoring: only observe the event (death) if it occurs before a certain time (2008).

Scenario 2: respondent 3 move away; loss follow up

ID	Transplant	Death	Time to death
1	2000	2005	5
2	2000	> 2008	> 8
3	2001	> 2005	> 4
4	2003	2004	1
5	2004	> 2008	> 4

How many patients are right censored?

1.4 Challenges in Survival Analysis (L)

Right Censoring: only observe the event if it occurs before a certain time.
Left Censoring: event has occurred prior to the start of a research
- Follow up every 3 years.
- Event has occurred -> event happened sometime before follow up.
Left Truncation: delayed entry; respondents are included only if they survived long enough.
- Start follow up with patients 100 days after the transplant
- Patients dies within 100 day wouldn’t be included in the dataset
Right Truncation: respondents are included only if they have already experienced the event.
- Retrospective analysis from deceased patients between 1990 and 2000.
- Patients not dead before 2000 are not included in the dataset.

2 Method & Result

2.1 Explain Dataset (M)

2.2 Table 1 (L)

2.2.1 HLA match `hla.match`


   0    1    2    3    4    5    6 <NA> 
 784 1504 1553 3778 1278  365  279  234

2.2.2 Donor age `age.donor`

mean(dat$age.donor, na.rm = TRUE)

[1] 31.29952

median(dat$age.donor, na.rm = TRUE)

[1] 33

Warning: Removed 113 rows containing non-finite outside the scale range
(`stat_count()`).

Warning: Removed 113 rows containing non-finite outside the scale range
(`stat_count()`).

Warning: Removed 113 rows containing non-finite outside the scale range
(`stat_density()`).

2.2.3 Recipient Age `age.rec`

mean(dat$age.rec, na.rm = TRUE)

[1] 11.64653

median(dat$age.rec, na.rm = TRUE)

[1] 13

Warning: Removed 9 rows containing non-finite outside the scale range
(`stat_count()`).

Warning: Removed 113 rows containing non-finite outside the scale range
(`stat_boxplot()`).

Warning: Removed 9 rows containing non-finite outside the scale range
(`stat_count()`).

Warning: Removed 9 rows containing non-finite outside the scale range
(`stat_density()`).

Warning: Removed 2250 rows containing non-finite outside the scale range
(`stat_boxplot()`).

2.2.4 Cold Ischemia Time `cold.isc`

mean(dat$cold.isc, na.rm = TRUE)

[1] 10.85967

median(dat$cold.isc, na.rm = TRUE)

[1] 7

Warning: Removed 2250 rows containing non-finite outside the scale range
(`stat_density()`).

Warning: Removed 2250 rows containing non-finite outside the scale range
(`stat_density()`).

2.2.5 Transplant Type `tx.type`

dat$tx.type |> table()


Cadaveric    Living 
     5148      4627

2.2.6 Year `year`


1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 
 728  734  666  736  787  813  789  745  740  842  706  834  655

Warning: Removed 9 rows containing non-finite outside the scale range
(`stat_boxplot()`).

Warning: Removed 113 rows containing non-finite outside the scale range
(`stat_boxplot()`).

2.3 Overall Kaplan-Meier

2.4 Hazard Functions

2.4.1 `death` Distribution (?)

2.4.2 Overall (?)

# A tibble: 7 × 5
  fu.interval n.censored n.event n.at.risk hazard.rate
        <dbl>      <int>   <int>     <dbl>       <dbl>
1       0.333        830     152      9775     0.0155 
2       0.667        491      37      8793     0.00421
3       1            613      27      8265     0.00327
4       2           1164      62      7625     0.00813
5       3           1143      41      6399     0.00641
6       4           1053      40      5215     0.00767
7       5            919      31      4122     0.00752

2.4.3 `tx.type` (M)

2.5 Cox Model

2.5.1 `age` (continuous) (L)

2.5.2 `age` (categorical) (M)

2.5.3 Full model (L)

Call:
coxph(formula = Surv(follow.up, death) ~ hla.match + tx.type, 
    data = dat)

  n= 9541, number of events= 451 
   (234 observations deleted due to missingness)

                  coef exp(coef) se(coef)      z Pr(>|z|)   
hla.match1    -0.08845   0.91535  0.17308 -0.511  0.60933   
hla.match2    -0.36340   0.69531  0.17919 -2.028  0.04256 * 
hla.match3    -0.50810   0.60164  0.18298 -2.777  0.00549 **
hla.match4    -0.65410   0.51991  0.22221 -2.944  0.00324 **
hla.match5    -0.39445   0.67405  0.29410 -1.341  0.17985   
hla.match6    -0.51491   0.59755  0.31489 -1.635  0.10201   
tx.typeLiving  0.39049   1.47771  0.12688  3.078  0.00209 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

              exp(coef) exp(-coef) lower .95 upper .95
hla.match1       0.9153     1.0925    0.6520    1.2850
hla.match2       0.6953     1.4382    0.4894    0.9879
hla.match3       0.6016     1.6621    0.4203    0.8612
hla.match4       0.5199     1.9234    0.3363    0.8037
hla.match5       0.6740     1.4836    0.3787    1.1996
hla.match6       0.5976     1.6735    0.3224    1.1077
tx.typeLiving    1.4777     0.6767    1.1524    1.8949

Concordance= 0.613  (se = 0.014 )
Likelihood ratio test= 56.79  on 7 df,   p=7e-10
Wald test            = 57.08  on 7 df,   p=6e-10
Score (logrank) test = 59.51  on 7 df,   p=2e-10

Call:
coxph(formula = Surv(follow.up, death) ~ hla.match * tx.type, 
    data = dat)

  n= 9541, number of events= 451 
   (234 observations deleted due to missingness)

                            coef exp(coef) se(coef)      z Pr(>|z|)  
hla.match1                0.8094    2.2467   0.8165  0.991   0.3215  
hla.match2                1.1371    3.1176   0.7530  1.510   0.1311  
hla.match3                0.4217    1.5245   0.7140  0.591   0.5548  
hla.match4                0.1094    1.1156   0.7341  0.149   0.8816  
hla.match5                0.4754    1.6087   0.7820  0.608   0.5432  
hla.match6                0.5836    1.7925   0.8021  0.728   0.4669  
tx.typeLiving             1.3801    3.9752   0.7218  1.912   0.0559 .
hla.match1:tx.typeLiving -0.9614    0.3823   0.8355 -1.151   0.2498  
hla.match2:tx.typeLiving -1.6578    0.1905   0.7762 -2.136   0.0327 *
hla.match3:tx.typeLiving -1.0096    0.3644   0.7461 -1.353   0.1760  
hla.match4:tx.typeLiving -0.5175    0.5960   0.7863 -0.658   0.5105  
hla.match5:tx.typeLiving -0.8601    0.4231   0.8805 -0.977   0.3286  
hla.match6:tx.typeLiving -1.3342    0.2634   0.9113 -1.464   0.1432  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

                         exp(coef) exp(-coef) lower .95 upper .95
hla.match1                  2.2467     0.4451   0.45341   11.1322
hla.match2                  3.1176     0.3208   0.71257   13.6400
hla.match3                  1.5245     0.6559   0.37617    6.1787
hla.match4                  1.1156     0.8964   0.26460    4.7032
hla.match5                  1.6087     0.6216   0.34737    7.4498
hla.match6                  1.7925     0.5579   0.37212    8.6342
tx.typeLiving               3.9752     0.2516   0.96600   16.3580
hla.match1:tx.typeLiving    0.3823     2.6155   0.07435    1.9662
hla.match2:tx.typeLiving    0.1905     5.2480   0.04162    0.8723
hla.match3:tx.typeLiving    0.3644     2.7444   0.08443    1.5726
hla.match4:tx.typeLiving    0.5960     1.6778   0.12763    2.7836
hla.match5:tx.typeLiving    0.4231     2.3635   0.07533    2.3763
hla.match6:tx.typeLiving    0.2634     3.7971   0.04414    1.5712

Concordance= 0.615  (se = 0.014 )
Likelihood ratio test= 66.78  on 13 df,   p=3e-09
Wald test            = 65.9  on 13 df,   p=5e-09
Score (logrank) test = 69.92  on 13 df,   p=8e-10

Analysis of Deviance Table
 Cox model: response is  Surv(follow.up, death)
 Model 1: ~ hla.match + tx.type
 Model 2: ~ hla.match * tx.type
   loglik  Chisq Df Pr(>|Chi|)
1 -3848.5                     
2 -3843.5 9.9891  6     0.1251

Slides

1 Introduction: research question

1.1 Survival after transplantation (M)

1.2 Identify Predictors (M)

1.3 Why Survival Analysis (L)

1.4 Challenges in Survival Analysis (L)

2 Method & Result

2.1 Explain Dataset (M)

2.2 Table 1 (L)

2.2.1 HLA match `hla.match`

2.2.2 Donor age `age.donor`

2.2.3 Recipient Age `age.rec`

2.2.4 Cold Ischemia Time `cold.isc`

2.2.5 Transplant Type `tx.type`

2.2.6 Year `year`

2.3 Overall Kaplan-Meier

2.4 Hazard Functions

2.4.1 `death` Distribution (?)

2.4.2 Overall (?)

2.4.3 `tx.type` (M)

2.5 Cox Model

2.5.1 `age` (continuous) (L)

2.5.2 `age` (categorical) (M)

2.5.3 Full model (L)

2.5.4 Discussion: include `year` or not?

2.6 Assumption Testing

3 Conclusion (M + L)

1 Introduction: research question

1.1 Survival after transplantation (M)

1.2 Identify Predictors (M)

1.3 Why Survival Analysis (L)

1.4 Challenges in Survival Analysis (L)

2 Method & Result

2.1 Explain Dataset (M)

2.2 Table 1 (L)

2.2.1 HLA match hla.match

2.2.2 Donor age age.donor

2.2.3 Recipient Age age.rec

2.2.4 Cold Ischemia Time cold.isc

2.2.5 Transplant Type tx.type

2.2.6 Year year

2.3 Overall Kaplan-Meier

2.4 Hazard Functions

2.4.1 death Distribution (?)

2.4.2 Overall (?)

2.4.3 tx.type (M)

2.5 Cox Model

2.5.1 age (continuous) (L)

2.5.2 age (categorical) (M)

2.5.3 Full model (L)

2.5.4 Discussion: include year or not?

2.6 Assumption Testing

3 Conclusion (M + L)

2.2.1 HLA match `hla.match`

2.2.2 Donor age `age.donor`

2.2.3 Recipient Age `age.rec`

2.2.4 Cold Ischemia Time `cold.isc`

2.2.5 Transplant Type `tx.type`

2.2.6 Year `year`

2.4.1 `death` Distribution (?)

2.4.3 `tx.type` (M)

2.5.1 `age` (continuous) (L)

2.5.2 `age` (categorical) (M)

2.5.4 Discussion: include `year` or not?