dat[dat$follow.up == 0, ] |>
group_by(death) |>
summarise(n = n())# A tibble: 2 × 2
death n
<int> <int>
1 0 21
2 1 4Slides
Warning
There are 25 repondents with follow.up = 0, 21 of which with death = 0 and other 4 with death = 1.
Reasons to omit them, that is, have dat <- dat[dat$follow.up > 0,] at the very beginning:
- If
death = 1, transplant failed. They shouldn’t be included in the research question “survival rate of a successful tranplant”. - If
death = 0, transplant was successful, but they were immediately removed from populatoin at risk, contributiong nothing to the analysis.
1 Title
2 Table of Contents (?)
3 Introduction (M)
4 Method & Result
4.1 Identify Predictors (M)
- Data source
- Study population
[1] 9775- Covariates
- Outcome
[1] "hla.match" "age.donor" "age.rec" "cold.isc" "death" "year"
[7] "sex" "tx.type" "follow.up"4.2 Table (L)
| Characteristic | Overall N = 9,775 |
Cadaveric N = 5,148 |
Living N = 4,627 |
|---|---|---|---|
| Sex, n(%) | |||
| Female | 4,014 (41%) | 2,105 (41%) | 1,909 (41%) |
| Male | 5,761 (59%) | 3,043 (59%) | 2,718 (59%) |
| Recipient age, median (IQR) | 13 (8, 16) | 13 (7, 16) | 14 (9, 16) |
| Unknown | 9 | 7 | 2 |
| Donor age, median (IQR) | 33 (21, 41) | 37 (31, 42) | 22 (15, 37) |
| Unknown | 113 | 112 | 1 |
| HLA matches, n(%) | |||
| 0 | 784 (8.2%) | 125 (2.5%) | 659 (14%) |
| 1 | 1,504 (16%) | 160 (3.2%) | 1,344 (29%) |
| 2 | 1,553 (16%) | 228 (4.6%) | 1,325 (29%) |
| 3 | 3,778 (40%) | 3,039 (61%) | 739 (16%) |
| 4 | 1,278 (13%) | 1,007 (20%) | 271 (5.9%) |
| 5 | 365 (3.8%) | 241 (4.9%) | 124 (2.7%) |
| 6 | 279 (2.9%) | 153 (3.1%) | 126 (2.7%) |
| Unknown | 234 | 195 | 39 |
| Cold ischemic time (hours), median (IQR), | 7 (1, 19) | 1 (0, 1) | 19 (13, 25) |
| Unknown | 2,250 | 1,490 | 760 |
5 Elementary Analysis: Transplant Type (L)
5.1 Transplant Type vs Sex
sex are similar across tx.type
5.2 Transplant Type vs Recipient Age
6 Elementary Analysis: Transplant Type (Cont.)
6.1 Transplant Type vs HLA Match
Living donors have less hla.match then cadaveric donors.
6.2 Transplant Type vs Donor Age
7 Overall Kaplan-Meier Curve
| Overall Kaplan-Meier Survival Summary | ||
|---|---|---|
| Follow-up Time | Number at Risk | Survival Probability |
| 0 | 9775 | 1.000 |
| 2 | 6451 | 0.968 |
| 4 | 4150 | 0.953 |
| 6 | 2398 | 0.937 |
| 8 | 1197 | 0.919 |
| 10 | 492 | 0.901 |
| 12 | 48 | 0.888 |
| Overall Kaplan-Meier Survival Summary | ||
|---|---|---|
| Follow-up Time | Number at Risk | Survival Probability |
| 0 | 9775 | 1.000 |
| 2 | 6451 | 0.968 |
| 4 | 4150 | 0.953 |
| 6 | 2398 | 0.937 |
| 8 | 1197 | 0.919 |
| 10 | 492 | 0.901 |
| 12 | 48 | 0.888 |
8 Cox Model (with intervals?)
9 Cox Model: Transplant Type
Call:
coxph(formula = Surv(follow.up, death) ~ tx.type, data = dat)
n= 9775, number of events= 465
coef exp(coef) se(coef) z Pr(>|z|)
tx.typeLiving 0.64469 1.90539 0.09558 6.745 1.53e-11 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
tx.typeLiving 1.905 0.5248 1.58 2.298
Concordance= 0.586 (se = 0.012 )
Likelihood ratio test= 47.1 on 1 df, p=7e-12
Wald test = 45.5 on 1 df, p=2e-11
Score (logrank) test = 47.09 on 1 df, p=7e-12
Call: survfit(formula = Surv(follow.up, death) ~ tx.type, data = dat)
tx.type=Cadaveric
time n.risk n.event survival std.err lower 95% CI upper 95% CI
0 5148 0 1.000 0.00000 1.000 1.000
4 2297 130 0.967 0.00294 0.961 0.973
8 711 34 0.943 0.00514 0.933 0.953
12 27 13 0.908 0.01135 0.886 0.931
tx.type=Living
time n.risk n.event survival std.err lower 95% CI upper 95% CI
0 4627 4 0.999 0.000432 0.998 1.000
4 1853 225 0.937 0.004212 0.928 0.945
8 486 52 0.891 0.007720 0.876 0.906
12 21 7 0.864 0.013257 0.838 0.89010 Cox Model: Age Recipient (Continuous)
Call:
coxph(formula = Surv(follow.up, death) ~ age.rec, data = dat)
n= 9766, number of events= 464
(9 observations deleted due to missingness)
coef exp(coef) se(coef) z Pr(>|z|)
age.rec -0.042550 0.958342 0.008434 -5.045 4.53e-07 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
age.rec 0.9583 1.043 0.9426 0.9743
Concordance= 0.586 (se = 0.016 )
Likelihood ratio test= 24.81 on 1 df, p=6e-07
Wald test = 25.45 on 1 df, p=5e-07
Score (logrank) test = 25.75 on 1 df, p=4e-07
View life table
Call: survfit(formula = Surv(follow.up, death) ~ age.rec, data = dat)
9 observations deleted due to missingness
age.rec=0
time n.risk n.event survival std.err lower 95% CI upper 95% CI
0 93 1 0.989 0.0107 0.969 1.00
4 34 17 0.784 0.0460 0.699 0.88
8 13 0 0.784 0.0460 0.699 0.88
age.rec=1
time n.risk n.event survival std.err lower 95% CI upper 95% CI
0 394 0 1.000 0.0000 1.000 1.000
4 158 32 0.900 0.0173 0.866 0.934
8 60 3 0.876 0.0217 0.835 0.920
12 1 0 0.876 0.0217 0.835 0.920
age.rec=2
time n.risk n.event survival std.err lower 95% CI upper 95% CI
0 381 0 1.000 0.0000 1.000 1.000
4 161 26 0.917 0.0160 0.886 0.949
8 73 2 0.899 0.0204 0.859 0.939
12 1 2 0.860 0.0336 0.797 0.929
age.rec=3
time n.risk n.event survival std.err lower 95% CI upper 95% CI
0 277 0 1.000 0.0000 1.000 1.000
4 130 15 0.938 0.0157 0.907 0.969
8 44 2 0.915 0.0223 0.872 0.960
age.rec=4
time n.risk n.event survival std.err lower 95% CI upper 95% CI
0 279 0 1.000 0.0000 1.000 1.000
4 129 14 0.942 0.0151 0.913 0.972
8 44 5 0.895 0.0257 0.846 0.947
12 1 0 0.895 0.0257 0.846 0.947
age.rec=5
time n.risk n.event survival std.err lower 95% CI upper 95% CI
0 286 0 1.000 0.0000 1.000 1.000
4 133 10 0.955 0.0141 0.928 0.983
8 49 3 0.920 0.0251 0.872 0.971
12 4 1 0.898 0.0330 0.835 0.965
age.rec=6
time n.risk n.event survival std.err lower 95% CI upper 95% CI
0 302 0 1.000 0.0000 1.000 1.000
4 140 10 0.961 0.0123 0.937 0.985
8 53 0 0.961 0.0123 0.937 0.985
12 2 0 0.961 0.0123 0.937 0.985
age.rec=7
time n.risk n.event survival std.err lower 95% CI upper 95% CI
0 333 0 1.000 0.0000 1.000 1.000
4 162 14 0.948 0.0140 0.921 0.976
8 62 2 0.932 0.0177 0.898 0.967
12 3 2 0.859 0.0546 0.759 0.973
age.rec=8
time n.risk n.event survival std.err lower 95% CI upper 95% CI
0 371 0 1.000 0.0000 1.000 1.000
4 183 12 0.964 0.0102 0.944 0.984
8 72 3 0.935 0.0199 0.897 0.975
12 1 3 0.869 0.0416 0.791 0.954
age.rec=9
time n.risk n.event survival std.err lower 95% CI upper 95% CI
0 388 0 1.000 0.0000 1.000 1.000
4 171 12 0.953 0.0137 0.927 0.980
8 63 7 0.891 0.0268 0.840 0.945
12 5 0 0.891 0.0268 0.840 0.945
age.rec=10
time n.risk n.event survival std.err lower 95% CI upper 95% CI
0 430 1 0.998 0.00232 0.993 1.000
4 203 7 0.976 0.00876 0.959 0.993
8 69 4 0.943 0.01874 0.907 0.981
12 3 2 0.846 0.06716 0.725 0.989
age.rec=11
time n.risk n.event survival std.err lower 95% CI upper 95% CI
0 443 0 1.000 0.0000 1.000 1.000
4 198 8 0.976 0.0087 0.959 0.993
8 63 6 0.921 0.0242 0.875 0.970
12 4 1 0.881 0.0455 0.796 0.975
age.rec=12
time n.risk n.event survival std.err lower 95% CI upper 95% CI
0 541 1 0.998 0.00185 0.995 1.000
4 246 15 0.963 0.00943 0.944 0.981
8 69 5 0.933 0.01684 0.900 0.966
12 2 1 0.904 0.03300 0.841 0.971
age.rec=13
time n.risk n.event survival std.err lower 95% CI upper 95% CI
0 644 0 1.000 0.00000 1.000 1.000
4 282 22 0.954 0.00981 0.935 0.974
8 57 4 0.922 0.02042 0.883 0.963
12 4 2 0.872 0.04013 0.796 0.954
age.rec=14
time n.risk n.event survival std.err lower 95% CI upper 95% CI
0 685 0 1.000 0.00000 1.000 1.000
4 303 15 0.971 0.00753 0.957 0.986
8 59 4 0.945 0.01537 0.916 0.976
12 2 1 0.909 0.03860 0.837 0.988
age.rec=15
time n.risk n.event survival std.err lower 95% CI upper 95% CI
0 874 1 0.999 0.00114 0.997 1.000
4 336 36 0.945 0.00910 0.927 0.963
8 66 6 0.918 0.01432 0.890 0.946
12 3 0 0.918 0.01432 0.890 0.946
age.rec=16
time n.risk n.event survival std.err lower 95% CI upper 95% CI
0 935 0 1.000 0.0000 1.000 1.000
4 375 29 0.956 0.0082 0.941 0.973
8 86 8 0.924 0.0145 0.896 0.953
12 4 0 0.924 0.0145 0.896 0.953
age.rec=17
time n.risk n.event survival std.err lower 95% CI upper 95% CI
0 1073 0 1.000 0.00000 1.000 1.000
4 410 28 0.961 0.00753 0.946 0.976
8 90 16 0.889 0.02020 0.850 0.929
12 1 2 0.854 0.03175 0.794 0.918
age.rec=18
time n.risk n.event survival std.err lower 95% CI upper 95% CI
0 1037 0 1.000 0.00000 1.000 1.000
4 395 32 0.955 0.00802 0.940 0.971
8 104 6 0.931 0.01309 0.906 0.957
12 7 2 0.905 0.02210 0.863 0.94911 Cox Model: Age Recipient (Categorical)
cox.age.rec.groupCall:
coxph(formula = Surv(follow.up, death) ~ age.rec.group, data = dat.age.rec.group)
coef exp(coef) se(coef) z p
age.rec.group2-5 -0.5947 0.5517 0.1767 -3.366 0.000764
age.rec.group6-10 -1.0578 0.3472 0.1776 -5.956 2.59e-09
age.rec.group11-18 -1.0192 0.3609 0.1513 -6.736 1.63e-11
Likelihood ratio test=44.93 on 3 df, p=9.559e-10
n= 9766, number of events= 464
(9 observations deleted due to missingness)eme.age.rec.group <- emmeans(cox.age.rec.group, pairwise ~ age.rec.group,
type = "response")
eme.age.rec.group$emmeans
age.rec.group response SE df asymp.LCL asymp.UCL
0-1 1.000 0.0000 Inf 1.000 1.000
2-5 0.552 0.0975 Inf 0.390 0.780
6-10 0.347 0.0617 Inf 0.245 0.492
11-18 0.361 0.0546 Inf 0.268 0.485
Confidence level used: 0.95
Intervals are back-transformed from the log scale
$contrasts
contrast ratio SE df null z.ratio p.value
(0-1) / (2-5) 1.813 0.320 Inf 1 3.366 0.0043
(0-1) / (6-10) 2.880 0.511 Inf 1 5.956 <.0001
(0-1) / (11-18) 2.771 0.419 Inf 1 6.736 <.0001
(2-5) / (6-10) 1.589 0.251 Inf 1 2.928 0.0179
(2-5) / (11-18) 1.529 0.196 Inf 1 3.314 0.0051
(6-10) / (11-18) 0.962 0.124 Inf 1 -0.298 0.9908
P value adjustment: tukey method for comparing a family of 4 estimates
Tests are performed on the log scale 
12 Identifying Predictors
12.1 Start with nothing
old.vars <- c()
test.new.var(old.vars, "tx.type", 1)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
tx.typeLiving 0.64466 1.90534 0.09558 6.745 1.53e-11
Likelihood ratio test=47.1 on 1 df, p=6.742e-12
n= 9775, number of events= 465 Wald df p
1 45.49594 1 1.529554e-11test.new.var(old.vars, "hla.match", 1)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
hla.match -0.2096 0.8109 0.0351 -5.972 2.34e-09
Likelihood ratio test=35.87 on 1 df, p=2.112e-09
n= 9541, number of events= 451
(234 observations deleted due to missingness) Wald df p
1 35.6703 1 2.336993e-09test.new.var(old.vars, "age.donor", 1)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
age.donor -0.021738 0.978497 0.003467 -6.269 3.62e-10
Likelihood ratio test=38.96 on 1 df, p=4.319e-10
n= 9662, number of events= 464
(113 observations deleted due to missingness) Wald df p
1 39.30567 1 3.62387e-10test.new.var(old.vars, "age.rec", 1)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
age.rec -0.042545 0.958348 0.008434 -5.045 4.54e-07
Likelihood ratio test=24.8 on 1 df, p=6.362e-07
n= 9766, number of events= 464
(9 observations deleted due to missingness) Wald df p
1 25.44856 1 4.543371e-07test.new.var(old.vars, "cold.isc", 1)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
cold.isc 0.019094 1.019278 0.004052 4.712 2.45e-06
Likelihood ratio test=20.88 on 1 df, p=4.888e-06
n= 7525, number of events= 357
(2250 observations deleted due to missingness) Wald df p
1 22.20574 1 2.449402e-06test.new.var(old.vars, "sex", 1)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
sexMale -0.08422 0.91923 0.09374 -0.898 0.369
Likelihood ratio test=0.8 on 1 df, p=0.37
n= 9775, number of events= 465 Wald df p
1 0.8072063 1 0.3689475test.new.var(old.vars, "year", 1)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
year -0.03667 0.96399 0.01550 -2.366 0.018
Likelihood ratio test=5.62 on 1 df, p=0.0178
n= 9775, number of events= 465 Wald df p
1 5.595603 1 0.0180056112.2 tx.type + ?
old.vars <- c("tx.type")
# test.new.var(old.vars, "hla.match", 2:7)
test.new.var(old.vars, "hla.match", 2)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
tx.typeLiving 0.45336 1.57359 0.11146 4.068 4.75e-05
hla.match -0.12459 0.88286 0.03959 -3.147 0.00165
Likelihood ratio test=52.43 on 2 df, p=4.122e-12
n= 9541, number of events= 451
(234 observations deleted due to missingness) Wald df p
1 9.905243 1 0.001648086test.new.var(old.vars, "age.donor", 2)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
tx.typeLiving 0.478191 1.613154 0.106660 4.483 7.35e-06
age.donor -0.013523 0.986568 0.003745 -3.611 0.000305
Likelihood ratio test=59.05 on 2 df, p=1.506e-13
n= 9662, number of events= 464
(113 observations deleted due to missingness) Wald df p
1 13.0412 1 0.0003047131test.new.var(old.vars, "age.rec", 2)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
tx.typeLiving 0.693775 2.001255 0.096157 7.215 5.39e-13
age.rec -0.049081 0.952104 0.008536 -5.750 8.92e-09
Likelihood ratio test=78.72 on 2 df, p=< 2.2e-16
n= 9766, number of events= 464
(9 observations deleted due to missingness) Wald df p
1 33.06396 1 8.917646e-09test.new.var(old.vars, "cold.isc", 2)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
tx.typeLiving 0.477550 1.612119 0.171044 2.792 0.00524
cold.isc 0.004924 1.004937 0.006728 0.732 0.46420
Likelihood ratio test=28.72 on 2 df, p=5.811e-07
n= 7525, number of events= 357
(2250 observations deleted due to missingness) Wald df p
1 0.5357515 1 0.4641988test.new.var(old.vars, "sex", 2)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
tx.typeLiving 0.64499 1.90597 0.09558 6.748 1.5e-11
sexMale -0.08657 0.91707 0.09374 -0.924 0.356
Likelihood ratio test=47.95 on 2 df, p=3.87e-11
n= 9775, number of events= 465 Wald df p
1 0.8528929 1 0.3557352test.new.var(old.vars, "year", 2)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
tx.typeLiving 0.63806 1.89281 0.09563 6.672 2.52e-11
year -0.03310 0.96744 0.01548 -2.138 0.0325
Likelihood ratio test=51.69 on 2 df, p=5.967e-12
n= 9775, number of events= 465 Wald df p
1 4.573044 1 0.0324788age.rec has the smallest \(p\)-value
12.3 tx.type + age.rec + ?
old.vars <- c("tx.type", "age.rec")
test.new.var(old.vars, "hla.match", 3)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
tx.typeLiving 0.512664 1.669733 0.112875 4.542 5.58e-06
age.rec -0.048268 0.952878 0.008688 -5.556 2.77e-08
hla.match -0.113954 0.892299 0.039929 -2.854 0.00432
Likelihood ratio test=81.92 on 3 df, p=< 2.2e-16
n= 9535, number of events= 450
(240 observations deleted due to missingness) Wald df p
1 8.144605 1 0.004318947test.new.var(old.vars, "age.donor", 3)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
tx.typeLiving 0.571325 1.770611 0.107377 5.321 1.03e-07
age.rec -0.044085 0.956872 0.008718 -5.057 4.26e-07
age.donor -0.009688 0.990359 0.003797 -2.551 0.0107
Likelihood ratio test=83.3 on 3 df, p=< 2.2e-16
n= 9655, number of events= 463
(120 observations deleted due to missingness) Wald df p
1 6.508773 1 0.01073435test.new.var(old.vars, "cold.isc", 3)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
tx.typeLiving 0.489852 1.632075 0.170731 2.869 0.00412
age.rec -0.025545 0.974779 0.010019 -2.550 0.01078
cold.isc 0.005637 1.005653 0.006709 0.840 0.40085
Likelihood ratio test=35.04 on 3 df, p=1.194e-07
n= 7519, number of events= 357
(2256 observations deleted due to missingness) Wald df p
1 0.705761 1 0.4008547test.new.var(old.vars, "sex", 3)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
tx.typeLiving 0.695122 2.003953 0.096164 7.229 4.88e-13
age.rec -0.050004 0.951226 0.008569 -5.835 5.37e-09
sexMale -0.130416 0.877730 0.094113 -1.386 0.166
Likelihood ratio test=80.63 on 3 df, p=< 2.2e-16
n= 9766, number of events= 464
(9 observations deleted due to missingness) Wald df p
1 1.920263 1 0.1658277test.new.var(old.vars, "year", 3)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
tx.typeLiving 0.687661 1.989059 0.096215 7.147 8.86e-13
age.rec -0.049029 0.952153 0.008537 -5.743 9.30e-09
year -0.033478 0.967076 0.015475 -2.163 0.0305
Likelihood ratio test=83.42 on 3 df, p=< 2.2e-16
n= 9766, number of events= 464
(9 observations deleted due to missingness) Wald df p
1 4.680122 1 0.03051359hla.match has the smallest \(p\)-value
12.4 tx.type + age.rec + hla.match + ?
old.vars <- c("tx.type", "age.rec", "hla.match")
test.new.var(old.vars, "age.donor", 4)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
tx.typeLiving 0.408204 1.504113 0.121727 3.353 0.000798
age.rec -0.043716 0.957225 0.008875 -4.926 8.39e-07
hla.match -0.107306 0.898251 0.040127 -2.674 0.007491
age.donor -0.008810 0.991229 0.003859 -2.283 0.022443
Likelihood ratio test=85.46 on 4 df, p=< 2.2e-16
n= 9433, number of events= 449
(342 observations deleted due to missingness) Wald df p
1 5.211086 1 0.02244331test.new.var(old.vars, "cold.isc", 4)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
tx.typeLiving 0.349594 1.418492 0.182948 1.911 0.0560
age.rec -0.023299 0.976970 0.010119 -2.302 0.0213
hla.match -0.071073 0.931394 0.044036 -1.614 0.1065
cold.isc 0.006428 1.006449 0.006717 0.957 0.3386
Likelihood ratio test=33.68 on 4 df, p=8.682e-07
n= 7381, number of events= 352
(2394 observations deleted due to missingness) Wald df p
1 0.9158816 1 0.3385587test.new.var(old.vars, "sex", 4)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
tx.typeLiving 0.515100 1.673806 0.112902 4.562 5.06e-06
age.rec -0.048944 0.952234 0.008721 -5.612 2.00e-08
hla.match -0.112940 0.893204 0.039936 -2.828 0.00468
sexMale -0.094882 0.909480 0.095783 -0.991 0.32188
Likelihood ratio test=82.9 on 4 df, p=< 2.2e-16
n= 9535, number of events= 450
(240 observations deleted due to missingness) Wald df p
1 0.9812753 1 0.3218842test.new.var(old.vars, "year", 4)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
tx.typeLiving 0.494911 1.640353 0.113403 4.364 1.28e-05
age.rec -0.048129 0.953011 0.008689 -5.539 3.04e-08
hla.match -0.122559 0.884654 0.040341 -3.038 0.00238
year -0.035190 0.965422 0.015831 -2.223 0.02622
Likelihood ratio test=86.88 on 4 df, p=< 2.2e-16
n= 9535, number of events= 450
(240 observations deleted due to missingness) Wald df p
1 4.941346 1 0.02622161age.donor has the smallest \(p\)-value
12.5 tx.type + age.rec + hla.match + age.donor + ?
old.vars <- c("tx.type", "age.rec", "hla.match", "age.donor")
test.new.var(old.vars, "cold.isc", 5)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
tx.typeLiving 0.355582 1.427011 0.188970 1.882 0.0599
age.rec -0.021491 0.978738 0.010298 -2.087 0.0369
hla.match -0.067709 0.934532 0.044223 -1.531 0.1257
age.donor -0.003508 0.996499 0.004230 -0.829 0.4069
cold.isc 0.004519 1.004530 0.006774 0.667 0.5047
Likelihood ratio test=34.04 on 5 df, p=2.333e-06
n= 7347, number of events= 351
(2428 observations deleted due to missingness) Wald df p
1 0.445129 1 0.5046572test.new.var(old.vars, "sex", 5)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
tx.typeLiving 0.411943 1.509748 0.121783 3.383 0.000718
age.rec -0.044462 0.956512 0.008914 -4.988 6.1e-07
hla.match -0.106379 0.899084 0.040136 -2.650 0.008038
age.donor -0.008675 0.991362 0.003862 -2.246 0.024677
sexMale -0.092710 0.911458 0.095941 -0.966 0.333885
Likelihood ratio test=86.39 on 5 df, p=< 2.2e-16
n= 9433, number of events= 449
(342 observations deleted due to missingness) Wald df p
1 0.9337712 1 0.3338849test.new.var(old.vars, "year", 5)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
tx.typeLiving 0.397903 1.488700 0.121903 3.264 0.00110
age.rec -0.043831 0.957116 0.008875 -4.939 7.87e-07
hla.match -0.115648 0.890789 0.040537 -2.853 0.00433
age.donor -0.008325 0.991709 0.003854 -2.160 0.03078
year -0.033109 0.967433 0.015844 -2.090 0.03664
Likelihood ratio test=89.84 on 5 df, p=< 2.2e-16
n= 9433, number of events= 449
(342 observations deleted due to missingness) Wald df p
1 4.367128 1 0.0366387year has the smallest \(p\)-value
12.6 tx.type + age.rec + hla.match + age.donor + year
old.vars <- c("tx.type", "age.rec", "hla.match", "age.donor", "year")
test.new.var(old.vars, "cold.isc", 6)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
tx.typeLiving 0.381855 1.464999 0.189901 2.011 0.0443
age.rec -0.021369 0.978857 0.010295 -2.076 0.0379
hla.match -0.073960 0.928708 0.044568 -1.660 0.0970
age.donor -0.002981 0.997024 0.004226 -0.705 0.4806
year -0.037375 0.963315 0.019227 -1.944 0.0519
cold.isc 0.002743 1.002747 0.006825 0.402 0.6878
Likelihood ratio test=37.86 on 6 df, p=1.195e-06
n= 7347, number of events= 351
(2428 observations deleted due to missingness) Wald df p
1 0.1615171 1 0.6877636test.new.var(old.vars, "sex", 6)Call:
coxph(formula = new.surv, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
tx.typeLiving 0.401829 1.494556 0.121963 3.295 0.000985
age.rec -0.044602 0.956378 0.008916 -5.003 5.66e-07
hla.match -0.114750 0.891589 0.040546 -2.830 0.004653
age.donor -0.008185 0.991848 0.003857 -2.122 0.033833
year -0.033222 0.967324 0.015844 -2.097 0.036011
sexMale -0.094191 0.910109 0.095956 -0.982 0.326293
Likelihood ratio test=90.8 on 6 df, p=< 2.2e-16
n= 9433, number of events= 449
(342 observations deleted due to missingness) Wald df p
1 0.9635531 1 0.3262933Neither has \(p\)-value less than 0.05. Stop.
13 Full Model
cox.fullCall:
coxph(formula = surv.full, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
tx.typeLiving 0.0734859 1.0762534 0.0253724 2.896 0.00378
age.rec 0.0154431 1.0155630 0.0020497 7.534 4.9e-14
hla.match -0.0210826 0.9791381 0.0086077 -2.449 0.01431
age.donor -0.0009874 0.9990131 0.0008841 -1.117 0.26405
year 0.4429003 1.5572171 0.0057586 76.910 < 2e-16
Likelihood ratio test=8440 on 5 df, p=< 2.2e-16
n= 9433, number of events= 9433
(342 observations deleted due to missingness)
13.1 What happened to year?
Probabily want to omit it
13.2 Without year
cox.full.1Call:
coxph(formula = surv.full.1, data = dat, method = "breslow")
coef exp(coef) se(coef) z p
tx.typeLiving 0.0037484 1.0037554 0.0254736 0.147 0.8830
age.rec 0.0139177 1.0140150 0.0020498 6.790 1.12e-11
hla.match -0.0854685 0.9180821 0.0089546 -9.545 < 2e-16
age.donor 0.0023071 1.0023098 0.0008657 2.665 0.0077
Likelihood ratio test=172.9 on 4 df, p=< 2.2e-16
n= 9433, number of events= 9433
(342 observations deleted due to missingness)
TODO: write a more generic function (UPDATE: FAILED)
14 Old
14.1 Death & Censored
dat |>
group_by(death) |>
summarise(
n = n(),
percentage = n() / nrow(dat) * 100
)# A tibble: 2 × 3
death n percentage
<int> <int> <dbl>
1 0 9310 95.2
2 1 465 4.76Most of the patients (95.24%) were censored.
hla.match age.donor age.rec cold.isc death year sex tx.type follow.up
1 2 25 15 6 0 2002 Female Living 0
2 1 42 10 11 1 1999 Female Living 0
3 1 9 14 8 0 2002 Male Living 0
4 0 35 12 26 0 2002 Male Living 0
5 2 17 12 28 1 1997 Male Living 0
6 3 44 13 1 0 2002 Male Cadaveric 0
7 1 34 12 1 0 2002 Male Cadaveric 0
8 4 38 11 2 0 2002 Female Cadaveric 0
9 0 23 10 6 0 2002 Female Living 0
10 3 26 1 1 0 2002 Male Cadaveric 0
11 NA 42 18 NA 0 2002 Male Cadaveric 0
12 4 39 9 NA 0 2002 Female Cadaveric 0
13 3 34 13 NA 0 2002 Male Cadaveric 0
14 3 NA 14 2 0 2002 Female Cadaveric 0
15 3 40 17 1 0 2002 Female Cadaveric 0
16 3 22 14 2 0 2002 Female Cadaveric 0
17 0 17 17 12 0 2002 Male Living 0
18 3 NA 7 NA 0 2002 Male Cadaveric 0
19 0 4 0 NA 1 2001 Female Living 0
20 4 NA NA 1 0 2002 Female Cadaveric 0
21 3 42 18 1 0 2002 Male Cadaveric 0
22 1 19 15 21 1 1994 Male Living 0
23 3 44 10 NA 0 2002 Male Cadaveric 0
24 0 13 14 NA 0 2001 Male Living 0
25 1 NA 6 1 0 2002 Male Cadaveric 0 hla.match age.donor age.rec cold.isc death year sex tx.type follow.up
1 2 25 15 6 0 2002 Female Living 0
2 1 42 10 11 1 1999 Female Living 0
3 1 9 14 8 0 2002 Male Living 0
4 0 35 12 26 0 2002 Male Living 0
5 2 17 12 28 1 1997 Male Living 0
6 3 44 13 1 0 2002 Male Cadaveric 014.2 Cox Model - Age as a Continuous Variable
cox.age.rec <- coxph(Surv(follow.up, death) ~ age.rec, data = dat)
cox.age.rec |> summary()Call:
coxph(formula = Surv(follow.up, death) ~ age.rec, data = dat)
n= 9766, number of events= 464
(9 observations deleted due to missingness)
coef exp(coef) se(coef) z Pr(>|z|)
age.rec -0.042550 0.958342 0.008434 -5.045 4.53e-07 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
age.rec 0.9583 1.043 0.9426 0.9743
Concordance= 0.586 (se = 0.016 )
Likelihood ratio test= 24.81 on 1 df, p=6e-07
Wald test = 25.45 on 1 df, p=5e-07
Score (logrank) test = 25.75 on 1 df, p=4e-0714.3 Cox Model - Age as a Categorical Variable
chisq df p
tx.type 2.46 1 0.11654
age.rec 36.09 1 1.9e-09
hla.match 8.63 1 0.00331
age.donor 10.88 1 0.00097
year 3.96 1 0.04666
GLOBAL 52.00 5 5.4e-10
14.4 Table 1 (L)
14.4.1 HLA match hla.match
0 1 2 3 4 5 6 <NA>
784 1504 1553 3778 1278 365 279 234 
Warning
TODO: change to percentage within each tx.type
Note
Living donors have less hla.match then cadaveric donors
14.4.2 Donor age age.donor
mean(dat$age.donor, na.rm = TRUE)[1] 31.29952median(dat$age.donor, na.rm = TRUE)[1] 33
Note
There is age difference between different tx.type
14.4.3 Recipient Age age.rec
mean(dat$age.rec, na.rm = TRUE)[1] 11.64653median(dat$age.rec, na.rm = TRUE)[1] 13
14.4.4 Cold Ischemia Time cold.isc
mean(dat$cold.isc, na.rm = TRUE)[1] 10.85967median(dat$cold.isc, na.rm = TRUE)[1] 7
14.4.5 Transplant Type tx.type
Cadaveric Living <NA>
5148 4627 0 dat$tx.type |> table()
Cadaveric Living
5148 4627 
14.4.6 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 
14.5 Kaplan-Meier
14.5.1 Overall (L)
Call: survfit(formula = Surv(follow.up, death) ~ 1, data = dat)
time n.risk n.event survival std.err lower 95% CI upper 95% CI
4 4150 359 0.953 0.00252 0.948 0.958
8 1197 86 0.919 0.00449 0.910 0.928
12 48 20 0.888 0.00867 0.871 0.905
14.5.2 tx.type (M)
14.6 Cox Model
14.6.1 death Distribution (?)
14.6.2 Overall (L)
Note
skip overall, jump directly to tx.type
# 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 0.0075214.6.3 tx.type (M)
14.6.4 age (continuous) (L)
14.6.5 age (categorical) (M)
14.6.6 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.match -0.12460 0.88285 0.03959 -3.148 0.00165 **
tx.typeLiving 0.45336 1.57359 0.11146 4.068 4.75e-05 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
hla.match 0.8828 1.1327 0.8169 0.9541
tx.typeLiving 1.5736 0.6355 1.2648 1.9578
Concordance= 0.611 (se = 0.014 )
Likelihood ratio test= 52.43 on 2 df, p=4e-12
Wald test = 51.54 on 2 df, p=6e-12
Score (logrank) test = 53.06 on 2 df, p=3e-12Call:
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.match -0.08577 0.91780 0.07828 -1.096 0.2732
tx.typeLiving 0.59989 1.82192 0.27919 2.149 0.0317 *
hla.match:tx.typeLiving -0.05232 0.94902 0.09086 -0.576 0.5647
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
hla.match 0.9178 1.0896 0.7873 1.070
tx.typeLiving 1.8219 0.5489 1.0541 3.149
hla.match:tx.typeLiving 0.9490 1.0537 0.7942 1.134
Concordance= 0.611 (se = 0.014 )
Likelihood ratio test= 52.77 on 3 df, p=2e-11
Wald test = 52.64 on 3 df, p=2e-11
Score (logrank) test = 54.76 on 3 df, p=8e-12Analysis 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 -3850.6
2 -3850.5 0.3327 1 0.564114.6.7 Discussion: include year or not?
14.7 Assumption Testing
## 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?
14.8 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.