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R Under development (unstable) (2024-04-17 r86441) -- "Unsuffered Consequences"
Copyright (C) 2024 The R Foundation for Statistical Computing
Platform: aarch64-unknown-linux-gnu
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> options(na.action=na.exclude) # preserve missings
> options(contrasts=c('contr.treatment', 'contr.poly'),
+ show.signif.stars=FALSE) #ensure constrast type
> library(survival)
>
> #
> # Fit the models found in Wei et. al.
> #
> wfit <- coxph(Surv(stop, event) ~ (rx + size + number)* strata(enum),
+ cluster=id, bladder, ties='breslow')
> wfit
Call:
coxph(formula = Surv(stop, event) ~ (rx + size + number) * strata(enum),
data = bladder, ties = "breslow", cluster = id)
coef exp(coef) se(coef) robust se z p
rx -0.51762 0.59594 0.31576 0.30750 -1.683 0.09231
size 0.06789 1.07025 0.10125 0.08529 0.796 0.42602
number 0.23599 1.26617 0.07608 0.07208 3.274 0.00106
rx:strata(enum)enum=2 -0.10182 0.90319 0.50427 0.32654 -0.312 0.75518
rx:strata(enum)enum=3 -0.18226 0.83339 0.55790 0.39162 -0.465 0.64165
rx:strata(enum)enum=4 -0.13317 0.87531 0.65813 0.49680 -0.268 0.78865
size:strata(enum)enum=2 -0.14401 0.86588 0.16800 0.11190 -1.287 0.19812
size:strata(enum)enum=3 -0.27920 0.75639 0.20862 0.15115 -1.847 0.06472
size:strata(enum)enum=4 -0.27106 0.76257 0.25146 0.18563 -1.460 0.14422
number:strata(enum)enum=2 -0.09843 0.90625 0.11931 0.11439 -0.861 0.38951
number:strata(enum)enum=3 -0.06616 0.93598 0.12983 0.11672 -0.567 0.57083
number:strata(enum)enum=4 0.09280 1.09724 0.14657 0.11754 0.790 0.42982
Likelihood ratio test=29.39 on 12 df, p=0.003443
n= 340, number of events= 112
>
> # Check the rx coefs versus Wei, et al, JASA 1989
> rx <- c(1,4,5,6) # the treatment coefs above
> cmat <- diag(4); cmat[1,] <- 1; #contrast matrix
> wfit$coefficients[rx] %*% cmat # the coefs in their paper (table 5)
[,1] [,2] [,3] [,4]
[1,] -0.5176209 -0.6194404 -0.6998771 -0.6507935
> t(cmat) %*% wfit$var[rx,rx] %*% cmat # var matrix (eqn 3.2)
[,1] [,2] [,3] [,4]
[1,] 0.09455501 0.06017669 0.05677331 0.0437777
[2,] 0.06017669 0.13242834 0.13011557 0.1160420
[3,] 0.05677331 0.13011557 0.17235879 0.1590865
[4,] 0.04377770 0.11604200 0.15908650 0.2398112
>
> # Anderson-Gill fit
> fita <- coxph(Surv(start, stop, event) ~ rx + size + number, cluster=id,
+ bladder2, ties='breslow')
> summary(fita)
Call:
coxph(formula = Surv(start, stop, event) ~ rx + size + number,
data = bladder2, ties = "breslow", cluster = id)
n= 178, number of events= 112
coef exp(coef) se(coef) robust se z Pr(>|z|)
rx -0.45979 0.63142 0.19996 0.25801 -1.782 0.07474
size -0.04256 0.95833 0.06903 0.07555 -0.563 0.57317
number 0.17164 1.18726 0.04733 0.06131 2.799 0.00512
exp(coef) exp(-coef) lower .95 upper .95
rx 0.6314 1.5837 0.3808 1.047
size 0.9583 1.0435 0.8264 1.111
number 1.1873 0.8423 1.0528 1.339
Concordance= 0.634 (se = 0.032 )
Likelihood ratio test= 16.77 on 3 df, p=8e-04
Wald test = 11.76 on 3 df, p=0.008
Score (logrank) test = 18.57 on 3 df, p=3e-04, Robust = 11.44 p=0.01
(Note: the likelihood ratio and score tests assume independence of
observations within a cluster, the Wald and robust score tests do not).
>
> # Prentice fits. Their model 1 a and b are the same
> fit1p <- coxph(Surv(stop, event) ~ rx + size + number, bladder2,
+ subset=(enum==1), ties='breslow')
> fit2pa <- coxph(Surv(stop, event) ~ rx + size + number, bladder2,
+ subset=(enum==2), ties='breslow')
> fit2pb <- coxph(Surv(stop-start, event) ~ rx + size + number, bladder2,
+ subset=(enum==2), ties='breslow')
> fit3pa <- coxph(Surv(stop, event) ~ rx + size + number, bladder2,
+ subset=(enum==3), ties='breslow')
> #and etc.
> fit1p
Call:
coxph(formula = Surv(stop, event) ~ rx + size + number, data = bladder2,
subset = (enum == 1), ties = "breslow")
coef exp(coef) se(coef) z p
rx -0.51762 0.59594 0.31576 -1.639 0.10115
size 0.06789 1.07025 0.10125 0.671 0.50253
number 0.23599 1.26617 0.07608 3.102 0.00192
Likelihood ratio test=9.66 on 3 df, p=0.02164
n= 85, number of events= 47
> fit2pa
Call:
coxph(formula = Surv(stop, event) ~ rx + size + number, data = bladder2,
subset = (enum == 2), ties = "breslow")
coef exp(coef) se(coef) z p
rx -0.424214 0.654284 0.402200 -1.055 0.292
size -0.125033 0.882467 0.117088 -1.068 0.286
number 0.001987 1.001989 0.093760 0.021 0.983
Likelihood ratio test=2.02 on 3 df, p=0.5688
n= 46, number of events= 29
> fit2pb
Call:
coxph(formula = Surv(stop - start, event) ~ rx + size + number,
data = bladder2, subset = (enum == 2), ties = "breslow")
coef exp(coef) se(coef) z p
rx -0.259112 0.771737 0.405110 -0.640 0.522
size -0.116363 0.890152 0.119236 -0.976 0.329
number -0.005709 0.994307 0.096671 -0.059 0.953
Likelihood ratio test=1.27 on 3 df, p=0.7353
n= 46, number of events= 29
> fit3pa
Call:
coxph(formula = Surv(stop, event) ~ rx + size + number, data = bladder2,
subset = (enum == 3), ties = "breslow")
coef exp(coef) se(coef) z p
rx -0.89854 0.40716 0.55352 -1.623 0.105
size 0.08497 1.08869 0.20861 0.407 0.684
number -0.01716 0.98298 0.12802 -0.134 0.893
Likelihood ratio test=4.16 on 3 df, p=0.2452
n= 27, number of events= 22
> rm(rx, cmat, wfit, fita, fit1p, fit2pa, fit2pb, fit3pa)
>
> proc.time()
user system elapsed
0.398 0.023 0.419
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