426 lines
15 KiB
Plaintext
426 lines
15 KiB
Plaintext
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R Under development (unstable) (2020-02-28 r77869) -- "Unsuffered Consequences"
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Copyright (C) 2020 The R Foundation for Statistical Computing
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Platform: x86_64-pc-linux-gnu (64-bit)
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R is free software and comes with ABSOLUTELY NO WARRANTY.
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You are welcome to redistribute it under certain conditions.
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Type 'license()' or 'licence()' for distribution details.
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R is a collaborative project with many contributors.
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Type 'contributors()' for more information and
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'citation()' on how to cite R or R packages in publications.
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Type 'demo()' for some demos, 'help()' for on-line help, or
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'help.start()' for an HTML browser interface to help.
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Type 'q()' to quit R.
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> options(na.action=na.exclude) # preserve missings
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> options(contrasts=c('contr.treatment', 'contr.poly')) #ensure constrast type
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> library(survival)
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>
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> # Tests of the weighted Cox model
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> #
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> # Similar data set to test1, but add weights,
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> # a double-death/censor tied time
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> # a censored last subject
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> # The latter two are cases covered only feebly elsewhere.
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> #
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> # The data set testw2 has the same data, but done via replication
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> #
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> aeq <- function(x,y) all.equal(as.vector(x), as.vector(y))
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>
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> testw1 <- data.frame(time= c(1,1,2,2,2,2,3,4,5),
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+ status= c(1,0,1,1,1,0,0,1,0),
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+ x= c(2,0,1,1,0,1,0,1,0),
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+ wt = c(1,2,3,4,3,2,1,2,1))
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> xx <- c(1,2,3,4,3,2,1,2,1)
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> testw2 <- data.frame(time= rep(c(1,1,2,2,2,2,3,4,5), xx),
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+ status= rep(c(1,0,1,1,1,0,0,1,0), xx),
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+ x= rep(c(2,0,1,1,0,1,0,1,0), xx),
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+ id= rep(1:9, xx))
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> indx <- match(1:9, testw2$id)
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> testw2 <- data.frame(time= rep(c(1,1,2,2,2,2,3,4,5), xx),
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+ status= rep(c(1,0,1,1,1,0,0,1,0), xx),
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+ x= rep(c(2,0,1,1,0,1,0,1,0), xx),
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+ id= rep(1:9, xx))
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> indx <- match(1:9, testw2$id)
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>
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> fit0 <- coxph(Surv(time, status) ~x, testw1, weights=wt,
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+ method='breslow', iter=0)
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> fit0b <- coxph(Surv(time, status) ~x, testw2, ties='breslow', iter=0)
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> fit <- coxph(Surv(time, status) ~x, testw1, weights=wt, ties='breslow')
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> fitb <- coxph(Surv(time, status) ~x, testw2, ties='breslow')
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>
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> texp <- function(beta) { # expected, Breslow estimate
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+ r <- exp(beta)
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+ temp <- cumsum(c(1/(r^2 + 11*r +7), 10/(11*r +5), 2/(2*r+1)))
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+ c(r^2, 1,r,r,1,r,1,r,1)* temp[c(1,1,2,2,2,2,2,3,3)]
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+ }
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> aeq(texp(0), c(1/19, 1/19, rep(103/152, 5), rep(613/456,2))) #verify texp()
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[1] TRUE
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>
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> xbar <- function(beta) { # xbar, Breslow estimate
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+ r <- exp(beta)
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+ temp <- r* rep(c(2*r + 11, 11/10, 1), c(2, 5, 2))
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+ temp * texp(beta)
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+ }
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>
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> fit0
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Call:
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coxph(formula = Surv(time, status) ~ x, data = testw1, weights = wt,
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method = "breslow", iter = 0)
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coef exp(coef) se(coef) z p
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x 0.0000 1.0000 0.5858 0 1
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Likelihood ratio test=0 on 1 df, p=1
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n= 9, number of events= 5
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> summary(fit)
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Call:
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coxph(formula = Surv(time, status) ~ x, data = testw1, weights = wt,
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ties = "breslow")
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n= 9, number of events= 5
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coef exp(coef) se(coef) z Pr(>|z|)
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x 0.8596 2.3621 0.7131 1.205 0.228
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exp(coef) exp(-coef) lower .95 upper .95
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x 2.362 0.4233 0.5839 9.556
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Concordance= 0.637 (se = 0.161 )
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Likelihood ratio test= 1.69 on 1 df, p=0.2
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Wald test = 1.45 on 1 df, p=0.2
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Score (logrank) test = 1.52 on 1 df, p=0.2
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> aeq(resid(fit0), testw1$status - texp(0))
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[1] TRUE
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> resid(fit0, type='score')
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1 2 3 4 5 6
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1.24653740 0.03601108 0.10056700 0.10056700 -0.22180142 -0.21193300
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7 8 9
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0.46569858 -0.10082189 0.91014302
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> resid(fit0, type='scho')
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1 2 2 2 4
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1.3157895 0.3125000 0.3125000 -0.6875000 0.3333333
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>
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> aeq(resid(fit0, type='mart'), (resid(fit0b, type='mart'))[indx])
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[1] TRUE
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> aeq(resid(fit0, type='scor'), (resid(fit0b, type='scor'))[indx])
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[1] TRUE
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> aeq(unique(resid(fit0, type='scho')), unique(resid(fit0b, type='scho')))
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[1] TRUE
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>
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>
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> aeq(resid(fit, type='mart'), testw1$status - texp(fit$coef))
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[1] TRUE
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> resid(fit, type='score')
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1 2 3 4 5 6
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0.88681615 0.02497653 0.03608964 0.03608964 -0.54297652 -0.12528780
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7 8 9
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0.29564605 -0.09476911 0.58400064
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> resid(fit, type='scho')
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1 2 2 2 4
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1.0368337 0.1613774 0.1613774 -0.8386226 0.1746960
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> aeq(resid(fit, type='mart'), (resid(fitb, type='mart'))[indx])
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[1] TRUE
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> aeq(resid(fit, type='scor'), (resid(fitb, type='scor'))[indx])
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[1] TRUE
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> aeq(unique(resid(fit, type='scho')), unique(resid(fitb, type='scho')))
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[1] TRUE
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> rr1 <- resid(fit, type='mart')
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> rr2 <- resid(fit, type='mart', weighted=T)
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> aeq(rr2/rr1, testw1$wt)
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[1] TRUE
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>
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> rr1 <- resid(fit, type='score')
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> rr2 <- resid(fit, type='score', weighted=T)
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> aeq(rr2/rr1, testw1$wt)
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[1] TRUE
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>
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> fit <- coxph(Surv(time, status) ~x, testw1, weights=wt, ties='efron')
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> fit
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Call:
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coxph(formula = Surv(time, status) ~ x, data = testw1, weights = wt,
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ties = "efron")
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coef exp(coef) se(coef) z p
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x 0.8726 2.3931 0.7126 1.225 0.221
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Likelihood ratio test=1.75 on 1 df, p=0.1858
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n= 9, number of events= 5
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> resid(fit, type='mart')
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1 2 3 4 5 6
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0.85334536 -0.02560716 0.32265266 0.32265266 0.71696234 -1.07772629
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7 8 9
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-0.45034077 -0.90490339 -0.79598658
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> resid(fit, type='score')
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1 2 3 4 5 6
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0.88116056 0.02477248 0.06057806 0.06057806 -0.59724033 -0.16737066
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7 8 9
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0.38040295 -0.13750290 0.66631324
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> resid(fit, type='scho')
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1 2 2 2 4
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1.0325955 0.1621759 0.1621759 -0.8378241 0.1728229
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>
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> # Tests of the weighted Cox model, AG form of the data
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> # Same solution as doweight1.s
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> #
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> testw3 <- data.frame(id = c( 1, 1, 2, 3, 3, 3, 4, 5, 5, 6, 7, 8, 8, 9),
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+ begin= c( 0, 5, 0, 0,10,15, 0, 0,14, 0, 0, 0,23, 0),
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+ time= c( 5,10,10,10,15,20,20,14,20,20,30,23,40,50),
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+ status= c( 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0),
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+ x= c( 2, 2, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0),
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+ wt = c( 1, 1, 2, 3, 3, 3, 4, 3, 3, 2, 1, 2, 2, 1))
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>
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> fit0 <- coxph(Surv(begin,time, status) ~x, testw3, weights=wt,
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+ ties='breslow', iter=0)
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> fit <- coxph(Surv(begin,time, status) ~x, testw3, weights=wt, ties='breslow')
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> fit0
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Call:
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coxph(formula = Surv(begin, time, status) ~ x, data = testw3,
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weights = wt, ties = "breslow", iter = 0)
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coef exp(coef) se(coef) z p
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x 0.0000 1.0000 0.5858 0 1
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Likelihood ratio test=0 on 1 df, p=1
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n= 14, number of events= 5
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> summary(fit)
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Call:
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coxph(formula = Surv(begin, time, status) ~ x, data = testw3,
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weights = wt, ties = "breslow")
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n= 14, number of events= 5
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coef exp(coef) se(coef) z Pr(>|z|)
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x 0.8596 2.3621 0.7131 1.205 0.228
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exp(coef) exp(-coef) lower .95 upper .95
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x 2.362 0.4233 0.5839 9.556
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Concordance= 0.637 (se = 0.172 )
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Likelihood ratio test= 1.69 on 1 df, p=0.2
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Wald test = 1.45 on 1 df, p=0.2
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Score (logrank) test = 1.52 on 1 df, p=0.2
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> resid(fit0, type='mart', collapse=testw3$id)
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1 2 3 4 5 6
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0.94736842 -0.05263158 0.32236842 0.32236842 0.32236842 -0.67763158
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7 8 9
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-0.67763158 -0.34429825 -1.34429825
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> resid(fit0, type='score', collapse=testw3$id)
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1 2 3 4 5 6
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1.24653740 0.03601108 0.10056700 0.10056700 -0.22180142 -0.21193300
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7 8 9
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0.46569858 -0.10082189 0.91014302
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> resid(fit0, type='scho')
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10 20 20 20 40
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1.3157895 0.3125000 0.3125000 -0.6875000 0.3333333
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>
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> resid(fit, type='mart', collapse=testw3$id)
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1 2 3 4 5 6
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0.85531186 -0.02593169 0.17636221 0.17636221 0.65131344 -0.82363779
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7 8 9
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-0.34868656 -0.64894181 -0.69807852
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> resid(fit, type='score', collapse=testw3$id)
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1 2 3 4 5 6
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0.88681615 0.02497653 0.03608964 0.03608964 -0.54297652 -0.12528780
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7 8 9
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0.29564605 -0.09476911 0.58400064
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> resid(fit, type='scho')
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10 20 20 20 40
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1.0368337 0.1613774 0.1613774 -0.8386226 0.1746960
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> fit0 <- coxph(Surv(begin, time, status) ~x,testw3, weights=wt, iter=0)
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> resid(fit0, 'mart', collapse=testw3$id)
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1 2 3 4 5 6
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0.94736842 -0.05263158 0.44454887 0.44454887 0.44454887 -0.88126566
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7 8 9
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-0.88126566 -0.54793233 -1.54793233
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> resid(coxph(Surv(begin, time, status) ~1, testw3, weights=wt)
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+ , collapse=testw3$id) #Null model
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1 2 3 4 5 6
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0.94736842 -0.05263158 0.44454887 0.44454887 0.44454887 -0.88126566
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7 8 9
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-0.88126566 -0.54793233 -1.54793233
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>
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> fit <- coxph(Surv(begin,time, status) ~x, testw3, weights=wt, ties='efron')
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> fit
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Call:
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coxph(formula = Surv(begin, time, status) ~ x, data = testw3,
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weights = wt, ties = "efron")
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coef exp(coef) se(coef) z p
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x 0.8726 2.3931 0.7126 1.225 0.221
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Likelihood ratio test=1.75 on 1 df, p=0.1858
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n= 14, number of events= 5
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> resid(fit, type='mart', collapse=testw3$id)
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1 2 3 4 5 6
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0.85334536 -0.02560716 0.32265266 0.32265266 0.71696234 -1.07772629
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7 8 9
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-0.45034077 -0.90490339 -0.79598658
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> resid(fit, type='score', collapse=testw3$id)
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1 2 3 4 5 6
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0.88116056 0.02477248 0.06057806 0.06057806 -0.59724033 -0.16737066
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7 8 9
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0.38040295 -0.13750290 0.66631324
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> resid(fit, type='scho')
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10 20 20 20 40
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1.0325955 0.1621759 0.1621759 -0.8378241 0.1728229
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> #
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> # Check out the impact of weights on the dfbetas
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> # Am I computing them correctly?
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> #
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> wtemp <- rep(1,26)
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> wtemp[c(5,10,15)] <- 2:4
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> fit <- coxph(Surv(futime, fustat) ~ age + ecog.ps, ovarian, weights=wtemp)
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> rr <- resid(fit, 'dfbeta')
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>
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> fit1 <- coxph(Surv(futime, fustat) ~ age + ecog.ps, ovarian, weights=wtemp,
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+ subset=(-5))
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> fit2 <- coxph(Surv(futime, fustat) ~ age + ecog.ps, ovarian, weights=wtemp,
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+ subset=(-10))
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> fit3 <- coxph(Surv(futime, fustat) ~ age + ecog.ps, ovarian, weights=wtemp,
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+ subset=(-15))
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>
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> #
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> # Effect of case weights on expected survival curves post Cox model
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> #
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> fit0 <- coxph(Surv(time, status) ~x, testw1, weights=wt, ties='breslow',
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+ iter=0)
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> fit0b <- coxph(Surv(time, status) ~x, testw2, ties='breslow', iter=0)
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>
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> surv1 <- survfit(fit0, newdata=list(x=0))
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> surv2 <- survfit(fit0b, newdata=list(x=0))
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> aeq(surv1$surv, surv2$surv)
|
||
|
|
[1] TRUE
|
||
|
|
> #
|
||
|
|
> # Check out the Efron approx.
|
||
|
|
> #
|
||
|
|
>
|
||
|
|
> fit0 <- coxph(Surv(time, status) ~x,testw1, weights=wt, iter=0)
|
||
|
|
> fit <- coxph(Surv(time, status) ~x,testw1, weights=wt)
|
||
|
|
> resid(fit0, 'mart')
|
||
|
|
1 2 3 4 5 6
|
||
|
|
0.94736842 -0.05263158 0.44454887 0.44454887 0.44454887 -0.88126566
|
||
|
|
7 8 9
|
||
|
|
-0.88126566 -0.54793233 -1.54793233
|
||
|
|
> resid(coxph(Surv(time, status) ~1, testw1, weights=wt)) #Null model
|
||
|
|
1 2 3 4 5 6
|
||
|
|
0.94736842 -0.05263158 0.44454887 0.44454887 0.44454887 -0.88126566
|
||
|
|
7 8 9
|
||
|
|
-0.88126566 -0.54793233 -1.54793233
|
||
|
|
>
|
||
|
|
> # lfun is the known log-likelihood for this data set, worked out in the
|
||
|
|
> # appendix of Therneau and Grambsch
|
||
|
|
> # ufun is the score vector and ifun the information matrix
|
||
|
|
> lfun <- function(beta) {
|
||
|
|
+ r <- exp(beta)
|
||
|
|
+ a <- 7*r +3
|
||
|
|
+ b <- 4*r +2
|
||
|
|
+ 11*beta - ( log(r^2 + 11*r +7) +
|
||
|
|
+ (10/3)*(log(a+b) + log(2*a/3 +b) + log(a/3 +b)) + 2*log(2*r +1))
|
||
|
|
+ }
|
||
|
|
> aeq(fit0$log[1], lfun(0))
|
||
|
|
[1] TRUE
|
||
|
|
> aeq(fit$log[2], lfun(fit$coef))
|
||
|
|
[1] TRUE
|
||
|
|
>
|
||
|
|
> ufun <- function(beta, efron=T) { #score statistic
|
||
|
|
+ r <- exp(beta)
|
||
|
|
+ xbar1 <- (2*r^2+11*r)/(r^2+11*r +7)
|
||
|
|
+ xbar2 <- 11*r/(11*r +5)
|
||
|
|
+ xbar3 <- 2*r/(2*r +1)
|
||
|
|
+ xbar2b<- 26*r/(26*r+12)
|
||
|
|
+ xbar2c<- 19*r/(19*r + 9)
|
||
|
|
+ temp <- 11 - (xbar1 + 2*xbar3)
|
||
|
|
+ if (efron) temp - (10/3)*(xbar2 + xbar2b + xbar2c)
|
||
|
|
+ else temp - 10*xbar2
|
||
|
|
+ }
|
||
|
|
> print(ufun(fit$coef) < 1e-4) # Should be true
|
||
|
|
x
|
||
|
|
TRUE
|
||
|
|
>
|
||
|
|
> ifun <- function(beta, efron=T) { # information matrix
|
||
|
|
+ r <- exp(beta)
|
||
|
|
+ xbar1 <- (2*r^2+11*r)/(r^2+11*r +7)
|
||
|
|
+ xbar2 <- 11*r/(11*r +5)
|
||
|
|
+ xbar3 <- 2*r/(2*r +1)
|
||
|
|
+ xbar2b<- 26*r/(26*r+12)
|
||
|
|
+ xbar2c<- 19*r/(19*r + 9)
|
||
|
|
+ temp <- ((4*r^2 + 11*r)/(r^2+11*r +7) - xbar1^2) +
|
||
|
|
+ 2*(xbar3 - xbar3^2)
|
||
|
|
+ if (efron) temp + (10/3)*((xbar2- xbar2^2) + (xbar2b - xbar2b^2) +
|
||
|
|
+ (xbar2c -xbar2c^2))
|
||
|
|
+ else temp + 10 * (xbar2- xbar2^2)
|
||
|
|
+ }
|
||
|
|
>
|
||
|
|
> aeq(fit0$var, 1/ifun(0))
|
||
|
|
[1] TRUE
|
||
|
|
> aeq(fit$var, 1/ifun(fit$coef))
|
||
|
|
[1] TRUE
|
||
|
|
>
|
||
|
|
>
|
||
|
|
>
|
||
|
|
> # Make sure that the weights pass through the residuals correctly
|
||
|
|
> rr1 <- resid(fit, type='mart')
|
||
|
|
> rr2 <- resid(fit, type='mart', weighted=T)
|
||
|
|
> aeq(rr2/rr1, testw1$wt)
|
||
|
|
[1] TRUE
|
||
|
|
> rr1 <- resid(fit, type='score')
|
||
|
|
> rr2 <- resid(fit, type='score', weighted=T)
|
||
|
|
> aeq(rr2/rr1, testw1$wt)
|
||
|
|
[1] TRUE
|
||
|
|
>
|
||
|
|
> #
|
||
|
|
> # Look at the individual components
|
||
|
|
> #
|
||
|
|
> dt0 <- coxph.detail(fit0)
|
||
|
|
> dt <- coxph.detail(fit)
|
||
|
|
> aeq(sum(dt$score), ufun(fit$coef)) #score statistic
|
||
|
|
[1] TRUE
|
||
|
|
> aeq(sum(dt0$score), ufun(0))
|
||
|
|
[1] TRUE
|
||
|
|
> aeq(dt0$hazard, c(1/19, (10/3)*(1/16 + 1/(6+20/3) + 1/(6+10/3)), 2/3))
|
||
|
|
[1] TRUE
|
||
|
|
>
|
||
|
|
>
|
||
|
|
>
|
||
|
|
> rm(fit, fit0, rr1, rr2, dt, dt0)
|
||
|
|
> #
|
||
|
|
> # Effect of weights on the robust variance
|
||
|
|
> #
|
||
|
|
> test1 <- data.frame(time= c(9, 3,1,1,6,6,8),
|
||
|
|
+ status=c(1,NA,1,0,1,1,0),
|
||
|
|
+ x= c(0, 2,1,1,1,0,0),
|
||
|
|
+ wt= c(3,0,1,1,1,1,1),
|
||
|
|
+ id= 1:7)
|
||
|
|
> testx <- data.frame(time= c(4,4,4,1,1,2,2,3),
|
||
|
|
+ status=c(1,1,1,1,0,1,1,0),
|
||
|
|
+ x= c(0,0,0,1,1,1,0,0),
|
||
|
|
+ wt= c(1,1,1,1,1,1,1,1),
|
||
|
|
+ id= 1:8)
|
||
|
|
>
|
||
|
|
> fit1 <- coxph(Surv(time, status) ~x, cluster=id, test1, ties='breslow',
|
||
|
|
+ weights=wt)
|
||
|
|
> fit2 <- coxph(Surv(time, status) ~x, cluster=id, testx, ties='breslow')
|
||
|
|
>
|
||
|
|
> db1 <- resid(fit1, 'dfbeta', weighted=F)
|
||
|
|
> db1 <- db1[-2] #toss the missing
|
||
|
|
> db2 <- resid(fit2, 'dfbeta')
|
||
|
|
> aeq(db1, db2[3:8])
|
||
|
|
[1] TRUE
|
||
|
|
>
|
||
|
|
> W <- c(3,1,1,1,1,1) #Weights, after removal of the missing value
|
||
|
|
> aeq(fit2$var, sum(db1*db1*W))
|
||
|
|
[1] TRUE
|
||
|
|
> aeq(fit1$var, sum(db1*db1*W*W))
|
||
|
|
[1] TRUE
|
||
|
|
>
|
||
|
|
>
|
||
|
|
> proc.time()
|
||
|
|
user system elapsed
|
||
|
|
0.986 0.049 1.028
|