Solution to Series 1

Transcription

1 Dr A Hauser Analysis AS 2018 Solution to Series 1 1 a) We can compute the cumulative hazard function: We know that H(t) = t 0 h(u)du = t H(t) = log S(t) ( S(t) = exp t ) The following plot shows the survivor function for different : S(t) Survivor function = 2 = 1 = b) The density f(t) is given by f(t) = h(t) S(t) f(t) = 1 ( exp t ) t The cumulative distribution function F (t) is given by F (t) = 1 S(t) ( F (t) = 1 exp t ) c) f(t) t is the unconditional ( a priori ) probability that the nucleus decays in the small time interval [t, t + t] h(t) t, in contrast, is the probability that the nucleus decays in the interval [t, t + t], given that is has not decayed yet at time t 2 x i d i n i p i Ŝ(x i ) = P(X > x i ) se(ŝ(x i)) /8 7/8 = = /8 = /6 7/8 5/6 = 35/48 = = /5 35/48 3/5 = 7/16 = = /16 = /2 7/16 1/2 = 7/32 = =

2 Calculation in R: > x <- c(1, 1, 2, 4, 4, 4, 6, 9) > status <- c(1, 0, 1, 1, 1, 0, 1, 1) # 0 for censored > rsf <- survfit(surv(x, status) 1, conftype = "none") > summary(rsf) survfit(formula = Surv(x, status) ~ 1, conftype = "none") time nrisk nevent survival stderr NA R-Code for drawing the survival curve: > plot(rsf, xlab = "", ylab = "") 3 a) > library(survival) > turl <- " > ddiabetes <- readtable(turl, header = TRUE, sep = ",") > rsv <- survfit(surv(lzeit, tod) ~ 1, type = "kaplan-meier", ddiabetes) > plot(rsv, confint = FALSE, xlab = "", ylab = "") b) The groups (diabetic, non diabetic) have different KM-curves The people in the diabetes group, seem to die sooner > rsv2 <- survfit(surv(lzeit, tod) ~ diab, data = ddiabetes) > plot(rsv2, confint = FALSE, lty = 1:2, xlab = "", ylab = "") > legend(3000, 08, c("no diabetes", "diabetes"), lty = 1:2)

3 3 no diabetes diabetes c) > rsv0 <- summary(survfit(surv(lzeit, tod) ~ 1, subset = diab == 0, conftype = "n", data = ddiabetes), censor = TRUE) > ttime0 <- rsv0$time > tweight0 <- rsv0$nevent / rsv0$nrisk > tsmooth0 <- density(ttime0, kernel = "epanechnikov", weights = tweight0, n = 100, from = 0, to = 4000) > plot(tsmooth0$x, tsmooth0$y, type = "l", ylim = c(0, 00018), main = "Kernel estimators of hazards", xlab = "time", ylab = "Smoothed hazard") > rsv1 <- summary(survfit(surv(lzeit, tod) ~ 1, subset = diab == 1, conftype = "n", data = ddiabetes), censor = TRUE) > ttime1 <- rsv1$time > tweight1 <- rsv1$nevent / rsv1$nrisk > tsmooth1 <- density(ttime1, kernel = "epanechnikov", weights = tweight1, n = 100, from = 0, to = 4000) > lines(tsmooth1$x, tsmooth1$y, lty = 2) > legend(0, 00018, c('no diabetes', 'diabetes'), lty = 1:2) Kernel estimators of hazards Smoothed hazard no diabetes diabetes time In the graph above, we can see, that the risk for the people not having diabetes is estimated to be zero after approx 3000 days The ratio is then no longer meaningful > plot(tsmooth0$x, tsmooth1$y / tsmooth0$y, xlim = c(0, 2800), ylim = c(0, 10), type = "l", main = "Ratio of Smoothed Hazards", xlab = "time", ylab = "hazard ratio")

4 4 hazard ratio Ratio of Smoothed Hazards time d) The Log-Rank-Test (p-value=002) confirms the difference seen by eye: > survdiff(surv(lzeit, tod) ~ diab, data = ddiabetes) survdiff(formula = Surv(lzeit, tod) ~ diab, data = ddiabetes) diab= diab= Chisq= 55 on 1 degrees of freedom, p= 002 e) The plot shows a difference between diabetics and non diabetics Especially the young non diabetics have a considerable larger chance of survival compared to the other groups There is an interesting structure in the diabetics group The young diabetics have a clearly larger chance of survival in the first years after the operation Then their curves cross and the young diabetics die sooner Here we should check why this can be like that > rsv4b <- survfit(surv(lzeit, tod) ~ diab + strata(altgr), data = ddiabetes) > summary(rsv4b) > plot(rsv4b, lty = 1:4) > legend(3000, 095, legend = c("(0,0)", "(0,1)", "(1,0)", "(1,1)"), lty = 1:4) > text(3500, 099, "(diab,altgr)") (diab,altgr) (0,0) (0,1) (1,0) (1,1) When we stratify by gender, there is a group with only 3 observations, 2 of which censored (the censored observations are displayed by a + in the survival curves, this can be done with the argument marktime = TRUE in the plot()-function) This curve (female, non diabetics) is thus little meaningful

5 5 > rsv4a <- survfit(surv(lzeit, tod) ~ diab + strata(sex), data = ddiabetes) > summary(rsv4a) > plot(rsv4a, marktime = TRUE, lty = 1:4) > legend(3000, 09, legend = c("(0,0)", "(0,1)", "(1,0)", "(1,1)"), lty = 1:4) > text(3500, 095, "(diab,sex)") (diab,sex) (0,0) (0,1) (1,0) (1,1) f) Stratifying by age: > survdiff(surv(lzeit,tod) ~ diab + strata(altgr), data = ddiabetes) survdiff(formula = Surv(lzeit, tod) ~ diab + strata(altgr), data = ddiabetes) diab= diab= Chisq= 43 on 1 degrees of freedom, p= 004 The difference in survival between diabetics and non-diabetics is also significant, if we stratify by age Stratifying by gender: > survdiff(surv(lzeit,tod) ~ diab + strata(sex), data = ddiabetes) survdiff(formula = Surv(lzeit, tod) ~ diab + strata(sex), data = ddiabetes) diab= diab= Chisq= 49 on 1 degrees of freedom, p= 003 Again the difference is significant Age and sex seem to play a secondary role in this case This may be surprising and we will come back to this later 4 a) Type I censored data because animals enter the study at the same time and are observed for a fixed period of time b) Type II censored data because animals enter the study at the same time but are observed until a fixed portion of the subjects have died