# 第15章 生存分析

## 15.1 Concepts

• examines and models the time it takes for events to occur
• the distribution of survival times
• Popular: Cox proportional-hazards regression model

## 15.2 Notation

• T as a random variable with cumulative distribution function $$P (t) = Pr(T ≤ t)$$ and probability density function $$p(t) = \frac{dP (t)}{dt}$$
• survival function S(t) is the complement of the distribution function, $$S(t) = Pr(T > t) = 1 − P (t)$$
• hazard function $$log h(t) = ν + ρt$$

## 15.3 Cox proportional-hazards regression model

• $$log h_i(t)=α+_1x_{i1} +β_2x_{ik} +···+β_kx_{ik}$$
• Cox model
• $$log h_i(t)=α(t)+β_1x_{i1} +β_2x_{ik} +···+β_kx_{ik}$$
• the Cox model is a proportional-hazards model
• $$\frac{h_i(t)}{h_{i'}(t)} = \frac{e^{\eta_i}}{e^{\eta'}}$$

## 15.4 Case: Recidivism

• Target: recidivism of 432 male prisoners, who were observed for a year after being released from prison
• arrest means the male prisoners who rearrested
• 52 weeks
• factors: financial aid after release from prison, affected，release ages，race，work experience，marriage，parole，prior convictions, education
library(survival)
library(car)
# perform survival analysis
data("Rossi")
Rossi[1:5, 1:10]
##   week arrest fin age  race wexp         mar paro prio educ
## 1   20      1  no  27 black   no not married  yes    3    3
## 2   17      1  no  18 black   no not married  yes    8    4
## 3   25      1  no  19 other  yes not married  yes   13    3
## 4   52      0 yes  23 black  yes     married  yes    1    5
## 5   52      0  no  19 other  yes not married  yes    3    3
mod.allison <- coxph(Surv(week, arrest) ~ fin + age + race + wexp + mar + paro + prio, data=Rossi)
summary(mod.allison)
## Call:
## coxph(formula = Surv(week, arrest) ~ fin + age + race + wexp +
##     mar + paro + prio, data = Rossi)
##
##   n= 432, number of events= 114
##
##                   coef exp(coef) se(coef)     z Pr(>|z|)
## finyes         -0.3794    0.6843   0.1914 -1.98   0.0474 *
## age            -0.0574    0.9442   0.0220 -2.61   0.0090 **
## raceother      -0.3139    0.7306   0.3080 -1.02   0.3081
## wexpyes        -0.1498    0.8609   0.2122 -0.71   0.4803
## marnot married  0.4337    1.5430   0.3819  1.14   0.2561
## paroyes        -0.0849    0.9186   0.1958 -0.43   0.6646
## prio            0.0915    1.0958   0.0286  3.19   0.0014 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##                exp(coef) exp(-coef) lower .95 upper .95
## finyes             0.684      1.461     0.470     0.996
## age                0.944      1.059     0.904     0.986
## raceother          0.731      1.369     0.399     1.336
## wexpyes            0.861      1.162     0.568     1.305
## marnot married     1.543      0.648     0.730     3.261
## paroyes            0.919      1.089     0.626     1.348
## prio               1.096      0.913     1.036     1.159
##
## Concordance= 0.64  (se = 0.027 )
## Likelihood ratio test= 33.3  on 7 df,   p=2e-05
## Wald test            = 32.1  on 7 df,   p=4e-05
## Score (logrank) test = 33.5  on 7 df,   p=2e-05
# plot time vs survival prob
plot(survfit(mod.allison), ylim=c(.7, 1), xlab='Weeks', ylab='Proportion Not Rearrested')

### 15.4.1 result

• The covariates age and prio (prior convictions) have highly statistically significant coefficients, while the coefficient for fin (financial aid) is marginally significant
• holding the other covariates constant, an additional year of age reduces the weekly hazard of rearrest by a factor of $$e^b = 0.944$$ on average – that is, by 5.6
• likelihood-ratio, Wald, and score chi-square statistics: null hypothesis all of the β’s are zero.

## 15.5 further

• assess the impact of financial aid on rearrest
• new data frame with two rows, one for each value of fin; the other covariates are fixed to their average values
Rossi.fin <- data.frame(fin=c(0,1), age=rep(mean(Rossi$age),2), race=rep(mean(as.numeric(Rossi$race)),2), wexp=rep(mean(as.numeric(Rossi$wexp)),2), mar=rep(mean(as.numeric(Rossi$mar)),2), paro=rep(mean(as.numeric(Rossi$paro)),2), prio=rep(mean(as.numeric(Rossi$prio)),2))
plot(survfit(mod.allison, newdata=Rossi.fin), conf.int=T, lty=c(1,2), ylim=c(.6, 1))
legend("bottomleft", legend=c('fin = 0', 'fin = 1'), lty=c(1,2))

• the higher estimated ‘survival’ of those receiving financial aid, but the two confidence envelopes overlap substantially, even after 52 weeks

## 15.6 Time-Dependent Covariates

• treat the employed variable as a tim-dependent covariates with 52 weeks’ record
sum(!is.na(Rossi[,11:62])) # record count
## [1] 19809
Rossi2 <- matrix(0, 19809, 14) # to hold new data set
colnames(Rossi2) <- c('start', 'stop', 'arresttime', names(Rossi)[1:10], 'employed')

row<-0
for (i in 1:nrow(Rossi)) {
for (j in 11:62) {
if (is.na(Rossi[i, j])) next
else {
row <- row + 1 # increment row counter
start <- j - 11 # start time (previous week)
stop <- start + 1 # stop time (current week)
arresttime <- if (stop == Rossi[i, 1] && Rossi[i, 2] ==1) 1 else 0
Rossi2[row,] <- c(start, stop, arresttime, unlist(Rossi[i, c(1:10, j)]))
}
}
}
Rossi2 <- as.data.frame(Rossi2)
remove(i, j, row, start, stop, arresttime)
modallison2 <- coxph(Surv(start, stop, arresttime) ~ fin + age + race + wexp + mar + paro + prio + employed, data=Rossi2)
summary(modallison2)
## Call:
## coxph(formula = Surv(start, stop, arresttime) ~ fin + age + race +
##     wexp + mar + paro + prio + employed, data = Rossi2)
##
##   n= 19809, number of events= 114
##
##             coef exp(coef) se(coef)     z Pr(>|z|)
## fin      -0.3567    0.7000   0.1911 -1.87   0.0620 .
## age      -0.0463    0.9547   0.0217 -2.13   0.0330 *
## race     -0.3387    0.7127   0.3096 -1.09   0.2740
## wexp     -0.0256    0.9748   0.2114 -0.12   0.9038
## mar       0.2937    1.3414   0.3830  0.77   0.4431
## paro     -0.0642    0.9378   0.1947 -0.33   0.7416
## prio      0.0851    1.0889   0.0290  2.94   0.0033 **
## employed -1.3283    0.2649   0.2507 -5.30  1.2e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##          exp(coef) exp(-coef) lower .95 upper .95
## fin          0.700      1.429     0.481     1.018
## age          0.955      1.047     0.915     0.996
## race         0.713      1.403     0.388     1.308
## wexp         0.975      1.026     0.644     1.475
## mar          1.341      0.745     0.633     2.842
## paro         0.938      1.066     0.640     1.374
## prio         1.089      0.918     1.029     1.152
## employed     0.265      3.775     0.162     0.433
##
## Concordance= 0.708  (se = 0.023 )
## Likelihood ratio test= 68.7  on 8 df,   p=9e-12
## Wald test            = 56.1  on 8 df,   p=3e-09
## Score (logrank) test = 64.5  on 8 df,   p=6e-11

## 15.7 Model Diagnostics

• Checking Proportional Hazards
modallison3 <- coxph(Surv(week, arrest) ~ fin + age + prio, data=Rossi)
modallison3
## Call:
## coxph(formula = Surv(week, arrest) ~ fin + age + prio, data = Rossi)
##
##         coef exp(coef) se(coef)  z     p
## finyes -0.35      0.71     0.19 -2 0.068
## age    -0.07      0.94     0.02 -3 0.001
## prio    0.10      1.10     0.03  4 4e-04
##
## Likelihood ratio test=29  on 3 df, p=2e-06
## n= 432, number of events= 114
cox.zph(modallison3)
##         chisq df     p
## fin    0.0638  1 0.801
## age    6.3255  1 0.012
## prio   0.5187  1 0.471
## GLOBAL 7.1367  3 0.068
par(mfrow=c(2,2))
plot(cox.zph(modallison3))

• there appears to be a trend in the plot for age, with the age effect declining with time
modallison4 <- coxph(Surv(start,stop,arresttime)~fin+age+age:stop:stop+prio, data = Rossi2)
modallison4
## Call:
## coxph(formula = Surv(start, stop, arresttime) ~ fin + age + age:stop:stop +
##     prio, data = Rossi2)
##
##            coef exp(coef) se(coef)    z     p
## fin      -0.349     0.706    0.190 -1.8 0.067
## age       0.032     1.033    0.039  0.8 0.413
## prio      0.098     1.103    0.027  3.6 3e-04
## age:stop -0.004     0.996    0.001 -2.6 0.009
##
## Likelihood ratio test=36  on 4 df, p=3e-07
## n= 19809, number of events= 114
• the coefficient for the interaction is negative and highly statistically significant: The effect of age declines with time

• use residual to find influential observations