15-shengcun.Rmd

# 生存分析 {#surv}

## Concepts

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

## 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 $log h(t) = ν + ρt$

## 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
    - $\frac{h_i(t)}{h_{i'}(t)} = \frac{e^{\eta_i}}{e^{\eta'}}$
    
## 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

```{r}
library(survival)
library(car)
# perform survival analysis
data("Rossi")
Rossi[1:5, 1:10]
mod.allison <- coxph(Surv(week, arrest) ~ fin + age + race + wexp + mar + paro + prio, data=Rossi)
summary(mod.allison)
# plot time vs survival prob
plot(survfit(mod.allison), ylim=c(.7, 1), xlab='Weeks', ylab='Proportion Not Rearrested')
```

### 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.

## 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

```{r}
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

## Time-Dependent Covariates

- treat the employed variable as a tim-dependent covariates with 52 weeks' record

```{r}
sum(!is.na(Rossi[,11:62])) # record count
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)
```

## Model Diagnostics

- Checking Proportional Hazards

```{r}
modallison3 <- coxph(Surv(week, arrest) ~ fin + age + prio, data=Rossi)
modallison3
cox.zph(modallison3)
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

```{r}
modallison4 <- coxph(Surv(start,stop,arresttime)~fin+age+age:stop:stop+prio, data = Rossi2)
modallison4
```

- the coefficient for the interaction is negative and highly statistically significant: The effect of age declines with time

- use residual to find influential observations

## Reference

- [Cox Proportional-Hazards Regression for Survival Data](http://cran.r-project.org/doc/contrib/Fox-Companion/appendix-cox-regression.pdf)
- [Real Cases](http://ehp.niehs.nih.gov/1104049/)