1. Survival time treatment effects
Ting-Ting Chung
2017/09/12

2. Outline When to use survival analysis Introduction of Cox PH model
The Problem of Cox PH model
Auxiliary model
IPW
IPWA

3. When to use survival analysis
Time (not must be time) to event
Will smoking reduce the time to a second heart-attack among men aged who have already had a heart attack?
What is the effect of participation in a supported work program after release from prison on the time until a subsequent arrest?

4. Survival analysis concepts
Time (not must be time) to event
Censored
Time origin

5. Compare with conventional model
Logistic regression
Ignores time of events
Can’t handle time-dependent variable
Linear regression
Can’t handle censored, time-dependent variable
Time to event can always to be unusual distribution

6. Data type Censoring Right lifetest, Phreg Left  ICLifetest, ICPHreg
If the observation is terminated before the event occurs
Left  ICLifetest, ICPHreg
When the observation experience the event before the start of follow-up
Interval  ICLifetest, ICPHreg
You know the survival time is that is between the values t and t+k B C E F
End of study
subject
event
Start of study time t time t+k
Withdrew or lost of follow-up

7. Right censored Type 1 Type 2 Radom (個人的離開不在控制範圍)
Subjects survived until end of study某個時間點結束
Type 2
Subjects survived until end of study (when a pre-specified numbers of events have occurred)EX:試驗收案，收案狀況 已超過預期，停止收案)
Radom (個人的離開不在控制範圍)
Uninformative ex : 搬離台灣
Informative  ex : 藥物試驗，無法承受副作用，因此離開， sensitivity analysis

8. Which model should be used in the survival-time analysis?

9. Cox Proportional Hazard Model
The Cox model models the probability that the event will occur in the next moment given that it has not yet happened as a function of covariates
hi(t|x)= ℎ0 𝑡 𝑒 {𝛽1𝑋1A+…+𝛽𝑘𝑋𝑖𝑘}

10. The hazard ratio that smoking raises the hazard of a second heart attack by a factor of 1:5 relative to not smoking
Hazard ratio = ℎ𝑎𝑧𝑎𝑟𝑑 𝑖𝑛 𝑔𝑟𝑜𝑢𝑝 𝐴 ℎ𝑎𝑧𝑎𝑟𝑑 𝑖𝑛 𝑔𝑟𝑜𝑢𝑝 B = ℎ0 𝑡 𝑒 {𝛽1𝑋1A+…+𝛽𝑘𝑋𝑖𝑘} ℎ0 𝑡 𝑒 {𝛽1𝑋1B+…+𝛽𝑘𝑋𝑖𝑘} = 𝑒 𝛽1𝑋1A 𝑒 𝛽1𝑋1𝐵 = 𝑒 𝛽1(𝑋1𝐴−𝑋1𝐵)

11. Compared with Parametric models
Parametric models (exponential, weibull,…)
Distribution of survival time is known
The hazard function is completely specified
Semi-parametric models (cox)
Distribution of survival time is unknown
The hazard function is unspecified

12. Stata command-1
stset atime, failure(fail)
(David M. Drukker, 2015)

13. Stata command-2 stcox smoke age exercise diet , nolog noshow
(David M. Drukker, 2015)

14. Problems with the Cox model

15. Two problems with the Cox model
It is hard to understand the units of the hazard ratio
How bad is it that smoking raises the hazard ratio by 1.5?
This interpretation is only useful if the treatment enters the x term linearly
If the treatment is interacted with other covariates, the effect of the treatment varies over individuals
(David M. Drukker, 2015)

16. Problems with the Cox model
The average difference in time to second heart attack when everyone smokes instead of when no one smokes
For each individual, the effect of the treatment is a contrast of what would happen if the individual received the treatment versus what would happen if the individual did not receive the treatment
The hazard-ratio measure of the treatment effect is the ratio of the hazard of the smoking potential outcome to the hazard nonsmoking potential outcome
(David M. Drukker, 2015)

17. Problems with the Cox model
Ratios of unconditional hazards are harder to estimate and more difficult to interpret than the average difference in time to second heart attack when everyone smokes instead of no one smokes
ATE在現實中是不太可能的: 一個人不可能同時屬於抽菸組，又 屬於不抽菸組
potential-outcome mean (POM)
(David M. Drukker, 2015)

18. Problems with the Cox model
The “fundamental problem of causal inference" is that we only observe one of the potential outcomes
We can use the tricks of missing-data analysis to estimate treatment effects
(David M. Drukker, 2015)

19. Random-assignment case
If smoking were randomly assigned, the missing potential outcome would be missing completely at random
If the time to second heart attack was never censored and smoking was randomly assigned
The average time to second heart attack among smokers would estimate the smoking POM
The average time to second heart attack among nonsmokers would estimate the nonsmoking POM
(David M. Drukker, 2015)

20. Random-assignment case
Instead of assuming that the treatment is randomly assigned, we assume that the treatment is as good as randomly assigned after conditioning on covariates
Formally, this assumption is known as conditional independence
The auxiliary model is how we condition on covariates so that the treatment is as good as randomly assigned
(David M. Drukker, 2015)

21. Auxiliary model

23. Inverse-probability-weighted (IPW) estimators-1
IPW estimators weight observations on the observed outcome variable by the inverse of the probability that it is observed to account for the missingness process
Observations that are not likely to contain missing data get a weight close to one; observations that are likely to contain missing data get a weight larger than one, potentially much larger
(David M. Drukker, 2015)

24. Inverse-probability-weighted (IPW) estimators-2
IPW estimators use estimates from models for the probability of treatment and the probability of censoring to correct for the missing potential outcome and the observations lost to censoring
(David M. Drukker, 2015)

25. 預測censoring機率的變項
預測治療機率的變項
stteffects ipw (smoke age exercise diet) (age exercise diet), nolog noshow
The average time to second heart attack is 1.7 years sooner when everyone in the population smokes instead of no one smokes
The average time to second heart attack is 4.2 years when no one smokes
(David M. Drukker, 2015)

26. teffects ipw (fail) (smoke age exercise diet, probit)
結果變項
預測治療變項
teffects ipw (fail) (smoke age exercise diet, probit)
stteffects ipw (smoke age exercise diet) (age exercise diet), nolog noshow

27. Inverse-probability-weighted regression-adjustment (IPWRA)
IPWRA estimators use the inverse of the estimated treatment-probability weights to estimate missing-data-corrected regression coefficients that are subsequently used to estimate the POMs
Censoring can be handled in the log likelihood function or by modeling the censoring process
(David M. Drukker, 2015)

28. 預測結果機率的變項
預測治療機率的變項
stteffects ipwra (age exercise diet) (smoke age exercise diet) (age exercise diet) 可寫可不寫
預測censoring機率的變項
The average time to second heart attack is 1.5 years sooner when everyone in the population smokes instead of no one smokes
The average time to second heart attack is 4.1 years when no one smokes
(David M. Drukker, 2015)

29. Reference
Drukker, D. M. (2015, November 12) . Estimating survival- time treatment effects from observational data.

30. Thanks~