Treatment effects for survival data

Let's see it work

Let's consider continued smoking and time to second heart attack. We have (fictional) data, which we have already stset. stset is Stata's way of registering the survival aspects of your data.

We will assume the following models:

• Treatment (smoking) is determined by age, exercise, and education.

• Outcome (time to second heart attack) is Weibull and determined by age, exercise, quality of diet, and education.

• Censoring, if we need to specify it, is determined by the same distribution and variables as Outcome.

Here's how we estimate the treatment effect based solely on outcome:

. stteffects ra (age exercise diet education)      /* outcome */
                 (smoke)                           /* whether treated */ 

Survival treatment-effects estimation           Number of obs     =      2,000
Estimator      : regression adjustment
Outcome model  : Weibull
Treatment model: none
Censoring model: none

             
 
               Robust
          _t 
 
 Coefficient  std. err.      z    P>|z|     [95% conf. interval]

ATE          
 
    
       smoke 
 
    
    (Smoker  
 
    
         vs  
 
    
 Nonsmoker)  
 
  -1.956657   .3331787    -5.87   0.000    -2.609676   -1.303639

POmean       
 
     
       smoke 
 
    
  Nonsmoker  
 
   4.243974   .2620538    16.20   0.000     3.730358     4.75759

We find the average treatment effect of smoking is to hasten the time to second heart attack by 1.96 years.

Here's how we estimate the treatment effect based solely on treatment. When we specify solely the treatment model, we must also specify the censoring model:

. stteffects ipw (smoke age exercise diet education)      /* treatment */
                  (age exercise diet education)           /* censoring */

Survival treatment-effects estimation           Number of obs     =      2,000
Estimator      : inverse-probability weights
Outcome model  : weighted mean
Treatment model: logit
Censoring model: Weibull

             
 
               Robust
          _t 
 
 Coefficient  std. err.      z    P>|z|     [95% conf. interval]

ATE          
 
    
       smoke 
 
    
    (Smoker  
 
    
         vs  
 
    
 Nonsmoker)  
 
  -2.187297   .6319837    -3.46   0.001    -3.425962   -.9486314

POmean       
 
     
       smoke 
 
    
  Nonsmoker  
 
   4.225331    .517501     8.16   0.000     3.211047    5.239614

We find that the ATE is now -2.19 years (treatment) rather than -1.96 years (outcome). All these results are within roughly a standard error. They are in agreement.

What would have happened had we specified the censoring models, too, by typing,

. stteffects ipwra (age exercise diet education)        /* outcome */
                    (smoke age exercise education)      /* treatment */
                    (age exercise diet education)       /* censoring */

         failure _d:  fail
   analysis time _t:  atime

Iteration 0:   EE criterion =  1.217e-17
Iteration 1:   EE criterion =  2.459e-30

Survival treatment-effects estimation           Number of obs     =      2,000
Estimator      : IPW regression adjustment
Outcome model  : Weibull
Treatment model: logit
Censoring model: Weibull

             
 
               Robust
          _t 
 
 Coefficient  std. err.      z    P>|z|     [95% conf. interval]

ATE          
 
    
       smoke 
 
    
    (Smoker  
 
    
         vs  
 
    
 Nonsmoker)  
 
  -2.285057   .7318456    -3.12   0.002    -3.719448   -.8506656

POmean       
 
     
       smoke 
 
    
  Nonsmoker  
 
   4.385841    .6427521     6.82   0.000     3.12607    5.645612

stteffects ipwra with three models is a different estimator, and thus infinite samples produce different results. Here we estimate the ATE to be -2.29 instead of -1.96 (the outcome only estimates) or -2.19 (the treatment only estimates).