Parametric survival models for interval-censored data

Let's see it work

We want to investigate the effect of the stage of AIDS patients on their time to resistance to the drug zidovudine. The exact times are not available, but they are known to be within time intervals recorded in variables ltime and rtime. Some patients developed drug resistance before their first follow-up examination time. Those observations are called left-censored. Others developed resistance between two follow-up visits. Those are called interval-censored. Yet others show no resistance even at the last follow-up visit. Those are called right-censored.

We can use stintreg to fit a Weibull model to these data.

. stintreg i.stage, interval(ltime rtime) distribution(weibull)

Weibull PH regression                               Number of obs     =     31
                                                           Uncensored =      0
                                                        Left-censored =     15
                                                       Right-censored =     13
                                                       Interval-cens. =      3

                                                    LR chi2(1)        =  10.02
Log likelihood = -13.27946                          Prob > chi2       = 0.0016



             
 
 Haz. ratio   Std. err.      z    P>|z|     [95% conf. interval]

       stage 
 

       late  
 
   6.757496   4.462933     2.89   0.004     1.851896    24.65783
       _cons 
 
   .0003517   .0010552    -2.65   0.008     9.82e-07    .1259506

       /ln_p 
 
   1.036663   .3978294     2.61   0.009     .2569315    1.816394

           p 
 
   2.819791   1.121796                      1.292957    6.149644
         1/p 
 
   .3546362   .1410847                       .162611    .7734212

Note: _cons estimates baseline hazard.

We find that the hazard of resisting zidovudine for patients in their late stage is approximately seven times the hazard for patients in their early stage.

We can plot the hazard functions for those two stages.

. stcurve, hazard at(stage = (0 1))
note: function evaluated at specified covariate values.

After about two months, the hazard of resisting zidovudine for late-stage patients (stage=1) increases rapidly compared with the hazard for early-stage patients (stage=0).

Suppose we believe that the hazard depends on the dosage level, low or high. We use the strata() option to stratify on dose.

. stintreg i.stage, interval(ltime rtime) distribution(weibull) strata(dose)
note: option nohr is implied if option strata() or ancillary() is specified.

Weibull PH regression                               Number of obs     =     31
                                                           Uncensored =      0
                                                        Left-censored =     15
                                                       Right-censored =     13
                                                       Interval-cens. =      3

                                                    LR chi2(2)        =  12.40
Log likelihood = -11.115197                         Prob > chi2       = 0.0020



             
 
 Coefficient  Std. err.      z    P>|z|     [95% conf. interval]

ltime        
 

       stage 
 

       late  
 
   2.711532   1.084146     2.50   0.012     .5866443    4.836419
             
 

        dose 
 

       high  
 
  -2.661869   5.883971    -0.45   0.651    -14.19424    8.870503
       _cons 
 
     -9.143   4.930796    -1.85   0.064    -18.80718    .5211828

ln_p         
 

        dose 
 

       high  
 
   .4538938   .6700989     0.68   0.498    -.8594759    1.767263
       _cons 
 
   1.051935   .6190548     1.70   0.089    -.1613905     2.26526

In the stratified model, the stage coefficients are the same across the two dosage levels, but the intercept and the shape parameter or, more precisely, the log-shape parameter ln_p are different for low and high doses. Stratifying on dose does not seem to improve our model.

We can consider a simpler model in which only the shape parameter is dose specific. We do this by specifying the ancillary() option.

. stintreg i.stage, interval(ltime rtime) distribution(weibull) ancillary(i.dose)
note: option nohr is implied if option strata() or ancillary() is specified.

Weibull PH regression                                    Number of obs     =     31
                                                                Uncensored =      0
                                                             Left-censored =     15
                                                            Right-censored =     13
                                                            Interval-cens. =      3

                                                         LR chi2(1)        =  12.20
Log likelihood = -11.214877                              Prob > chi2       = 0.0005



             
 
 Coefficient   Std. err.      z       P>|z|      [95% conf. interval]

ltime        
 

       stage 
 

       late  
 
    2.795072   1.167495      2.39     0.017      .5068246    5.083319
       _cons 
 
    -10.8462   4.233019     -2.56     0.010     -19.14276   -2.549632

ln_p         
 

        dose 
 

       high  
 
    .1655302     .08745      1.89     0.058     -.0058688    .3369291
       _cons 
 
     1.25236   .4143214      3.02     0.003      .4403054    2.064415

There is no statistical evidence, at least at the 5% significance level, that dosage levels affect the shape parameter of the Weibull model.