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Bayesian parametric survival models

Highlights

  • Simply prefix your survival command with bayes:
  • Exponential, Weibull, lognormal, and more survival distributions
  • Proportional-hazards and accelerated failure-time metrics
  • Stratified estimation
  • Flexible modeling of ancillary parameters
  • Frailty models
  • Multilevel survival models
  • Default and custom prior distributions
  • Full Bayesian-features support

What's this about?

You can fit parametric survival models in Stata using streg. You can fit multilevel parametric survival models using mestreg. You can now fit Bayesian parametric survival models by simply typing bayes: in front of streg and mestreg!

Let's see it work

Let's look at several examples.

Parametric survival models

Consider a dataset in which we model the time until hip fracture as a function of age and whether the patient wears a hip-protective device (variable protect). Let's fit a Bayesian Weibull model to these data and compare the results with the classical analysis.

First, we declare our survival data.

. stset time1, id(id) failure(fracture) time0(time0)

Then, we fit a Weibull survival model using streg.

. streg protect age, distribution(weibull)

         failure _d:  fracture
   analysis time _t:  time1
                 id:  id

Weibull PH regression

No. of subjects =          148                  Number of obs    =         206
No. of failures =           37
Time at risk    =         1703
                                                LR chi2(2)       =       49.97
Log likelihood  =   -77.446477                  Prob > chi2      =      0.0000

_t Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval]
protect .0922046 .0321722 -6.83 0.000 .0465318 .1827072
age 1.101041 .038173 2.78 0.005 1.028709 1.178459
_cons .000024 .0000624 -4.09 0.000 1.48e-07 .0039042
/ln_p .4513032 .1265975 3.56 0.000 .2031767 .6994297
p 1.570357 .1988033 1.225289 2.012605
1/p .6367977 .080617 .4968686 .816134
Note: Estimates are transformed only in the first equation. Note: _cons estimates baseline hazard.

Finally, to fit a Bayesian survival model, we simply prefix the above streg command with bayes:.

. bayes: streg protect age, distribution(weibull)

Model summary
Likelihood:
_t ~ streg_weibull(xb__t,{ln_p})
Priors:
{_t:protect age _cons} ~ normal(0,10000) (1)
{ln_p} ~ normal(0,10000)
(1) Parameters are elements of the linear form xb__t.
Bayesian Weibull PH regression MCMC iterations = 12,500 Random-walk Metropolis-Hastings sampling Burn-in = 2,500 MCMC sample size = 10,000 No. of subjects = 148 Number of obs = 206 No. of failures = 37 No. at risk = 1703 Acceptance rate = .368 Efficiency: min = .05571 avg = .09994 Log marginal likelihood = -107.88854 max = .1767
Equal-tailed
Haz. Ratio Std. Dev. MCSE Median [95% Cred. Interval]
_t
protect .0956023 .0338626 .001435 .0899154 .0463754 .1787249
age 1.103866 .0379671 .001313 1.102685 1.033111 1.180283
_cons .0075815 .0411427 .000979 .000567 4.02e-06 .0560771
ln_p .4473869 .1285796 .004443 .4493192 .1866153 .6912467
Note: Estimates are transformed only in the first equation. Note: _cons estimates baseline hazard. Note: Default priors are used for model parameters.

Because the default priors used are noninformative for these data, the above results are similar to those obtained from streg. Instead of the default priors, you can specify your own; see Custom priors.

The hazard ratios are reported by default, but you can use the nohr option with bayes, during estimation or on replay, to report coefficients. Alternatively, you can specify this option with streg during estimation.

. bayes, nohr

Equal-tailed
Mean Std. Dev. MCSE Median [95% Cred. Interval]
_t
protect -2.407909 .3482806 .015077 -2.408886 -3.070986 -1.721908
age .0982285 .0343418 .001189 .0977484 .0325748 .165754
_cons -7.561389 2.474563 .084712 -7.475201 -12.42343 -2.881028
ln_p .4473869 .1285796 .004443 .4493192 .1866153 .6912467

Unlike streg, bayes: streg reports only the log of the shape parameter. We can use the bayesstats summary command ([BAYES] bayesstats summary) to obtain the estimates of the shape parameter and its reciprocal.

. bayesstats summary (p: exp({ln_p})) (sigma: 1/exp({ln_p}))

Posterior summary statistics                      MCMC sample size =    10,000

           p : exp({ln_p})
       sigma : 1/exp({ln_p})

Equal-tailed
Mean Std. Dev. MCSE Median [95% Cred. Interval]
p 1.577122 .201685 .006993 1.567245 1.205164 1.996203
sigma .6446338 .0839366 .002879 .6380624 .5009511 .8297629

Tell me more

Learn more about the general features of the bayes prefix.

Learn more about Stata's Bayesian analysis and survival-time features.

Read more about the bayes prefix and Bayesian analysis in the Stata Bayesian Analysis Reference Manual.


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