Calculating standard deviations to use in approximating beta distributions

[Emily McPherson]

Hello all, my name is Emily McPherson. I sent this question to Statalist, but the more is merrier, yes? ;)


My question relates to calculating the standard deviation (SD) of transition probabilities derived from coefficients estimated through Weibull regression in Stata.

The transition probabilities are being used to model disease progression of leukemia patients over 40 cycles of 90 days (about 10 years). I need the SDs of the probabilities (which change over the run of the Markov model) to create beta distributions whose parameters can be approximated using the corresponding Markov cycle probability and its SD. These distributions are then used to do Probabilistic sensitivity analysis, i.e., they are substituted for the simple probabilities (one for each cycle) and random draws from them can evaluate the robustness of the model’s cost-effectiveness results.

Anyway, using time to event survival data, I’ve used regression analysis to estimate coefficients that can be plugged into an equation to generate transition probabilities. For example...


. streg, nohr dist(weibull)

        failure _d:  event
   analysis time _t:  time

Fitting constant-only model:

Iteration 0:   log likelihood = -171.82384
Iteration 1:   log likelihood = -158.78902
Iteration 2:   log likelihood = -158.64499
Iteration 3:   log likelihood = -158.64497
Iteration 4:   log likelihood = -158.64497

Fitting full model:
Iteration 0:   log likelihood = -158.64497 

Weibull regression -- log relative-hazard form

No. of subjects =           93                     Number of obs   =        93
No. of failures =           62
Time at risk    =        60250
                                                   LR chi2(0)      =     -0.00
Log likelihood  =   -158.64497                     Prob > chi2     =         .

------------------------------------------------------------------------------
          _t |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------
-------------+------
       _cons |  -4.307123   .4483219    -9.61   0.000    -5.185818   -3.428429
-------------+----------------------------------------------------------
-------------+------
       /ln_p |  -.4638212   .1020754    -4.54   0.000    -.6638854    -.263757
-------------+----------------------------------------------------------
-------------+------
           p |    .628876   .0641928                      .5148471    .7681602
         1/p |   1.590139   .1623141                      1.301812    1.942324


We then create the probabilities with an equation () that uses p and _cons as well as t for time (i.e., Markov cycle number) and u for cycle length (usually a year, mine is 90 days since I’m working with leukemia patients who are very likely to have an event, i.e., relapse or die).

So where lambda = p, gamma = (exp(_cons))

gen result = (exp((lambda*((t-u)^ (gamma)))-(lambda*(t^(gamma)))))

gen transitions = 1-result

Turning to the variability, I first calculate the standard errors for the coefficients


. nlcom (exp(_b[_cons])) (exp(_b[/ln_p]))

       _nl_1:  exp(_b[_cons])
       _nl_2:  exp(_b[/ln_p])

------------------------------------------------------------------------------
          _t |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------
-------------+------
       _nl_1 |   .0116539   .0044932     2.59   0.009     .0028474    .0204604
       _nl_2 |   .6153864    .054186    11.36   0.000     .5091838     .721589


But what I’m really after is the standard errors on the transitions values, e.g.,

nlcom (_b[transitions])

But this doesn’t work. Any feedback on how to get closer to this idea would be much appreciated. Thanks in advance!

Emily