St with shared frailty only

| Gabriel |

Several Stata commands in the xt family allow you to specify a random model (i.e., structured error terms) with no fixed model (i.e., independent variables). For instance:

xtreg y, re i(clustervar)
xtmixed y || clustervar:
gllamm y, i(clustervar)

This is very useful if the only thing you’re interested in is rho, the proportion of variance clustered within groups. To take a classic example of multilevel modeling, you might have test score data on students by classroom and you may be interested simply in how much good performance clusters by classroom (rho) before you get to independent variables like whether teacher credentials or class size matter.

In the [st] syntax, shared frailty is closely analogous to random effects (and strata are analogous to fixed effects). However unlike most xt commands, the st syntax expects there to be independent variables and it chokes if it doesn’t get them. Fortunately this is not a limitation of the model, only the syntax parsing, and you can trick Stata by feeding it a constant. It drops the constant from the model and estimates only the shared frailty. For instance, this model shows only the extent to which radio station adoptions of a particular song clustered by the stations’ corporate owners:

. gen x1=1

. streg x1, shared(owner_n) distribution(exponential)

Note: frailty(gamma) assumed.

         failure _d:  add
   analysis time _t:  (fpdate-origin)
             origin:  time firstevent
                 id:  station_n
note: x1 dropped because of collinearity

Fitting exponential model:

Iteration 0:   log likelihood =  -260.2503
Iteration 1:   log likelihood = -252.85211
Iteration 2:   log likelihood = -246.83872
Iteration 3:   log likelihood = -246.28281
Iteration 4:   log likelihood = -246.10157
Iteration 5:   log likelihood = -246.10117
Iteration 6:   log likelihood = -246.10117  

Exponential regression --
         log relative-hazard form               Number of obs      =       171
         Gamma shared frailty                   Number of groups   =        46
Group variable: owner_n

No. of subjects =          171                  Obs per group: min =         1
No. of failures =          164                                 avg =  3.717391
Time at risk    =         3739                                 max =        58

                                                F(   0,      .)    =         .
Log likelihood  =   -246.10117                  Prob > F           =         .

------------------------------------------------------------------------------
          _t | Haz. Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
-------------+----------------------------------------------------------------
     /ln_the |  -1.792063   .4458721    -4.02   0.000    -2.665956   -.9181693
-------------+----------------------------------------------------------------
       theta |   .1666161   .0742895                      .0695329    .3992493
------------------------------------------------------------------------------
Likelihood-ratio test of theta=0: chibar2(01) =    15.60 Prob>=chibar2 = 0.000

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