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st: unobs. heterogeneity in discrete hazard models


From   Jenkins S P <stephenj@essex.ac.uk>
To   Statalist <statalist@hsphsun2.harvard.edu>
Subject   st: unobs. heterogeneity in discrete hazard models
Date   Sat, 24 Sep 2005 18:43:49 +0100 (BST)

Gijs Dekkers asked about the interpretation of his estimates (reproduced below). My inclination would be to interpret these as he does, i.e. as suggesting that there is no 'significant' heterogeneity. If you added the trace option to the -pgmhaz8- command, you'll probably see the -ml- evaluator trying to estimate smaller and smaller values of the gamma variance but zero can never be reached given the way the model is parameterised (you'll see the logvariance becoming a larger and larger negative number). Further reassurance that the results are not a quirk might be gained by experimenting with different starting values for the gamma variance and seeing whether get same behaviour. You could also see what happens if you model frailty using discrete mass point approach (see -hshaz-)

I suspect that part of the "problem" is related to the short length of the ECHP panel which is used to create the spell data set used here. I conjecture that this makes it harder to distinguish frailty from duration dependence.

Stephen
=============================================
Professor Stephen P. Jenkins <stephenj@essex.ac.uk>
Institute for Social and Economic Research (ISER)
University of Essex, Colchester CO4 3SQ, UK
Phone: +44 1206 873374. Fax: +44 1206 873151.
http://www.iser.essex.ac.uk
Survival Analysis Using Stata: http://www.iser.essex.ac.uk/teaching/degree/stephenj/ec968/index.php




Date: Fri, 23 Sep 2005 11:55:51 +0200
From: Gijs Dekkers <gd@plan.be>
Subject: st: unobserved hetereogeneity and duration: interpreting pgmhaz8 and xtclog

Dear fellow Stata-users,

I am estimating a discrete duration model, explaining the probability
that a cohabiting (unmarried) individual (cohab=1) separates
i.e. no longer consensual union and not married after a certain time
(the variable 'duration'). The dataset is the European Comunity
Household Panel ECHP.

The variables are
   pid: unique person identifier
   duration: time (years)
   cosep: 0 if the individual lives in consensual union, 1=if (s)he
does not live in consensual union (and is not married)

The data is of the following form:
    +----------------------------+
    |     pid   duration   cosep |
    |----------------------------|
 1. | 1028101          1       0 |
 2. | 1028101          2       0 |
 3. | 1028105          1       0 |
 4. | 1028105          2       0 |
 5. | 2053101          1       0 |
    |----------------------------|
 6. | 2053102          1       0 |
 7. | 3023101          1       0 |
 8. | 3023101          2       0 |
 9. | 3023101          3       1 |
etc...

A first analysis (somewhat dissapointingly) showed that the only
significant explanatory variables are a function of duration. In fact,
the best model explains 'cosep' using 'duration' and its quadrature
'duration2'

. cloglog cosep duration duration2

Iteration 0:   log likelihood = -377.28398
Iteration 1:   log likelihood = -377.24866
Iteration 2:   log likelihood = -377.24865

Complementary log-log regression                Number of obs
=       2410
                                               Zero outcomes
=       2319
                                               Nonzero outcomes
=         91

                                               LR chi2(2)        =
20.35
Log likelihood = -377.24865                     Prob > chi2       =
0.0000

- ------------------------------------------------------------------------------
      cosep |      Coef.   Std. Err.      z    P>|z|     [95% Conf.
Interval]
- -------------+----------------------------------------------------------------
   duration |   .8310887   .2263603     3.67   0.000     .3874307
1.274747
  duration2 |  -.0825692   .0272601    -3.03   0.002     -.135998
- -.0291405
      _cons |  -4.782091   .4091147   -11.69   0.000    -5.583941
- -3.980241
- ------------------------------------------------------------------------------

However, I want to test for various parametric forms of frailty, using
Jenkins' Lesson 7 on 'unobserved heterogeneity'
(http://www.iser.essex.ac.uk/teaching/degree/stephenj/ec968/#_Toc520705914).

First, he suggests to test for heterogeneity assuming a normally
distributed frailty term (page 14).
. xtclog cosep duration duration2, nolog i(pid)

Random-effects complementary log-log model      Number of obs
=      2410
Group variable (i): pid                         Number of groups
=       739

Random effects u_i ~ Gaussian                   Obs per group: min
=         1
                                                              avg
=       3.3
                                                              max
=         8

                                               Wald chi2(2)       =
18.20
Log likelihood  = -377.24865                    Prob > chi2        =
0.0001

- ------------------------------------------------------------------------------
      cosep |      Coef.   Std. Err.      z    P>|z|     [95% Conf.
Interval]
- -------------+----------------------------------------------------------------
   duration |   .8310886   .2263602     3.67   0.000     .3874307
1.274747
  duration2 |  -.0825692   .0272601    -3.03   0.002     -.135998
- -.0291404
      _cons |  -4.782091   .4091147   -11.69   0.000    -5.583941
- -3.980241
- -------------+----------------------------------------------------------------
   /lnsig2u |        -14          .
.           .
- -------------+----------------------------------------------------------------
    sigma_u |   .0009119          .
.           .
        rho |   5.06e-07          .
.           .
- ------------------------------------------------------------------------------
Likelihood-ratio test of rho=0: chibar2(01) =     0.00 Prob >= chibar2 =
1.000

Now this already looks pretty strange to me, or is it my suspicious
mind? Can I safely coclude that the hypothesis of normally distributed
unobserved heterogeneity shoud (very much) be rejected?

Secondly, I used pgmhaz8 to test for gamma-distributed unobserved
heterogeneity. I found the pgmhaz8-manual at
http://ideas.repec.org/c/boc/bocode/s438501.html
If I understand this manual correctly (but I am not quite sure), the
model should be

. pgmhaz8 duration2, id(pid) dead(cosep) seq(duration)

(anyway, the model pgmhaz8 duration duration2 etc. does not converge)

The results are:

PGM hazard model without gamma frailty

Generalized linear models                          No. of obs
=      2410
Optimization     : ML                              Residual df
=      2408
                                                  Scale parameter
=         1
Deviance         =   769.387838                    (1/df) Deviance =
.3195132
Pearson          =  2400.608032                    (1/df) Pearson  =
.9969302

Variance function: V(u) = u*(1-u)                  [Bernoulli]
Link function    : g(u) = ln(-ln(1-u))             [Complementary log-log]

                                                  AIC             =
.3209078
Log likelihood   =  -384.693919                    BIC             =
- -17982.63

- ------------------------------------------------------------------------------
            |                 OIM
      cosep |      Coef.   Std. Err.      z    P>|z|     [95% Conf.
Interval]
- -------------+----------------------------------------------------------------
  duration2 |   .0135626   .0055006     2.47   0.014     .0027817
.0243435
      _cons |  -3.456552   .1401122   -24.67   0.000    -3.731167
- -3.181937
- ------------------------------------------------------------------------------

Iteration 0:   log likelihood = -385.00279
Iteration 1:   log likelihood = -384.79069
Iteration 2:   log likelihood = -384.73062
Iteration 3:   log likelihood = -384.70334
Iteration 4:   log likelihood = -384.69612
Iteration 5:   log likelihood =  -384.6944
Iteration 6:   log likelihood = -384.69403
Iteration 7:   log likelihood = -384.69394
Iteration 8:   log likelihood = -384.69392
Iteration 9:   log likelihood = -384.69392
Iteration 10:  log likelihood = -384.69392

PGM hazard model with gamma frailty               Number of obs
=       2410
                                                 LR chi2()
=          .
Log likelihood = -384.69392                       Prob > chi2
=          .

- ------------------------------------------------------------------------------
      cosep |      Coef.   Std. Err.      z    P>|z|     [95% Conf.
Interval]
- -------------+----------------------------------------------------------------
hazard       |
  duration2 |   .0135593   .0055347     2.45   0.014     .0027114
.0244072
      _cons |  -3.456778    .141079   -24.50   0.000    -3.733287
- -3.180268
- -------------+----------------------------------------------------------------
ln_varg      |
      _cons |  -13.77345    952.569    -0.01   0.988    -1880.774
1853.228
- -------------+----------------------------------------------------------------
 Gamma var. |   1.04e-06   .0009935     0.00   0.999
0           .
- ------------------------------------------------------------------------------
LR test of Gamma var. = 0: chibar2(01) =  -8.9e-06  Prob.>=chibar2
=       .5

And here it is again: analogous to the results from the xtclog, the
hypothesis of gamma-distributed unobserved heterogeneity should be
rejected. However, again like the xtclog results, the above results of
pgmhaz8 suspiciously look like some sort of corner solution, or an
artefact.

And this (finally!) brings me to my question: can I trust these results
and safely conclude that the hypotheses of unobserved hetereogeneity
(both normally and gamma-distributed) should be rejected? Or is there
something else going on? If so, any suggestions?

Any help would be appreciated!

Gijs

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