st: Mysterious Output Given by GLLAMM for Multiple-Equation Generalized Linear Model

["Weihao Yin" <alexyin@vt.edu>]

Hi, 

I am learning to use GLLAMM to estimate a multiple-equation Generalized
Linear Model, which jointly models binary and count data. The  dataset I use
is about the condition and length of hospital stay for 32 herniorrhaphy
patients. The dataset is freely available online. The response variables are
[leave] and [los], which denote the condition of the patient upon leaving
the operating room and the length of hospital stay after the operation. One
of the two covariate is [OKstatus] which distinguishes patients based on
their post-operative physical status with "1" indicating better status. The
other is patient's age. 

The model consists of two equations: one is logit model for [leave] and the
other is a Poisson model for [los]. The correlation between the two is
captured using a common random effect. Mathematically, it is written as:

g1([leave]) = b01+b11*[age]+b12*[OKstatus]+u+epsilon1
g2([los])   = b02+b12*[age]+b22*[OKstatus]+u+epsilon2

where u is the random effect and epsilon1&2 are errors. This example is also
used in the SAS GLIMMIX procedure documentation (pp. 203-209). I just try to
reproduce the results using GLLAMM. Since the random effect is used as an
intercept, I use a constraint to make the factor loadings for both equations
on the random effect equal to 1. The GLLAMM gives the following output, in
which [d1] is the dummy variable for logit and [d2] is Poisson. 

-------- Start of the GLLAMM Output --------------

gllamm model with constraints

log likelihood = -101.4974201636495

 ( 1)  [pat1_1l]d2 = 1
----------------------------------------------------------------------------
--
        resp |      Coef.   Std. Err.      z    P>|z|     [95% Conf.
Interval]
-------------+--------------------------------------------------------------
--
      age_d1 |  -.0774867   .0381989    -2.03   0.043    -.1523552
-.0026181
      age_d2 |   .0192609   .0073045     2.64   0.008     .0049444
.0335775
okstatus_d1  |  -.4720305   1.132692    -0.42   0.677    -2.692066
1.748005
okstatus_d2  |   -.193629   .2960872    -0.65   0.513    -.7739493
.3866912
          d1 |   5.904778   2.926447     2.02   0.044      .169047
11.64051
          d2 |   .7702758    .565793     1.36   0.173    -.3386582
1.87921
----------------------------------------------------------------------------
--

 
Variances and covariances of random effects
----------------------------------------------------------------------------
--


***level 2 (patient)

    var(1): .27889144 (.10479844)

    loadings for random effect 1
    d1: 1 (fixed)
    d2: 1 (0)

----------------------End of the GLLAMM
Output--------------------------------------------------------

The output generally is the same as given by the SAS GLIMMIX. However, when
I rerun the model using "allc" option to list all the estimated parameters,
here is what I got. 

----------------------------------------------------------------------------
--
        resp |      Coef.   Std. Err.      z    P>|z|     [95% Conf.
Interval]
-------------+--------------------------------------------------------------
--
resp         |
      age_d1 |  -.0774867   .0381989    -2.03   0.043    -.1523552
-.0026181
      age_d2 |   .0192609   .0073045     2.64   0.008     .0049444
.0335775
okstatus_d1  |  -.4720305   1.132692    -0.42   0.677    -2.692066
1.748005
okstatus_d2  |   -.193629   .2960872    -0.65   0.513    -.7739493
.3866912
          d1 |   5.904778   2.926447     2.02   0.044      .169047
11.64051
          d2 |   .7702758    .565793     1.36   0.173    -.3386582
1.87921
-------------+--------------------------------------------------------------
--
pat1_1l      |
          d2 |          1          .        .       .            .
.
-------------+--------------------------------------------------------------
--
pat1_1       |
          d1 |   .5281017   .0992218     5.32   0.000     .3336305
.7225729
----------------------------------------------------------------------------
--


The last parameter [pat1_1]d1 = 0.5281017. Does anyone know what this
parameter is? The first loading "d1" is supposed to be 1 and it is according
to the previous output. By the name of it, it seems to be the factor
loading. Does GLLAMM estimate the variance of the two epsilons? If it does
not, how can I calculate the correlation between the two responses, which is
the whole point of this?

I know it is a long post but I really need some help. Any input is greatly
appreciated. Thanks!


Weihao Yin
Ph.D Candidate
Department of Civil and Environmental Engineering
Virginia Polytechnic and State University


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