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## Using gsem to combine estimation results

gsem is a very flexible command that allows us to fit very sophisticated models. However, it is also useful in situations that involve simple models.

For example, when we want to compare parameters among two or more models, we usually use suest, which combines the estimation results under one parameter vector and creates a simultaneous covariance matrix of the robust type. This covariance estimate is described in the Methods and formulas of [R] suest as the robust variance from a “stacked model”. Actually, gsem can estimate these kinds of “stacked models”, even if the estimation samples are not the same and eventually overlap. By using the option vce(robust), we can replicate the results from suest if the models are available for gsem. In addition, gsem allows us to combine results from some estimation commands that are not supported by suest, like models including random effects.

### Example: Comparing parameters from two models

Let’s consider the childweight dataset, described in [ME] mixed. Consider the following models, where weights of boys and girls are modeled using the age and the age-squared:

. webuse childweight, clear
(Weight data on Asian children)

. regress  weight age c.age#c.age if girl == 0, noheader
------------------------------------------------------------------------------
weight |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
age |   7.985022   .6343855    12.59   0.000     6.725942    9.244101
|
c.age#c.age |   -1.74346   .2374504    -7.34   0.000    -2.214733   -1.272187
|
_cons |   3.684363   .3217223    11.45   0.000     3.045833    4.322893
------------------------------------------------------------------------------

. regress  weight age c.age#c.age if girl == 1, noheader
------------------------------------------------------------------------------
weight |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
age |   7.008066   .5164687    13.57   0.000     5.982746    8.033386
|
c.age#c.age |  -1.450582   .1930318    -7.51   0.000    -1.833798   -1.067365
|
_cons |   3.480933   .2616616    13.30   0.000     2.961469    4.000397
------------------------------------------------------------------------------


To test whether birthweights are the same for the two groups, we need to test whether the intercepts in the two regressions are the same. Using suest, we would proceed as follows:

. quietly regress weight age c.age#c.age if girl == 0, noheader

. estimates store boys

. quietly regress weight age c.age#c.age if girl == 1, noheader

. estimates store girls

. suest boys girls

Simultaneous results for boys, girls

Number of obs   =        198

------------------------------------------------------------------------------
|               Robust
|      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
boys_mean    |
age |   7.985022   .4678417    17.07   0.000     7.068069    8.901975
|
c.age#c.age |   -1.74346   .2034352    -8.57   0.000    -2.142186   -1.344734
|
_cons |   3.684363   .1719028    21.43   0.000      3.34744    4.021286
-------------+----------------------------------------------------------------
boys_lnvar   |
_cons |   .4770289   .1870822     2.55   0.011     .1103546    .8437032
-------------+----------------------------------------------------------------
girls_mean   |
age |   7.008066   .4166916    16.82   0.000     6.191365    7.824766
|
c.age#c.age |  -1.450582   .1695722    -8.55   0.000    -1.782937   -1.118226
|
_cons |   3.480933   .1556014    22.37   0.000      3.17596    3.785906
-------------+----------------------------------------------------------------
girls_lnvar  |
_cons |   .0097127   .1351769     0.07   0.943    -.2552292    .2746545
------------------------------------------------------------------------------


Invoking an estimation command with the option coeflegend will give us a legend we can use to refer to the parameters when we use postestimation commands like test.

. suest, coeflegend

Simultaneous results for boys, girls

Number of obs   =        198

------------------------------------------------------------------------------
|      Coef.  Legend
-------------+----------------------------------------------------------------
boys_mean    |
age |   7.985022  _b[boys_mean:age]
|
c.age#c.age |   -1.74346  _b[boys_mean:c.age#c.age]
|
_cons |   3.684363  _b[boys_mean:_cons]
-------------+----------------------------------------------------------------
boys_lnvar   |
_cons |   .4770289  _b[boys_lnvar:_cons]
-------------+----------------------------------------------------------------
girls_mean   |
age |   7.008066  _b[girls_mean:age]
|
c.age#c.age |  -1.450582  _b[girls_mean:c.age#c.age]
|
_cons |   3.480933  _b[girls_mean:_cons]
-------------+----------------------------------------------------------------
girls_lnvar  |
_cons |   .0097127  _b[girls_lnvar:_cons]
------------------------------------------------------------------------------

. test  _b[boys_mean:_cons] = _b[girls_mean:_cons]

( 1)  [boys_mean]_cons - [girls_mean]_cons = 0

chi2(  1) =    0.77
Prob > chi2 =    0.3803


We find no evidence that the intercepts are different.

Now, let’s replicate those results by using the gsem command. We generate the variable weightboy, a copy of weight for boys and missing otherwise, and the variable weightgirl, a copy of weight for girls and missing otherwise.

. quietly generate weightboy = weight if girl == 0

. quietly generate weightgirl = weight if girl == 1

. gsem (weightboy <- age c.age#c.age) (weightgirl <- age c.age#c.age), ///
>      nolog vce(robust)

Generalized structural equation model             Number of obs   =        198
Log pseudolikelihood =  -302.2308

-------------------------------------------------------------------------------
|              Robust
|      Coef.  Std. Err.     z   P>|z|     [95% Conf. Interval]
-----------------+-------------------------------------------------------------
weightboy <-     |
age |   7.985022  .4678417   17.07  0.000     7.068069    8.901975
|
c.age#c.age |   -1.74346  .2034352   -8.57  0.000    -2.142186   -1.344734
|
_cons |   3.684363  .1719028   21.43  0.000      3.34744    4.021286
-----------------+-------------------------------------------------------------
weightgirl <-    |
age |   7.008066  .4166916   16.82  0.000     6.191365    7.824766
|
c.age#c.age |  -1.450582  .1695722   -8.55  0.000    -1.782937   -1.118226
|
_cons |   3.480933  .1556014   22.37  0.000      3.17596    3.785906
-----------------+-------------------------------------------------------------
var(e.weightboy)|   1.562942  .3014028                    1.071012    2.280821
var(e.weightgirl)|    .978849  .1364603                    .7448187    1.286414
-------------------------------------------------------------------------------

. gsem, coeflegend

Generalized structural equation model             Number of obs   =        198
Log pseudolikelihood =  -302.2308

-------------------------------------------------------------------------------
|      Coef.  Legend
-----------------+-------------------------------------------------------------
weightboy <-     |
age |   7.985022  _b[weightboy:age]
|
c.age#c.age |   -1.74346  _b[weightboy:c.age#c.age]
|
_cons |   3.684363  _b[weightboy:_cons]
-----------------+-------------------------------------------------------------
weightgirl <-    |
age |   7.008066  _b[weightgirl:age]
|
c.age#c.age |  -1.450582  _b[weightgirl:c.age#c.age]
|
_cons |   3.480933  _b[weightgirl:_cons]
-----------------+-------------------------------------------------------------
var(e.weightboy)|   1.562942  _b[var(e.weightboy):_cons]
var(e.weightgirl)|    .978849  _b[var(e.weightgirl):_cons]
-------------------------------------------------------------------------------

. test  _b[weightgirl:_cons]=  _b[weightboy:_cons]

( 1)  - [weightboy]_cons + [weightgirl]_cons = 0

chi2(  1) =    0.77
Prob > chi2 =    0.3803


gsem allowed us to fit models on different subsets simultaneously. By default, the model is assumed to be a linear regression, but several links and families are available; for example, you can combine two Poisson models or a multinomial logistic model with a regular logistic model. See [SEM] sem and gsem for details.

Here, I use the vce(robust) option to replicate the results for suest. However, when estimation samples don’t overlap, results from both estimations are assumed to be independent, and thus the option vce(robust) is not needed. When performing the estimation without the vce(robust) option, the joint covariance matrix will contain two blocks with the covariances from the original models and 0s outside those blocks.

### An example with random effects

The childweight dataset contains repeated measures, and it is, in the documentation, analyzed used the mixed command, which allows us to account for the intra-individual correlation via random effects.

Now, let’s use the techniques described above to combine results from two random-effects models. Here are the two separate models:

. mixed weight age c.age#c.age if girl == 0 || id:, nolog

Mixed-effects ML regression                     Number of obs      =       100
Group variable: id                              Number of groups   =        34

Obs per group: min =         1
avg =       2.9
max =         5

Wald chi2(2)       =   1070.28
Log likelihood = -149.05479                     Prob > chi2        =    0.0000

------------------------------------------------------------------------------
weight |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
age |   8.328882   .4601093    18.10   0.000     7.427084    9.230679
|
c.age#c.age |  -1.859798   .1722784   -10.80   0.000    -2.197458   -1.522139
|
_cons |   3.525929   .2723617    12.95   0.000      2.99211    4.059749
------------------------------------------------------------------------------

------------------------------------------------------------------------------
Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Identity                 |
var(_cons) |   .7607779   .2439115      .4058409    1.426133
-----------------------------+------------------------------------------------
var(Residual) |   .7225673   .1236759      .5166365    1.010582
------------------------------------------------------------------------------
LR test vs. linear regression: chibar2(01) =    30.34 Prob >= chibar2 = 0.0000

. mixed weight age c.age#c.age if girl == 1 || id:, nolog

Mixed-effects ML regression                     Number of obs      =        98
Group variable: id                              Number of groups   =        34

Obs per group: min =         1
avg =       2.9
max =         5

Wald chi2(2)       =   2141.72
Log likelihood =  -114.3008                     Prob > chi2        =    0.0000

------------------------------------------------------------------------------
weight |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
age |   7.273082   .3167266    22.96   0.000     6.652309    7.893854
|
c.age#c.age |  -1.538309    .118958   -12.93   0.000    -1.771462   -1.305156
|
_cons |   3.354834   .2111793    15.89   0.000      2.94093    3.768738
------------------------------------------------------------------------------

------------------------------------------------------------------------------
Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Identity                 |
var(_cons) |   .6925554   .1967582       .396848    1.208606
-----------------------------+------------------------------------------------
var(Residual) |   .3034231   .0535359      .2147152    .4287799
------------------------------------------------------------------------------
LR test vs. linear regression: chibar2(01) =    47.42 Prob >= chibar2 = 0.0000


Random effects can be included in a gsem model by incorporating latent variables at the group level; these are the latent variables M1[id] and M2[id] below. By default, gsem will try to estimate a covariance when it sees two latent variables at the same level. This can be easily solved by restricting this covariance term to 0. Option vce(robust) should be used whenever we want to produce the mechanism used by suest.

. gsem (weightboy <- age c.age#c.age M1[id])   ///
>      (weightgirl <- age c.age#c.age M2[id]), ///
>      cov(M1[id]*M2[id]@0) vce(robust) nolog

Generalized structural equation model             Number of obs   =        198
Log pseudolikelihood = -263.35559

( 1)  [weightboy]M1[id] = 1
( 2)  [weightgirl]M2[id] = 1
(Std. Err. adjusted for clustering on id)
-------------------------------------------------------------------------------
|              Robust
|      Coef.  Std. Err.     z   P>|z|     [95% Conf. Interval]
-----------------+-------------------------------------------------------------
weightboy <-     |
age |   8.328882  .4211157   19.78  0.000      7.50351    9.154253
|
c.age#c.age |  -1.859798  .1591742  -11.68  0.000    -2.171774   -1.547823
|
M1[id] |          1 (constrained)
|
_cons |   3.525929  .1526964   23.09  0.000      3.22665    3.825209
-----------------+-------------------------------------------------------------
weightgirl <-    |
age |   7.273082  .3067378   23.71  0.000     6.671887    7.874277
|
c.age#c.age |  -1.538309   .120155  -12.80  0.000    -1.773808    -1.30281
|
M2[id] |          1 (constrained)
|
_cons |   3.354834  .1482248   22.63  0.000     3.064319     3.64535
-----------------+-------------------------------------------------------------
var(M1[id])|   .7607774  .2255575                     .4254915    1.360268
var(M2[id])|   .6925553  .1850283                    .4102429    1.169144
-----------------+-------------------------------------------------------------
var(e.weightboy)|   .7225674  .1645983                     .4623572    1.129221
var(e.weightgirl)|   .3034231  .0667975                    .1970877    .4671298
-------------------------------------------------------------------------------


Above, we have the joint output from the two models, which would allow us to perform tests among parameters in both models. Notice that option vce(robust) implies that standard errors will be clustered on the groups determined by id.

gsem, when called with the vce(robust) option, will complain if there are inconsistencies among the groups in the models (for example, if the random effects in both models were crossed).

### Checking that you are fitting the same model

In the previous model, gsem‘s default covariance structure included a term that wasn’t in the original two models, so we needed to include an additional restriction. This can be easy to spot in a simple model, but if you don’t want to rely just on a visual inspection, you can write a small loop to make sure that all the estimates in the joint model are actually also in the original models.

Let’s see an example with random effects, this time with overlapping data.

. *fit first model and save the estimates
. gsem (weightboy <- age c.age#c.age M1[id]), nolog

Generalized structural equation model             Number of obs   =        100
Log likelihood = -149.05479

( 1)  [weightboy]M1[id] = 1
-------------------------------------------------------------------------------
|      Coef.  Std. Err.     z    P>|z|     [95% Conf. Interval]
----------------+--------------------------------------------------------------
weightboy <-    |
age |   8.328882  .4609841   18.07   0.000     7.425369    9.232394
|
c.age#c.age |  -1.859798  .1725233  -10.78   0.000    -2.197938   -1.521659
|
M1[id] |          1 (constrained)
|
_cons |   3.525929  .2726322   12.93   0.000      2.99158    4.060279
----------------+--------------------------------------------------------------
var(M1[id])|   .7607774  .2439114                     .4058407    1.426132
----------------+--------------------------------------------------------------
var(e.weightboy)|   .7225674  .1236759                     .5166366    1.010582
-------------------------------------------------------------------------------

. mat b1 = e(b)

. *fit second model and save the estimates
. gsem (weight <- age M2[id]), nolog

Generalized structural equation model             Number of obs   =        198
Log likelihood = -348.32402

( 1)  [weight]M2[id] = 1
------------------------------------------------------------------------------
|      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
weight <-    |
age |   3.389281   .1152211    29.42   0.000     3.163452    3.615111
|
M2[id] |          1  (constrained)
|
_cons |   5.156913   .1803059    28.60   0.000      4.80352    5.510306
-------------+----------------------------------------------------------------
var(M2[id])|   .6076662   .2040674                      .3146395    1.173591
-------------+----------------------------------------------------------------
var(e.weight)|   1.524052   .1866496                      1.198819    1.937518
------------------------------------------------------------------------------

. mat b2 = e(b)

. *stack estimates from first and second models
. mat stacked = b1, b2

. *estimate joint model and save results
. gsem (weightboy <- age c.age#c.age M1[id]) ///
>      (weight <- age M2[id]), cov(M1[id]*M2[id]@0) vce(robust) nolog

Generalized structural equation model             Number of obs   =        198
Log pseudolikelihood = -497.37881

( 1)  [weightboy]M1[id] = 1
( 2)  [weight]M2[id] = 1
(Std. Err. adjusted for clustering on id)
-------------------------------------------------------------------------------
|              Robust
|      Coef.  Std. Err.     z    P>|z|     [95% Conf. Interval]
----------------+--------------------------------------------------------------
weightboy <-    |
age |   8.328882  .4211157   19.78   0.000      7.50351    9.154253
|
c.age#c.age |  -1.859798  .1591742  -11.68   0.000    -2.171774   -1.547823
|
M1[id] |          1 (constrained)
|
_cons |   3.525929  .1526964   23.09   0.000      3.22665    3.825209
----------------+--------------------------------------------------------------
weight <-       |
age |   3.389281  .1157835   29.27   0.000      3.16235    3.616213
|
M2[id] |          1 (constrained)
|
_cons |   5.156913  .1345701   38.32   0.000      4.89316    5.420665
----------------+--------------------------------------------------------------
var(M1[id])|   .7607774  .2255575                     .4254915    1.360268
var(M2[id])|   .6076662     .1974                     .3214791    1.148623
----------------+--------------------------------------------------------------
var(e.weightboy)|   .7225674  .1645983                     .4623572    1.129221
var(e.weight)|   1.524052  .1705637                     1.223877    1.897849
-------------------------------------------------------------------------------

. mat b = e(b)

. *verify that estimates from the joint model are the same as
. *from models 1 and 2
. local stripes : colfullnames(b)

. foreach l of local stripes{
2.    matrix  r1 =  b[1,"l'"]
3.    matrix r2 = stacked[1,"l'"]
4.    assert reldif(el(r1,1,1), el(r2,1,1))<1e-5
5. }


The loop above verifies that all the labels in the second model correspond to estimates in the first and that the estimates are actually the same. If you omit the restriction for the variance in the joint model, the assert command will produce an error.

### Technical note

As documented in [U] 20.21.2 Correlated errors: Cluster-robust standard errors, the formula for the robust estimator of the variance is

$V_{robust} = \hat V(\sum_{j=1}^N u’_ju_j) \hat V$

where $$N$$ is the number of observations, $$\hat V$$ is the conventional estimator of the variance, and for each observation $$j$$, $$u_j$$ is a row vector (with as many columns as parameters), which represents the contribution of this observation to the gradient. (If we stack the rows $$u_j$$, the columns of this matrix are the scores.)

When we apply suest, the matrix $$\hat V$$ is constructed as the stacked block-diagonal conventional variance estimates from the original submodels; this is the variance you will see if you apply gsem to the joint model without the vce(robust) option. The $$u_j$$ values used by suest are now the values from both estimations, so we have as many $$u_j$$ values as the sum of observations in the two original models and each row contains as many columns as the total number of parameters in both models. This is the exact operation that gsem, vce(robust) does.

When random effects are present, standard errors will be clustered on groups. Instead of observation-level contributions to the gradient, we would use cluster-level contributions. This means that observations in the two models would need to be clustered in a consistent manner; observations that are common to the two estimations would need to be in the same cluster in the two estimations.

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