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How can I check measurement invariance using the sem command? (Stata 12)

Measurement invariance is a very important requisite in multiple group structural equation modeling. It attempts to verify that the factors are measuring the same underlying latent construct within each group.

This page will show the steps and the Stata code for checking invariance using a single factor example with two groups. The data are found in an ssd file named invariance.dta. There are five items in the data, cesd01, cesd02, cesd07, cesd11 and cesd20 along with a grouping variable grp. 

use http://www.ats.ucla.edu/stat/data/invariance.dta, clear
We will skip over several very necessary preliminary steps assuming that separate CFAs for each group have been estimated along with a combined CFA on all observations treated as a single group and have displayed no major problems. We are also going to assume that each group has the same number of factors, in this case, one.

We will begin by estimating a two group model in which all parameters are freely estimated. For model identification purposes we will fix the loading for the first item at one and the item intercepts at zero. We will estimate the factor variance for group 1 instead of sem's default of zero. This model with all parameters free gives us an idea of the best fit for the measurement model that we can obtain with these data. 

* model 1 -- free all parameters

sem (Somatic -> cesd01@1 cesd02 cesd07 cesd11 cesd20) ///
    (Somatic -> cesd01@1)       ///      /* set loading   to 1 in both groups */ 
    (cesd01 <- Somatic _cons@0) ///      /* set intercept to 0 in both groups */                         
    , group(grp)                ///
    ginvariant(none)            ///      /* no invariant parameters           */
    mean(1: Somatic@m1)                  /* estimate mean for grp 1           */
    

Endogenous variables

Measurement:  cesd01 cesd02 cesd07 cesd11 cesd20

Exogenous variables

Latent:       Somatic

Fitting target model:

Iteration 0:   log likelihood = -13308.842  
Iteration 1:   log likelihood = -13051.707  (not concave)
[output omitted]
Iteration 47:  log likelihood = -12473.244  
Iteration 48:  log likelihood = -12473.244  

Structural equation model                       Number of obs      =      2252
Grouping variable  = grp                        Number of groups   =         2
Estimation method  = ml
Log likelihood     = -12473.244

 ( 1)  [cesd01]1bn.grp#c.Somatic = 1
 ( 2)  [cesd01]1bn.grp = 0
 ( 3)  [cesd01]2.grp#c.Somatic = 1
 ( 4)  [cesd01]2.grp = 0
------------------------------------------------------------------------------
             |                 OIM
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
Measurement  |
  cesd01 <-  |
    Somatic  |
         [*] |          1  (constrained)
    _cons    |
         [*] |          0  (constrained)
  -----------+----------------------------------------------------------------
  cesd02 <-  |
    Somatic  |
          1  |    .928435    .062672    14.81   0.000     .8056002     1.05127
          2  |   .8385611   .1891791     4.43   0.000     .4677769    1.209345
    _cons    |
          1  |  -.0648223   .0273122    -2.37   0.018    -.1183533   -.0112914
          2  |   .0525936   .0860614     0.61   0.541    -.1160837    .2212709
  -----------+----------------------------------------------------------------
  cesd07 <-  |
    Somatic  |
          1  |   1.677874   .1007259    16.66   0.000     1.480455    1.875293
          2  |   1.758985   .3554401     4.95   0.000     1.062335    2.455634
    _cons    |
          1  |  -.0653832   .0428129    -1.53   0.127    -.1492949    .0185285
          2  |   .1893793   .1541684     1.23   0.219    -.1127853    .4915438
  -----------+----------------------------------------------------------------
  cesd11 <-  |
    Somatic  |
          1  |   1.388108   .0901113    15.40   0.000     1.211494    1.564723
          2  |   1.341272   .2818361     4.76   0.000      .788883     1.89366
    _cons    |
          1  |   .1565085   .0391205     4.00   0.000     .0798337    .2331832
          2  |   .1181773   .1226974     0.96   0.335    -.1223051    .3586597
  -----------+----------------------------------------------------------------
  cesd20 <-  |
    Somatic  |
          1  |   1.474365    .090211    16.34   0.000     1.297555    1.651175
          2  |   1.491461    .282652     5.28   0.000     .9374732    2.045449
    _cons    |
          1  |  -.0469629   .0383135    -1.23   0.220     -.122056    .0281301
          2  |  -.0675705   .1211398    -0.56   0.577    -.3050001    .1698591
-------------+----------------------------------------------------------------
Mean         |
  Somatic    |
          1  |     .32834   .0157635    20.83   0.000     .2974441    .3592359
          2  |     .35081   .0446252     7.86   0.000     .2633461    .4382738
-------------+----------------------------------------------------------------
Variance     |
  e.cesd01   |
          1  |   .3762446   .0136731                      .3503779    .4040209
          2  |   .3686835   .0406348                      .2970555    .4575828
  e.cesd02   |
          1  |   .3219994   .0117202                      .2998284    .3458098
          2  |    .485269   .0485134                      .3989202    .5903084
  e.cesd07   |
          1  |   .4220553   .0197965                      .3849849    .4626952
          2  |   .8977649   .1035432                      .7161266    1.125474
  e.cesd11   |
          1  |   .5876808   .0221566                      .5458203    .6327516
          2  |   .6459501   .0705419                      .5214856     .800121
  e.cesd20   |
          1  |   .3656715   .0162806                      .3351148    .3990144
          2  |   .3808461   .0549203                      .2870787    .5052403
  Somatic    |
          1  |   .1217268    .012324                      .0998179    .1484445
          2  |   .1251867   .0377611                      .0693111    .2261066
------------------------------------------------------------------------------
Note: [*] identifies parameter estimates constrained to be equal across
      groups.
LR test of model vs. saturated: chi2(10)  =     25.11, Prob > chi2 = 0.0051
We will record the chi-square value (25.11) and degrees of freedom (10) for comparison with the next model. The p-value for this model (.0051) is indicates that this does not fit our particularly well. We will ignore this issue to focus on the steps in checking factor invariance.

Checking factor invariance involves a hierarchy of steps that progressively constrains more parameters to be equal across the groups. Here is a listing the steps in the hierarchy along with which parameters are being constrained.

model 1 -- all parameters free (above)
model 2 -- metric (pattern) invariance -- loadings are invariant
model 3 -- strong (scalar) invariance  -- loadings & intercepts are invariant
model 4 -- strict invariance           -- loadings, intercepts & residuals are invariant
model 5 -- strict invariance plus factor means are invariant
model 6 -- strict invariance plus factor means & variances are invariant
Now we can check for metric invariance. 
* model 2 -- metric invariance -- loadings invariant
sem (Somatic -> cesd01 cesd02 cesd07 cesd11 cesd20) ///
    (cesd01 <- Somatic _cons@0)  ///     /* fix intercept at 0 both groups */
    (Somatic->cesd07@l1)         ///     /* constrain loadings   */
    (1: cesd07@i1)(2: cesd07@i2) ///     /* free intercepts      */
    (Somatic->cesd11@l2)         ///     /* constrain loadings   */
    (1: cesd11@i3)(2: cesd07@i4) ///     /* free intercepts      */
    (Somatic->cesd20@l3)         ///     /* constrain loadings   */
    (1: cesd20@i5)(2: cesd20@i6) ///     /* free intercepts      */
    (Somatic->cesd02@l4)         ///     /* constrain loadings   */
    (1: cesd02@i7)(2: cesd02@i8) ///     /* free intercepts      */
    , group(grp)                 ///
    mean(1: Somatic@m1)                  /* estimate factor mean for grp 1   */

Endogenous variables

Observed:     cesd02 cesd07 cesd11 cesd20
Measurement:  cesd01

Exogenous variables

Latent:       Somatic

Fitting target model:

Iteration 0:   log likelihood = -13284.565  
Iteration 1:   log likelihood = -12652.048  
Iteration 2:   log likelihood = -12484.832  
Iteration 3:   log likelihood = -12474.294  
Iteration 4:   log likelihood = -12473.477  
Iteration 5:   log likelihood = -12473.472  
Iteration 6:   log likelihood = -12473.472  

Structural equation model                       Number of obs      =      2252
Grouping variable  = grp                        Number of groups   =         2
Estimation method  = ml
Log likelihood     = -12473.472

 ( 1)  [cesd01]1bn.grp#c.Somatic = 1
 ( 2)  [cesd02]1bn.grp#c.Somatic - [cesd02]2.grp#c.Somatic = 0
 ( 3)  [cesd07]1bn.grp#c.Somatic - [cesd07]2.grp#c.Somatic = 0
 ( 4)  [cesd11]1bn.grp#c.Somatic - [cesd11]2.grp#c.Somatic = 0
 ( 5)  [cesd20]1bn.grp#c.Somatic - [cesd20]2.grp#c.Somatic = 0
 ( 6)  [cesd01]1bn.grp = 0
 ( 7)  [cesd01]2.grp#c.Somatic = 1
 ( 8)  [cesd01]2.grp = 0
------------------------------------------------------------------------------
             |                 OIM
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
Structural   |
  cesd02 <-  |
    Somatic  |
         [*] |   .9209276   .0593449    15.52   0.000     .8046138    1.037242
    _cons    |
          1  |  -.0623574   .0264546    -2.36   0.018    -.1142075   -.0105072
          2  |   .0236994   .0601589     0.39   0.694    -.0942099    .1416087
  -----------+----------------------------------------------------------------
  cesd07 <-  |
    Somatic  |
         [*] |   1.681261   .0965874    17.41   0.000     1.491953    1.870568
    _cons    |
          1  |  -.0664951   .0417905    -1.59   0.112     -.148403    .0154128
          2  |    .216647   .0949812     2.28   0.023     .0304873    .4028067
  -----------+----------------------------------------------------------------
  cesd11 <-  |
    Somatic  |
         [*] |   1.381923   .0856666    16.13   0.000     1.214019    1.549826
    _cons    |
          1  |   .1585395   .0379905     4.17   0.000     .0840794    .2329995
          2  |   .1039177   .0794803     1.31   0.191    -.0518609    .2596962
  -----------+----------------------------------------------------------------
  cesd20 <-  |
    Somatic  |
         [*] |   1.474055    .085681    17.20   0.000     1.306123    1.641987
    _cons    |
          1  |  -.0468612   .0371712    -1.26   0.207    -.1197153     .025993
          2  |  -.0614632   .0753769    -0.82   0.415    -.2091993    .0862729
-------------+----------------------------------------------------------------
Measurement  |
  cesd01 <-  |
    Somatic  |
         [*] |          1  (constrained)
    _cons    |
         [*] |          0  (constrained)
-------------+----------------------------------------------------------------
Mean         |
  Somatic    |
          1  |     .32834    .015768    20.82   0.000     .2974353    .3592447
          2  |     .35081   .0445244     7.88   0.000     .2635438    .4380762
-------------+----------------------------------------------------------------
Variance     |
  e.cesd01   |
          1  |   .3762228   .0136197                      .3504536    .4038867
          2  |   .3673419   .0376073                      .3005572    .4489665
  e.cesd02   |
          1  |   .3225452   .0116904                      .3004274    .3462914
          2  |   .4785012   .0466075                      .3953424    .5791521
  e.cesd07   |
          1  |   .4209217   .0196516                      .3841149    .4612554
          2  |   .9142389   .0951236                      .7455811    1.121049
  e.cesd11   |
          1  |   .5883498   .0220535                      .5466753    .6332013
          2  |   .6411438   .0662853                      .5235438    .7851594
  e.cesd20   |
          1  |   .3654038   .0161423                      .3350965    .3984522
          2  |   .3868222   .0466565                      .3053824    .4899806
  Somatic    |
          1  |   .1220313   .0118595                      .1008668    .1476367
          2  |   .1242983   .0198831                      .0908457    .1700693
------------------------------------------------------------------------------
Note: [*] identifies parameter estimates constrained to be equal across
      groups.
LR test of model vs. saturated: chi2(14)  =     25.57, Prob > chi2 = .97727633
For this model chi-square equals 25.57 with 10 degrees of freedom. We can compare this with model 1 by computing the differences in chi-square and degrees of freedom. 
chi2 = 25.57 - 25.11 = .46 with df(14-10 = 4), p-value = 2.463e-09
The extremely small change in chi-square tells us that the model in which the factor loadings are constrained to be equal fits equally as well as the model with all parameters free to vary. The bottom line is that we have demonstrated metric factor invariance for this single factor model.

Next, we can check for strong invariance in which both factor loadings and intercepts are constrained to be equal across groups. By default, the sem estimates a model with strong invariance whenever you include the group option. Our version of strong invariance fits the same but is parameterized slightly differently. 

* model 3 -- strong invariance -- loadings & intercepts invariant
sem (Somatic -> cesd01 cesd02 cesd07 cesd11 cesd20) ///
    (cesd01 <- Somatic _cons@0) ///  /* fix intercept at 0 both groups   */
    , group(grp)                ///
    mean(1: Somatic@m1)              /* estimate factor mean for grp 1   */

Endogenous variables

Measurement:  cesd01 cesd02 cesd07 cesd11 cesd20

Exogenous variables

Latent:       Somatic

Fitting target model:

Iteration 0:   log likelihood = -13516.826  (not concave)
Iteration 1:   log likelihood = -12634.906  (not concave)
[output omitted]
Iteration 9:   log likelihood = -12483.842  
Iteration 10:  log likelihood = -12483.841  

Structural equation model                       Number of obs      =      2252
Grouping variable  = grp                        Number of groups   =         2
Estimation method  = ml
Log likelihood     = -12483.841

 ( 1)  [cesd01]1bn.grp#c.Somatic = 1
 ( 2)  [cesd02]1bn.grp#c.Somatic - [cesd02]2.grp#c.Somatic = 0
 ( 3)  [cesd07]1bn.grp#c.Somatic - [cesd07]2.grp#c.Somatic = 0
 ( 4)  [cesd11]1bn.grp#c.Somatic - [cesd11]2.grp#c.Somatic = 0
 ( 5)  [cesd20]1bn.grp#c.Somatic - [cesd20]2.grp#c.Somatic = 0
 ( 6)  [cesd01]1bn.grp = 0
 ( 7)  [cesd02]1bn.grp - [cesd02]2.grp = 0
 ( 8)  [cesd07]1bn.grp - [cesd07]2.grp = 0
 ( 9)  [cesd11]1bn.grp - [cesd11]2.grp = 0
 (10)  [cesd20]1bn.grp - [cesd20]2.grp = 0
 (11)  [cesd01]2.grp#c.Somatic = 1
 (12)  [cesd01]2.grp = 0
------------------------------------------------------------------------------
             |                 OIM
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
Measurement  |
  cesd01 <-  |
    Somatic  |
         [*] |          1  (constrained)
    _cons    |
         [*] |          0  (constrained)
  -----------+----------------------------------------------------------------
  cesd02 <-  |
    Somatic  |
         [*] |   .9249795   .0595205    15.54   0.000     .8083215    1.041638
    _cons    |
         [*] |  -.0560251   .0260192    -2.15   0.031    -.1070218   -.0050285
  -----------+----------------------------------------------------------------
  cesd07 <-  |
    Somatic  |
         [*] |   1.688943   .0971202    17.39   0.000     1.498591    1.879295
    _cons    |
         [*] |  -.0513029   .0412593    -1.24   0.214    -.1321696    .0295638
  -----------+----------------------------------------------------------------
  cesd11 <-  |
    Somatic  |
         [*] |   1.376843   .0854916    16.11   0.000     1.209283    1.544404
    _cons    |
         [*] |   .1552514    .037177     4.18   0.000     .0823858    .2281171
  -----------+----------------------------------------------------------------
  cesd20 <-  |
    Somatic  |
         [*] |   1.471068   .0854962    17.21   0.000     1.303498    1.638637
    _cons    |
         [*] |  -.0469581   .0364197    -1.29   0.197    -.1183394    .0244232
-------------+----------------------------------------------------------------
Mean         |
  Somatic    |
          1  |   .3246982   .0152092    21.35   0.000     .2948888    .3545076
          2  |   .3795509   .0299712    12.66   0.000     .3208084    .4382935
-------------+----------------------------------------------------------------
Variance     |
  e.cesd01   |
          1  |   .3763018   .0136198                      .3505323    .4039657
          2  |   .3675061   .0376885                      .3005883    .4493214
  e.cesd02   |
          1  |   .3223506   .0116913                      .3002315    .3460993
          2  |   .4804669   .0469221                      .3967671    .5818235
  e.cesd07   |
          1  |   .4199315   .0196947                      .3830515    .4603622
          2  |   .9738935   .1014744                      .7940008    1.194544
  e.cesd11   |
          1  |   .5892406   .0220642                      .5475444     .634112
          2  |   .6518477   .0673462                      .5323576    .7981579
  e.cesd20   |
          1  |    .366112   .0161388                      .3358084    .3991501
          2  |   .3924233   .0473101                      .3098381    .4970209
  Somatic    |
          1  |   .1218173   .0118432                      .1006825    .1473885
          2  |   .1229574    .019835                      .0896277    .1686813
------------------------------------------------------------------------------
Note: [*] identifies parameter estimates constrained to be equal across
      groups.
LR test of model vs. saturated: chi2(18)  =     46.31, Prob > chi2 = 0.0003
We can see how this model fits relative to metric invariance by differencing the chi-square from the previous model. 
chi2 = 46.31 - 25.57 = 20.74 with df(18-14=4), p-value = .00035656
The fit of the strong invariance model is significantly poorer than the fit of the metric invariance model. So, we cannot assume strong invariance.

There are still three steps remaining in the invariance testing hierarchy. Even though we did not meet the invariance requirement for step 3, we will demonstrate how to run the remaining steps beginning with strict factor invariance which constrains loadings, intercepts and residuals. 

* model 4 -- strict invariance -- loadings, intercepts & residuals invariant
sem (Somatic -> cesd01 cesd02 cesd07 cesd11 cesd20) ///
    (cesd01 <- Somatic _cons@0) ///  /* fix intercept at 0 both groups   */
    , group(grp)                ///
    variance(e.cesd01@v1)       ///  /* constrain residuals              */
    variance(e.cesd11@v2)       ///  /* constrain residuals              */
    variance(e.cesd20@v3)       ///  /* constrain residuals              */
    variance(e.cesd02@v4)       ///  /* constrain residuals              */
    variance(e.cesd07@v5)       ///  /* constrain residuals              */
    mean(1: Somatic@m1)              /* estimate factor mean for grp 1   */
    

Exogenous variables

Latent:       Somatic

Fitting target model:

Iteration 0:   log likelihood = -13493.497  (not concave)
[output omitted] 
Iteration 9:   log likelihood = -12524.619  

Structural equation model                       Number of obs      =      2252
Grouping variable  = grp                        Number of groups   =         2
Estimation method  = ml
Log likelihood     = -12524.619

 ( 1)  [cesd01]1bn.grp#c.Somatic = 1
 ( 2)  [cesd02]1bn.grp#c.Somatic - [cesd02]2.grp#c.Somatic = 0
 ( 3)  [cesd07]1bn.grp#c.Somatic - [cesd07]2.grp#c.Somatic = 0
 ( 4)  [cesd11]1bn.grp#c.Somatic - [cesd11]2.grp#c.Somatic = 0
 ( 5)  [cesd20]1bn.grp#c.Somatic - [cesd20]2.grp#c.Somatic = 0
 ( 6)  [var(e.cesd01)]1bn.grp - [var(e.cesd01)]2.grp = 0
 ( 7)  [var(e.cesd02)]1bn.grp - [var(e.cesd02)]2.grp = 0
 ( 8)  [var(e.cesd07)]1bn.grp - [var(e.cesd07)]2.grp = 0
 ( 9)  [var(e.cesd11)]1bn.grp - [var(e.cesd11)]2.grp = 0
 (10)  [var(e.cesd20)]1bn.grp - [var(e.cesd20)]2.grp = 0
 (11)  [cesd01]1bn.grp = 0
 (12)  [cesd02]1bn.grp - [cesd02]2.grp = 0
 (13)  [cesd07]1bn.grp - [cesd07]2.grp = 0
 (14)  [cesd11]1bn.grp - [cesd11]2.grp = 0
 (15)  [cesd20]1bn.grp - [cesd20]2.grp = 0
 (16)  [cesd01]2.grp#c.Somatic = 1
 (17)  [cesd01]2.grp = 0
------------------------------------------------------------------------------
             |                 OIM
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
Measurement  |
  cesd01 <-  |
    Somatic  |
         [*] |          1  (constrained)
    _cons    |
         [*] |          0  (constrained)
  -----------+----------------------------------------------------------------
  cesd02 <-  |
    Somatic  |
         [*] |   .9265332   .0600708    15.42   0.000     .8087966     1.04427
    _cons    |
         [*] |  -.0547348   .0262507    -2.09   0.037    -.1061853   -.0032844
  -----------+----------------------------------------------------------------
  cesd07 <-  |
    Somatic  |
         [*] |   1.714776     .09999    17.15   0.000     1.518799    1.910753
    _cons    |
         [*] |  -.0464016   .0423768    -1.09   0.274    -.1294587    .0366554
  -----------+----------------------------------------------------------------
  cesd11 <-  |
    Somatic  |
         [*] |   1.377827   .0859593    16.03   0.000      1.20935    1.546304
    _cons    |
         [*] |   .1538792   .0372906     4.13   0.000      .080791    .2269674
  -----------+----------------------------------------------------------------
  cesd20 <-  |
    Somatic  |
         [*] |   1.465235   .0856664    17.10   0.000     1.297331    1.633138
    _cons    |
         [*] |  -.0455513   .0364271    -1.25   0.211    -.1169472    .0258446
-------------+----------------------------------------------------------------
Mean         |
  Somatic    |
          1  |   .3225318   .0151328    21.31   0.000      .292872    .3521916
          2  |   .3977438   .0304149    13.08   0.000     .3381317     .457356
-------------+----------------------------------------------------------------
Variance     |
  e.cesd01   |
         [*] |   .3759644   .0129793                      .3513671    .4022837
  e.cesd02   |
         [*] |   .3397691   .0116453                      .3176945    .3633775
  e.cesd07   |
         [*] |   .4745692   .0208849                      .4353513      .51732
  e.cesd11   |
         [*] |   .5968166   .0212554                      .5565775    .6399648
  e.cesd20   |
         [*] |   .3724319   .0157235                      .3428549    .4045605
  Somatic    |
          1  |   .1185613   .0116213                      .0978382    .1436737
          2  |    .141167   .0208813                       .105639    .1886437
------------------------------------------------------------------------------
Note: [*] identifies parameter estimates constrained to be equal across
      groups.
LR test of model vs. saturated: chi2(23)  =    127.86, Prob > chi2 = 0.0000
As you can see the fit is becoming even poorer. 
chi2 = 127.86 - 46.31 = 81.55 with df(23-18=5), p-value = 3.976e-16
Next up is a demonstration of strict factor invariance with factor means constrained to be equal across groups. 
* model 5 -- strict invariance plus equal factor means

sem (Somatic -> cesd01 cesd02 cesd07 cesd11 cesd20) ///
    (1: Somatic -> cesd01@1)    ///  /* fix loading at 1 for group 1     */
    (2: Somatic -> cesd01@1)    ///  /* fix loading at 1 for group 2     */
    (cesd01 <- Somatic _cons@0) ///  /* fix intercept at 0 both groups   */
    , group(grp)                ///
    variance(e.cesd01@v1)       ///  /* constrain residuals              */
    variance(e.cesd11@v2)       ///  /* constrain residuals              */
    variance(e.cesd20@v3)       ///  /* constrain residuals              */
    variance(e.cesd02@v4)       ///  /* constrain residuals              */
    variance(e.cesd07@v5)       ///  /* constrain residuals              */
    mean(Somatic@m1)                 /* constrain factor means           */
    

Endogenous variables

Measurement:  cesd01 cesd02 cesd07 cesd11 cesd20

Exogenous variables

Latent:       Somatic

Fitting target model:

Iteration 0:   log likelihood = -13696.198  (not concave)
Iteration 1:   log likelihood = -12625.682  (not concave)
Iteration 2:   log likelihood = -12582.626  
Iteration 3:   log likelihood = -12552.249  
Iteration 4:   log likelihood = -12528.646  
Iteration 5:   log likelihood = -12527.916  
Iteration 6:   log likelihood = -12527.879  
Iteration 7:   log likelihood = -12527.879  

Structural equation model                       Number of obs      =      2252
Grouping variable  = grp                        Number of groups   =         2
Estimation method  = ml
Log likelihood     = -12527.879

 ( 1)  [cesd01]1bn.grp#c.Somatic = 1
 ( 2)  [cesd02]1bn.grp#c.Somatic - [cesd02]2.grp#c.Somatic = 0
 ( 3)  [cesd07]1bn.grp#c.Somatic - [cesd07]2.grp#c.Somatic = 0
 ( 4)  [cesd11]1bn.grp#c.Somatic - [cesd11]2.grp#c.Somatic = 0
 ( 5)  [cesd20]1bn.grp#c.Somatic - [cesd20]2.grp#c.Somatic = 0
 ( 6)  [var(e.cesd01)]1bn.grp - [var(e.cesd01)]2.grp = 0
 ( 7)  [var(e.cesd02)]1bn.grp - [var(e.cesd02)]2.grp = 0
 ( 8)  [var(e.cesd07)]1bn.grp - [var(e.cesd07)]2.grp = 0
 ( 9)  [var(e.cesd11)]1bn.grp - [var(e.cesd11)]2.grp = 0
 (10)  [var(e.cesd20)]1bn.grp - [var(e.cesd20)]2.grp = 0
 (11)  [cesd01]1bn.grp = 0
 (12)  [cesd02]1bn.grp - [cesd02]2.grp = 0
 (13)  [cesd07]1bn.grp - [cesd07]2.grp = 0
 (14)  [cesd11]1bn.grp - [cesd11]2.grp = 0
 (15)  [cesd20]1bn.grp - [cesd20]2.grp = 0
 (16)  [mean(Somatic)]1bn.grp - [mean(Somatic)]2.grp = 0
 (17)  [cesd01]2.grp#c.Somatic = 1
 (18)  [cesd01]2.grp = 0
------------------------------------------------------------------------------
             |                 OIM
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
Measurement  |
  cesd01 <-  |
    Somatic  |
         [*] |          1  (constrained)
    _cons    |
         [*] |          0  (constrained)
  -----------+----------------------------------------------------------------
  cesd02 <-  |
    Somatic  |
         [*] |   .9243261   .0599267    15.42   0.000      .806872     1.04178
    _cons    |
         [*] |  -.0540047   .0261998    -2.06   0.039    -.1053554    -.002654
  -----------+----------------------------------------------------------------
  cesd07 <-  |
    Somatic  |
         [*] |   1.702982   .0992192    17.16   0.000     1.508516    1.897448
    _cons    |
         [*] |     -.0425   .0421114    -1.01   0.313    -.1250368    .0400367
  -----------+----------------------------------------------------------------
  cesd11 <-  |
    Somatic  |
         [*] |    1.37989   .0859505    16.05   0.000      1.21143     1.54835
    _cons    |
         [*] |   .1531968   .0372876     4.11   0.000     .0801145    .2262791
  -----------+----------------------------------------------------------------
  cesd20 <-  |
    Somatic  |
         [*] |   1.466815   .0857704    17.10   0.000     1.298708    1.634922
    _cons    |
         [*] |  -.0460741   .0364502    -1.26   0.206    -.1175152     .025367
-------------+----------------------------------------------------------------
Mean         |
  Somatic    |
         [*] |   .3297945   .0148722    22.18   0.000     .3006456    .3589434
-------------+----------------------------------------------------------------
Variance     |
  e.cesd01   |
         [*] |   .3755483   .0129791                      .3509522    .4018683
  e.cesd02   |
         [*] |   .3399106   .0116496                      .3178278    .3635278
  e.cesd07   |
         [*] |   .4782652   .0208425                      .4391107    .5209111
  e.cesd11   |
         [*] |   .5953331   .0212457                      .5551153    .6384647
  e.cesd20   |
         [*] |   .3709739   .0157399                      .3413723    .4031425
  Somatic    |
          1  |     .11907   .0116526                       .098288    .1442461
          2  |   .1458758   .0214926                      .1092878    .1947129
------------------------------------------------------------------------------
Note: [*] identifies parameter estimates constrained to be equal across
      groups.
LR test of model vs. saturated: chi2(24)  =    134.38, Prob > chi2 = 0.0000
Next, we check the fit of model 5 versus model 4. 
chi2 = 134.38 - 127.86 = 6.52 with df(24-23=1), p-value = .0106668
Finally, we will check for strict invariance with both factor means and variances constrained to be equal across groups. 
* model 6 -- strict invariance plus equal factor means & variances
sem (Somatic -> cesd01 cesd02 cesd07 cesd11 cesd20) ///
    (1: Somatic -> cesd01@1)    ///  /* fix loading at 1 for group 1     */
    (2: Somatic -> cesd01@1)    ///  /* fix loading at 1 for group 2     */
    (cesd01 <- Somatic _cons@0) ///  /* fix intercept at 0 both groups   */
    , group(grp)                ///
    variance(e.cesd01@v1)       ///  /* constrain residuals              */
    variance(e.cesd11@v2)       ///  /* constrain residuals              */
    variance(e.cesd20@v3)       ///  /* constrain residuals              */
    variance(e.cesd02@v4)       ///  /* constrain residuals              */
    variance(e.cesd07@v5)       ///  /* constrain residuals              */
    mean(Somatic@m1)            ///  /* constrain factor means           */
    variance(Somatic@v6)             /* constrain factor variances       */
    
Endogenous variables

Measurement:  cesd01 cesd02 cesd07 cesd11 cesd20

Exogenous variables

Latent:       Somatic

Fitting target model:

Iteration 0:   log likelihood = -13636.745  (not concave)
Iteration 1:   log likelihood = -12645.581  (not concave)
Iteration 2:   log likelihood = -12588.687  
Iteration 3:   log likelihood = -12535.656  (not concave)
Iteration 4:   log likelihood = -12530.377  
Iteration 5:   log likelihood = -12529.118  
Iteration 6:   log likelihood = -12529.105  
Iteration 7:   log likelihood = -12529.105  

Structural equation model                       Number of obs      =      2252
Grouping variable  = grp                        Number of groups   =         2
Estimation method  = ml
Log likelihood     = -12529.105

 ( 1)  [cesd01]1bn.grp#c.Somatic = 1
 ( 2)  [cesd02]1bn.grp#c.Somatic - [cesd02]2.grp#c.Somatic = 0
 ( 3)  [cesd07]1bn.grp#c.Somatic - [cesd07]2.grp#c.Somatic = 0
 ( 4)  [cesd11]1bn.grp#c.Somatic - [cesd11]2.grp#c.Somatic = 0
 ( 5)  [cesd20]1bn.grp#c.Somatic - [cesd20]2.grp#c.Somatic = 0
 ( 6)  [var(e.cesd01)]1bn.grp - [var(e.cesd01)]2.grp = 0
 ( 7)  [var(e.cesd02)]1bn.grp - [var(e.cesd02)]2.grp = 0
 ( 8)  [var(e.cesd07)]1bn.grp - [var(e.cesd07)]2.grp = 0
 ( 9)  [var(e.cesd11)]1bn.grp - [var(e.cesd11)]2.grp = 0
 (10)  [var(e.cesd20)]1bn.grp - [var(e.cesd20)]2.grp = 0
 (11)  [var(Somatic)]1bn.grp - [var(Somatic)]2.grp = 0
 (12)  [cesd01]1bn.grp = 0
 (13)  [cesd02]1bn.grp - [cesd02]2.grp = 0
 (14)  [cesd07]1bn.grp - [cesd07]2.grp = 0
 (15)  [cesd11]1bn.grp - [cesd11]2.grp = 0
 (16)  [cesd20]1bn.grp - [cesd20]2.grp = 0
 (17)  [mean(Somatic)]1bn.grp - [mean(Somatic)]2.grp = 0
 (18)  [cesd01]2.grp#c.Somatic = 1
 (19)  [cesd01]2.grp = 0
------------------------------------------------------------------------------
             |                 OIM
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
Measurement  |
  cesd01 <-  |
    Somatic  |
         [*] |          1  (constrained)
    _cons    |
         [*] |          0  (constrained)
  -----------+----------------------------------------------------------------
  cesd02 <-  |
    Somatic  |
         [*] |   .9235701   .0598179    15.44   0.000     .8063291    1.040811
    _cons    |
         [*] |  -.0537546   .0261636    -2.05   0.040    -.1050343   -.0024748
  -----------+----------------------------------------------------------------
  cesd07 <-  |
    Somatic  |
         [*] |   1.692658    .098248    17.23   0.000     1.500096    1.885221
    _cons    |
         [*] |  -.0390848   .0418017    -0.94   0.350    -.1210146     .042845
  -----------+----------------------------------------------------------------
  cesd11 <-  |
    Somatic  |
         [*] |   1.380176   .0858521    16.08   0.000     1.211909    1.548443
    _cons    |
         [*] |   .1531023   .0372545     4.11   0.000     .0800847    .2261198
  -----------+----------------------------------------------------------------
  cesd20 <-  |
    Somatic  |
         [*] |   1.467051   .0857967    17.10   0.000     1.298893     1.63521
    _cons    |
         [*] |  -.0461523   .0364472    -1.27   0.205    -.1175874    .0252829
-------------+----------------------------------------------------------------
Mean         |
  Somatic    |
         [*] |   .3308145   .0148642    22.26   0.000     .3016811    .3599479
-------------+----------------------------------------------------------------
Variance     |
  e.cesd01   |
         [*] |    .375135   .0129755                      .3505464    .4014483
  e.cesd02   |
         [*] |   .3397284   .0116417                      .3176604    .3633295
  e.cesd07   |
         [*] |   .4813582    .020682                       .442482      .52365
  e.cesd11   |
         [*] |   .5944493   .0212324                      .5542579    .6375552
  e.cesd20   |
         [*] |   .3699995   .0157375                      .3404053    .4021665
  Somatic    |
         [*] |   .1224343   .0117591                      .1014263    .1477937
------------------------------------------------------------------------------
Note: [*] identifies parameter estimates constrained to be equal across
      groups.
LR test of model vs. saturated: chi2(25)  =    136.84, Prob > chi2 = 0.0000
The chi-square here is very close to the previous model. 
chi2 = 136.84 - 134.38 = 2.46 with df(25-24=1), p-value = .11677878
It may be moot but the fit of model 6 is not significantly worst than the fit for model 5.

In conclusion, we have evidence for metric invariance but not for any of the invariances higher in the invariance hierarchy.

Reference

Gregorich, S.E. 2006. Do Self-Report Instruments Allow Meaningful Comparisons Across Diverse Population Groups? Testing Measurement Invariance Using the Confirmatory Factor Analysis Framework. Medical Care, Volume 44, Number 11, Suppl 3.

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