Moderation in SEM: Difference between creating interaction term & conducting group analysis

[Kwok-wing Sum]

Dear all,

I have tried to conduct moderation analysis in SEM by 2 different methods, i.e. creating interaction term and conducting group analysis.
However, I have obtained results which seem to lead to different conclusions.
Here are the variables that I have used:
ilo1 - ilo6: observed variables
Ilo: latent factor (and DV)
pre_post_r: categorical IV (0 or 1)
sex_r: categorical moderator (0 or 1)

By creating interaction term between pre_post_r and sex_r (sex_pp), I run the following SEM model:

sem (Ilo -> ilo1 ilo2 ilo3 ilo4 ilo5 ilo6) (pre_post_r -> Ilo) (sex_r -> Ilo) (sex_pp -> Ilo), cov(e.ilo2*e.ilo3) cov(e.ilo1*e.ilo2) cov(e.ilo1*e.ilo3)

Below are the results. The interaction term (sex_pp) is insignificant. So I conclude that there is no moderation effect between the IV and the moderator.

Structural equation model                                Number of obs = 2,977
Estimation method: ml

Log likelihood = -28216.79

 ( 1)  [ilo1]Ilo = 1
-----------------------------------------------------------------------------------
                  |                 OIM
                  | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
------------------+----------------------------------------------------------------
Structural        |
  Ilo             |
       pre_post_r |   .6922629   .0669853    10.33   0.000     .5609741    .8235516
            sex_r |  -.6654334   .0517889   -12.85   0.000    -.7669378   -.5639291
           sex_pp |   .0830214   .0918951     0.90   0.366    -.0970898    .2631325
------------------+----------------------------------------------------------------
Measurement       |
  ilo1            |
              Ilo |          1  (constrained)
            _cons |    4.44004   .0393967   112.70   0.000     4.362824    4.517256
  ----------------+----------------------------------------------------------------
  ilo2            |
              Ilo |   .7338301   .0175133    41.90   0.000     .6995047    .7681554
            _cons |   4.852122    .032443   149.56   0.000     4.788535    4.915709
  ----------------+----------------------------------------------------------------
  ilo3            |
              Ilo |   .8969586   .0160456    55.90   0.000     .8655098    .9284074
            _cons |   4.723906   .0354097   133.41   0.000     4.654504    4.793308
  ----------------+----------------------------------------------------------------
  ilo4            |
              Ilo |   1.071829   .0188535    56.85   0.000     1.034877    1.108781
            _cons |   4.508011   .0408367   110.39   0.000     4.427973    4.588049
  ----------------+----------------------------------------------------------------
  ilo5            |
              Ilo |   1.150123   .0191861    59.95   0.000     1.112519    1.187727
            _cons |   4.127608   .0430686    95.84   0.000     4.043196    4.212021
  ----------------+----------------------------------------------------------------
  ilo6            |
              Ilo |    1.05216   .0180955    58.14   0.000     1.016693    1.087626
            _cons |   4.348902   .0397949   109.28   0.000     4.270905    4.426899
------------------+----------------------------------------------------------------
       var(e.ilo1)|   .7572459   .0229944                      .7134926    .8036823
       var(e.ilo2)|   1.046165   .0289415                      .9909515    1.104456
       var(e.ilo3)|   .6229266   .0189183                      .5869293    .6611316
       var(e.ilo4)|   .5212302   .0174359                      .4881526    .5565491
       var(e.ilo5)|   .4115313    .015937                      .3814513    .4439832
       var(e.ilo6)|   .4391127   .0153359                      .4100605    .4702233
        var(e.Ilo)|   1.242897   .0473465                      1.153479    1.339246
------------------+----------------------------------------------------------------
cov(e.ilo1,e.ilo2)|   .1561735   .0188641     8.28   0.000     .1192004    .1931465
cov(e.ilo1,e.ilo3)|    .113366   .0154121     7.36   0.000     .0831588    .1435731
cov(e.ilo2,e.ilo3)|   .3653889   .0185338    19.71   0.000     .3290634    .4017144
-----------------------------------------------------------------------------------
LR test of model vs. saturated: chi2(21) = 223.95              Prob > chi2 = 0.0000

Afterwards, I run the following group analysis with another SEM model:

sem (Ilo ->  ilo1 ilo2 ilo3 ilo4 ilo5 ilo6) (pre_post_r -> Ilo), group(sex_r) ginvariant(mcoef mcons serrvar) cov( e.ilo1*e.ilo2 e.ilo1*e.ilo3 e.ilo2*e.ilo3)

Structural equation model                             Number of obs    = 2,977
Grouping variable: sex_r                              Number of groups =     2
Estimation method: ml

Log likelihood = -26260.007

 ( 1)  [ilo1]0bn.sex_r#c.Ilo = 1
 ( 2)  [ilo2]0bn.sex_r#c.Ilo - [ilo2]1.sex_r#c.Ilo = 0
 ( 3)  [ilo3]0bn.sex_r#c.Ilo - [ilo3]1.sex_r#c.Ilo = 0
 ( 4)  [ilo4]0bn.sex_r#c.Ilo - [ilo4]1.sex_r#c.Ilo = 0
 ( 5)  [ilo5]0bn.sex_r#c.Ilo - [ilo5]1.sex_r#c.Ilo = 0
 ( 6)  [ilo6]0bn.sex_r#c.Ilo - [ilo6]1.sex_r#c.Ilo = 0
 ( 7)  [/]var(e.Ilo)#0bn.sex_r - [/]var(e.Ilo)#1.sex_r = 0
 ( 8)  [ilo1]0bn.sex_r - [ilo1]1.sex_r = 0
 ( 9)  [ilo2]0bn.sex_r - [ilo2]1.sex_r = 0
 (10)  [ilo3]0bn.sex_r - [ilo3]1.sex_r = 0
 (11)  [ilo4]0bn.sex_r - [ilo4]1.sex_r = 0
 (12)  [ilo5]0bn.sex_r - [ilo5]1.sex_r = 0
 (13)  [ilo6]0bn.sex_r - [ilo6]1.sex_r = 0
 (14)  [ilo1]1.sex_r#c.Ilo = 1

Group: Male                                              Number of obs = 1,448

-----------------------------------------------------------------------------------
                  |                 OIM
                  | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
------------------+----------------------------------------------------------------
Structural        |
  Ilo             |
       pre_post_r |   1.031074   .0642268    16.05   0.000     .9051915    1.156956
------------------+----------------------------------------------------------------
Measurement       |
  ilo1            |
              Ilo |          1  (constrained)
            _cons |   4.102239   .0303745   135.06   0.000     4.042706    4.161772
  ----------------+----------------------------------------------------------------
  ilo2            |
              Ilo |    .735468   .0174669    42.11   0.000     .7012335    .7697025
            _cons |    4.59934   .0268293   171.43   0.000     4.546756    4.651925
  ----------------+----------------------------------------------------------------
  ilo3            |
              Ilo |   .9008603   .0160247    56.22   0.000     .8694524    .9322681
            _cons |   4.415707   .0273912   161.21   0.000     4.362021    4.469393
  ----------------+----------------------------------------------------------------
  ilo4            |
              Ilo |   1.071535   .0189074    56.67   0.000     1.034478    1.108593
            _cons |   4.136944   .0307681   134.46   0.000      4.07664    4.197248
  ----------------+----------------------------------------------------------------
  ilo5            |
              Ilo |   1.150142   .0192086    59.88   0.000     1.112494     1.18779
            _cons |   3.737108   .0318121   117.47   0.000     3.674757    3.799459
  ----------------+----------------------------------------------------------------
  ilo6            |
              Ilo |   1.052952   .0181181    58.12   0.000     1.017441    1.088462
            _cons |   3.992495   .0296573   134.62   0.000     3.934368    4.050622
------------------+----------------------------------------------------------------
       var(e.ilo1)|    .759053    .033199                      .6966951    .8269924
       var(e.ilo2)|   1.018051   .0406341                      .9414452    1.100891
       var(e.ilo3)|   .5764516   .0254992                      .5285792    .6286596
       var(e.ilo4)|   .5715825   .0268171                      .5213662    .6266354
       var(e.ilo5)|   .4250043   .0230977                      .3820614    .4727739
       var(e.ilo6)|   .4385163   .0218616                      .3976951    .4835275
        var(e.Ilo)|   1.318692   .0500662                      1.224126    1.420563
------------------+----------------------------------------------------------------
cov(e.ilo1,e.ilo2)|   .1849651    .027043     6.84   0.000     .1319617    .2379684
cov(e.ilo1,e.ilo3)|   .1135263   .0215603     5.27   0.000     .0712689    .1557838
cov(e.ilo2,e.ilo3)|   .2892495   .0248841    11.62   0.000     .2404776    .3380215
-----------------------------------------------------------------------------------

Group: Female                                            Number of obs = 1,529

-----------------------------------------------------------------------------------
                  |                 OIM
                  | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
------------------+----------------------------------------------------------------
Structural        |
  Ilo             |
       pre_post_r |   .4492335   .0604778     7.43   0.000     .3306992    .5677677
------------------+----------------------------------------------------------------
Measurement       |
  ilo1            |
              Ilo |          1  (constrained)
            _cons |   4.102239   .0303745   135.06   0.000     4.042706    4.161772
  ----------------+----------------------------------------------------------------
  ilo2            |
              Ilo |    .735468   .0174669    42.11   0.000     .7012335    .7697025
            _cons |    4.59934   .0268293   171.43   0.000     4.546756    4.651925
  ----------------+----------------------------------------------------------------
  ilo3            |
              Ilo |   .9008603   .0160247    56.22   0.000     .8694524    .9322681
            _cons |   4.415707   .0273912   161.21   0.000     4.362021    4.469393
  ----------------+----------------------------------------------------------------
  ilo4            |
              Ilo |   1.071535   .0189074    56.67   0.000     1.034478    1.108593
            _cons |   4.136944   .0307681   134.46   0.000      4.07664    4.197248
  ----------------+----------------------------------------------------------------
  ilo5            |
              Ilo |   1.150142   .0192086    59.88   0.000     1.112494     1.18779
            _cons |   3.737108   .0318121   117.47   0.000     3.674757    3.799459
  ----------------+----------------------------------------------------------------
  ilo6            |
              Ilo |   1.052952   .0181181    58.12   0.000     1.017441    1.088462
            _cons |   3.992495   .0296573   134.62   0.000     3.934368    4.050622
------------------+----------------------------------------------------------------
       var(e.ilo1)|   .7504911   .0314915                      .6912388    .8148224
       var(e.ilo2)|   1.068583   .0410555                      .9910704    1.152157
       var(e.ilo3)|   .6594714   .0275581                      .6076112    .7157578
       var(e.ilo4)|   .4853136   .0231607                      .4419777    .5328986
       var(e.ilo5)|   .3990046   .0214138                      .3591662    .4432619
       var(e.ilo6)|   .4357954   .0209641                       .396584    .4788836
        var(e.Ilo)|   1.318692   .0500662                      1.224126    1.420563
------------------+----------------------------------------------------------------
cov(e.ilo1,e.ilo2)|   .1241603   .0261024     4.76   0.000     .0730005    .1753201
cov(e.ilo1,e.ilo3)|   .1067533   .0216582     4.93   0.000     .0643041    .1492026
cov(e.ilo2,e.ilo3)|   .4318641   .0272252    15.86   0.000     .3785036    .4852245
-----------------------------------------------------------------------------------
LR test of model vs. saturated: chi2(34) = 422.95              Prob > chi2 = 0.0000

When testing for group invariance, the Wald test shows that the coefficient of the path pre_post_r -> Ilo is significant, indicating that the effect of IV on DV is different for the different levels of moderator (0 or 1):

Tests for group invariance of parameters

-----------------------------------------------------------------------------------
                  |            Wald test                       Score test
                  |      chi2         df    P>chi2       chi2          df    P>chi2
------------------+----------------------------------------------------------------
Structural        |
  Ilo             |
       pre_post_r |    54.418          1    0.0000          .           .         .
------------------+----------------------------------------------------------------

Why is there such a contradictory result? Is there any constraint that I should add or release in the group analysis? Or should I use categorical IV and moderator in SEM for testing moderation?
Thank you very much.

[Kwok-wing Sum]

Dear all,

Is my post not detailed enough? It seems that my post(s) (including my last one) is not that effective and they receive no responses so far. As I am only a novice in Statalist, may anyone who can point out in what way I should amend my posts so that I may receive feedback from other users?
Thank you so much.

[Richard Williams]

Notice how all those variances and covariances differ across groups. Also notice how the DF differ between the two approaches. I think your ginvariant option is wrong. I think you want to make everything ginvariant EXCEPT scoef. Example 23 in the sem manual shows ways to do this.

At least, I think that is what you should do if you want the two approaches to produce identical results. But in practice you usually just want the measurement coefficients to be the same across groups, I.e. you want something like what you did.

in other words, you are not running equivalent models here. I think you can make them equivalent. But, the first model isn’t necessarily what you want. It imposes a bunch of equality constraints across groups that may not be necessary or desirable.

[Joseph Coveney]

pre_post_r: categorical IV (0 or 1)

This implies that you have repeated measurements over time (two time points) on each of your male and female whatevers.

If so, is there a reason why you didn't set up your model to accommodates that study design, e.g., like a latent growth curve SEM?

Or else use gsem with random effects.

[Kwok-wing Sum]

Thank you Prof. Williams and Dr. Coveney for your replies.

The reason why I have added error covariances between certain items (ilo1, ilo2, ilo3) is that previously I have run CFA for the whole sample and multigroup CFA for pre- and post- groups (to test for measurement invariance), and the modification indices show that the model fit would be much better with these error covariances.
As Prof. Williams said, I have tried to make the models more equivalent (see below). But still the coefficient of the path "pre_post_r -> Ilo" is significant. The result is still different from the SEM with interaction term.
In this case, may I simply use the SEM model with the interaction term? Is the model correct enough?

sem (Ilo ->  ilo1 ilo2 ilo3 ilo4 ilo5 ilo6 ) (pre_post_r -> Ilo), group(sex_r) ginvariant(scons mcoef mcons serrvar merrvar smerrcov meanex covex) cov( e.ilo1*e.ilo2 e.ilo1*e.ilo3 e.ilo2*e.ilo3)

Structural equation model                             Number of obs    = 2,977
Grouping variable: sex_r                              Number of groups =     2
Estimation method: ml

Log likelihood = -26277.853

Group: Male                                              Number of obs = 1,448

-----------------------------------------------------------------------------------
                  |                 OIM
                  | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
------------------+----------------------------------------------------------------
Structural        |
  Ilo             |
       pre_post_r |     1.0322   .0642604    16.06   0.000     .9062523    1.158148
------------------+----------------------------------------------------------------
Measurement       |
  ilo1            |
              Ilo |          1  (constrained)
            _cons |   4.101196   .0303532   135.12   0.000     4.041704    4.160687
  ----------------+----------------------------------------------------------------
  ilo2            |
              Ilo |   .7341535   .0174991    41.95   0.000     .6998558    .7684512
            _cons |   4.603395   .0267991   171.77   0.000      4.55087     4.65592
  ----------------+----------------------------------------------------------------
  ilo3            |
              Ilo |   .8972643   .0160334    55.96   0.000     .8658395    .9286891
            _cons |   4.419908   .0272956   161.93   0.000     4.366409    4.473406
  ----------------+----------------------------------------------------------------
  ilo4            |
              Ilo |   1.068358   .0188077    56.80   0.000     1.031496     1.10522
            _cons |   4.145609   .0305574   135.67   0.000     4.085717      4.2055
  ----------------+----------------------------------------------------------------
  ilo5            |
              Ilo |   1.148001   .0191365    59.99   0.000     1.110494    1.185507
            _cons |   3.738373   .0317925   117.59   0.000     3.676061    3.800686
  ----------------+----------------------------------------------------------------
  ilo6            |
              Ilo |   1.051796    .018041    58.30   0.000     1.016436    1.087155
            _cons |   3.992465   .0296579   134.62   0.000     3.934337    4.050594
------------------+----------------------------------------------------------------
       var(e.ilo1)|   .7534025   .0229478                      .7097417     .799749
       var(e.ilo2)|   1.043402   .0288968                      .9882747    1.101604
       var(e.ilo3)|   .6190326   .0188627                      .5831447    .6571292
       var(e.ilo4)|   .5276721   .0175686                      .4943377    .5632543
       var(e.ilo5)|    .413576   .0160233                      .3833336    .4462042
       var(e.ilo6)|   .4359774   .0153031                      .4069923    .4670266
        var(e.Ilo)|    1.32266   .0501173                      1.227991    1.424628
------------------+----------------------------------------------------------------
cov(e.ilo1,e.ilo2)|   .1528803   .0188259     8.12   0.000     .1159822    .1897783
cov(e.ilo1,e.ilo3)|   .1094717   .0153727     7.12   0.000     .0793418    .1396016
cov(e.ilo2,e.ilo3)|    .362107   .0184853    19.59   0.000     .3258764    .3983376
-----------------------------------------------------------------------------------

Group: Female                                            Number of obs = 1,529

-----------------------------------------------------------------------------------
                  |                 OIM
                  | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
------------------+----------------------------------------------------------------
Structural        |
  Ilo             |
       pre_post_r |   .4496628   .0606062     7.42   0.000     .3308769    .5684487
------------------+----------------------------------------------------------------
Measurement       |
  ilo1            |
              Ilo |          1  (constrained)
            _cons |   4.101196   .0303532   135.12   0.000     4.041704    4.160687
  ----------------+----------------------------------------------------------------
  ilo2            |
              Ilo |   .7341535   .0174991    41.95   0.000     .6998558    .7684512
            _cons |   4.603395   .0267991   171.77   0.000      4.55087     4.65592
  ----------------+----------------------------------------------------------------
  ilo3            |
              Ilo |   .8972643   .0160334    55.96   0.000     .8658395    .9286891
            _cons |   4.419908   .0272956   161.93   0.000     4.366409    4.473406
  ----------------+----------------------------------------------------------------
  ilo4            |
              Ilo |   1.068358   .0188077    56.80   0.000     1.031496     1.10522
            _cons |   4.145609   .0305574   135.67   0.000     4.085717      4.2055
  ----------------+----------------------------------------------------------------
  ilo5            |
              Ilo |   1.148001   .0191365    59.99   0.000     1.110494    1.185507
            _cons |   3.738373   .0317925   117.59   0.000     3.676061    3.800686
  ----------------+----------------------------------------------------------------
  ilo6            |
              Ilo |   1.051796    .018041    58.30   0.000     1.016436    1.087155
            _cons |   3.992465   .0296579   134.62   0.000     3.934337    4.050594
------------------+----------------------------------------------------------------
       var(e.ilo1)|   .7534025   .0229478                      .7097417     .799749
       var(e.ilo2)|   1.043402   .0288968                      .9882747    1.101604
       var(e.ilo3)|   .6190326   .0188627                      .5831447    .6571292
       var(e.ilo4)|   .5276721   .0175686                      .4943377    .5632543
       var(e.ilo5)|    .413576   .0160233                      .3833336    .4462042
       var(e.ilo6)|   .4359774   .0153031                      .4069923    .4670266
        var(e.Ilo)|    1.32266   .0501173                      1.227991    1.424628
------------------+----------------------------------------------------------------
cov(e.ilo1,e.ilo2)|   .1528803   .0188259     8.12   0.000     .1159822    .1897783
cov(e.ilo1,e.ilo3)|   .1094717   .0153727     7.12   0.000     .0793418    .1396016
cov(e.ilo2,e.ilo3)|    .362107   .0184853    19.59   0.000     .3258764    .3983376
-----------------------------------------------------------------------------------
LR test of model vs. saturated: chi2(43) = 458.64              Prob > chi2 = 0.0000

Tests for group invariance of parameters

-----------------------------------------------------------------------------------
                  |            Wald test                       Score test
                  |      chi2         df    P>chi2       chi2          df    P>chi2
------------------+----------------------------------------------------------------
Structural        |
  Ilo             |
       pre_post_r |    54.419          1    0.0000          .           .         .

I considered Dr. Coveney's advice of running a latent growth curve SEM before, but it seems that the way that it treats the intercept and slope of the growth curve is rather complicated. Also, I have only 2 time points, which I am not sure if it can be regarded as a "growth" model.
May Dr. Coveney suggest how I may run it?
Besides, what difference would it make if I run the above model with gsem? Any changes that I should make in the above Stata command?

Thank you very much again.

[Richard Williams]

I have a feeling your last model is still a bit off. Note that the parameters are slightly different from your original interaction model. What happens if you drop scons from the ginvariant option? It seems the original interaction model allowed the structural intercepts to differ by sex, but the last model constrains them to be the same.

" In this case, may I simply use the SEM model with the interaction term? Is the model correct enough?"

Well, that model certainly doesn't fit perfectly. I think you'd still need to explore whether that is the best you can do, e.g. should you allow more differences across groups?

I think my preferred model would be your original two group model, with ginvariant left off completely, which would default to ginvariant(mcoef mcons). When comparing groups you usually want the coefficients of the measurement model to be the same (but you should test if this is actually true). You can test other equalities across groups if you want to but imposing them may not gain you that much (besides a more parsimonious model).

I haven't done models like this in a while so I may be missing something. But, try what I suggest above (especially dropping scons from ginvariant) and see what happens.

[Richard Williams]

You might add the showg option to your sem commands. It will show the estimated means for the latent variable and will make it easy to see what the equality constraints are.

[Kwok-wing Sum]

Thank you Prof. Williams again for your reply.
As advised, I have run the two group models by dropping scons from ginvariant:

sem (Ilo ->  ilo1 ilo2 ilo3 ilo4 ilo5 ilo6 ) (pre_post_r -> Ilo), group(sex_r) ginvariant(mcoef mcons serrvar merrvar smerrcov meanex covex) cov( e.ilo1*e.ilo2 e.ilo1*e.ilo3 e.ilo2*e.ilo3)

Below are the results. But it seems that the path "pre_post_r -> Ilo" is still significant.
I am thinking about whether it is the properties of the interaction term (e.g., its error covariance) in my original interaction model which makes it different from the two group models. If this is the case, should I take the results of the two group models as correct, and interpret that there is no gender difference in the effect of "pre_post_r" on the latent variable (though it is contradictory to the interaction model by now)?

Thank you so much again.

Structural equation model                             Number of obs    = 2,977
Grouping variable: sex_r                              Number of groups =     2
Estimation method: ml

Log likelihood = -26277.853

Group: Male                                              Number of obs = 1,448

-----------------------------------------------------------------------------------
                  |                 OIM
                  | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
------------------+----------------------------------------------------------------
Structural        |
  Ilo             |
       pre_post_r |     1.0322   .0642604    16.06   0.000     .9062523    1.158148
------------------+----------------------------------------------------------------
Measurement       |
  ilo1            |
              Ilo |          1  (constrained)
            _cons |   4.101196   .0303532   135.12   0.000     4.041704    4.160687
  ----------------+----------------------------------------------------------------
  ilo2            |
              Ilo |   .7341535   .0174991    41.95   0.000     .6998558    .7684512
            _cons |   4.603395   .0267991   171.77   0.000      4.55087     4.65592
  ----------------+----------------------------------------------------------------
  ilo3            |
              Ilo |   .8972643   .0160334    55.96   0.000     .8658395    .9286891
            _cons |   4.419908   .0272956   161.93   0.000     4.366409    4.473406
  ----------------+----------------------------------------------------------------
  ilo4            |
              Ilo |   1.068358   .0188077    56.80   0.000     1.031496     1.10522
            _cons |   4.145609   .0305574   135.67   0.000     4.085717      4.2055
  ----------------+----------------------------------------------------------------
  ilo5            |
              Ilo |   1.148001   .0191365    59.99   0.000     1.110494    1.185507
            _cons |   3.738373   .0317925   117.59   0.000     3.676061    3.800686
  ----------------+----------------------------------------------------------------
  ilo6            |
              Ilo |   1.051796    .018041    58.30   0.000     1.016436    1.087155
            _cons |   3.992465   .0296579   134.62   0.000     3.934337    4.050594
------------------+----------------------------------------------------------------
       var(e.ilo1)|   .7534025   .0229478                      .7097417     .799749
       var(e.ilo2)|   1.043402   .0288968                      .9882747    1.101604
       var(e.ilo3)|   .6190326   .0188627                      .5831447    .6571292
       var(e.ilo4)|   .5276721   .0175686                      .4943377    .5632543
       var(e.ilo5)|    .413576   .0160233                      .3833336    .4462042
       var(e.ilo6)|   .4359774   .0153031                      .4069923    .4670266
        var(e.Ilo)|    1.32266   .0501173                      1.227991    1.424628
------------------+----------------------------------------------------------------
cov(e.ilo1,e.ilo2)|   .1528803   .0188259     8.12   0.000     .1159822    .1897783
cov(e.ilo1,e.ilo3)|   .1094717   .0153727     7.12   0.000     .0793418    .1396016
cov(e.ilo2,e.ilo3)|    .362107   .0184853    19.59   0.000     .3258764    .3983376
-----------------------------------------------------------------------------------

Group: Female                                            Number of obs = 1,529

-----------------------------------------------------------------------------------
                  |                 OIM
                  | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
------------------+----------------------------------------------------------------
Structural        |
  Ilo             |
       pre_post_r |   .4496628   .0606062     7.42   0.000     .3308769    .5684487
------------------+----------------------------------------------------------------
Measurement       |
  ilo1            |
              Ilo |          1  (constrained)
            _cons |   4.101196   .0303532   135.12   0.000     4.041704    4.160687
  ----------------+----------------------------------------------------------------
  ilo2            |
              Ilo |   .7341535   .0174991    41.95   0.000     .6998558    .7684512
            _cons |   4.603395   .0267991   171.77   0.000      4.55087     4.65592
  ----------------+----------------------------------------------------------------
  ilo3            |
              Ilo |   .8972643   .0160334    55.96   0.000     .8658395    .9286891
            _cons |   4.419908   .0272956   161.93   0.000     4.366409    4.473406
  ----------------+----------------------------------------------------------------
  ilo4            |
              Ilo |   1.068358   .0188077    56.80   0.000     1.031496     1.10522
            _cons |   4.145609   .0305574   135.67   0.000     4.085717      4.2055
  ----------------+----------------------------------------------------------------
  ilo5            |
              Ilo |   1.148001   .0191365    59.99   0.000     1.110494    1.185507
            _cons |   3.738373   .0317925   117.59   0.000     3.676061    3.800686
  ----------------+----------------------------------------------------------------
  ilo6            |
              Ilo |   1.051796    .018041    58.30   0.000     1.016436    1.087155
            _cons |   3.992465   .0296579   134.62   0.000     3.934337    4.050594
------------------+----------------------------------------------------------------
       var(e.ilo1)|   .7534025   .0229478                      .7097417     .799749
       var(e.ilo2)|   1.043402   .0288968                      .9882747    1.101604
       var(e.ilo3)|   .6190326   .0188627                      .5831447    .6571292
       var(e.ilo4)|   .5276721   .0175686                      .4943377    .5632543
       var(e.ilo5)|    .413576   .0160233                      .3833336    .4462042
       var(e.ilo6)|   .4359774   .0153031                      .4069923    .4670266
        var(e.Ilo)|    1.32266   .0501173                      1.227991    1.424628
------------------+----------------------------------------------------------------
cov(e.ilo1,e.ilo2)|   .1528803   .0188259     8.12   0.000     .1159822    .1897783
cov(e.ilo1,e.ilo3)|   .1094717   .0153727     7.12   0.000     .0793418    .1396016
cov(e.ilo2,e.ilo3)|    .362107   .0184853    19.59   0.000     .3258764    .3983376
-----------------------------------------------------------------------------------
LR test of model vs. saturated: chi2(43) = 458.64              Prob > chi2 = 0.0000

.

Tests for group invariance of parameters

-----------------------------------------------------------------------------------
                  |            Wald test                       Score test
                  |      chi2         df    P>chi2       chi2          df    P>chi2
------------------+----------------------------------------------------------------
Structural        |
  Ilo             |
       pre_post_r |    54.419          1    0.0000          .           .         .
------------------+----------------------------------------------------------------

[Joseph Coveney]

I considered . . . a latent growth curve SEM before, but it seems that the way that it treats the intercept and slope of the growth curve is rather complicated. Also, I have only 2 time points, which I am not sure if it can be regarded as a "growth" model.

Yes, latent growth curve SEMs can get hard to fathom. With multiple indicator variables, you'd first form confirmatory factor analysis-like latent factors at each time point and then treat the intercept and slope latent factors like the second level of a two-level CFA. It's illustrated as Example 6.14 in Mplus's user's manual.

But with only two time points, it's liable to get dicey and you might be better off just to omit the captive-intermediate first-level CFA latent factors and load the indicators on the intercept and slope latent factors directly, applying the measurement-invariance constraints there. The approach is illustrated in the output below as the first model fit—it's at the "SEM (growth-curve like)" comment. (The stuff at the beginning is just to create a fictitious dataset for illustration. I've used only three indicator variables for brevity, but the approach will be the same with your six indicator variables.)

. . . how I may run it?
Besides, what difference would it make if I run the above model with gsem? Any changes that I should make in the above Stata command?

I illustrate a random-slope, within-and-between formulation using gsem below at the "-gsem- random-slope model, within-and-between formulation" comment. The model is slightly different from that of the latent growth curve formulation above in that it allows for intercepts for each of the indicator variables (growth curve formulations typically set them at zero for identification of the means of the slope and intercept latent factors); nevertheless, the results are essentially the same for the fixed effects of sex, time and sex × time interaction.

Below last I fit an analogous model using mixed, with the indicator variables included as another fixed effect. Again the results for the effects of interest and for the variances and covariance of intercept and random slope are essentially the same within estimation error, as are the residual variance estimates for the indicator variables. Principal estimates are tabulated below for ease of comparison. (SEM-GC is the growth curve-like formulation using sem, SEM-WB is the random slope, within-and-between formulation using gsem and MIXED is with mixed)

Main effects of sex (coefficient ± standard error)
SEM-GC 0.54 ± 0.03
SEM-WB 0.54 ± 0.05
MIXED 0.54 ± 0.05

Main effects of time
SEM-GC 0.51 ± 0.03
SEM-WB 0.51 ± 0.04 (avg.)
MIXED 0.51 ± 0.04

Sex × time interaction
SEM-GC 0.46 ± 0.06
SEM-WB 0.50 ± 0.06
MIXED 0.49 ± 0.06

Intercept Variance (coefficient only)
SEM-GC 1.6
SEM-WB 1.6
MIXED 1.6

Random Slope Variance
SEM-GC 1.9
SEM-WB 2.0
MIXED 1.9

Intercept*Slope Covariance
SEM-GC -0.76
SEM-WB -0.77
MIXED -0.75

Here is the output. The corresponding do-file and log file are attached.

.ÿ
.ÿversionÿ18.0

.ÿ
.ÿlogÿcloseÿ_all

.ÿlogÿusingÿ"GrowthÿCurve.smcl",ÿnomsgÿname(lo)

.ÿ
.ÿclearÿ*

.ÿ
.ÿ//ÿseedem
.ÿsetÿseedÿ960930312

.ÿ
.ÿquietlyÿdrawnormÿinterceptÿslope,ÿdoubleÿcorr(1ÿ-0.25ÿ\ÿ-0.25ÿ1)ÿn(3000)

.ÿgenerateÿbyteÿsexÿ=ÿmod(_n,ÿ2)

.ÿ
.ÿtempnameÿCorr

.ÿmatrixÿdefineÿ`Corr'ÿ=ÿJ(3,ÿ3,ÿ0.5)ÿ+ÿI(3)ÿ*ÿ0.5

.ÿforvaluesÿtimÿ=ÿ0/1ÿ{
ÿÿ2.ÿÿÿÿÿquietlyÿdrawnormÿe1`tim'ÿe2`tim'ÿe3`tim',ÿdoubleÿcorr(`Corr')
ÿÿ3.ÿÿÿÿÿforvaluesÿiÿ=ÿ1/3ÿ{
ÿÿ4.ÿÿÿÿÿÿÿÿÿgenerateÿdoubleÿout`i'`tim'ÿ=ÿ///
>ÿÿÿÿÿÿÿÿÿÿÿÿÿ0ÿ+ÿinterceptÿ+ÿ///
>ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿsexÿ/ÿ2ÿ+ÿ///
>ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ`tim'ÿ*ÿ(1/2ÿ+ÿslope)ÿ+ÿ///
>ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿsexÿ*ÿ`tim'ÿ/ÿ2ÿ+ÿ///
>ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿe`i'`tim'
ÿÿ5.ÿÿÿÿÿ}
ÿÿ6.ÿ}

.ÿ
.ÿ*
.ÿ*ÿSEMÿ(growth-curveÿlike)
.ÿ*
.ÿconstraintÿdefineÿÿ1ÿ[out10]Interceptÿ=ÿ1

.ÿconstraintÿdefineÿÿ2ÿ[out11]Interceptÿ=ÿ1

.ÿconstraintÿdefineÿÿ3ÿ[out11]Slopeÿ=ÿ1

.ÿ
.ÿconstraintÿdefineÿÿ4ÿ[out20]Interceptÿ=ÿ[out21]Intercept

.ÿconstraintÿdefineÿÿ5ÿ[out30]Interceptÿ=ÿ[out31]Intercept

.ÿconstraintÿdefineÿÿ6ÿ[out21]Interceptÿ=ÿ[out21]Slope

.ÿconstraintÿdefineÿÿ7ÿ[out31]Interceptÿ=ÿ[out31]Slope

.ÿ
.ÿconstraintÿdefineÿÿ8ÿ[/]var(e.out10)ÿ=ÿ[/]var(e.out11)

.ÿconstraintÿdefineÿÿ9ÿ[/]var(e.out20)ÿ=ÿ[/]var(e.out21)

.ÿconstraintÿdefineÿ10ÿ[/]var(e.out30)ÿ=ÿ[/]var(e.out31)

.ÿ
.ÿgenerateÿbyteÿtimÿ=ÿ!sex

.ÿ
.ÿconstraintÿdefineÿ11ÿ[Intercept]timÿ=ÿ0

.ÿ
.ÿsemÿ///
>ÿÿÿÿÿÿÿÿÿ(out*ÿ<-ÿIntercept,ÿnoconstant)ÿ///
>ÿÿÿÿÿÿÿÿÿ(out?1ÿ<-ÿSlope,ÿnoconstant)ÿ///
>ÿÿÿÿÿÿÿÿÿ(InterceptÿSlopeÿ<-ÿtimÿsex,ÿnoconstant),ÿ///
>ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿconstraints(1/11)ÿ///
>ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿcovariance((e.Intercept*e.Slope))ÿ///
>ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿnocnsreportÿnodescribeÿnolog

StructuralÿequationÿmodelÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿNumberÿofÿobsÿ=ÿ3,000
Estimationÿmethod:ÿml

Logÿlikelihoodÿ=ÿ-30753.295

-----------------------------------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿOIM
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|ÿCoefficientÿÿstd.ÿerr.ÿÿÿÿÿÿzÿÿÿÿP>|z|ÿÿÿÿÿ[95%ÿconf.ÿinterval]
------------------------+----------------------------------------------------------------
Structuralÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿInterceptÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿsexÿ|ÿÿÿ.5429784ÿÿÿ.0341708ÿÿÿÿ15.89ÿÿÿ0.000ÿÿÿÿÿÿ.476005ÿÿÿÿ.6099519
ÿÿ----------------------+----------------------------------------------------------------
ÿÿSlopeÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿtimÿ|ÿÿÿ.5102283ÿÿÿ.0346671ÿÿÿÿ14.72ÿÿÿ0.000ÿÿÿÿÿ.4422821ÿÿÿÿ.5781746
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿsexÿ|ÿÿÿ1.003576ÿÿÿ.0393589ÿÿÿÿ25.50ÿÿÿ0.000ÿÿÿÿÿ.9264335ÿÿÿÿ1.080718
------------------------+----------------------------------------------------------------
Measurementÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿout10ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿInterceptÿ|ÿÿÿÿÿÿÿÿÿÿ1ÿÿ(constrained)
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ_consÿ|ÿÿÿÿÿÿÿÿÿÿ0ÿÿ(constrained)
ÿÿ----------------------+----------------------------------------------------------------
ÿÿout20ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿInterceptÿ|ÿÿÿ.9973584ÿÿÿ.0088294ÿÿÿ112.96ÿÿÿ0.000ÿÿÿÿÿ.9800532ÿÿÿÿ1.014664
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ_consÿ|ÿÿÿÿÿÿÿÿÿÿ0ÿÿ(constrained)
ÿÿ----------------------+----------------------------------------------------------------
ÿÿout30ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿInterceptÿ|ÿÿÿ.9946205ÿÿÿ.0087487ÿÿÿ113.69ÿÿÿ0.000ÿÿÿÿÿ.9774734ÿÿÿÿ1.011768
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ_consÿ|ÿÿÿÿÿÿÿÿÿÿ0ÿÿ(constrained)
ÿÿ----------------------+----------------------------------------------------------------
ÿÿout11ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿInterceptÿ|ÿÿÿÿÿÿÿÿÿÿ1ÿÿ(constrained)
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿSlopeÿ|ÿÿÿÿÿÿÿÿÿÿ1ÿÿ(constrained)
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ_consÿ|ÿÿÿÿÿÿÿÿÿÿ0ÿÿ(constrained)
ÿÿ----------------------+----------------------------------------------------------------
ÿÿout21ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿInterceptÿ|ÿÿÿ.9973584ÿÿÿ.0088294ÿÿÿ112.96ÿÿÿ0.000ÿÿÿÿÿ.9800532ÿÿÿÿ1.014664
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿSlopeÿ|ÿÿÿ.9973584ÿÿÿ.0088294ÿÿÿ112.96ÿÿÿ0.000ÿÿÿÿÿ.9800532ÿÿÿÿ1.014664
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ_consÿ|ÿÿÿÿÿÿÿÿÿÿ0ÿÿ(constrained)
ÿÿ----------------------+----------------------------------------------------------------
ÿÿout31ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿInterceptÿ|ÿÿÿ.9946205ÿÿÿ.0087487ÿÿÿ113.69ÿÿÿ0.000ÿÿÿÿÿ.9774734ÿÿÿÿ1.011768
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿSlopeÿ|ÿÿÿ.9946205ÿÿÿ.0087487ÿÿÿ113.69ÿÿÿ0.000ÿÿÿÿÿ.9774734ÿÿÿÿ1.011768
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ_consÿ|ÿÿÿÿÿÿÿÿÿÿ0ÿÿ(constrained)
------------------------+----------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿÿÿÿvar(e.out10)|ÿÿÿ.4961152ÿÿÿ.0139092ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ.4695891ÿÿÿÿ.5241397
ÿÿÿÿÿÿÿÿÿÿÿÿvar(e.out20)|ÿÿÿ.5169276ÿÿÿ.0141649ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ.4898974ÿÿÿÿ.5454493
ÿÿÿÿÿÿÿÿÿÿÿÿvar(e.out30)|ÿÿÿÿ.494592ÿÿÿ.0138505ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ.468177ÿÿÿÿ.5224974
ÿÿÿÿÿÿÿÿÿÿÿÿvar(e.out11)|ÿÿÿ.4961152ÿÿÿ.0139092ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ.4695891ÿÿÿÿ.5241397
ÿÿÿÿÿÿÿÿÿÿÿÿvar(e.out21)|ÿÿÿ.5169276ÿÿÿ.0141649ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ.4898974ÿÿÿÿ.5454493
ÿÿÿÿÿÿÿÿÿÿÿÿvar(e.out31)|ÿÿÿÿ.494592ÿÿÿ.0138505ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ.468177ÿÿÿÿ.5224974
ÿÿÿÿÿÿÿÿvar(e.Intercept)|ÿÿÿ1.572194ÿÿÿ.0477378ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ1.481359ÿÿÿÿ1.668598
ÿÿÿÿÿÿÿÿÿÿÿÿvar(e.Slope)|ÿÿÿ1.949152ÿÿÿ.0624624ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ1.830493ÿÿÿÿ2.075502
------------------------+----------------------------------------------------------------
cov(e.Intercept,e.Slope)|ÿÿ-.7580683ÿÿÿ.0409676ÿÿÿ-18.50ÿÿÿ0.000ÿÿÿÿ-.8383634ÿÿÿ-.6777733
-----------------------------------------------------------------------------------------

.ÿ//ÿsexÿ×ÿtimÿinteraction
.ÿlincomÿ[Slope]sexÿ-ÿ[Intercept]sex

ÿ(ÿ1)ÿÿ-ÿ[Intercept]sexÿ+ÿ[Slope]sexÿ=ÿ0

------------------------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿÿÿÿÿ|ÿCoefficientÿÿStd.ÿerr.ÿÿÿÿÿÿzÿÿÿÿP>|z|ÿÿÿÿÿ[95%ÿconf.ÿinterval]
-------------+----------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿ(1)ÿ|ÿÿÿ.4605971ÿÿÿ.0626489ÿÿÿÿÿ7.35ÿÿÿ0.000ÿÿÿÿÿ.3378075ÿÿÿÿ.5833866
------------------------------------------------------------------------------

.ÿ
.ÿ*
.ÿ*ÿ-gsem-ÿrandom-slopeÿmodel,ÿwithin-and-betweenÿformulation
.ÿ*
.ÿkeepÿsexÿout*

.ÿgenerateÿintÿpidÿ=ÿ_n

.ÿquietlyÿreshapeÿlongÿout1ÿout2ÿout3,ÿi(pid)ÿj(tim)

.ÿ
.ÿgsemÿ///
>ÿÿÿÿÿÿÿÿÿ(out?ÿ<-ÿi.timÿM0[pid]ÿ1.tim#M1[pid])ÿ///
>ÿÿÿÿÿÿÿÿÿ(M0[pid]ÿM1[pid]ÿ<-ÿi.sex),ÿ///
>ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿcovariance(e.M0[pid]*e.M1[pid])ÿ///
>ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿnocnsreportÿnodvheaderÿdifficultÿnolog

GeneralizedÿstructuralÿequationÿmodelÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿNumberÿofÿobsÿ=ÿ6,000
Logÿlikelihoodÿ=ÿ-26397.1

------------------------------------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|ÿCoefficientÿÿStd.ÿerr.ÿÿÿÿÿÿzÿÿÿÿP>|z|ÿÿÿÿÿ[95%ÿconf.ÿinterval]
-------------------------+----------------------------------------------------------------
out1ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ1.timÿ|ÿÿÿ.5105775ÿÿÿÿ.042022ÿÿÿÿ12.15ÿÿÿ0.000ÿÿÿÿÿ.4282159ÿÿÿÿ.5929392
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿM0[pid]ÿ|ÿÿÿÿÿÿÿÿÿÿ1ÿÿ(constrained)
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿtim#M1[pid]ÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿ|ÿÿÿÿÿÿÿÿÿÿ1ÿÿ(constrained)
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ_consÿ|ÿÿ-.0118826ÿÿÿ.0357285ÿÿÿÿ-0.33ÿÿÿ0.739ÿÿÿÿ-.0819093ÿÿÿÿÿ.058144
-------------------------+----------------------------------------------------------------
out2ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ1.timÿ|ÿÿÿ.5121864ÿÿÿ.0417436ÿÿÿÿ12.27ÿÿÿ0.000ÿÿÿÿÿ.4303704ÿÿÿÿ.5940024
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿM0[pid]ÿ|ÿÿÿ.9926379ÿÿÿ.0112807ÿÿÿÿ87.99ÿÿÿ0.000ÿÿÿÿÿ.9705282ÿÿÿÿ1.014748
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿtim#M1[pid]ÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿ|ÿÿÿ.9850132ÿÿÿ.0149378ÿÿÿÿ65.94ÿÿÿ0.000ÿÿÿÿÿ.9557358ÿÿÿÿ1.014291
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ_consÿ|ÿÿÿ.0066585ÿÿÿ.0356315ÿÿÿÿÿ0.19ÿÿÿ0.852ÿÿÿÿ-.0631779ÿÿÿÿÿ.076495
-------------------------+----------------------------------------------------------------
out3ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ1.timÿ|ÿÿÿ.5042191ÿÿÿ.0415053ÿÿÿÿ12.15ÿÿÿ0.000ÿÿÿÿÿ.4228702ÿÿÿÿ.5855679
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿM0[pid]ÿ|ÿÿÿ.9923839ÿÿÿ.0111816ÿÿÿÿ88.75ÿÿÿ0.000ÿÿÿÿÿ.9704684ÿÿÿÿ1.014299
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿtim#M1[pid]ÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿ|ÿÿÿ.9837346ÿÿÿ.0148352ÿÿÿÿ66.31ÿÿÿ0.000ÿÿÿÿÿ.9546582ÿÿÿÿ1.012811
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ_consÿ|ÿÿÿÿ.005367ÿÿÿ.0355074ÿÿÿÿÿ0.15ÿÿÿ0.880ÿÿÿÿ-.0642262ÿÿÿÿ.0749601
-------------------------+----------------------------------------------------------------
M0[pid]ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ1.sexÿ|ÿÿÿÿ.544308ÿÿÿ.0484154ÿÿÿÿ11.24ÿÿÿ0.000ÿÿÿÿÿ.4494156ÿÿÿÿ.6392004
-------------------------+----------------------------------------------------------------
M1[pid]ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ1.sexÿ|ÿÿÿ.4970093ÿÿÿ.0557831ÿÿÿÿÿ8.91ÿÿÿ0.000ÿÿÿÿÿ.3876764ÿÿÿÿ.6063422
-------------------------+----------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿÿÿvar(e.M0[pid])|ÿÿÿ1.579528ÿÿÿ.0495803ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ1.485282ÿÿÿÿ1.679755
ÿÿÿÿÿÿÿÿÿÿÿvar(e.M1[pid])|ÿÿÿ1.979694ÿÿÿ.0690054ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ1.848962ÿÿÿÿ2.119669
-------------------------+----------------------------------------------------------------
ÿcov(e.M0[pid],e.M1[pid])|ÿÿ-.7658207ÿÿÿ.0417177ÿÿÿ-18.36ÿÿÿ0.000ÿÿÿÿ-.8475859ÿÿÿ-.6840554
-------------------------+----------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿvar(e.out1)|ÿÿÿ.4920182ÿÿÿ.0142119ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ.4649372ÿÿÿÿ.5206765
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿvar(e.out2)|ÿÿÿ.5193569ÿÿÿ.0143494ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ.4919804ÿÿÿÿ.5482567
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿvar(e.out3)|ÿÿÿÿ.495891ÿÿÿÿ.014079ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ.4690503ÿÿÿÿ.5242675
------------------------------------------------------------------------------------------

.ÿ//ÿAverageÿtimÿeffectÿ(ignoringÿindicatorÿconstants)
.ÿlincomÿ([out1]1.timÿ+ÿ[out2]1.timÿ+ÿ[out3]1.tim)ÿ/ÿ3

ÿ(ÿ1)ÿÿ.3333333*[out1]1.timÿ+ÿ.3333333*[out2]1.timÿ+ÿ.3333333*[out3]1.timÿ=ÿ0

------------------------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿÿÿÿÿ|ÿCoefficientÿÿStd.ÿerr.ÿÿÿÿÿÿzÿÿÿÿP>|z|ÿÿÿÿÿ[95%ÿconf.ÿinterval]
-------------+----------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿ(1)ÿ|ÿÿÿ.5089943ÿÿÿ.0389318ÿÿÿÿ13.07ÿÿÿ0.000ÿÿÿÿÿ.4326895ÿÿÿÿ.5852992
------------------------------------------------------------------------------

.ÿ
.ÿ*
.ÿ*ÿ-mixed-
.ÿ*
.ÿquietlyÿreshapeÿlongÿout,ÿi(pidÿtim)ÿj(ind)

.ÿmixedÿoutÿi.sex##i.timÿi.indÿ||ÿpid:ÿtim,ÿcovariance(unstructured)ÿ///
>ÿÿÿÿÿÿÿÿÿresiduals(independent,ÿby(ind))ÿnolrtestÿnolog

Mixed-effectsÿMLÿregressionÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿNumberÿofÿobsÿÿÿÿ=ÿÿ18,000
Groupÿvariable:ÿpidÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿNumberÿofÿgroupsÿ=ÿÿÿ3,000
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿObsÿperÿgroup:
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿminÿ=ÿÿÿÿÿÿÿ6
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿavgÿ=ÿÿÿÿÿ6.0
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿmaxÿ=ÿÿÿÿÿÿÿ6
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿWaldÿchi2(5)ÿÿÿÿÿ=ÿ1136.50
Logÿlikelihoodÿ=ÿ-26398.004ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿProbÿ>ÿchi2ÿÿÿÿÿÿ=ÿÿ0.0000

--------------------------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿÿÿoutÿ|ÿCoefficientÿÿStd.ÿerr.ÿÿÿÿÿÿzÿÿÿÿP>|z|ÿÿÿÿÿ[95%ÿconf.ÿinterval]
---------------+----------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿ1.sexÿ|ÿÿÿ.5414935ÿÿÿ.0480434ÿÿÿÿ11.27ÿÿÿ0.000ÿÿÿÿÿ.4473302ÿÿÿÿ.6356569
ÿÿÿÿÿÿÿÿÿ1.timÿ|ÿÿÿ.5088338ÿÿÿ.0389306ÿÿÿÿ13.07ÿÿÿ0.000ÿÿÿÿÿ.4325313ÿÿÿÿ.5851363
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿsex#timÿ|
ÿÿÿÿÿÿÿÿÿÿ1ÿ1ÿÿ|ÿÿÿ.4920562ÿÿÿ.0550561ÿÿÿÿÿ8.94ÿÿÿ0.000ÿÿÿÿÿ.3841482ÿÿÿÿ.5999642
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿindÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿ2ÿÿ|ÿÿÿ.0154798ÿÿÿ.0130061ÿÿÿÿÿ1.19ÿÿÿ0.234ÿÿÿÿ-.0100116ÿÿÿÿ.0409712
ÿÿÿÿÿÿÿÿÿÿÿÿ3ÿÿ|ÿÿÿ.0099766ÿÿÿ.0128493ÿÿÿÿÿ0.78ÿÿÿ0.437ÿÿÿÿ-.0152076ÿÿÿÿ.0351609
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿ_consÿ|ÿÿ-.0083652ÿÿÿ.0347733ÿÿÿÿ-0.24ÿÿÿ0.810ÿÿÿÿ-.0765197ÿÿÿÿ.0597892
--------------------------------------------------------------------------------

------------------------------------------------------------------------------
ÿÿRandom-effectsÿparametersÿÿ|ÿÿÿEstimateÿÿÿStd.ÿerr.ÿÿÿÿÿ[95%ÿconf.ÿinterval]
-----------------------------+------------------------------------------------
pid:ÿUnstructuredÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿvar(tim)ÿ|ÿÿÿ1.938513ÿÿÿ.0588627ÿÿÿÿÿÿÿ1.82651ÿÿÿÿ2.057384
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿvar(_cons)ÿ|ÿÿÿ1.563692ÿÿÿ.0447514ÿÿÿÿÿÿ1.478396ÿÿÿÿÿ1.65391
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿcov(tim,_cons)ÿ|ÿÿ-.7538533ÿÿÿ.0399979ÿÿÿÿÿ-.8322478ÿÿÿ-.6754588
-----------------------------+------------------------------------------------
Residual:ÿIndependent,ÿÿÿÿÿÿÿ|
ÿÿÿÿbyÿindÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ1:ÿvar(e)ÿ|ÿÿÿ.4980293ÿÿÿ.0133712ÿÿÿÿÿÿ.4724998ÿÿÿÿ.5249382
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ2:ÿvar(e)ÿ|ÿÿÿ.5169173ÿÿÿ.0136577ÿÿÿÿÿÿÿÿ.49083ÿÿÿÿ.5443911
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ3:ÿvar(e)ÿ|ÿÿÿ.4926044ÿÿÿÿ.013329ÿÿÿÿÿÿ.4671606ÿÿÿÿ.5194339
------------------------------------------------------------------------------

.ÿ
.ÿquietlyÿlogÿcloseÿlo

.ÿ
.ÿexit

endÿofÿdo-file


.

For the latent growth curve formulation with sem above, I've coded the constraints explicitly beforehand. In earlier experience, I've found that the syntax parser occasionally gets confused when I try to impose constraints simultaneously on both indicator intercepts and factor loadings using path notation syntax.

The attached do-file contains code for the random slope, single-equation formulation with gsem (see Example 38g in the user's manual). It follows the exit command and so it's not run and shown in the output above, but it does give the same results as the three approaches above.

[Richard Williams]

This has been driving me crazy. As I suspected, your 2 approaches were not equivalent. To make them equivalent, I think the groups model would have to allow the mean of Ilo to differ between groups. But, I can't figure out how to do that.

So instead, I changed your single goup model to NOT allow the means to differ by groups. Which is not a good model, but it is equivalent to the groups model you have been running (I think). That is, if you run this with your own data, I think both approaches will give you the same results.

* Createv a mockup of data
webuse nhanes2f, clear
renvars houssiz age height weight bpsystol bpdiast black sex \ ///
    ilo1 ilo2 ilo3 ilo4 ilo5 ilo6 pre_post_r sex_r
gen sex_pp = pre_post_r * sex_r

* Run single group model, excluding sex dummy
sem (Ilo -> ilo1 ilo2 ilo3 ilo4 ilo5 ilo6) (pre_post_r -> Ilo) (sex_r -> Ilo@0) (sex_pp -> Ilo), ///
    cov(e.ilo2*e.ilo3) cov(e.ilo1*e.ilo2) cov(e.ilo1*e.ilo3) ///
    nolog nocnsr

* This runs the equivalent multi-group model
sem (Ilo ->  ilo1 ilo2 ilo3 ilo4 ilo5 ilo6 ) (pre_post_r -> Ilo), /// 
    group(sex_r) ginvariant(mcoef mcons serrvar merrvar smerrcov meanex covex) /// 
    cov( e.ilo1*e.ilo2 e.ilo1*e.ilo3 e.ilo2*e.ilo3) ///
    nolog showg nocnsr

estat ginvariant, showp(scoef)

That still leaves open which is the best model to use, since neither of the above models is very good! My inclination is to do what I said earlier, and just use the ginvariant defaults.

* Best model?
sem (Ilo ->  ilo1 ilo2 ilo3 ilo4 ilo5 ilo6 ) (pre_post_r -> Ilo), /// 
    group(sex_r)  /// 
    cov( e.ilo1*e.ilo2 e.ilo1*e.ilo3 e.ilo2*e.ilo3) ///
    nolog showg nocnsr

estat ginvariant, showp(scoef scons)

With my fake data, you would conclude that the effect of pre_post_r does not significantly differ by group. I don't know what your real data will show.

[Joseph Coveney]

It dawned on me afterwards that the postestimation computation above of the sex × tim interaction for the growth curve-like SEM incorrectly used [Intercept]sex for the baseline when that had already been accounted for by loading the second time point's indicators on the Intercept latent factor's sex predictor. It's the time effect that's not been deducted from the [Slope]sex coefficient (due to the noconstant specification). The interaction term is is correctly

lincom [Slope]sex - [Slope]tim

and the corrected comparison table:

Main effects of sex (coefficient ± standard error)
SEM-GC 0.54 ± 0.03
SEM-WB 0.54 ± 0.05
MIXED 0.54 ± 0.05

Main effects of time
SEM-GC 0.51 ± 0.03
SEM-WB 0.51 ± 0.04 (avg.)
MIXED 0.51 ± 0.04

Sex × time interaction
SEM-GC 0.49 ± 0.05
SEM-WB 0.50 ± 0.06
MIXED 0.49 ± 0.06

Intercept Variance (coefficient only)
SEM-GC 1.6
SEM-WB 1.6
MIXED 1.6

Random Slope Variance
SEM-GC 1.9
SEM-WB 2.0
MIXED 1.9

Intercept*Slope Covariance
SEM-GC -0.76
SEM-WB -0.77
MIXED -0.75

Revised do-file and log file are attached.

[Kwok-wing Sum]

Thank you Prof. Williams for your tryout.
Yes, you are right in saying that the revised interaction model produces the exactly same result as the groups model. I have tried your codes and the results are below.
However, setting sex_r -> Ilo@0 in the interaction model seems to mean that we drop this path, i.e., the main effect of sex, right?
As far as I know, to test moderation effect, the main effects of both the IV (pre & post) and the moderator (sex) and the interaction term have to be included. Would it be correct if we drop the main effect of sex?
BTW, it can be seen from the results below that the interaction term has become significant (insignificant in the original model). But I wonder if it has something to do with the dropping the path sex_r -> Ilo

Thank you Dr. Coveney for your advice also. I think it may take some time for me to digest and try out the growth model that you mentioned. There are in fact quite many syntaxes that I don't understand (those run before gsem or mixed).

Revised interaction model (by setting sex_r -> Ilo@0):

-----------------------------------------------------------------------------------
                  |                 OIM
                  | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
------------------+----------------------------------------------------------------
Structural        |
  Ilo             |
       pre_post_r |     1.0322   .0642604    16.06   0.000     .9062523    1.158148
           sex_pp |  -.5825375   .0789673    -7.38   0.000    -.7373106   -.4277644
------------------+----------------------------------------------------------------
Measurement       |
  ilo1            |
              Ilo |          1  (constrained)
            _cons |   4.101196   .0303532   135.12   0.000     4.041704    4.160687
  ----------------+----------------------------------------------------------------
  ilo2            |
              Ilo |   .7341535   .0174991    41.95   0.000     .6998558    .7684512
            _cons |   4.603395   .0267991   171.77   0.000      4.55087     4.65592
  ----------------+----------------------------------------------------------------
  ilo3            |
              Ilo |   .8972643   .0160334    55.96   0.000     .8658395    .9286891
            _cons |   4.419908   .0272956   161.93   0.000     4.366409    4.473406
  ----------------+----------------------------------------------------------------
  ilo4            |
              Ilo |   1.068358   .0188077    56.80   0.000     1.031496     1.10522
            _cons |   4.145609   .0305574   135.67   0.000     4.085717      4.2055
  ----------------+----------------------------------------------------------------
  ilo5            |
              Ilo |   1.148001   .0191365    59.99   0.000     1.110494    1.185507
            _cons |   3.738373   .0317925   117.59   0.000     3.676061    3.800686
  ----------------+----------------------------------------------------------------
  ilo6            |
              Ilo |   1.051796    .018041    58.30   0.000     1.016436    1.087155
            _cons |   3.992465   .0296579   134.62   0.000     3.934337    4.050594
------------------+----------------------------------------------------------------
       var(e.ilo1)|   .7534025   .0229478                      .7097417     .799749
       var(e.ilo2)|   1.043402   .0288968                      .9882747    1.101604
       var(e.ilo3)|   .6190326   .0188627                      .5831447    .6571292
       var(e.ilo4)|   .5276721   .0175686                      .4943377    .5632543
       var(e.ilo5)|    .413576   .0160233                      .3833336    .4462042
       var(e.ilo6)|   .4359774   .0153031                      .4069923    .4670266
        var(e.Ilo)|    1.32266   .0501173                      1.227991    1.424628
------------------+----------------------------------------------------------------
cov(e.ilo1,e.ilo2)|   .1528803   .0188259     8.12   0.000     .1159822    .1897783
cov(e.ilo1,e.ilo3)|   .1094717   .0153727     7.12   0.000     .0793417    .1396016
cov(e.ilo2,e.ilo3)|    .362107   .0184853    19.59   0.000     .3258764    .3983376
-----------------------------------------------------------------------------------
LR test of model vs. saturated: chi2(22) = 388.62              Prob > chi2 = 0.0000

Groups model:

-----------------------------------------------------------------------------------
                  |                 OIM
                  | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
------------------+----------------------------------------------------------------
Structural        |
  Ilo             |
       pre_post_r |
            Male  |     1.0322   .0642604    16.06   0.000     .9062523    1.158148
          Female  |   .4496628   .0606062     7.42   0.000     .3308769    .5684487
------------------+----------------------------------------------------------------
Measurement       |
  ilo1            |
              Ilo |[*] |          1  (constrained)
            _cons |[*] |   4.101196   .0303532   135.12   0.000     4.041704    4.160687
  ----------------+----------------------------------------------------------------
  ilo2            |
              Ilo |[*] |   .7341535   .0174991    41.95   0.000     .6998558    .7684512
            _cons |[*] |   4.603395   .0267991   171.77   0.000      4.55087     4.65592
  ----------------+----------------------------------------------------------------
  ilo3            |
              Ilo |[*] |   .8972643   .0160334    55.96   0.000     .8658395    .9286891
            _cons |[*] |   4.419908   .0272956   161.93   0.000     4.366409    4.473406
  ----------------+----------------------------------------------------------------
  ilo4            |
              Ilo |[*] |   1.068358   .0188077    56.80   0.000     1.031496     1.10522
            _cons |[*] |   4.145609   .0305574   135.67   0.000     4.085717      4.2055
  ----------------+----------------------------------------------------------------
  ilo5            |
              Ilo |[*] |   1.148001   .0191365    59.99   0.000     1.110494    1.185507
            _cons |[*] |   3.738373   .0317925   117.59   0.000     3.676061    3.800686
  ----------------+----------------------------------------------------------------
  ilo6            |
              Ilo |[*] |   1.051796    .018041    58.30   0.000     1.016436    1.087155
            _cons |[*] |   3.992465   .0296579   134.62   0.000     3.934337    4.050594
------------------+----------------------------------------------------------------
       var(e.ilo1)|[*] |   .7534025   .0229478                      .7097417     .799749
       var(e.ilo2)|[*] |   1.043402   .0288968                      .9882747    1.101604
       var(e.ilo3)|[*] |   .6190326   .0188627                      .5831447    .6571292
       var(e.ilo4)|[*] |   .5276721   .0175686                      .4943377    .5632543
       var(e.ilo5)|[*] |    .413576   .0160233                      .3833336    .4462042
       var(e.ilo6)|[*] |   .4359774   .0153031                      .4069923    .4670266
        var(e.Ilo)|[*] |    1.32266   .0501173                      1.227991    1.424628
------------------+----------------------------------------------------------------
cov(e.ilo1,e.ilo2)|[*] |   .1528803   .0188259     8.12   0.000     .1159822    .1897783
cov(e.ilo1,e.ilo3)|[*] |   .1094717   .0153727     7.12   0.000     .0793418    .1396016
cov(e.ilo2,e.ilo3)|[*] |    .362107   .0184853    19.59   0.000     .3258764    .3983376
-----------------------------------------------------------------------------------
Note:[*] identifies parameter estimates constrained to be equal across groups.
LR test of model vs. saturated: chi2(43) = 458.64              Prob > chi2 = 0.0000

[Richard Williams]

It would NOT be correct to drop the main effect of sex. Basically I was showing how to get a single group model and a two group model to give identical Incorrect effects. If I could have cloned your original single group model I would have but I couldn’t figure out how.

My own inclination is to run the last model I showed, using only the default settings of ginvariant. What result does it give you?

Getting back to basics, there is still the question of whether this is a good latent variable model at all. Do these 6 indicators really reflect a single latent variable?

[Kwok-wing Sum]

Thank you Prof. Williams again.
I did try your last two groups model. Below are the results. The path pre_post_r -> Ilo is significant, so it contradicts with the original interaction model.
Even though you have shown identical incorrect effects, it is thought-provoking and makes me think the other way around, i.e., what constraint(s) I should release in your two groups model, so that it may produce identical correct results with the interaction model (though adjustment to the interaction model is still needed). I have tried to release some of the constraints but failed to get identical results.

Yes, these 6 indicators do load on a single latent variable when I run the SEM model with all observations without grouping. I think the goodness-of-fit indices are acceptable (though not very good and some modifications are needed). Please see the table at the bottom.

. sem (Ilo ->  ilo1 ilo2 ilo3 ilo4 ilo5 ilo6 ) (pre_post_r -> Ilo), group(sex_r)  cov( e.ilo1*e.ilo2 e.ilo1*e.ilo3 e.ilo2*e.ilo3) nolog showg nocnsr
(3987 observations with missing values excluded)

Endogenous variables
  Measurement: ilo1 ilo2 ilo3 ilo4 ilo5 ilo6
  Latent:      Ilo

Exogenous variables
  Observed: pre_post_r

Structural equation model                             Number of obs    = 2,977
Grouping variable: sex_r                              Number of groups =     2
Estimation method: ml

Log likelihood = -26255.936

-----------------------------------------------------------------------------------
                  |                 OIM
                  | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
------------------+----------------------------------------------------------------
Structural        |
  Ilo             |
       pre_post_r |
            Male  |   1.057029   .0668884    15.80   0.000     .9259299    1.188128
          Female  |    .475181   .0593236     8.01   0.000     .3589088    .5914531
------------------+----------------------------------------------------------------
Measurement       |
  ilo1            |
              Ilo |[*] |          1  (constrained)
            _cons |[*] |   4.076275   .0316695   128.71   0.000     4.014204    4.138346
  ----------------+----------------------------------------------------------------
  ilo2            |
              Ilo |[*] |   .7354143   .0174511    42.14   0.000     .7012108    .7696178
            _cons |[*] |   4.580237   .0276641   165.57   0.000     4.526016    4.634457
  ----------------+----------------------------------------------------------------
  ilo3            |
              Ilo |[*] |   .9005706   .0160093    56.25   0.000     .8691929    .9319483
            _cons |[*] |   4.392349    .028554   153.83   0.000     4.336385    4.448314
  ----------------+----------------------------------------------------------------
  ilo4            |
              Ilo |[*] |   1.071573   .0189109    56.66   0.000     1.034508    1.108638
            _cons |[*] |   4.109271   .0322031   127.60   0.000     4.046155    4.172388
  ----------------+----------------------------------------------------------------
  ilo5            |
              Ilo |[*] |   1.150107   .0191952    59.92   0.000     1.112485    1.187729
            _cons |[*] |   3.707391   .0334018   110.99   0.000     3.641925    3.772857
  ----------------+----------------------------------------------------------------
  ilo6            |
              Ilo |[*] |   1.052774   .0181001    58.16   0.000     1.017299     1.08825
            _cons |[*] |   3.965231    .031095   127.52   0.000     3.904285    4.026176
------------------+----------------------------------------------------------------
       var(e.ilo1)|
            Male  |   .7583969   .0331749                      .6960847    .8262871
          Female  |   .7504299   .0314838                      .6911917    .8147451
       var(e.ilo2)|
            Male  |    1.01689   .0405871                       .940373    1.099634
          Female  |    1.06846   .0410409                      .9909743    1.152004
       var(e.ilo3)|
            Male  |   .5753334   .0254611                      .5275335    .6274644
          Female  |   .6600116   .0275646                       .608138    .7163101
       var(e.ilo4)|
            Male  |   .5715683   .0268245                      .5213389    .6266371
          Female  |    .486231   .0231963                      .4428279    .5338883
       var(e.ilo5)|
            Male  |   .4239916   .0230802                      .3810848    .4717293
          Female  |   .3998188    .021425                      .3599565    .4440956
       var(e.ilo6)|
            Male  |   .4376258    .021837                      .3968524    .4825884
          Female  |   .4365623    .020979                      .3973213     .479679
        var(e.Ilo)|
            Male  |   1.433753   .0694416                       1.30391    1.576525
          Female  |   1.211685   .0577822                      1.103565    1.330397
------------------+----------------------------------------------------------------
cov(e.ilo1,e.ilo2)|
            Male  |   .1839323    .027015     6.81   0.000     .1309838    .2368809
          Female  |   .1240563   .0260917     4.75   0.000     .0729175    .1751951
cov(e.ilo1,e.ilo3)|
            Male  |   .1126119   .0215364     5.23   0.000     .0704013    .1548224
          Female  |   .1070265   .0216561     4.94   0.000     .0645814    .1494716
cov(e.ilo2,e.ilo3)|
            Male  |   .2880909   .0248446    11.60   0.000     .2393964    .3367854
          Female  |    .432002    .027221    15.87   0.000     .3786499    .4853541
-----------------------------------------------------------------------------------
Note:[*] identifies parameter estimates constrained to be equal across groups.
LR test of model vs. saturated: chi2(33) = 414.81              Prob > chi2 = 0.0000

. estat ginvariant, showp(scoef scons)

Tests for group invariance of parameters

------------------------------------------------------------------------------
             |            Wald test                       Score test
             |      chi2         df    P>chi2       chi2          df    P>chi2
-------------+----------------------------------------------------------------
Structural   |
  Ilo        |
  pre_post_r |    54.061          1    0.0000          .           .         .
------------------------------------------------------------------------------

Goodness-of-fit indices of the original model (without grouping):

----------------------------------------------------------------------------
Fit statistic        |      Value   Description
---------------------+------------------------------------------------------
Likelihood ratio     |
          chi2_ms(6) |     69.971   model vs. saturated
            p > chi2 |      0.000
         chi2_bs(15) |  14706.878   baseline vs. saturated
            p > chi2 |      0.000
---------------------+------------------------------------------------------
Population error     |
               RMSEA |      0.060   Root mean squared error of approximation
 90% CI, lower bound |      0.048
         upper bound |      0.073
              pclose |      0.089   Probability RMSEA <= 0.05
---------------------+------------------------------------------------------
Information criteria |
                 AIC |  49020.749   Akaike's information criterion
                 BIC |  49146.721   Bayesian information criterion
---------------------+------------------------------------------------------
Baseline comparison  |
                 CFI |      0.996   Comparative fit index
                 TLI |      0.989   Tucker–Lewis index
---------------------+------------------------------------------------------
Size of residuals    |
                SRMR |      0.011   Standardized root mean squared residual
                  CD |      0.937   Coefficient of determination
----------------------------------------------------------------------------

[Joseph Coveney]

I think it may take some time for me to digest and try out the growth model that you mentioned.

Yeah, me too, for example, I mistakenly thought that constraining its intercept to zero is necessary for identification of the sex and time effects. Not so, and it's better for it to be freed in order to facilitate estimation of a nonzero constant (fixed-effects intercept).

Updates to code in red below; corresponding do-file and log file again attached.

quietly drawnorm intercept slope, double corr(1 -0.25 \ -0.25 1) n(3000)
generate byte sex = mod(_n, 2)

tempname Corr
matrix define `Corr' = J(3, 3, 0.5) + I(3) * 0.5
forvalues tim = 0/1 {
    quietly drawnorm e1`tim' e2`tim' e3`tim', double corr(`Corr')
    forvalues i = 1/3 {
        generate double out`i'`tim' = ///
            10 + intercept + ///
                sex / 2 + ///
                `tim' * (1/2 + slope) + ///
                    sex * `tim' / 2 + ///
                        e`i'`tim'
    }
}

*
* SEM (growth-curve like)
*
constraint define  1 [out10]Intercept = 1
constraint define  2 [out11]Intercept = 1
constraint define  3 [out11]Slope = 1

constraint define  4 [out20]Intercept = [out21]Intercept
constraint define  5 [out30]Intercept = [out31]Intercept
constraint define  6 [out21]Intercept = [out21]Slope
constraint define  7 [out31]Intercept = [out31]Slope

constraint define  8 [/]var(e.out10) = [/]var(e.out11)
constraint define  9 [/]var(e.out20) = [/]var(e.out21)
constraint define 10 [/]var(e.out30) = [/]var(e.out31)

generate byte tim = !sex

* constraint define 11 [Intercept]tim = 0

sem ///
    (out* <- Intercept, noconstant) ///
    (out?1 <- Slope, noconstant) ///
    (Intercept Slope <- tim sex, noconstant), ///
        constraints(1/10) ///
        covariance((e.Intercept*e.Slope)) ///
            nocnsreport nodescribe nolog
// sex
lincom [Intercept]sex - [Intercept]tim
// sex × tim interaction
lincom [Slope]sex - [Slope]tim

Updated coefficient-comparison table below reflects the changes, including a new section for fixed-effects intercept (constant) and slight changes in standard errors for growth curve-like SEM due to relaxation of constraints. (The growth curve-like SEM's standard errors now match the other two methods' more closely than in #11 above.)

Intercept (coefficient ± standard error)
SEM-GC 9.99 ± 0.03
SEM-WB 9.99 ± 0.04
MIXED 9.99 ± 0.03

Main effects of sex
SEM-GC 0.54 ± 0.05
SEM-WB 0.54 ± 0.05
MIXED 0.54 ± 0.05

Main effects of time
SEM-GC 0.51 ± 0.03
SEM-WB 0.51 ± 0.04 (avg.)
MIXED 0.51 ± 0.04

Sex × time interaction
SEM-GC 0.49 ± 0.06
SEM-WB 0.50 ± 0.06
MIXED 0.49 ± 0.06

Intercept Variance (coefficient only)
SEM-GC 1.6
SEM-WB 1.6
MIXED 1.6

Random Slope Variance
SEM-GC 1.9
SEM-WB 2.0
MIXED 1.9

Intercept*Slope Covariance
SEM-GC -0.76
SEM-WB -0.77
MIXED -0.75