When becoming nearly any mannequin, we could also be considering investigating whether or not parameters differ throughout teams equivalent to time intervals, age teams, gender, or faculty attended. In different phrases, we could want to carry out assessments of moderation when the moderator variable is categorical. For regression fashions, this may be so simple as together with group indicators within the mannequin and interacting them with different predictors.
We naturally have hypotheses relating to variations in parameters throughout teams when becoming structural equation fashions as nicely. When these fashions contain latent variables and the corresponding noticed measurements, we are able to take a look at whether or not these measurements are invariant throughout teams. Analysis of measurement invariance sometimes entails a sequence of assessments for equality of measurement coefficients (issue loadings), equality of intercepts, and equality of error variances throughout teams.
On this publish, I reveal how one can use the sem command’s group() and ginvariant() choices in addition to the postestimation command estat ginvariant to simply carry out assessments of measurement invariance.
Measurement invariance instance
I take advantage of knowledge from Holzinger and Swineford (1939), which information college students’ scores on quite a few exams designed to measure various kinds of talents. The scholars on this dataset got here from two totally different colleges, the Pasteur faculty and the Grant-White faculty, and I need to take a look at for variations throughout colleges. Right here I give attention to three exams that have been meant to measure spatial talents. I’ll match the confirmatory issue mannequin akin to the next path diagram and carry out a sequence of assessments for measurement invariance. Though this instance makes use of the sem command, I may have equivalently drawn this diagram within the Builder and chosen group evaluation to suit all of the fashions mentioned beneath.
To start, I match a mannequin with all parameters estimated individually throughout teams. There are numerous methods to set the required figuring out constraints that present a scale and placement for the latent variable. Right here I set the imply of the Spatial latent variable to 0 and the variance to 1 in each teams.
. sem (Spatial -> visible cubes paper),
> variance(Spatial@1) imply(Spatial@0) ginvariant(none) group(faculty)
Endogenous variables
Measurement: visible cubes paper
Exogenous variables
Latent: Spatial
Becoming goal mannequin:
Iteration 0: log chance = -2603.5782
Iteration 1: log chance = -2603.5782
Structural equation mannequin Variety of obs = 301
Grouping variable = faculty Variety of teams = 2
Estimation methodology = ml
Log chance = -2603.5782
( 1) [var(Spatial)]1bn.faculty = 1
( 2) [mean(Spatial)]1bn.faculty = 0
( 3) [var(Spatial)]2.faculty = 1
( 4) [mean(Spatial)]2.faculty = 0
-------------------------------------------------------------------------------
| OIM
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
--------------+----------------------------------------------------------------
Measurement |
visible <- |
Spatial |
Pasteur | 4.264065 .8600633 4.96 0.000 2.578372 5.949759
Grant-Wh~e | 5.49895 1.190435 4.62 0.000 3.165739 7.83216
_cons |
Pasteur | 29.64744 .5674293 52.25 0.000 28.53529 30.75958
Grant-Wh~e | 29.57931 .5721785 51.70 0.000 28.45786 30.70076
------------+----------------------------------------------------------------
cubes <- |
Spatial |
Pasteur | 2.26321 .5214501 4.34 0.000 1.241187 3.285234
Grant-Wh~e | 1.808245 .5031516 3.59 0.000 .8220861 2.794404
_cons |
Pasteur | 23.9359 .3927222 60.95 0.000 23.16618 24.70562
Grant-Wh~e | 24.8 .3678649 67.42 0.000 24.079 25.521
------------+----------------------------------------------------------------
paper <- |
Spatial |
Pasteur | 1.695466 .3429472 4.94 0.000 1.023302 2.36763
Grant-Wh~e | 1.311235 .3413206 3.84 0.000 .6422592 1.980211
_cons |
Pasteur | 14.16026 .227089 62.36 0.000 13.71517 14.60534
Grant-Wh~e | 14.30345 .2335324 61.25 0.000 13.84573 14.76116
--------------+----------------------------------------------------------------
imply(Spatial)|
[*] | 0 (constrained)
--------------+----------------------------------------------------------------
var(e.visible)|
Pasteur | 32.04601 6.912718 20.9971 48.90898
Grant-White | 17.23285 12.18676 4.309258 68.91467
var(e.cubes)|
Pasteur | 18.93787 2.710244 14.30585 25.06967
Grant-White | 16.35232 2.318816 12.38443 21.59149
var(e.paper)|
Pasteur | 5.170226 1.09911 3.408453 7.842631
Grant-White | 6.188581 .9975804 4.512114 8.487938
var(Spatial)|
[*] | 1 (constrained)
-------------------------------------------------------------------------------
Observe: [*] identifies parameter estimates constrained to be equal throughout teams.
LR take a look at of mannequin vs. saturated: chi2(0) = 0.00, Prob > chi2 = .
Glancing by this output, we see that most of the parameter estimates are very related for the 2 colleges. The estat ginvariant command gives assessments of invariance throughout teams.
. estat ginvariant, showpclass(mcoef) class
Assessments for group invariance of parameters
------------------------------------------------------------------------------
| Wald Check Rating Check
| chi2 df p>chi2 chi2 df p>chi2
-------------+----------------------------------------------------------------
Measurement |
visible <- |
Spatial | 0.707 1 0.4004 . . .
-----------+----------------------------------------------------------------
cubes <- |
Spatial | 0.394 1 0.5301 . . .
-----------+----------------------------------------------------------------
paper <- |
Spatial | 0.631 1 0.4271 . . .
------------------------------------------------------------------------------
Joint assessments for every parameter class
------------------------------------------------------------------------------
| Wald Check Rating Check
| chi2 df p>chi2 chi2 df p>chi2
-------------+----------------------------------------------------------------
mcoef | 1.097 3 0.7778 . . .
------------------------------------------------------------------------------
The showpclass(mcoef) and class choices restricted the outcomes to assessments relating to measurement coefficients and requested a joint take a look at for the speculation that every one measurement coefficients are equal throughout teams. The primary desk within the output stories separate assessments of equality of the measurement coefficients throughout teams. My focus now, nonetheless, is on the joint Wald take a look at proven within the second desk, and we fail to reject the speculation of equality throughout teams for all measurement coefficients.
I now embrace the ginvariant(mcoef) possibility as a way to match a mannequin with the measurement coefficients constrained to be equal throughout teams by typing
. sem (Spatial -> visible cubes paper), variance(Spatial@1) ///
imply(Spatial@0) ginvariant(mcoef) group(faculty)
after which take a look at whether or not the intercepts will be constrained:
. estat ginvariant, showpclass(mcons) class
Assessments for group invariance of parameters
------------------------------------------------------------------------------
| Wald Check Rating Check
| chi2 df p>chi2 chi2 df p>chi2
-------------+----------------------------------------------------------------
Measurement |
visible <- |
_cons | 0.007 1 0.9326 . . .
-----------+----------------------------------------------------------------
cubes <- |
_cons | 2.580 1 0.1082 . . .
-----------+----------------------------------------------------------------
paper <- |
_cons | 0.193 1 0.6605 . . .
------------------------------------------------------------------------------
Joint assessments for every parameter class
------------------------------------------------------------------------------
| Wald Check Rating Check
| chi2 df p>chi2 chi2 df p>chi2
-------------+----------------------------------------------------------------
mcons | 3.011 3 0.3900 . . .
------------------------------------------------------------------------------
We fail to reject the null speculation that every one intercepts are equal throughout teams, so I match the mannequin with these equality constraints by specifying the ginvariant(mcoef mcons) possibility.
. sem (Spatial -> visible cubes paper), variance(Spatial@1) ///
imply(Spatial@0) ginvariant(mcoef mcons) group(faculty)
Then, I take a look at the equality of the error variances.
. estat ginvariant, showpclass(merrvar) class
Assessments for group invariance of parameters
------------------------------------------------------------------------------
| Wald Check Rating Check
| chi2 df p>chi2 chi2 df p>chi2
-------------+----------------------------------------------------------------
var(e.visible)| 0.359 1 0.5493 . . .
var(e.cubes)| 1.413 1 0.2345 . . .
var(e.paper)| 0.014 1 0.9052 . . .
------------------------------------------------------------------------------
Joint assessments for every parameter class
------------------------------------------------------------------------------
| Wald Check Rating Check
| chi2 df p>chi2 chi2 df p>chi2
-------------+----------------------------------------------------------------
merrvar | 1.857 3 0.6027 . . .
------------------------------------------------------------------------------
As soon as once more, we fail to reject the null speculation of invariance throughout teams. I now impose constraints on the coefficients, intercepts, and error variances whereas permitting the imply and variance of the latent variable to vary throughout teams. To do that, I take away the imply(Spatial@0) possibility and exchange the variance(Spatial@1) with variance(1:Spatial@1). With this variation, the imply and variance of Spatial can be set to 0 and 1, respectively, within the first group however estimated freely within the second group.
. sem (Spatial -> visible cubes paper),
> variance(1:Spatial@1) ginvariant(mcoef mcons merrvar) group(faculty)
Endogenous variables
Measurement: visible cubes paper
Exogenous variables
Latent: Spatial
Becoming goal mannequin:
Iteration 0: log chance = -5357.6935 (not concave)
Iteration 1: log chance = -4792.5814 (not concave)
Iteration 2: log chance = -4316.3827 (not concave)
Iteration 3: log chance = -2769.069 (not concave)
Iteration 4: log chance = -2662.2605
Iteration 5: log chance = -2645.7652
Iteration 6: log chance = -2629.1987
Iteration 7: log chance = -2622.83 (not concave)
Iteration 8: log chance = -2622.3555
Iteration 9: log chance = -2622.3227
Iteration 10: log chance = -2621.9007
Iteration 11: log chance = -2621.8931
Iteration 12: log chance = -2621.893
Structural equation mannequin Variety of obs = 301
Grouping variable = faculty Variety of teams = 2
Estimation methodology = ml
Log chance = -2621.893
( 1) [cubes]1bn.faculty#c.Spatial - [cubes]2.faculty#c.Spatial = 0
( 2) [paper]1bn.faculty#c.Spatial - [paper]2.faculty#c.Spatial = 0
( 3) [var(e.visual)]1bn.faculty - [var(e.visual)]2.faculty = 0
( 4) [var(e.cubes)]1bn.faculty - [var(e.cubes)]2.faculty = 0
( 5) [var(e.paper)]1bn.faculty - [var(e.paper)]2.faculty = 0
( 6) [var(Spatial)]1bn.faculty = 1
( 7) [visual]1bn.faculty - [visual]2.faculty = 0
( 8) [cubes]1bn.faculty - [cubes]2.faculty = 0
( 9) [paper]1bn.faculty - [paper]2.faculty = 0
(10) [visual]2.faculty#c.Spatial = 1
(11) [mean(Spatial)]1bn.faculty = 0
-------------------------------------------------------------------------------
| OIM
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
--------------+----------------------------------------------------------------
Measurement |
visible <- |
Spatial |
Pasteur | 5.472561 1.129916 4.84 0.000 3.257966 7.687156
Grant-Wh~e | 1 (constrained)
_cons |
[*] | 29.32102 .4932735 59.44 0.000 28.35422 30.28782
------------+----------------------------------------------------------------
cubes <- |
Spatial |
[*] | .3968564 .1833049 2.17 0.030 .0375854 .7561274
_cons |
[*] | 24.26618 .2890016 83.97 0.000 23.69975 24.83262
------------+----------------------------------------------------------------
paper <- |
Spatial |
[*] | .2953686 .137265 2.15 0.031 .0263341 .5644031
_cons |
[*] | 14.16525 .1786194 79.30 0.000 13.81516 14.51533
--------------+----------------------------------------------------------------
imply(Spatial)|
Pasteur | 0 (constrained)
Grant-White | .4140109 .6928933 0.60 0.550 -.9440351 1.772057
--------------+----------------------------------------------------------------
var(e.visible)|
[*] | 19.50062 12.09195 5.784095 65.74481
var(e.cubes)|
[*] | 20.08682 1.784905 16.87617 23.90829
var(e.paper)|
[*] | 6.864085 .691005 5.634982 8.361281
var(Spatial)|
Pasteur | 1 (constrained)
Grant-White | 25.44848 15.33031 7.814351 82.87636
-------------------------------------------------------------------------------
Observe: [*] identifies parameter estimates constrained to be equal throughout teams.
LR take a look at of mannequin vs. saturated: chi2(7) = 36.63, Prob > chi2 = 0.0000
The imply of 0.414 for Spatial within the Grant-White faculty represents the distinction in technique of this latent variable throughout colleges, and we discover the distinction in means throughout colleges is just not considerably totally different from 0.
Abstract
Assessments of hypotheses relating to the equality of parameters throughout teams are simply carried out utilizing the sem command and estat ginvariant. Whereas there are minor variations all through structural equation modeling literature in suggestions for setting figuring out constraints and for the order of assessments for invariance, the instruments that I’ve demonstrated will be tailored to accommodate any type of assessments for measurement invariance. These similar instruments can be used to check for parameter invariance throughout teams in different varieties of structural equation fashions.
Reference
Holzinger, Ok.~J., and F. Swineford. 1939. A examine in issue evaluation: The soundness of a bi-factor answer. Supplementary Academic Monographs, 48. Chicago, IL: College of Chicago.
