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I am using LISREL 8 to do path analysis or structural equation modeling and I would like to compare my model for different groups. What syntax would I have to use in LISREL to examine whether differences in the model exist between the groups? How do I read the output to determine if differences exist?
This FAQ assumes that you know how to interpret and run a single group analysis using LISREL. If you do not, see our online LISREL tutorial.
LISREL allows you to test whether your groups meet the assumption that they are equal by examining whether different matrices in your model (which represent sets of path coefficients) are "invariant". In other words, you will be testing whether the matrices in your model are equal for your groups. The following examples will test whether two groups, boys academic or boys non-academic are equal. You can find this example in the LISREL 8 users guide in Chapter 9.
Note: You must use covariance matrices for this analysis, not correlation matrices.
Different assumptions of group equality can be tested and they are usually tested in a particular order (Bollen, 1989). Table 1 indicates the models we will test and their related hypotheses. We will not test Problem A in Chapter 9, but we will start testing the equality for Problem B. You should note that the following procedures impose more stringent restrictions of equality at each step. The order in which these restrictions are imposed will depend upon the model you are testing. If you are using path modeling, for example, you would not test for hypothesis B or C as you would not have any measurement model and thus no LX parameter estimates.
Table 1. Testing for the Equality of Two Groups
| Problem | Hypothesis | X2 | df | p-value | Decision |
| B | two factors exist for both of the groups | 1.52 | 2 | 0.47 | Accepted |
| C | the two groups' factor loadings are equal | 8.77 | 4 | 0.07 | Accepted |
| D | there are equal factor loadings for both groups AND the errors are equal for both groups | 21.5 | 8 | 0.006 | Rejected |
| E | there are equal factor loadings for both groups AND the errors are equal for both groups AND the groups have equal factor variances and covariances | 38.2 | 11 | 0.000 | Rejected |
In addition to testing invariance hypotheses, you can also use LISREL
EQ commands to set specific parameter estimates to be equal across
groups. See the LISREL 8 User's Guide for more documentation on the EQ
command.
Before we begin testing the equality of the two groups, we will first run the two groups separately to examine the output for each group. The following syntax tests the hypothesized model for the boys academic group.
!Testing the boys academic model.DA NI=4 NO=373
MO NX=4 NK=2
LA
read-gr5 writ-gr5 read-gr7 writ-gr7
CM SY
281.35
184.22 182.82
216.74 171.70 283.29
198.38 153.20 208.84 246.07
FR LX(2,1) LX(4,2)
VA 1 LX(1,1) LX(3,2)
OU
The following output should be familiar to you, as it is essentially the same as what you get when you examine a model for one group. Important findings to note are that there are 9 parameters estimated for this model and the Chi-square for this group is .86 and is nonsignificant at p=.35. The fit indices are also excellent: Goodness of Fit Index (GFI) = 1.00, Normed Fit Index (NFI) = 1.00, and Non-normed Fit Index (NNFI) = 1.00.
Note that the parts discussed above are shown in red in the following output.
*****The LISREL output begins here******
The following lines were read from file multigroup1.ls8 multigroup1.out:
!testing the boys' academic model
da ni=4 no=373
mo nx=4 nk=2
LA
read-gr5 writ-gr5 read-gr7 writ-gr7
CM SY
281.349
184.219 182.821
216.739 171.699 283.289
198.376 153.201 208.837 246.069
fr lx(2,1) lx(4,2)
va 1 lx(1,1) lx(3,2)
ou
!testing the boys' academic model
NUMBER OF INPUT VARIABLES 4
NUMBER OF Y - VARIABLES 0
NUMBER OF X - VARIABLES 4
NUMBER OF ETA - VARIABLES 0
NUMBER OF KSI - VARIABLES 2
NUMBER OF OBSERVATIONS 373
COVARIANCE MATRIX TO BE ANALYZED
read-gr5 writ-gr5 read-gr7 writ-gr7
-------- -------- -------- --------
read-gr5 281.35
writ-gr5 184.22 182.82
read-gr7 216.74 171.70 283.29
writ-gr7 198.38 153.20 208.84
246.07
!testing the boys' academic model
PARAMETER SPECIFICATIONS
LAMBDA-X
KSI 1 KSI 2
-------- --------
read-gr5
0 0
writ-gr5
1 0
read-gr7
0 0
writ-gr7
0 2
PHI
KSI 1 KSI 2
-------- --------
KSI 1
3
KSI 2
4 5
THETA-DELTA
read-gr5 writ-gr5 read-gr7 writ-gr7
-------- -------- -------- --------
6 7
8 9
!testing the boys' academic model
Number of Iterations = 4
LISREL ESTIMATES (MAXIMUM LIKELIHOOD)
LAMBDA-X
KSI 1 KSI 2
-------- --------
read-gr5 1.00 - -
writ-gr5
.78 - -
(.03)
24.93
read-gr7 - - 1.00
writ-gr7
- - .91
(.04)
23.49
PHI
KSI 1 KSI 2
-------- --------
KSI 1
235.11
(21.02)
11.18
KSI 2 217.75
230.65
(18.45) (21.14)
11.80 10.91
THETA-DELTA
read-gr5 writ-gr5 read-gr7 writ-gr7
-------- -------- -------- --------
46.24
38.48 52.64 56.98
(6.28)
(4.30) (6.74) (6.16)
7.37
8.95 7.81 9.25
SQUARED MULTIPLE CORRELATIONS
FOR X - VARIABLES
read-gr5 writ-gr5 read-gr7 writ-gr7
-------- -------- -------- --------
.84
.79 .81 .77
GOODNESS OF FIT STATISTICS
CHI-SQUARE WITH 1 DEGREE
OF FREEDOM = 0.86 (P = 0.35)
ESTIMATED NON-CENTRALITY PARAMETER
(NCP) = 0.0
90 PERCENT CONFIDENCE INTERVAL
FOR NCP = (0.0 ; 6.61)
MINIMUM FIT FUNCTION VALUE =
0.0023
POPULATION DISCREPANCY FUNCTION
VALUE (F0) = 0.0
90 PERCENT CONFIDENCE INTERVAL
FOR F0 = (0.0 ; 0.018)
ROOT MEAN SQUARE ERROR OF APPROXIMATION
(RMSEA) = 0.0
90 PERCENT CONFIDENCE INTERVAL
FOR RMSEA = (0.0 ; 0.13)
P-VALUE FOR TEST OF CLOSE FIT
(RMSEA < 0.05) = 0.54
EXPECTED CROSS-VALIDATION INDEX
(ECVI) = 0.051
90 PERCENT CONFIDENCE INTERVAL
FOR ECVI = (0.051 ; 0.069)
ECVI FOR SATURATED MODEL = 0.054
ECVI FOR INDEPENDENCE MODEL
= 3.27
CHI-SQUARE FOR INDEPENDENCE MODEL
WITH 6 DEGREES OF FREEDOM = 1209.45
INDEPENDENCE AIC = 1217.45
MODEL AIC = 18.86
SATURATED AIC = 20.00
INDEPENDENCE CAIC = 1237.14
MODEL CAIC = 63.15
SATURATED CAIC = 69.22
ROOT MEAN SQUARE RESIDUAL (RMR)
= 0.73
STANDARDIZED RMR = 0.0031
GOODNESS OF FIT INDEX (GFI)
= 1.00
ADJUSTED GOODNESS OF FIT
INDEX (AGFI) = 0.99
PARSIMONY GOODNESS OF FIT INDEX
(PGFI) = 0.100
NORMED FIT INDEX (NFI) = 1.00
NON-NORMED FIT INDEX (NNFI)
= 1.00
PARSIMONY NORMED FIT INDEX (PNFI)
= 0.17
COMPARATIVE FIT INDEX (CFI)
= 1.00
INCREMENTAL FIT INDEX (IFI)
= 1.00
RELATIVE FIT INDEX (RFI) = 1.00
CRITICAL N (CN) = 2870.39
[OUTPUT DELETED]
Now, we will test the same model for the non-academic boys. This syntax is essentially the same as that for the boys’ academic model except that we are testing the model for a different sample. Therefore, the covariance matrix is different as is the number of participants (NO=249).
!Testing the boys non-academic model.DA NI=4 NO=249
MO NX=4 NK=2
LA
read-gr5 writ-gr5 read-gr7 writ-gr7
CM SY
174.48
134.47 161.87
129.84 118.84 228.45
102.19 97.77 136.06 180.46
OU
This output should also look familiar as we are testing the same model with a different sample. Again, we have 9 parameter estimates. The Chi-square for this model is .66 and is also nonsignificant, with p = .42. The fit indices for this model also indicate an excellent fit, Goodness of Fit Index (GFI) = 1.00, Normed Fit Index (NFI) = 1.00, and Non-normed Fit Index (NNFI) = 1.00.
*****The LISREL output starts here*****
!testing boys non-academic model
da ni=4 no=249
mo nx=4 nk=2
LA
read-gr5 writ-gr5 read-gr7 write-gr7
CM SY
174.485
134.468 161.869
129.840 118.836 228.449
102.194 97.767 136.058 180.460
fr lx(2,1) lx(4,2)
va 1 lx(1,1) lx(3,2)
ou
!testing boys non-academic model
NUMBER OF INPUT VARIABLES 4
NUMBER OF Y - VARIABLES 0
NUMBER OF X - VARIABLES 4
NUMBER OF ETA - VARIABLES 0
NUMBER OF KSI - VARIABLES 2
NUMBER OF OBSERVATIONS 249
!testing boys non-academic model
COVARIANCE MATRIX TO BE ANALYZED
read-gr5 writ-gr5 read-gr7 write-gr
-------- -------- -------- --------
read-gr5 174.49
writ-gr5 134.47 161.87
read-gr7 129.84 118.84 228.45
write-gr 102.19 97.77 136.06
180.46
!testing boys non-academic model
PARAMETER SPECIFICATIONS
LAMBDA-X
KSI 1 KSI 2
-------- --------
read-gr5
0 0
writ-gr5
1 0
read-gr7
0 0
write-gr
0 2
PHI
KSI 1 KSI 2
-------- --------
KSI1
3
KSI2
4 5
THETA-DELTA
read-gr5 writ-gr5 read-gr7 write-gr
-------- -------- -------- --------
6 7
8 9
!testing boys non-academic model
Number of Iterations = 4
LISREL ESTIMATES (MAXIMUM LIKELIHOOD)
LAMBDA-X
KSI 1 KSI 2
-------- --------
read-gr5 1.00 - -
writ-gr5
.93 - -
(.06)
16.54
read-gr7 - - 1.00
write-gr
- - .80
(.07)
12.24
PHI
KSI 1 KSI 2
-------- --------
KSI1 144.68
(16.66)
8.68
KSI2 129.03
169.74
(15.06) (22.26)
8.57 7.63
THETA-DELTA
read-gr5 writ-gr5 read-gr7 write-gr
-------- -------- -------- --------
29.80
36.89 58.71 71.40
(6.82) (6.35)
(11.41) (9.13)
4.37
5.81 5.15 7.82
SQUARED MULTIPLE CORRELATIONS
FOR X - VARIABLES
read-gr5 writ-gr5 read-gr7
write-gr
-------- -------- --------
--------
.83 .77 .74
.60
GOODNESS OF FIT STATISTICS
CHI-SQUARE WITH 1 DEGREE OF
FREEDOM = 0.66 (P = 0.42)
ESTIMATED NON-CENTRALITY PARAMETER
(NCP) = 0.0
90 PERCENT CONFIDENCE INTERVAL
FOR NCP = (0.0 ; 6.01)
MINIMUM FIT FUNCTION VALUE =
0.0027
POPULATION DISCREPANCY FUNCTION
VALUE (F0) = 0.0
90 PERCENT CONFIDENCE INTERVAL
FOR F0 = (0.0 ; 0.024)
ROOT MEAN SQUARE ERROR OF APPROXIMATION
(RMSEA) = 0.0
90 PERCENT CONFIDENCE INTERVAL
FOR RMSEA = (0.0 ; 0.16)
P-VALUE FOR TEST OF CLOSE FIT
(RMSEA < 0.05) = 0.54
EXPECTED CROSS-VALIDATION INDEX
(ECVI) = 0.075
90 PERCENT CONFIDENCE INTERVAL
FOR ECVI = (0.077 ; 0.10)
ECVI FOR SATURATED MODEL = 0.081
ECVI FOR INDEPENDENCE MODEL
= 2.33
CHI-SQUARE FOR INDEPENDENCE MODEL
WITH 6 DEGREES OF FREEDOM = 570.44
INDEPENDENCE AIC = 578.44
MODEL AIC = 18.66
SATURATED AIC = 20.00
INDEPENDENCE CAIC = 596.51
MODEL CAIC = 59.32
SATURATED CAIC = 65.17
ROOT MEAN SQUARE RESIDUAL (RMR)
= 0.78
STANDARDIZED RMR = 0.0043
GOODNESS OF FIT INDEX (GFI)
= 1.00
ADJUSTED GOODNESS OF FIT
INDEX (AGFI) = 0.99
PARSIMONY GOODNESS OF FIT INDEX
(PGFI) = 0.100
NORMED FIT INDEX (NFI) = 1.00
NON-NORMED FIT INDEX (NNFI)
= 1.00
PARSIMONY NORMED FIT INDEX (PNFI)
= 0.17
COMPARATIVE FIT INDEX (CFI)
= 1.00
INCREMENTAL FIT INDEX (IFI)
= 1.00
RELATIVE FIT INDEX (RFI) = 0.99
CRITICAL N (CN) = 2495.96
[OUTPUT DELETED]
The next test (see Problem B, Table 1) that we conduct is a group comparison analysis. Note the NG command which indicates that we are testing the model for two groups. Essentially, we combine our syntax for the separate models into one syntax file and indicate the constraints that we are testing in the syntax for the second group.
This first hypothesis (B) puts a constraint on the two models. Specifically, we are testing whether two factors exist for each of the groups. Note that the MO line in the syntax for the second group indicates MO LX=PS.
You can place different constraints on the MO line. For example, the PS command means that the same pattern and starting values should be found as in the corresponding matrices for the previous group. IN indicates that a particular matrix should be equal for the two groups.
For the DA line, NG stands for number of groups and NI indicates number of items and these appear only once in the first group's model specification. Each group has a separate NO (number of observations) specification.
Thus, there are 4 items, regardless of which group’s data are analyzed, yet each group has its own sample size.
For the MO line, you specify the number of x, y, ksi, and eta variables in the first group’s specification. Subsequent group MO lines specify the type of equality constraint you are imposing on the two groups.
This next analysis is essentially "adding" the two separate analyses we did above together—we will get the same results and the same Chi-squares for each of the groups. In other words, whereas model B puts a constraint on the two models to test whether each group has two factors, there are no cross-group equality constraints, thus our results are additive.
It is also important to understand what happens when you standardize the variables within each group in a multi-sample analysis with constraints across groups, (e.g., adding SS or SC to the OU line). Neither the observed nor the latent variables should be standardized as the meaning of a standardized solution in a multi-sample analysis is different. If SS is included on the OU line, for example, the latent variables are scaled so that the "weighted average of their covariance matrix is a correlation matrix, thereby retaining a scale common to all groups" (p. 274 in the LISREL 8 manual).
Table 2 indicates the difference between using a standardized solution (SS) and a completely standardized solution (SC) for a group analysis and how these solutions differ in a group analysis, based on LISREL’s common metric standardized solution or the within group standardized solution.
Table 2. Standardized Solutions in a Group Analysis
| Standardized solution (SS) | Completely Standardized Solution (SC) | |
| common metric |
latent variables standardized only; not correlations, but you can compare coefficients ACROSS groups |
latent and observed variables standardized; not correlations, but you can compare coefficients ACROSS groups |
| within group |
latent variables standardized only; coefficient matrices are correlation matrices; compare coefficients WITHIN each group only |
latent variables and observed variables standardized;
coefficient matrices are correlation matrices; compare coefficients WITHIN each group only |
If you have a path model with only observed variables, you would obviously not conduct the following test as you would not have any LX parameter estimates.
!Testing equality of factor structures. hypothesis b.group: boys academic
DA NG=2 NI=4 NO=373
MO NX=4 NK=2
LA
read-gr5 writ-gr5 read-gr7 writ-gr7
CM SY
281.35
184.22 182.82
216.74 171.70 283.29
198.38 153.20 208.84 246.07
FR LX(2,1) LX(4,2)
VA 1 LX(1,1) LX(3,2)
OUgroup: boys non-academic
DA NO=249
MO LX=PS
LA
read-gr5 writ-gr5 read-gr7 writ-gr7
CM SY
174.48
134.47 161.87
129.84 118.84 228.45
102.19 97.77 136.06 180.46
OU
For the parameter estimates, we get values for LAMBDA-X, PHI, and THETA-DELTA for both of the groups; therefore, there are 18 parameters which are estimated (9 for each group).
The LISREL estimates that appear first are for the first group that we tested, boys academic. After these estimates, we get goodness of fit statistics for the boys academic group, specifically, a Contribution to Chi-square, which is equal to .86. The Chi-square Contribution statistic gives us information about how much each group’s model is contributing to the overall Chi-square.
Underneath the Chi-square contribution statistic, we get information which tells us that the boys' academic Chi-square contributes 56.60% to the overall Chi-square for this group analysis.
The next set of LISREL estimates is for the boys non-academic group. The goodness of fit statistics which follow are related to the fit of the model for both groups. Notice that underneath the overall Chi-square is the Contribution to Chi-square for the boys non-academic group, which is .66 and contributes 43.40% to the overall Chi-square.
The overall Chi-square we get has 2 degrees of freedom and is = 1.52. Note that if you add the two groups’ Contributions to Chi-square together (e.g., .86+.66), you will get the overall Chi-square of 1.52. Notice that these Chi-square values are the same as the model Chi-square values from the academic boys’ only analysis (.86) and the nonacademic boys’ only analysis (.66).
Thus, as discussed earlier, Model B may be considered an additive sum of the two separate models. This identity holds for Model B because Model B does not impose any invariance constraints on the parameter estimates across the two models.
The Chi-square is not significant, with a p value of .47. This means that the structure that we have specified is a good fit. We can therefore accept the equality constraint that we tested and conclude that each group has two factors. Examination of the fit indices also indicates an excellent fit, GFI = 1.00, NFI = 1.00, and NNFI = 1.00.
*****The LISREL output starts here*****
The following lines were read
from file multigroup.ls8 multigroup.out:
!testing equality of factor structures
hypothesis b. group: boys academic
da ng=2 ni=4 no=373
mo nx=4 nk=2
LA
read-gr5 writ-gr5 read-gr7 writ-gr7
CM SY
281.349
184.219 182.821
216.739 171.699 283.289
198.376 153.201 208.837 246.069
fr lx(2,1) lx(4,2)
va 1 lx(1,1) lx(3,2)
ou
!testing equality of factor
structures
NUMBER OF INPUT VARIABLES 4
NUMBER OF Y - VARIABLES 0
NUMBER OF X - VARIABLES 4
NUMBER OF ETA - VARIABLES 0
NUMBER OF KSI - VARIABLES 2
NUMBER OF OBSERVATIONS 373
NUMBER OF GROUPS 2
!testing equality of factor structures
hypothesis b. group: boys non-academic
da no=249
mo lx=ps
LA
read-gr5 writ-gr5 read-gr7 write-gr7
CM SY
174.485
134.468 161.869
129.840 118.836 228.449
102.194 97.767 136.058 180.460
ou
hypothesis b. group: boys non-academic
NUMBER OF INPUT VARIABLES 4
NUMBER OF Y - VARIABLES 0
NUMBER OF X - VARIABLES 4
NUMBER OF ETA - VARIABLES 0
NUMBER OF KSI - VARIABLES 2
NUMBER OF OBSERVATIONS 249
NUMBER OF GROUPS 2
!testing equality of factor structures
COVARIANCE MATRIX TO BE ANALYZED
read-gr5 writ-gr5 read-gr7 writ-gr7
-------- -------- -------- --------
read-gr5 281.35
writ-gr5 184.22
182.82
read-gr7 216.74
171.70 283.29
writ-gr7 198.38
153.20 208.84 246.07
hypothesis b. group: boys non-academic
COVARIANCE MATRIX TO BE ANALYZED
read-gr5 writ-gr5 read-gr7 write-gr
-------- -------- -------- --------
read-gr5 174.49
writ-gr5 134.47
161.87
read-gr7 129.84
118.84 228.45
write-gr 102.19
97.77 136.06 180.46
!testing equality of factor structures
PARAMETER SPECIFICATIONS
LAMBDA-X
KSI 1 KSI 2
-------- --------
read-gr5
0 0
writ-gr5
1 0
read-gr7
0 0
writ-gr7
0 2
PHI
KSI 1 KSI 2
-------- --------
KSI1
3
KSI2
4 5
THETA-DELTA
read-gr5 writ-gr5 read-gr7 writ-gr7
-------- -------- -------- --------
6 7
8 9
hypothesis b. group: boys non-academic
PARAMETER SPECIFICATIONS
LAMBDA-X
KSI 1 KSI 2
-------- --------
read-gr5
0 0
writ-gr5
10 0
read-gr7
0 0
write-gr
0 11
PHI
KSI 1 KSI 2
-------- --------
KSI1
12
KSI2
13 14
THETA-DELTA
read-gr5 writ-gr5 read-gr7 write-gr
-------- -------- -------- --------
15 16
17 18
!testing equality of factor structures
Number of Iterations = 4
LISREL ESTIMATES (MAXIMUM LIKELIHOOD)
LAMBDA-X
KSI 1 KSI 2
-------- --------
read-gr5 1.00 - -
writ-gr5
.78 - -
(.03)
24.93
read-gr7 - - 1.00
writ-gr7
- - .91
(.04)
23.49
PHI
KSI 1 KSI 2
-------- --------
KSI1 235.11
(21.02)
11.18
KSI2 217.75
230.65
(18.45) (21.14)
11.80 10.91
THETA-DELTA
read-gr5 writ-gr5 read-gr7 writ-gr7
-------- -------- -------- --------
46.24
38.48 52.64 56.98
(6.28) (4.30)
(6.74) (6.16)
7.37
8.95 7.81 9.25
SQUARED MULTIPLE CORRELATIONS FOR X - VARIABLES
read-gr5 writ-gr5 read-gr7 writ-gr7
-------- -------- -------- --------
.84
.79 .81 .77
GOODNESS OF FIT STATISTICS
CONTRIBUTION TO CHI-SQUARE
= 0.86
PERCENTAGE CONTRIBUTION TO
CHI-SQUARE = 56.60
[OUTPUT DELETED]
hypothesis b. group: boys non-academic
Number of Iterations = 4
LISREL ESTIMATES (MAXIMUM LIKELIHOOD)
LAMBDA-X
KSI 1 KSI 2
-------- --------
read-gr5
1.00 - -
writ-gr5
.93 - -
(.06)
16.54
read-gr7 - - 1.00
write-gr
- - .80
(.07)
12.24
PHI
KSI 1 KSI 2
-------- --------
KSI 1
144.68
(16.66)
8.68
KSI 2
129.03 169.74
(15.06) (22.26)
8.57 7.63
THETA-DELTA
read-gr5 writ-gr5 read-gr7 write-gr
-------- -------- -------- --------
29.80
36.89 58.71 71.40
(6.82) (6.35)
(11.41) (9.13)
4.37
5.81 5.15 7.82
SQUARED MULTIPLE CORRELATIONS FOR X - VARIABLES
read-gr5 writ-gr5 read-gr7 write-gr
-------- -------- -------- --------
.83
.77 .74 .60
GOODNESS OF FIT STATISTICS
CHI-SQUARE WITH 2 DEGREES
OF FREEDOM = 1.52 (P = 0.47)
CONTRIBUTION TO CHI-SQUARE
= 0.66
PERCENTAGE CONTRIBUTION TO
CHI-SQUARE = 43.40
ESTIMATED NON-CENTRALITY PARAMETER
(NCP) = 0.0
90 PERCENT CONFIDENCE INTERVAL
FOR NCP = (0.0 ; 6.68)
MINIMUM FIT FUNCTION VALUE =
0.0025
POPULATION DISCREPANCY FUNCTION
VALUE (F0) = 0.0
90 PERCENT CONFIDENCE INTERVAL
FOR F0 = (0.0 ; 0.011)
ROOT MEAN SQUARE ERROR OF APPROXIMATION
(RMSEA) = 0.0
90 PERCENT CONFIDENCE INTERVAL
FOR RMSEA = (0.0 ; 0.10)
P-VALUE FOR TEST OF CLOSE FIT
(RMSEA < 0.05) = 0.82
EXPECTED CROSS-VALIDATION INDEX
(ECVI) = 0.061
90 PERCENT CONFIDENCE INTERVAL
FOR ECVI = (0.045 ; 0.056)
ECVI FOR SATURATED MODEL = 0.032
ECVI FOR INDEPENDENCE MODEL
= 2.88
CHI-SQUARE FOR INDEPENDENCE MODEL
WITH 12 DEGREES OF FREEDOM = 1779.89
INDEPENDENCE AIC = 1795.89
MODEL AIC = 37.52
SATURATED AIC = 40.00
INDEPENDENCE CAIC = 1839.36
MODEL CAIC = 135.31
SATURATED CAIC = 148.66
ROOT MEAN SQUARE RESIDUAL (RMR)
= 0.78
STANDARDIZED RMR = 0.0043
GOODNESS OF FIT INDEX (GFI)
= 1.00
PARSIMONY GOODNESS OF FIT
INDEX (PGFI) = 0.20
NORMED FIT INDEX (NFI) = 1.00
NON-NORMED FIT INDEX (NNFI)
= 1.00
PARSIMONY NORMED FIT INDEX (PNFI)
= 0.17
COMPARATIVE FIT INDEX (CFI)
= 1.00
INCREMENTAL FIT INDEX (IFI)
= 1.00
RELATIVE FIT INDEX (RFI) = 0.99
CRITICAL N (CN) = 3758.89
[OUTPUT DELETED]
The next model we are testing examines hypothesis C (see Table 1), that the two groups have equal factor loadings. Again, if you have a path model with only observed variables, you would not conduct this test because you do not have any LX parameter estimates. Note the change on the MO line in the second group’s syntax. We now write MO LX=IN, which indicates that the LX parameters should be equal or invariant.
!Testing equality of factor structures. hypothesis c.group: boys academic
DA NG=2 NI=4 NO=373
MO NX=4 NK=2
LA
read-gr5 writ-gr5 read-gr7 writ-gr7
CM SY
281.35
184.22 182.82
216.74 171.70 283.29
198.38 153.20 208.84 246.07
FR LX(2,1) LX(4,2)
VA 1 LX(1,1) LX(3,2)
OUgroup: boys non-academic
DA NO=249
MO LX=IN
LA
read-gr5 writ-gr5 read-gr7 writ-gr7
CM SY
174.48
134.47 161.87
129.84 118.84 228.45
102.19 97.77 136.06 180.46
OU
You will notice a few differences with this output versus the output from the previous example. First, we get a message before the parameter specifications which says "LAMBDA-X EQUALS LAMBDA-X IN THE FOLLOWING GROUP." We first get the parameter specifications for PHI and THETA-DELTA for the first group. Because we constrained the LAMBDA-X estimates to be equal for the two groups, we only get one set of LAMBDA-X parameter estimates. We therefore estimate 2 less parameters than Model B, so the total number of parameters we estimate now equals 16 instead of 18.
The PHI and THETA-DELTA parameter estimates for the second group follow the single LX parameter estimates.
The boys academic LISREL estimates and goodness of fit statistics again come first, with a Chi-square contribution of 2.84, which contributes 32.37% to the overall Chi-square.
The goodness of fit statistics are then presented for the fit of the whole model (i.e., that the factor loadings are equal) and we get an overall Chi-square = 8.77.
Underneath the overall Chi-square, we get the boys' nonacademic Contribution to Chi-square, which equals 5.93 and contributes 67.63% to the overall Chi-square. Note that if we add the two group Chi-squares together (e.g., 2.84 and 5.93), we get the overall Chi-square of 8.77. Fit indices also indicate an excellent fit, GFI = .99, NFI = 1.00, NNFI = .99.
The overall Chi-square is not significant (p = .067), which indicates that the groups' factor loadings do not differ. Therefore, we have gone one step further than hypothesis B. We can conclude that not only does the covariance structure fit when the factor loadings are allowed to be different across the two groups (i.e., MO LX=PS), but the fit is still good when we constrain those factor loading values to be the same across the same groups. Hypothesis C is not rejected and tells us that we cannot claim the groups have unequal factor loadings.
*****The LISREL output starts here*****
The following lines were read from file multigroup.ls8 multigroup.out:
!testing equality of factor structures
hypothesis c. group: boys academic
da ng=2 ni=4 no=373
mo nx=4 nk=2
LA
read-gr5 writ-gr5 read-gr7 writ-gr7
CM SY
281.349
184.219 182.821
216.739 171.699 283.289
198.376 153.201 208.837 246.069
fr lx(2,1) lx(4,2)
va 1 lx(1,1) lx(3,2)
ou
!testing equality of factor structures NUMBER OF INPUT VARIABLES 4
NUMBER OF Y - VARIABLES 0
NUMBER OF X - VARIABLES 4
NUMBER OF ETA - VARIABLES 0
NUMBER OF KSI - VARIABLES 2
NUMBER OF OBSERVATIONS 373
NUMBER OF GROUPS 2
!testing equality of factor structures
hypothesis c. group: boys non-academic
da no=249
mo lx=in
LA
read-gr5 writ-gr5 read-gr7 write-gr7
CM SY
174.485
134.468 161.869
129.840 118.836 228.449
102.194 97.767 136.058 180.460
ou
hypothesis c. group: boys non-academic
NUMBER OF INPUT VARIABLES 4
NUMBER OF Y - VARIABLES 0
NUMBER OF X - VARIABLES 4
NUMBER OF ETA - VARIABLES 0
NUMBER OF KSI - VARIABLES 2
NUMBER OF OBSERVATIONS 249
NUMBER OF GROUPS 2
!testing equality of factor structures
COVARIANCE MATRIX TO BE ANALYZED
read-gr5 writ-gr5 read-gr7 writ-gr7
-------- -------- -------- --------
read-gr5 281.35
writ-gr5 184.22 182.82
read-gr7 216.74 171.70 283.29
writ-gr7 198.38 153.20 208.84
246.07
hypothesis c. group: boys non-academic
COVARIANCE MATRIX TO BE ANALYZED
read-gr5 writ-gr5 read-gr7 write-gr
-------- -------- -------- --------
read-gr5 174.49
writ-gr5 134.47 161.87
read-gr7 129.84 118.84 228.45
write-gr 102.19 97.77 136.06
180.46
!testing equality of factor structures
PARAMETER SPECIFICATIONS
LAMBDA-X EQUALS LAMBDA-X IN THE FOLLOWING GROUP
PHI
KSI 1 KSI 2
--------
--------
KSI1
3
KSI2
4 5
THETA-DELTA
read-gr5 writ-gr5 read-gr7 writ-gr7
-------- -------- -------- --------
6 7
8 9
hypothesis c. group: boys non-academic
PARAMETER SPECIFICATIONS
LAMBDA-X
KSI 1 KSI 2
-------- --------
read-gr5
0 0
writ-gr5
1 0
read-gr7
0 0
write-gr
0 2
PHI
KSI 1 KSI 2
-------- --------
KSI1
10
KSI2
11 12
THETA-DELTA
read-gr5 writ-gr5 read-gr7 write-gr
-------- -------- -------- --------
13 14
15 16
!testing equality of factor structures
Number of Iterations = 6
LISREL ESTIMATES (MAXIMUM
LIKELIHOOD)
LAMBDA-X EQUALS LAMBDA-X
IN THE FOLLOWING GROUP
PHI
KSI 1 KSI 2
-------- --------
KSI1 224.06
(19.82)
11.31
KSI2 215.20
236.34
(18.07) (21.13)
11.91 11.19
THETA-DELTA
read-gr5 writ-gr5 read-gr7 writ-gr7
-------- -------- -------- --------
49.13
36.89 51.09 58.33
(6.09) (4.29)
(6.74) (6.09)
8.06
8.59 7.58 9.57
SQUARED MULTIPLE CORRELATIONS
FOR X - VARIABLES
read-gr5 writ-gr5 read-gr7 writ-gr7
-------- -------- -------- --------
.82
.80 .82 .76
GOODNESS OF FIT STATISTICS
CONTRIBUTION TO CHI-SQUARE
= 2.84
PERCENTAGE CONTRIBUTION TO
CHI-SQUARE = 32.37
hypothesis c. group: boys non-academic
Number of Iterations = 6
LISREL ESTIMATES (MAXIMUM
LIKELIHOOD)
LAMBDA-X
KSI 1 KSI 2
-------- --------
read-gr5
1.00 - -
writ-gr5
.82 - -
(.03)
29.45
read-gr7 - - 1.00
write-gr
- - .88
(.03)
26.43
PHI
KSI 1 KSI 2
-------- --------
KSI1
157.94
(16.44)
9.60
KSI 2
128.16 156.03
(14.41) (18.50)
8.90 8.43
THETA-DELTA
read-gr5 writ-gr5 read-gr7 write-gr
-------- -------- -------- --------
22.03
43.78 64.56 67.91
(6.42) (5.71)
(9.99) (8.77)
3.43
7.66 6.46 7.74
SQUARED MULTIPLE CORRELATIONS
FOR X - VARIABLES
read-gr5 writ-gr5 read-gr7 write-gr
-------- -------- -------- --------
.88
.71 .71 .64
GOODNESS OF FIT STATISTICS
CHI-SQUARE WITH 4 DEGREES
OF FREEDOM = 8.77 (P = 0.067)
CONTRIBUTION TO CHI-SQUARE
= 5.93
PERCENTAGE CONTRIBUTION TO
CHI-SQUARE = 67.63
ESTIMATED NON-CENTRALITY PARAMETER
(NCP) = 4.77
90 PERCENT CONFIDENCE INTERVAL
FOR NCP = (0.0 ; 17.44)
MINIMUM FIT FUNCTION VALUE =
0.014
POPULATION DISCREPANCY FUNCTION
VALUE (F0) = 0.0077
90 PERCENT CONFIDENCE INTERVAL
FOR F0 = (0.0 ; 0.028)
ROOT MEAN SQUARE ERROR OF APPROXIMATION
(RMSEA) = 0.062
90 PERCENT CONFIDENCE INTERVAL
FOR RMSEA = (0.0 ; 0.12)
P-VALUE FOR TEST OF CLOSE FIT
(RMSEA < 0.05) = 0.54
EXPECTED CROSS-VALIDATION INDEX
(ECVI) = 0.066
90 PERCENT CONFIDENCE INTERVAL
FOR ECVI = (0.042 ; 0.070)
ECVI FOR SATURATED MODEL = 0.032
ECVI FOR INDEPENDENCE MODEL
= 2.88
CHI-SQUARE FOR INDEPENDENCE MODEL
WITH 12 DEGREES OF FREEDOM = 1779.89
INDEPENDENCE AIC = 1795.89
MODEL AIC = 40.77
SATURATED AIC = 40.00
INDEPENDENCE CAIC = 1839.36
MODEL CAIC = 127.70
SATURATED CAIC = 148.66
ROOT MEAN SQUARE RESIDUAL (RMR)
= 7.83
STANDARDIZED RMR = 0.044
GOODNESS OF FIT INDEX (GFI)
= 0.99
PARSIMONY GOODNESS OF FIT
INDEX (PGFI) = 0.40
NORMED FIT INDEX (NFI) = 1.00
NON-NORMED FIT INDEX (NNFI)
= 0.99
PARSIMONY NORMED FIT INDEX (PNFI)
= 0.33
COMPARATIVE FIT INDEX (CFI)
= 1.00
INCREMENTAL FIT INDEX (IFI)
= 1.00
RELATIVE FIT INDEX (RFI) = 0.99
CRITICAL N (CN) = 939.09
As discussed earlier, each consecutive test that we conduct puts further constraints on the two groups. As we were able to accept hypothesis C, that the two groups had equal factor loadings, we now put another constraint on the model to determine whether the groups’ item error variances are also equal.
Therefore, the next test that we conduct examines whether the groups’ factor loadings are equal AND whether the groups have equal errors (see Problem D, Table 1). We therefore constrain the errors for each of the groups to be equal. If you are using path modeling, you could begin with this test of the TD matrices as you were not able to test the constraints on the LX matrices. You would only therefore have MO TD=IN whereas those who are testing a structural equation model will have the following syntax, MO LX=IN TD=IN.
!Testing equality of factor structures AND errors. hypothesis d.group: boys academic
DA NG=2 NI=4 NO=373
MO NX=4 NK=2
LA
read-gr5 writ-gr5 read-gr7 writ-gr7
CM SY
281.35
184.22 182.82
216.74 171.70 283.29
198.38 153.20 208.84 246.07
FR LX(2,1) LX(4,2)
VA 1 LX(1,1) LX(3,2)
OUgroup: boys non-academic
DA NO=249
MO LX=IN TD=IN
LA
read-gr5 writ-gr5 read-gr7 writ-gr7
CM SY
174.48
134.47 161.87
129.84 118.84 228.45
102.19 97.77 136.06 180.46
OU
Notice that the parameter specifications for this hypothesis begin with the PHI matrix and we receive two messages indicating that "LAMBDA-X EQUALS LAMBDA-X IN THE FOLLOWING GROUP" and "THETA-DELTA EQUALS THETA-DELTA IN THE FOLLOWING GROUP."
We therefore estimate only 12 parameters in this example versus the first example where there were no constraints and we estimated all 18.
The goodness of fit statistics for the boys academic model indicate a Contribution to Chi-square of 7.40 which contributes 34.35% to the overall Chi-square. The Contribution Chi-square for the boys non-academic model is 14.15 and it contributes 65.65% to the overall model.
The goodness of fit statistics for the whole analysis indicate a significant Chi-square = 21.55, p = .006. Again, note that if we add the two group Chi-squares together, we get the overall Chi-square (e.g., 7.40+14.15=21.55).
The significant Chi-square means that we must reject the hypothesis that the errors are equal for the two groups. Fit indices for the model are GFI = .97, NFI = .99, NNFI = .99.
Note that although we reject the model, we still obtained adequate fit
indices.
*****The LISREL output starts here*****
The following lines were read from file multigroup.ls8 multigroup.out:
!testing equality of factor structures
hypothesis d. group: boys academic
da ng=2 ni=4 no=373
mo nx=4 nk=2
LA
read-gr5 writ-gr5 read-gr7 writ-gr7
CM SY
281.349
184.219 182.821
216.739 171.699 283.289
198.376 153.201 208.837 246.069
fr lx(2,1) lx(4,2)
va 1 lx(1,1) lx(3,2)
ou
!testing equality of factor structures
NUMBER OF INPUT VARIABLES 4
NUMBER OF Y - VARIABLES 0
NUMBER OF X - VARIABLES 4
NUMBER OF ETA - VARIABLES 0
NUMBER OF KSI - VARIABLES 2
NUMBER OF OBSERVATIONS 373
NUMBER OF GROUPS 2
!testing equality of factor structures
hypothesis d. group: boys non-academic
da no=249
mo lx=in td=in
LA
read-gr5 writ-gr5 read-gr7 write-gr7
CM SY
174.485
134.468 161.869
129.840 118.836 228.449
102.194 97.767 136.058 180.460
ou
hypothesis d. group: boys non-academic
NUMBER OF INPUT VARIABLES 4
NUMBER OF Y - VARIABLES 0
NUMBER OF X - VARIABLES 4
NUMBER OF ETA - VARIABLES 0
NUMBER OF KSI - VARIABLES 2
NUMBER OF OBSERVATIONS 249
NUMBER OF GROUPS 2
!testing equality of factor structures
COVARIANCE MATRIX TO BE ANALYZED
read-gr5 writ-gr5 read-gr7 writ-gr7
-------- -------- -------- --------
read-gr5 281.35
writ-gr5 184.22 182.82
read-gr7 216.74 171.70 283.29
writ-gr7 198.38 153.20 208.84
246.07
hypothesis d. group: boys non-academic
COVARIANCE MATRIX TO BE ANALYZED
read-gr5 writ-gr5 read-gr7 write-gr
-------- -------- -------- --------
read-gr5 174.49
writ-gr5 134.47 161.87
read-gr7 129.84 118.84 228.45
write-gr 102.19 97.77 136.06
180.46
!testing equality of factor
structures
PARAMETER SPECIFICATIONS
LAMBDA-X EQUALS LAMBDA-X IN THE FOLLOWING GROUP
PHI
KSI 1 KSI 2
--------
--------
KSI1
3
KSI2
4 5
THETA-DELTA EQUALS THETA-DELTA
IN THE FOLLOWING GROUP
hypothesis d. group: boys non-academic
PARAMETER SPECIFICATIONS
LAMBDA-X
KSI 1 KSI 2
-------- --------
read-gr5
0 0
writ-gr5
1 0
read-gr7
0 0
write-gr
0 2
PHI
KSI 1 KSI 2
-------- --------
KSI1
10
KSI 2
11 12
THETA-DELTA
read-gr5 writ-gr5 read-gr7 write-gr
-------- -------- -------- --------
6 7
8 9
!testing equality of factor
structures
Number of Iterations = 6
LISREL ESTIMATES (MAXIMUM LIKELIHOOD)
LAMBDA-X EQUALS LAMBDA-X IN
THE FOLLOWING GROUP
PHI
KSI1 KSI2
-------- --------
KSI 1
228.30
(19.62)
11.63
KSI 2
216.21 234.15
(18.08) (21.19)
11.96 11.05
THETA-DELTA EQUALS THETA-DELTA IN THE FOLLOWING GROUP
GOODNESS OF FIT STATISTICS
CONTRIBUTION TO CHI-SQUARE
= 7.40
PERCENTAGE CONTRIBUTION TO
CHI-SQUARE = 34.35
Number of Iterations = 6
LISREL ESTIMATES (MAXIMUM LIKELIHOOD)
LAMBDA-X
KSI 1 KSI 2
-------- --------
read-gr5 1.00 - -
writ-gr5
.82 - -
(.03)
30.04
read-gr7 - - 1.00
write-gr
- - .88
(.03)
26.08
PHI
KSI 1 KSI 2
-------- --------
KSI1
156.25
(16.83)
9.29
KSI2
130.85 160.54
(14.72) (18.35)
8.89 8.75
THETA-DELTA
read-gr5 writ-gr5 read-gr7 write-gr
-------- -------- -------- --------
39.13
39.08 56.65 62.03
(4.64) (3.55)
(5.84) (5.15)
8.43
11.02 9.71 12.05
SQUARED MULTIPLE CORRELATIONS
FOR X - VARIABLES
read-gr5 writ-gr5 read-gr7 write-gr
-------- -------- -------- --------
.80
.73 .74 .67
GOODNESS OF FIT STATISTICS
CHI-SQUARE WITH 8 DEGREES
OF FREEDOM = 21.55 (P = 0.0058)
CONTRIBUTION TO CHI-SQUARE
= 14.15
PERCENTAGE CONTRIBUTION TO
CHI-SQUARE = 65.65
ESTIMATED NON-CENTRALITY PARAMETER
(NCP) = 13.55
90 PERCENT CONFIDENCE INTERVAL
FOR NCP = (3.41 ; 31.32)
MINIMUM FIT FUNCTION VALUE =
0.035
POPULATION DISCREPANCY FUNCTION
VALUE (F0) = 0.022
90 PERCENT CONFIDENCE INTERVAL
FOR F0 = (0.0055 ; 0.051)
ROOT MEAN SQUARE ERROR OF APPROXIMATION
(RMSEA) = 0.074
90 PERCENT CONFIDENCE INTERVAL
FOR RMSEA = (0.037 ; 0.11)
P-VALUE FOR TEST OF CLOSE FIT
(RMSEA < 0.05) = 0.40
EXPECTED CROSS-VALIDATION INDEX
(ECVI) = 0.073
90 PERCENT CONFIDENCE INTERVAL
FOR ECVI = (0.041 ; 0.086)
ECVI FOR SATURATED MODEL = 0.032
ECVI FOR INDEPENDENCE MODEL
= 2.88
CHI-SQUARE FOR INDEPENDENCE MODEL
WITH 12 DEGREES OF FREEDOM = 1779.89
INDEPENDENCE AIC = 1795.89
MODEL AIC = 45.55
SATURATED AIC = 40.00
INDEPENDENCE CAIC = 1839.36
MODEL CAIC = 110.74
SATURATED CAIC = 148.66
ROOT MEAN SQUARE RESIDUAL (RMR)
= 11.06
STANDARDIZED RMR = 0.062
GOODNESS OF FIT INDEX (GFI)
= 0.97
PARSIMONY GOODNESS OF FIT
INDEX (PGFI) = 0.78
NORMED FIT INDEX (NFI) = 0.99
NON-NORMED FIT INDEX (NNFI)
= 0.99
PARSIMONY NORMED FIT INDEX (PNFI)
= 0.66
COMPARATIVE FIT INDEX (CFI)
= 0.99
INCREMENTAL FIT INDEX (IFI)
= 0.99
RELATIVE FIT INDEX (RFI) = 0.98
CRITICAL N (CN) = 579.14
Typically, we would stop here as we found that while the groups have
equal factor loadings, they do not have equal errors. This last assumption
is usually much more difficult to meet in practice. For the sake of example,
however, we will examine the final hypothesis, which is even more stringent
and puts constraints on the LX, TD, and PHI matrices. Note the change on
the MO line for the second group, with the PHI matrix now also set
to be invariant. Setting the PHI matrix to be invariant means that the
model now assumes equal factor variances and covariances, as well as equal
factor loadings and measurement errors.
!Testing equality of factor structures AND errors AND factor variances and covariances. !hypothesis e.group: boys academic
DA NG=2 NI=4 NO=373
MO NX=4 NK=2
LA
read-gr5 writ-gr5 read-gr7 writ-gr7
CM SY
281.35
184.22 182.82
216.74 171.70 283.29
198.38 153.20 208.84 246.07
FR LX(2,1) LX(4,2)
VA 1 LX(1,1) LX(3,2)
OU
group: boys non-academic
DA NO=249
MO LX=IN TD=IN PHI=IN
LA
read-gr5 writ-gr5 read-gr7 writ-gr7
CM SY
174.48
134.47 161.87
129.84 118.84 228.45
102.19 97.77 136.06 180.46
OUNotice that for this model, as we have constrained all of the parameters to be equal across the two groups, there is only one set of estimates for each matrix so that we are only estimating 9 total parameters.
The goodness of fit statistics for the boys academic model indicate a Chi-square contribution of 13.60, which contributes 35.58% to the overall Chi-square. The boys’ non-academic Chi-square contribution is 24.62 (64.42% of the overall Chi-square). Again, these two contribution Chi-squares add up to the overall Chi-square of 38.22, which is statistically significant.
Although the fit indices suggest an adequate fit: GFI = .96, NFI = .98, NNFI = .98, the significant Chi-square indicates that we should revise our hypothesis of equal factor variances and covariances. We also know from our previous analysis that the groups also do not have equal errors.
We therefore conclude from these separate hypothesis tests that whereas the groups’ factor loadings appear to be equal, the rest of the matrices in the model are not--they do not have equal errors (TD) or factor variances and covariances (PHI). Therefore, we reject the null hypothesis that the errors and the factor variances and covariances are the same across the two groups.
Note that the parts discussed above are shown in red in the following output.
*****The LISREL output starts here *****
The following lines were read from file multigroup.ls8 multigroup.out:
!testing equality of factor structures
hypothesis e. group: boys academic
da ng=2 ni=4 no=373
mo nx=4 nk=2
LA
read-gr5 writ-gr5 read-gr7 writ-gr7
CM SY
281.349
184.219 182.821
216.739 171.699 283.289
198.376 153.201 208.837 246.069
fr lx(2,1) lx(4,2)
va 1 lx(1,1) lx(3,2)
ou!testing equality of factor structures
NUMBER OF INPUT VARIABLES 4
NUMBER OF Y - VARIABLES 0
NUMBER OF X - VARIABLES 4
NUMBER OF ETA - VARIABLES 0
NUMBER OF KSI - VARIABLES 2
NUMBER OF OBSERVATIONS 373
NUMBER OF GROUPS 2!testing equality of factor structures
hypothesis e. group: boys non-academic
da no=249
mo lx=in td=in ph=in
LA
read-gr5 writ-gr5 read-gr7 write-gr7
CM SY
174.485
134.468 161.869
129.840 118.836 228.449
102.194 97.767 136.058 180.460
ouhypothesis e. group: boys non-academic
NUMBER OF INPUT VARIABLES 4
NUMBER OF Y - VARIABLES 0
NUMBER OF X - VARIABLES 4
NUMBER OF ETA - VARIABLES 0
NUMBER OF KSI - VARIABLES 2
NUMBER OF OBSERVATIONS 249
NUMBER OF GROUPS 2!testing equality of factor structures
COVARIANCE MATRIX TO BE ANALYZED
read-gr5 writ-gr5 read-gr7 writ-gr7
-------- -------- -------- --------
read-gr5 281.35
writ-gr5 184.22 182.82
read-gr7 216.74 171.70 283.29
writ-gr7 198.38 153.20 208.84 246.07hypothesis e. group: boys non-academic
COVARIANCE MATRIX TO BE ANALYZED
read-gr5 writ-gr5 read-gr7 write-gr
-------- -------- -------- --------
read-gr5 174.49
writ-gr5 134.47 161.87