Appendix As
Annotated Mplus Syntax for Factorial Invariance Testing Across Within-Level Groups
Multilevel factor mixture model
TITLE: ML FMM for Configural Invariance
DATA: file is demo.dat;
VARIABLE: names are y1-y5 cluster g;
usevariables are y1-y5;
cluster = cluster ; !name of the cluster variable
classes = cg (2); !name of latent classes with the number of classes within parentheses
!number of groups with knownclass option
knownclass = cg (g=0 g=1); !latent classes are replaced with observed groups
!grouping variable g with 0 for group1 and 1 for group2
ANALYSIS: type = twolevel mixture;
MODEL:
%within%
%overall%
fw1 by y1-y5;
%cg#2%
fw1 by y2-y5; !factor loadings of group2 are relaxed except y1
y1-y5;
!residual variances of group2 are relaxed
%between%
%overall%
fb1 by y1-y5;
%cg#2%
[y2-y5];
!intercepts of group2 are relaxed except y1
TITLE: ML FMM for Weak Invariance
DATA: file is demo.dat;
VARIABLE: names are y1-y5 cluster g;
usevariables are y1-y5;
cluster = cluster ; !name of the cluster variable
classes = cg (2); !name of latent classes with the number of classes within parentheses
!number of groups with knownclass option
knownclass = cg (g=0 g=1); !latent classes are replaced with observed groups
!grouping variable g with 0 for group1 and 1 for group2
ANALYSIS: type = twolevel mixture;
MODEL:
%within%
%overall%
fw1 by y1-y5;
%cg#2%
y1-y5;
!residual variances of group2 are relaxed
%between%
%overall%
fb1 by y1-y5;
%cg#2%
[y2-y5];
!intercepts of group2 are relaxed except y1
TITLE: ML FMM for Strong Invariance
DATA: file is demo.dat;
VARIABLE: names are y1-y5 cluster g;
usevariables are y1-y5;
cluster = cluster ; !name of the cluster variable
classes = cg (2); !name of latent classes with the number of classes within parentheses
!number of groups with knownclass option
knownclass = cg (g=0 g=1); !latent classes are replaced with observed groups
!grouping variable g with 0 for group1 and 1 for group2
ANALYSIS: type = twolevel mixture;
MODEL:
%within%
%overall%
fw1 by y1-y5;
%cg#2%
y1-y5;
!residual variances of group2 are relaxed
%between%
%overall%
fb1 by y1-y5;
Multilevel MIMIC model
TITLE: ML MIMIC Relaxed Model for Each Item;
DATA: file is demo.dat;
VARIABLE: names are y1-y5 cluster g;
usevariables are y1-y5 g;
cluster = cluster ; !name of the cluster variable
within = g ; !grouping variable g is specified as a within-level covariate
ANALYSIS: type = twolevel random;
algorithm = integration;
MODEL:
%within%
fw1 by y1-y5;
fw1 on g; !test a group difference in a within-level factor
intract | fw1 XWITH g; !create an interaction between a latent variable
!and an observed grouping variable
y1 on g intract; !test invariance of intercept and factor loading of y1
!each item should be tested one by one in a separate run
%between%
fb1 by y1-y5;
TITLE: ML MIMIC Constrained Model;
DATA: file is demo.dat;
VARIABLE: names are y1-y5 cluster g;
usevariables are y1-y5 g;
cluster = cluster ; !name of the cluster variable
within = g ; !grouping variable g is specified as a within-level covariate
ANALYSIS: type = twolevel random;
algorithm = integration;
MODEL:
%within%
fw1 by y1-y5;
fw1 on g; !test a group difference in a within-level factor
intract | fw1 XWITH g; !create an interaction between a latent variable
!and an observed grouping variable
y1 on intract@0; !fix the interaction effect at zero, i.e., factor loading invariance of y1
!y1 on g also fixed at zero by default, i.e., intercept invariance of y1
!replacing y1 with other observed variables does not change the model fit
%between%
fb1 by y1-y5;
Appendix Bs
PISA 2003 Mathematics Self-Efficacy Iems
Items
Q31a
Q31b
Q31c
Q31d
Q31h
Using a train timetable to work out how long it would take to get from one place to another
Calculating how much cheaper A TV would be after a 30% discount
Calculating how many square metres of tiles you need to cover a floor
Understanding graphs presented in newspapers
Calculating the petrol consumption rate of a car
Note. The self-efficacy items asked “How confident do you feel about having to do the following
mathematics tasks?” on a 4-point Likert scale where 1 = very confident, 2 = confident, 3 = not
very confident, and 4 = not at all confident.
Appendix Cs
Parameter Estimates and Model Fit Indices of Multilevel Factor Mixture Model for PISA 2003
Mathematics Self-Efficacy Measure
Configural
Weak
Strong
Parameters
Female
Male
Female
Male
Female
Male
λ1
1.000
1.000
1.000
λ2
1.025
1.024
1.025
1.028
Factor
λ3
1.092
1.040
1.057
1.053
loadings
λ4
0.788
0.840
0.821
0.824
λ5
1.010
0.917
0.951
0.928
Within
Factor variance Φ
0.243
0.305
0.250
0.301
0.250
0.304
level
0.356
0.326
0.353
0.327
0.353
0.326
σ2 1
Residual
variance
Factor
loadings
Between
level
Intercepts
Model fit
indices
Factor means
Factor variance
Free parameters
Log likelihood
Correction Factor
AIC
BIC
SSBIC
SB LR
σ2 2
0.397
0.328
0.393
0.329
0.392
0.328
σ2 3
0.394
0.395
0.399
0.394
0.399
0.393
σ2 4
0.541
0.508
0.536
0.511
0.543
0.513
σ2 5
λ1
λ2
λ3
λ4
λ5
τ1
τ2
τ3
τ4
τ5
κ
Φ
0.450
0.535
1.000
0.980
0.925
0.912
0.783
2.183
1.943
1.937
2.092
2.058
2.006
2.106
2.453
2.222
0.244
0.000
0.074
36
-27086.597
1.310
54245.194
54474.132
54359.739
4.648
0.458
0.532
0.468
0.542
1.000
0.977
0.964
0.817
1.020
2.163
1.920
2.049
2.054
2.286
0.279
0.000
0.067
28
-27134.208
1.412
54324.417
54502.479
54413.507
98.475**
1.000
0.980
0.925
0.912
0.786
2.183
1.943
1.938
2.092
2.058
2.006
2.106
2.453
2.222
0.245
0.000
0.074
32
-27088.890
1.350
54241.779
54445.279
54343.596
Note. SB LR = Satorra-Bentler scaled likelihood ratio test, * < .05, ** < .001
Appendix Ds
Uniform and Nonuniform Factorial Invariance Testing Using Multilevel MIMIC Models for PISA 2003 Mathematical Self-Efficacy
Measure
Item tested for factorial invariance in a relaxed model
Item 1
Item 2
Item 3
Item 4
Item 5
Item regressed on
Interactiona
genderb
Model fit
Free parameters
Log likelihood
Correction factor
AIC
BIC
ssBIC
SB LR
Constrained
model
.053
.037
.008
.038
.035
.033
.056c
.107**c
.127*c
-.202**c
0
0
23
-24217.865
1.277
48481.729
48627.995
48554.910
3.746
23
-24218.164
1.275
48482.327
48628.593
48555.508
3.131
23
-24218.506
1.278
48483.012
48629.277
48556.193
2.274
23
-24208.191
1.274
48462.382
48608.648
48535.563
26.712**
23
-24185.848
1.265
48417.697
48563.962
48490.878
89.579**
21
-24219.523
1.314
48481.047
48614.594
48547.864
Note. aunstandardized regression coefficient as nonuniform invariance, bunstandardized regression coefficient as uniform invariance,
c
the parameter estimates are from a model with each parameter relaxed at a time from the constrained model, SB LR = Satorra-Bentler
scaled likelihood ratio test, * < .05, ** < .001