Structural Equation Modeling: Special Topics Gregory Hancock, Ph.D. Upcoming Seminar: May 13-15, 2021, Remote Seminar
Structural Equation Modeling: Special Topics
Gregory Hancock, Ph.D.
Upcoming Seminar: May 13-15, 2021, Remote Seminar
Latent Means Models
© 2021 Gregory R. Hancock
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LATENTM EANSMODELS
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Go Introduction; Group Code AnalysisMeasured Mean StructureLatent Mean Structure for Between‐Subjects Designs
Go
Go
Structured Means Models for Within‐Subjects Designs
Means Modeling ExtensionsSummary and Supplemental
Readings
GoGoGo
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Measurement error leads to:underestimation of the standardized effect size;decreased power to detect treatment/group differences.
Introduction:Measurement error and Effect Size
Diet 1 Diet 2
Y
Y2Y1Y
d
T
T2T1T
dTrue weight
Measured weight
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Measuredoutcomevariable
Group membershipvariable
?
Residual(prediction and
measurement error)
Introduction:GLM/ANOVA Outcome
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Latent Means Models
© 2021 Gregory R. Hancock
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Group membershipvariable
?
Residual(just prediction error)
Latentvariable
Measuredoutcomevariable
measurementerror
Introduction:Desired GLM/ANOVA Outcome
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To diminish the unreliability problem, the ANOVA paradigm can be circumvented: Population differences in the means of a latent variable can be modeled. For example …… Do 9th grade boys and girls differ in their Mathematics Proficiency, the latent variable believed to underlie scores on the Mathematics, Problem Solving, and Procedure subtests of the Stanford Achievement Test 9?
SATmathMath‐Prof SATprob
SATproc
Introduction:Example Research Question
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Similar to conducting a regression analysis with a dummy‐coded predictor to address an ANOVA‐type question:
the outcome variable now is a latent factor, not a measured variable;the analysis is conducted on a combined data set; homogeneity of dispersion across groups must be assumed; mean values on measured variables are not explicitly required.
Group Code Analysis
Special case of MIMIC (Multiple‐Indicator, MultIple‐Cause) modeling
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Latent Means Models
© 2021 Gregory R. Hancock
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This method assumes invariance of all covariance structure parameters, down to the last error variance.
Structured means analysis, although more complex, is also more flexible.The structured means analysis approach to assessing latent group differences uses separate group data, inferring a latent mean difference directly from differences in the observed means.
Group Code Analysis vs.Structured Means
FThe group code approach uses combined group data, inferring a latent mean difference by the degree of relation of the group code (dummy) variable to the measured outcomes.
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Structured Means Analysis
© Measured Mean Structure:A simple regression example
b21E2V2V1
V2 = b21V1 + E2
V2 = aV2(1) + b21V1 + E2
b21E2V2V1
aV11
aV1 is a mean
“structural equation”
aV2 is an intercept term
aV2
V2 = aV2 + b21V1 + E2
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Latent Means Models
© 2021 Gregory R. Hancock
© Measured Mean Structure:Mplus Syntax
b21E2V2satmath
V1tgoal
1aV1 aV2
Data from n=1000 9th grade girls on task goal orientation (tgoal) and Stanford Achievement Test math score (satmath).
DATA:FILE IS simple_data.csv;
VARIABLE:NAMES ARE tgoal satmath;
MODEL:satmath ON tgoal;tgoal; satmath; [tgoal]; [satmath];
OUTPUT:SAMPSTAT STDYX;
default
© Measured Mean Structure:Mplus Output
MODEL FIT INFORMATIONChi-Square Test of Model Fit
Value 0.000Degrees of Freedom 0P-Value 0.0000
RMSEA (Root Mean Square Error Of Approximation)Estimate 0.00090 Percent C.I. 0.000 0.000Probability RMSEA
Latent Means Models
© 2021 Gregory R. Hancock
© Measured Mean Structure:Mplus Output / SPSS Output
Two-TailedEstimate S.E. Est./S.E. P-Value
SATMATH ONTGOAL -1.428 0.920 -1.553 0.120
InterceptsSATMATH 690.063 3.259 211.711 0.000
STDYX StandardizationTwo-Tailed
Estimate S.E. Est./S.E. P-ValueSATMATH ON
TGOAL -0.049 0.032 -1.555 0.120
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Structural equationsV1 = bV1F1F1 + E1
V2 = bV2F1F1 + E2
V3 = 1 F1 + E3
aV3
aV3(1)+
represents the latent/factor mean
1
aV1(1)+ aV1
aV2(1)+
aV2
F1 = aF1(1) + D1
aF1 1 D1 cD1
V1
F1
V2 V3
1
bV1F1 bV2F1
E1
1
1E2
1E3
cE1 cE2 cE3
Latent Mean Structure:A simple one‐factor example
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aV31
aV1 aV2
aF1 1 D1 cD1
V1
F1
V2 V3
1
bV1F1 bV2F1
E1
1
1E2
1E3
cE1 cE2 cE3
Two Equivalent Representations
aV31
aV1 aV2
aF1
V1
F1
V2 V3
1
bV1F1 bV2F1
E1
1
1E2
1E3
cE1 cE2 cE3
cF1
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Latent Means Models
© 2021 Gregory R. Hancock
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Model‐Implied Covariance Matrix
E3F1F1V2F1F1V1F1
E2F121V2FF1V2F1V1F1
E1F12V1F1
cccbcbccbcbb
ccb
Model‐Implied Covariance Matrix with Six Unknowns
aV31
aV1 aV2
aF1
V1
F1
V2 V3
1
bV1F1 bV2F1
E1
1
1E2
1E3
cE1 cE2 cE3
Structural equationsV1 = bV1F1F1 + E1
V2 = bV2F1F1 + E2
V3 = 1 F1 + E3aV3(1)+
aV1(1)+
aV2(1)+
cF1
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Model‐Implied Mean Vector
Model‐Implied Mean Vector with Four Additional Unknowns
F1V3F1V2F1V2F1V1F1V1 aaabaaba
aV31
aV1 aV2
aF1
V1
F1
V2 V3
1
bV1F1 bV2F1
E1
1
1E2
1E3
cE1 cE2 cE3
Structural equationsV1 = bV1F1F1 + E1
V2 = bV2F1F1 + E2
V3 = 1 F1 + E3aV3(1)+
aV1(1)+
aV2(1)+
cF1
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Observed covariance matrix
233231
2221
21
sssss
s=
9 equations: 6 (covariance structure) + 3 (mean structure)10 unknowns: 6 (covariance structure) + 4 (mean structure)
Thus, the mean structure currently is under‐identified.
Model‐implied covariance matrix
E3F1F1V2F1F1V1F1
E2F12V2F1F1V2F1V1F1
E1F12V1F1
cccbcbccbcbb
ccb
Solving for the Unknown Parameters:An Identification Problem
Observed mean vector
V3V2V1=Model‐implied mean vector
F1V3F1V2F1V2F1V1F1V1 aaabaaba
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Latent Means Models
© 2021 Gregory R. Hancock
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When comparing means across two groups it is assumed, at least initially, that
loadings are equal groups/time (i.e., that equal amounts of change in the factor lead to equal amounts of changein the indicators) andintercepts are equal across groups/time (i.e., that equal amounts of the factor lead to equal amounts of the indicators).
Solving the Identification Problem—Partially
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0
aV1
V1
F1
same loadingsame intercept
0
aV1
V1
F1
same loadingdifferent intercept
0
aV1
V1
F1
different loadingdifferent intercept
Solving the Identification Problem—Partially
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These initial assumptions of invariant loadings and intercepts help, partially, to solve the under‐identification problem in the means portion of the model because now there are fewer unconstrained parameters to estimate.They will be testable (to a point).
Solving the Identification Problem—Partially
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Latent Means Models
© 2021 Gregory R. Hancock
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AaF1
1 1
Group A Group B
BaF1
=bV1F1 =bV1F1
=aV1 =aV1=aV2 =aV2=aV3 =aV3
BcE1 BcE2 BcE3AcE1 AcE2 AcE3
=bV2F1=bV2F1
1 1 1
V1
F1
V2 V3
E1 E2 E3
1
1 1 1
V1
F1
V2 V3
E1 E2 E3
1
The Constrained Means Model Across Two Groups
AcF1 BcF1
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0
1 1
Group A Group B
BaF1
=bV1F1 =bV1F1
=aV1 =aV1=aV2 =aV2=aV3 =aV3
BcE1 BcE2 BcE3AcE1 AcE2 AcE3
=bV2F1=bV2F1
1 1 1
V1
F1
V2 V3
E1 E2 E3
1
1 1 1
V1
F1
V2 V3
E1 E2 E3
1
Assessing Latent Mean Differencesby Using a Reference Group
AcF1 BcF1
©
1.14725.9917.11460.15757.1420
3.2105S
means = 709.47 696.67 669.17
9.15307.10175.11927.16255.1503
0.2323S
means = 692.36 680.63 654.31
Girls (nF = 1,000)SATvoc SATcomp SATlang
Boys (nM = 1,000)SATvoc SATcomp SATlang
Do Boys and Girls Differ onMean Latent Reading Proficiency?
SATvoc
Read‐Prof SATcomp
SATlang
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Latent Means Models
© 2021 Gregory R. Hancock
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Girls (nF = 1,000) Boys (nM = 1,000)
I. Are Factor Loadings Invariant?
SATvoc
Read‐Prof SATcomp
SATlang
SATvoc
Read‐Prof SATcomp
SATlang1
=bV1F1
=bV2F1
1
=bV1F1
=bV2F1
First, use a multi‐group CFA to assess the measurement model fit across groups. Here, data‐model fit is good with the loadings constrained to be equal across groups.
If a loading were judged to be noninvariant (e.g., through modification indices), its constraint should be released.
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Second, impose a model with mean structure and assess the intercept constraints (but do not impose intercept constraints on variables whose loadings are not constrained).
II. Are Intercepts Invariant?
0Girls
Read‐Prof
V1 V2 V3
1=bV1F1 =bV2F1
=aV1 =aV2
=aV31 FcF1
MaF1 Read‐Prof
Boys
V1 V2 V3
1=bV1F1 =bV2F1
=aV1 =aV2
=aV31 McF1
If an intercept is judged to be noninvariant (e.g., through modification indices), its constraint should be released.
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14 parameters
=bV1F1=bV2F1
loadings
FcE1FcE2FcE3
FcF1
McE1McE2McE3
McF1
variances
From covariance structure:10 unknown parameters
=aV1=aV2=aV3
intercepts
MaF1 latent mean
From mean structure: 4 unknown parameters
The Unknown Parameters
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Latent Means Models
© 2021 Gregory R. Hancock
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Model‐implied covariance matrices
Observed covariance matrices
Estimating the Covariance Structure
12 equations
23F32F31F
22F21F
21F
sssss
s
E3FF1FF1FV2F1F1FV1F1
E2FF1F2V2F1F1FV2F1V1F1
E1FF1F2V1F1
cccbcbccbcbb
ccb
=Girls
23M32M31M
22M21M
21M
sssss
s
E3MF1MF1MV2F1F1MV1F1
2MF1M2V2F1F1MV2F1V1F1
E1MF1M2V1F1
cccbcbccbcbb
ccb
E =Boys
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Model‐implied mean vectors
Observed mean vectors
Estimating the Mean Structure
6 equations
Thus, u = 18 = 12 (covariance structure) + 6 (mean structure)t = 14 parameters to be estimateddf = u – t = 18 – 14 = 4
= )0()0()0( V3V2F1V2V1F1V1 ababa V3V2V1 FFFGirls
Boys V3V2V1 MMM )()()( F1MV3F1MV2F1V2F1MV1F1V1 aaabaaba =
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The Model for the Software
0Girls
rprof
satvoc
1=bV1F1 =bV2F1
=aV1 =aV2
=aV31
MaF1rprof
Boys
1=bV1F1 =bV2F1
=aV1 =aV2
=aV31
satcomp satlang satvoc satcomp satlang
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Latent Means Models
© 2021 Gregory R. Hancock
© Running Mplus:Model Syntax (separate data files)
DATA:FILE (female) IS means_data_girls.txt;FILE (male) IS means_data_boys.txt;
VARIABLE:NAMES ARE satvoc satcomp satlang;
MODEL:rprof BY satvoc* satcomp* satlang@1;
MODEL female:[rprof@0];
MODEL male:[rprof];
OUTPUT:sampstat modindices(3.841);
© Running Mplus:Model Syntax (combined data file)
DATA:FILE IS means_data_girls_boys.txt;
VARIABLE:NAMES ARE gender satvoc satcomp satlang;GROUPING IS gender(1=female 2=male);
MODEL:rprof BY satvoc* satcomp* satlang@1;
MODEL female:[rprof@0];
MODEL male:[rprof];
OUTPUT:sampstat modindices(3.841);
©
Generally ill‐advised unless a null model is computed separately and comparative indices are hand‐derived.
Running Mplus:Data‐model fit
MODEL FIT INFORMATIONChi-Square Test of Model Fit
Value 5.174Degrees of Freedom 4P-Value 0.2699
Chi-Square Contributions From Each GroupGIRLS 2.142BOYS 3.031
RMSEA (Root Mean Square Error Of Approximation)Estimate 0.01790 Percent C.I. 0.000 0.053Probability RMSEA
Latent Means Models
© 2021 Gregory R. Hancock
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Two-TailedEstimate S.E. Est./S.E. P-Value
RPROF BYSATVOC 1.433 0.038 38.079 0.000SATCOMP 1.240 0.033 38.104 0.000SATLANG 1.000 0.000 999.000 999.000
MeansRPROF 0.000 0.000 999.000 999.000
InterceptsSATVOC 710.029 1.410 503.654 0.000SATCOMP 696.598 1.220 570.964 0.000SATLANG 668.183 1.101 606.882 0.000
VariancesRPROF 798.589 51.525 15.499 0.000
Residual VariancesSATVOC 463.593 39.922 11.612 0.000SATCOMP 345.174 29.923 11.536 0.000SATLANG 672.466 35.147 19.133 0.000
Running Mplus:Parameter estimates (girls)
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Two-TailedEstimate S.E. Est./S.E. P-Value
RPROF BYSATVOC 1.433 0.038 38.079 0.000SATCOMP 1.240 0.033 38.104 0.000SATLANG 1.000 0.000 999.000 999.000
MeansRPROF -12.816 1.383 -9.266 0.000
InterceptsSATVOC 710.029 1.410 503.654 0.000SATCOMP 696.598 1.220 570.964 0.000SATLANG 668.183 1.101 606.882 0.000
VariancesRPROF 838.099 54.031 15.512 0.000
Residual VariancesSATVOC 575.527 44.982 12.795 0.000SATCOMP 336.133 31.306 10.737 0.000SATLANG 720.011 37.711 19.093 0.000
Running Mplus:Parameter estimates (boys)
©
Model 2 = 5.17, df = 4, p = 0.27
RMSEA = 0.017, CI: (.00, .053)
Read‐Prof
0
Girls
798.591‐12.82* Read‐
Prof
Boys
838.101
Do Boys and Girls Differ onMean Latent Reading Proficiency?
The fit results implicitly support intercepts’ invariance across groupsThe mean latent reading proficiency is statistically significantly higher for girls than for boys.
*p
Latent Means Models
© 2021 Gregory R. Hancock
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Estimated standardized effect size:
1F
1FM
1F
1FM1FF
ˆˆ
ˆˆˆˆ
ca
caa
d
45.
)10001000()10.838(1000)59.798(1000
82.12ˆ
d
Estimated Standardized Effect Size:Hancock (2001)
MF
1FMM1FFF1F
)ˆ()ˆ(ˆnn
cncnc
= pooled variance estimate of F1
©
latent Reading Proficiency continuum
Girls are, on average, almost half a standard deviation higher than boys on the latent Reading Proficiency continuum.
Boys Girls
How Much do Boys and Girls Differ on Mean Latent Reading Proficiency?
© Side Note:Graphic Representation of Latent Effect Size
LATENT STRESS CONTINUUM
NL US
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Latent Means Models
© 2021 Gregory R. Hancock
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Conduct structured means model to assess the latent mean difference between Chinese and Korean adults on English Reading.
Data were collected on three measured indicators per factor from n = 86 Chinese adults and n = 96 Korean adults.Does there appear to be acceptable data‐model fit across both groups?Which group appears to be higher on average on the latent construct? Is the group difference statistically significant? How many standard deviations would you estimate separate the population means along the latent continuum? That is, what is the estimated standardized effect size? [This is a hand calculation.]Start with the partial syntax file
Structured Means Analysis:Hands‐on Exercise
©
The ModelKoreanChinese
aF1F1
V1 V2 V3
1
=bV3F1=bV2F1
=aV1 =aV2
=aV31
0F1
V1 V2 V3
1
=bV3F1=bV2F1
=aV1 =aV2
=aV31
STOP
© Running Mplus:Data‐model fit
MODEL FIT INFORMATIONChi-Square Test of Model Fit
Value 2.675Degrees of Freedom 4P-Value 0.6137
RMSEA (Root Mean Square Error Of Approximation)Estimate 0.00090 Percent C.I. 0.000 0.132Probability RMSEA
Latent Means Models
© 2021 Gregory R. Hancock
© Running Mplus:Unstandardized parameter estimates
Two-TailedEstimate S.E. Est./S.E. P-Value
Group CHINESEREAD BY
READ1 1.000 0.000 999.000 999.000READ2 1.001 0.061 16.516 0.000READ3 0.900 0.059 15.252 0.000
MeansREAD 0.000 0.000 999.000 999.000
InterceptsREAD1 16.208 0.720 22.521 0.000READ2 17.335 0.724 23.932 0.000READ3 18.180 0.676 26.898 0.000
VariancesREAD 37.287 6.810 5.475 0.000
© Running Mplus:Unstandardized parameter estimates
Two-TailedEstimate S.E. Est./S.E. P-Value
Group KOREANREAD BY
READ1 1.000 0.000 999.000 999.000READ2 1.001 0.061 16.516 0.000READ3 0.900 0.059 15.252 0.000
MeansREAD 2.948 0.927 3.178 0.001
InterceptsREAD1 16.208 0.720 22.521 0.000READ2 17.335 0.724 23.932 0.000READ3 18.180 0.676 26.898 0.000
VariancesREAD 32.034 5.605 5.716 0.000
©
Estimated standardized effect size:
1F
1
ˆˆˆ
cad F
499.
)9686()03.32(96)29.37(86
95.2ˆ
d
Computing the EstimatedStandardized Effect Size
KC
1FKK1FCC1F
)ˆ()ˆ(ˆnn
cncnc
= pooled variance estimate of F1
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Latent Means Models
© 2021 Gregory R. Hancock
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latent English Reading Ability continuum
Korean native speakers are, on average, about .50 of a standard deviation higher than Chinese native speakers on the latent English Reading Ability continuum.
Chinese Korean
How Much do Chinese & Korean AdultsDiffer on Latent English Reading Ability?
©
Similar analyses with other latent outcomes yielded the following results:
d≈.63 on latent English Listening Ability.d≈.26 on latent English Speaking Ability (NS).d≈.26 on latent English Writing Ability (NS).
How Much do Chinese & Korean AdultsDiffer on Other Factors?
Would it be possible to do all four outcomes (reading, listening, speaking, writing) simultaneously?
© Can We Do Multiple Factors at Once?Four‐factor CFA model, with mean structure
read1 read3read2
READ
list1 list3list2
LIST
speak1 speak3speak2
SPEAK
write1 write3write2
WRITE
10 0
00
1 1 1 1
Reference group:Chinese
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Latent Means Models
© 2021 Gregory R. Hancock
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read1 read3read2
READ
list1 list3list2
LIST
speak1 speak3speak2
SPEAK
write1 write3write2
WRITE
1
loadings and interceptsconstrained across groups
1 1 1 1
Non‐reference group:Korean
Can We Do Multiple Factors at Once?Four‐factor CFA model, with mean structure
© Running Mplus:Model Syntax
DATA:FILE (chinese) IS chinese_means.txt;FILE (korean) IS korean_means.txt;
VARIABLE:NAMES ARE read1-read3 list1-list3
speak1-speak3 write1-write3;MODEL:read BY read1-read3; list BY list1-list3;speak BY speak1-speak3; write BY write1-write3;read1-read3 PWITH list1-list3;read1-read3 PWITH speak1-speak3;read1-read3 PWITH write1-write3;list1-list3 PWITH speak1-speak3;list1-list3 PWITH write1-write3;speak1-speak3 PWITH write1-write3;
© Running Mplus:Model Syntax (cont.)
MODEL chinese:[read@0]; [list@0]; [speak@0]; [write@0];
MODEL korean:[read]; [list]; [speak]; [write];
OUTPUT:sampstat modindices(3.841);
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Latent Means Models
© 2021 Gregory R. Hancock
© Mean Structure Modeling Extensions:More Than Two Groups
Group kGroup 1
1
=bV3F1=bV2F1
1
=bV3F1=bV2F1
kaF1F1
0F1
V1 V2 V3
E1 E2 E3
=aV1 =aV2
=aV3
V1 V2 V3
E1 E2 E3
=aV1 =aV2
=aV31 1
……
©
2aF1
Mean Structure Modeling Extensions: Latent ANCOVA, Measured Covariate
Group 2Group 1
1
=bV3F1=bV2F1
1
=bV3F1=bV2F1
F1 D10
F1 D1
V1 V2 V3
=aV1 =aV2
=aV3
V1 V2 V3
=aV1 =aV2
=aV31 1
cov. cov.
The path from the pseudo‐variable (1) to the factor is the test of latent mean differences above and beyond the measured covariate; the assumption of homogeneity of slope is testable in this model.
©
2aF1
cov. cov.
Mean Structure Modeling Extensions: Latent ANCOVA, Latent Covariate
Group 2Group 1
1
=bV3F1=bV2F1
1
=bV3F1=bV2F1
F1 D10
F1 D1
V1 V2 V3
=aV1 =aV2
=aV3
V1 V2 V3
=aV1 =aV2
=aV31
The path from the pseudo‐variable (1) to the outcome factor is the test of latent mean differences above and beyond the latent covariate; the assumption of homogeneity of slope is testable in this model.
1
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Latent Means Models
© 2021 Gregory R. Hancock
© Mean Structure Modeling Extensions:What About MANOVA?
MANOVA: is often mistakenly used when researchers really should be using a latent means model;is a measured variable model, not a latent variable model, seeking population differences on any measured variable or linear combination of measured variables; makes strong assumptions about homogeneity of dispersion across populations;can be conducted better within the SEM framework (if you still insist on conducting something like a MANOVA).
© Mean Structure Modeling Extensions:MANOVA Using Structured Means Analysis
Group 2Group 1
V1 V2 V3
=aV1 =aV2
=aV3
V1 V2 V3
=aV1 =aV2
=aV31 1
The 2 for the above model, with intercept constraints, corresponds to the omnibus MANOVA test, but without requiring the assumption of homogeneity of dispersion.
©
The 2 for this model, with intercept constraints, corresponds to the between‐subjects omnibus test of means, but without requiring homogeneity of variance. It has also been shown to outperform ANOVA and adjusted forms of ANOVA (Fan & Hancock, 2012).
=aV1
1
V1
Same measured variable at different points in time, or under different conditions:
=aV1
1
V1
……
Mean Structure Modeling Extensions:ANOVA Using Structured Means Analysis
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© 2021 Gregory R. Hancock
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Does a population differ with respect to the average amount of a particular latent construct across time (or, more generally, across conditions)?Do populations of matched individuals (e.g., mothers/daughters) differ with respect to the average amount of a particular latent construct?
Repeated Measure Means Models:Latent Variable Research Questions
E6V6
F2 E5V5E4V4
E3 V3
F1E2 V2E1 V1
©
cF1
V1
F1
V2 V3
1
bV3F1bV2F1
E1
1
1E2
1E3
V1
F1
V2 V3
1
bV2F1
E11E2
1E3
cF1
bV3F11
aV31
aV1 aV2
0aV3
aV1 aV2
aF11
Between subjects
= == =
= == =
= =
Between vs. Within Subjects Designs
©
V1
F1
V2 V3
1
bV3F1bV2F1
E1
1
1E2
1E3E1 E2 E3
cF2F1
cF1
= =
V4
F2
V5 V6
1
bV5F2
E41E5
1E6
cF2
bV6F21 = =
Repeated Measure Designs:Covariance Structure
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© 2021 Gregory R. Hancock
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V1
F1
V2 V3
1
bV3F1bV2F1
E1
1
1E2
1E3
V4
F2
V5 V6
1
bV5F2
E41E5
1E6
cF2
bV6F21
E1 E2 E3
cF2F1
1cF1 aV1
aV2 aV3
aV6
aV5aV4
= =
====
= == =
Repeated Measure Designs:Mean Structure
©
= 0
V1
F1
V2 V3
1
bV3F1bV2F1
E1
1
1E2
1E3
V4
F2
V5 V6
1
bV5F2
E41E5
1E6
cF2
bV6F21
E1 E2 E3
cF2F1
1cF1
aF2aF1aV1
aV2 aV3
aV6
aV5aV4
= =
====
= == =
Repeated Measure Designs:Mean Structure
©
= 0
V1
F1
V2 V3
1
bV3F1bV2F1
E11E2
1E3
V4
F2
V5 V6
1
bV5F2
E41E5
1E6
cF2
bV6F2
E1 E2 E3
11
aF2aF1aV1
aV2 aV3
aV6
aV5aV4
= =
====
= == =
Repeated Measure Designs:Mean Structure (alternative)
bV1F1 bV4F2= =
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Latent Means Models
© 2021 Gregory R. Hancock
© Duncan & Stoolmiller (1993) Example:Model Set‐up
V1 V2 V3
bV3F1bV2F1
V4 V5 V6
bV5F2
cF2
bV6F2
cF2F1
1aF20
socsupp1 socsupp2aV1
aV2 aV3
aV6
aV5aV4
= =
====
= == =1
cF1
1
©
1676.819.719.640.612.1680.514.620.464.
1621.616.672.1752.819.
1812.1
R
n = 84
84.289.163.274.276.146.2s 12.1014.1203.1190.983.1196.10m
Duncan & Stoolmiller (1993) Example:Summary Statistics
© Running Mplus:Model Syntax
DATA:FILE IS repeated_means.txt;TYPE IS MEANS STD CORR;NOBS IS 84;
VARIABLE:NAMES ARE v1-v6;
MODEL:socsupp1 BY v1
v2 (a)v3 (b);
socsupp2 BY v4v5 (a)v6 (b);
v1-v3 PWITH v4-v6;
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Latent Means Models
© 2021 Gregory R. Hancock
©
[v1] (c); [v4] (c);[v2] (d); [v5] (d);[v3] (e); [v6] (e); [socsupp1@0];[socsupp2];
OUTPUT:sampstat modindices(3.841);
Running Mplus:Model Syntax (cont.)
© Running Mplus:Data‐model fit
MODEL FIT INFORMATIONChi-Square Test of Model Fit
Value 11.967Degrees of Freedom 9P-Value 0.2152
RMSEA (Root Mean Square Error Of Approximation)Estimate 0.06390 Percent C.I. 0.000 0.146Probability RMSEA
Latent Means Models
© 2021 Gregory R. Hancock
© Exercise Behavior Means Example:Key Results
5.6675.331
10F1SocialSupport
F2SocialSupport
4.136
.184NS
Model 2=11.97, df=9, p=.22RMSEA=.063; SRMR=0.044
078.
)8484()667.5(84)331.5(84
184.ˆ
d
©
Conduct a repeated latent means model to assess the latent mean difference from time 1 to time 3 for a sample of Chinese adults on English Ability.
Data were collected on four measured indicators per factor (reading, listening, speaking, writing) from n = 86 Chinese adults.Researchers have a theoretical reason (due to test administration) that the reading and listening measures’ errors should covary at time 3. Does there appear to be acceptable data‐model fit?Is the latent mean difference from time 1 to time 3 statistically significant? How many standard deviations would you estimate separate time 1 and time 3 along the latent continuum? That is, what is the estimated standardized effect size? [This is a hand calculation.]Start with the partial syntax file
Repeated Measure Latent Means:Hands‐on Exercise
© The Model:Constraints not shown
read1 list1 write1
1E1
1
1E2
1E4
read3 list3 write3
1E5
1E6
1E8
1
E1 E2
1ENGLISH1
(F1)ENGLISH3
(F2)
speak1
1E3
speak3
1E7
STOP
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Latent Means Models
© 2021 Gregory R. Hancock
© Running Mplus:Data‐model fit
MODEL FIT INFORMATIONChi-Square Test of Model Fit
Value 32.814Degrees of Freedom 20P-Value 0.0354
RMSEA (Root Mean Square Error Of Approximation)Estimate 0.08690 Percent C.I. 0.023 0.138Probability RMSEA
Latent Means Models
© 2021 Gregory R. Hancock
©
ENGLISH1
read1 list1 write1speak1
ENGLISH2
read2 list2 write2speak2
ENGLISH3
read3 list3 write3speak3
English language testing data from n=170 male non‐native adult speakers taken at three monthly intervals.
1
0
Plus loading and intercept constraints, and error covariances, across common items.
Mean Structure Modeling Extensions:More than two time points
©
DATA:FILE IS latent_means2_data.txt;
VARIABLE:NAMES ARE read1-read3 list1-list3
speak1-speak3 write1-write3;MODEL:
ENGLISH1 BY read1 list1 (a)speak1 (b)write1 (c);
ENGLISH2 BY read2 list2 (a)speak2 (b)write2 (c);
ENGLISH3 BY read3 list3 (a)speak3 (b)write3 (c);
Mean Structure Modeling Extensions:More than two time points
©
read1 WITH read2-read3; read2 WITH read3;list1 WITH list2-list3; list2 WITH list3;speak1 WITH speak2-speak3; speak2 WITH speak3; write1 WITH write2-write3; write2 WITH write3;[read1-read3] (d);[list1-list3] (e); [speak1-speak3] (f); [write1-write3] (g);
Mean Structure Modeling Extensions:More than two time points
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Latent Means Models
© 2021 Gregory R. Hancock
©
[ENGLISH1@0];[ENGLISH2] (eng2);[ENGLISH3] (eng3);
MODEL CONSTRAINT:NEW(engdiff);engdiff = eng2 – eng3;
OUTPUT:SAMPSTAT MODINDICES(3.841);
Mean Structure Modeling Extensions:More than two time points
©
Chi-Square Test of Model FitValue 108.271Degrees of Freedom 51P-Value 0.0000
RMSEA (Root Mean Square Error Of Approximation)Estimate 0.08190 Percent C.I. 0.060 0.103Probability RMSEA
Latent Means Models
© 2021 Gregory R. Hancock
©
=a =a=a
1
The 2 for this model, with intercept constraints, corresponds to the repeated measure omnibus test, but without requiring the assumption of sphericity.
V1 V2 V3
Same measured variables at different points in time, or under different conditions:
Mean Structure Modeling Extensions:Repeated ANOVA Using Means Analysis
©
Latent means models allow for inferences about population mean differences at the latent level.By parsing out measurement error, latent means models have more power than the measured variable techniques that they subsume (e.g., MANOVA, ANOVA).Multiple groups, multiple time points, as well as measured and latent covariates, may be accommodated within this analytic framework.Methods of modeling, effect size calculations, power and sample size estimations – exist for simple as well as more complex models.
Summary
©
Supplemental Readings
Hancock, G. R. (2004). Experimental, quasi‐experimental, and nonexperimental design and analysis with latent variables. In D. Kaplan (Ed.), SAGE handbook of quantitative methodology for the social sciences (pp. 317‐334). Thousand Oaks, CA: Sage.Thompson, M. S., & Green, S. B. (2013). Evaluating between‐group differences in latent variable means. In G. R. Hancock & R. O. Mueller (Eds.), Structural equation modeling: A second course (2nd ed.) (pp. 163‐218). Charlotte, NC: Information Age Publishing.Sörbom, D. (1974). A general method for studying differences in factor means and factor structures between groups. British Journal of Mathematical and Statistical Psychology, 27, 229‐239.
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