structural equation modeling: special topics...latent ancova, latent covariate group 1 group 2 1...

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

1

2

3

Latent Means Models


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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

F

4

5

6

Latent Means Models


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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

7

8

9

Latent Means Models


© 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


© 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

13

14

15

Latent Means Models


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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


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


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

16

17

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Latent Means Models


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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


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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).


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20

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Latent Means Models


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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

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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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23

25

Latent Means Models


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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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27

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Latent Means Models


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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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30

31

Latent Means Models


© 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];


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Generally ill‐advised unless a null model is computed separately and comparative indices are hand‐derived.

Running Mplus:Data‐model fit



Chi-Square Contributions From Each GroupGIRLS 2.142BOYS 3.031


Latent Means Models


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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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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)

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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


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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


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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

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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




Latent Means Models


© Running Mplus:Unstandardized parameter estimates


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


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

44

45

46

Latent Means Models


©

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


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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];


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Latent Means Models


© 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

53

54

55

Latent Means Models


© 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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Latent Means Models


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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

59

60

61

Latent Means Models


©

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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63

64

Latent Means Models


© 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


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[v1] (c); [v4] (c);[v2] (d); [v5] (d);[v3] (e); [v6] (e); [socsupp1@0];[socsupp2];


Running Mplus:Model Syntax (cont.)





Latent Means Models


© 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

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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






Latent Means Models


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ENGLISH1

read1 list1 write1speak1

ENGLISH2


ENGLISH3


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

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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);




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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);


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Latent Means Models


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[ENGLISH1@0];[ENGLISH2] (eng2);[ENGLISH3] (eng3);

MODEL CONSTRAINT:NEW(engdiff);engdiff = eng2 – eng3;

OUTPUT:SAMPSTAT MODINDICES(3.841);


©

Chi-Square Test of Model FitValue 108.271Degrees of Freedom 51P-Value 0.0000


Latent Means Models


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=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

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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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structural equation modeling: special topics...latent ancova, latent covariate group 1 group 2 1...

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