3.Measurement Model
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Introduction to Structural Equation Modelling
Joaqun Alds-ManzanoUniversitat de ValnciaDepartment of Marketing
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Measurement Model Reliability & Validity
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Psychometric properties: Reliability
Reliability An instrument is said to be reliableif it is shown to provide consistent
scores upon repeated administration, upon administration by alternateforms, and so forth.
But test-retest is not usually a feasible way to establish a scale reliability(time and economic constraints) so we rely in testing internalconsistency.
Internal consistency is the extent to which the individual items thatconstitute a test correlate with one another or with the test total
If items are highly correlated, that indicates a common LV is causingthem, not that is the LV we were trying to measure, so:
Reliability is a necessary but not sufficient condition to validity We use three reliability indicators:
Cronbachs alpha (Cronbach, 1951) Composite Reliability (Fornell & Larcker, 1981) Average Variance Extracted (Fornell & Larcker, 1981)
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Psychometric properties: Reliability
Cronbachs Starting point: covariance matrix among indicators Its standardization is the correlation matrix Total scale variance: sum of C elements Total variance=Common variance + Specific variance (unique)
Common variance: Variance among items provoked by the latentvariable (shared by the items). If LV changes, items change.
Specific variance: caused by item measurement errors: C matrixdiagonal
3
C =
!1
2!
12!
13! !
1k
!12
!2
2!
23! !
2k
!13
!23
!3
2! !
3k
" " " # "
!1k
!2k
!3k ! !k
2
!
"
#######
$
%
&&&&&&&
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Psychometric properties: Reliability
Cronbachs
ais the part of the total variance that can be attributed to the latent variable(common variance)
4
X1
X2
X3
Y
e1
e2
e3
Common variance sourceUnique variance source
! =
"y
2! !
i
2
"!
y
2=1!
!i
2
"!
y
2
Total variance: sum of all theelements of the covariancematrix Specific (unique) variance: sum ofthe elements in the diagonal of thecovariance matrix
Unique variance source
Unique variance source
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Psychometric properties: Reliability
Cronbachs We must correct the effect of the different number of elements in the
numerator and denominator of the previous expression, as we have k2elements in the matrix but only kin its diagonal
So k2-kelements can be found in the numerator and kin the denominator.So to make the ratio express relative magnitudes and not the number ofcases, we correct by k2/ (k2k) that si k/(k-1)
a can also be expressed in terms of correlations more than variances andcovariances (Crocker y Algina, 1986):
where r is the average correlation among scale items
5
! =
k
k!1 1!"i
2
""y2#
$%&
'(
! =k"
1+ k!1( )"
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Psychometric properties: Reliability
Cronbachs Benchmark values for Cronbachs:
Nunnally & Bernstein (1994; p.265-6): a.70 Carmines & Zeller (1979; p.51): a.80
Estimates in excess of .90 are suggestive of item redundancy or inordinatescale length (ORourke, Hatcher & Stepanski, 2005)
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Psychometric properties: Reliability Cronbachs . An annotated example (ORourke, Hatcher & Stepanski, 2005)
HELPING OTHERS X1. Went out of my way to do a favour for a co-worker. 1 2 3 4 5 6 7 X2. Went out of my way to do a favour for a relative. 1 2 3 4 5 6 7 X3. Went out of my way to do a favour for a friend. 1 2 3 4 5 6 7 FINANCIAL GIVING
X4. Gave money to a religious charity. 1 2 3 4 5 6 7 X5. Gave money to a charity not associated with a religion 1 2 3 4 5 6 7 X6. Gave money to a panhandler. 1 2 3 4 5 6 7
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Helpingothers
Financialgiving
X1
X2
X3
X4
X5
X6
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Psychometric properties: Reliability
Cronbachs . An annotated example (ORourke, Hatcher & Stepanski, 2005) I want to test the reliability of Helping others construct but I make a
mistake and add item X4 to the X1-X3 list
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CovarianceMatrix
X1X2X3X4
X11,9465X2,76331,2245
X31,2106,52241,4792
X4-,2604,1061-,05393,2147
CorrelationMatrix
X1X2X3X4
X11,0000
X2,49441,0000
X3,7134,38821,0000
X4-,1041,0535-,02471,0000
RELIABILITYANALYSIS-SCALE(ALPHA)
MeanStdDevCases1.X15,18001,395250,0
2.X25,40001,106650,0
3.X35,52001,216250,0
4.X43,64001,793050,0
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Psychometric properties: Reliability
Cronbachs . An annotated example (ORourke, Hatcher & Stepanski, 2005)
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NofStatisticsforMeanVarianceStdDevVariables
Scale19,740012,44123,52724
ItemMeansMeanMinimumMaximumRangeMax/MinVariance4,93503,64005,52001,88001,5165,7652
ItemVariancesMeanMinimumMaximumRangeMax/MinVariance
1,96621,22453,21471,99022,6253,7821Inter-item
CovariancesMeanMinimumMaximumRangeMax/MinVariance,3814-,26041,21061,4710-4,6489,2783
Inter-item
CorrelationsMeanMinimumMaximumRangeMax/MinVariance
,2535-,1041,7134,8176-6,8534,0969
Total scale variance
Average correlation among items
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Psychometric properties: Reliability
Cronbachs . An annotated example (ORourke, Hatcher & Stepanski, 2005)
10
ReliabilityCoefficients4items
Alpha=,4904Standardizeditemalpha=,5759
! =4
31!
7,8649
12, 4412
"#$
%&'= 0,4904
! =4! 0,2535
1+ 4 "1( )! 0,2535= 0,5759
!i
2=1,9465+1,2245+1, 4792 + 3,2147 = 7,8649!
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Psychometric properties: Reliability
Cronbachs . An annotated example (ORourke, Hatcher & Stepanski, 2005) I realize I committed a mistake, What happens if I delete X4? Sensibility analysis to item deletion
11
Item-totalStatistics
ScaleScaleCorrectedMeanVarianceItem-SquaredAlpha
ifItemifItemTotalMultipleifItem
DeletedDeletedCorrelationCorrelationDeleted
X114,56007,0678,4620,5753,2439X214,34008,4331,4331,2574,3189
X314,22007,6037,5007,5127,2403
X416,10009,6429-,0374,0295,7766
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Psychometric properties: Reliability
Composite reliability Takes into account all the LVs in the measurement model A CFA must be performed to get the necessary information A CR is calculated for each LV (Fornell y Larcker, 1981):
Being Lij the standardized loading of each of thejindicators of the LVi Var(Eij) is the variance of the error tem of each indicator that can be
calculated as follows:
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Var Eij( ) =1!Lij2
CR
L
L Var E
ij
j
ijj
ijj
=
+
( )
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2
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Psychometric properties: Reliability Composite reliability: An annotated example
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STANDARDIZEDSOLUTION:V1=V1=.963*F1+.270E1
V2=V2=.514*F1+.858E2V3=V3=.741*F1+.671E3
V4=V4=.945*F2+.326E4V5=V5=.657*F2+.754E5V6=V6=.673*F2+.740E6
Helpingothers
Financialgiving
X1
X2
X3
X4
X5
X6
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Psychometric properties: Reliability Composite reliability: An annotated example
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CR1=
Lijj
!"
#$%
&'
2
Lijj
!"#$%&'
2
+ Var Eij( )j
!=
2,218( )2
2,218( )2
+1,259
= 0, 796 CR2=
2,275( )2
2,275( )2
+1, 222
= 0,809
Benchmark:Same as Cronbachs a
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Psychometric properties: Reliability
Average Variance Extracted (AVE) Average variance that the LV can explain of all its indicators (Fornell y
Larcker, 1981):
Being all the notation known but ki that is the number of indicators of theith LV
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AVEi =
Lijj
!2
Lijj!
2
+ Var Eij( )j!=
Lijj
!2
ki
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Psychometric properties: Reliability Average Variance Extracted (AVE)
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AVE1 =
Lijj
!2
Lijj!
2
+ Var Eij( )j!
=1, 741
1, 741+1,259
= 0,580 AVE2 =1, 778
1, 778+1,222
= 0,592
Benchmark:AVE > .50
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Psychometric properties: Validity
Validity Construct validity is the extent to which a set of measured items actually
reflect the theoretical latent construct they are designed to measure.
Types of validity: Face validity: the extent to which the content of the items is consistent
with the construct definition, based solely on the researchersjudgment
Convergent validity: the extent to which indicators of a specificconstruct converge or share a high proportion of variance incommon.
Discriminant validity: the extent to which a construct is truly distinctfrom other constructs
Nomological validity: examines whether the correlations between theconstructs in the measurement theory make sense
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Published results from previous studies. Pre-test or pilot study findings
With CFA SEM
information
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Psychometric properties: Validity Validity. An annotated example (Rusbult, 1980; Hatcher, 1994)
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Psychometric properties: Validity
Convergent validity: Model goodness-of-fit must be adequate (same as for the rest of validity
criteria)
Check Lagrange multipliers as some indicators may be being caused formore than one LV (bad item design)
Loadings must be significant. Loadings size must be adequate:
Ideally 0.70 and higher. If some of them are not, the average of theloadings for each factor should be .70 or higher (Hair, Anderson,Tatham & Black, 1998)
At least .60 (Bagozzi y Yi, 1988) The rationale of the .70 benchmark is that .702 implies that approximately
50% of the item variance will be explained by the LV. Lower values implythat most of the variance in the indicator is error variance.
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Psychometric properties: Validity
Convergent validity. An annotated example (Rusbult, 1980; Hatcher, 1994). Estimating a CFA (measurement model)
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Psychometric properties: Validity Convergent validity. An annotated example (Rusbult, 1980; Hatcher, 1994)
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/SPECIFICATIONSVARIABLES=19;CASES=240;METHOD=ML;
ANALYSIS=COVARIANCE;MATRIX=COR;
/EQUATIONS
V1=*F1+E1;V2=*F1+E2;
V3=*F1+E3;
V4=*F1+E4;
V5=*F2+E5;
V6=*F2+E6;
V7=*F2+E7;V8=*F3+E8;
V9=*F3+E9;
V10=*F3+E10;V11=*F4+E11;
V12=*F4+E12;V13=*F4+E13;
V14=*F5+E14;
V15=*F5+E15;
V16=*F5+E16;
V17=*F6+E17;V18=*F6+E18;
V19=*F6+E19;
/VARIANCES
F1TOF6=1;
E1TOE19=*;/COVARIANCES
F1TOF6=*;/PRINT
FIT=ALL;
GOODNESSOFFITSUMMARYFORMETHOD=ML
INDEPENDENCEMODELCHI-SQUARE=2459.673ON171DEGREESOF
FREEDOM
INDEPENDENCEAIC=2117.67331INDEPENDENCECAIC=1351.48405
MODELAIC=-26.32656MODELCAIC=-640.17410
CHI-SQUARE=247.673BASEDON137DEGREESOFFREEDOM
PROBABILITYVALUEFORTHECHI-SQUARESTATISTICIS.00000
THENORMALTHEORYRLSCHI-SQUAREFORTHISMLSOLUTIONIS
234.506.
FITINDICES
-----------BENTLER-BONETTNORMEDFITINDEX=.899
BENTLER-BONETTNON-NORMEDFITINDEX=.940
COMPARATIVEFITINDEX(CFI)=.952
BOLLEN(IFI)FITINDEX=.952
MCDONALD(MFI)FITINDEX=.794LISRELGFIFITINDEX=.906
LISRELAGFIFITINDEX=.870
ROOTMEAN-SQUARERESIDUAL(RMR)=.237
STANDARDIZEDRMR=.047
ROOTMEAN-SQUAREERROROFAPPROXIMATION(RMSEA)=.05890%CONFIDENCEINTERVALOFRMSEA(.046,.069)
Model GoF
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Psychometric properties: Validity Convergent validity. An annotated example (Rusbult, 1980; Hatcher, 1994)
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CUMULATIVEMULTIVARIATESTATISTICSUNIVARIATEINCREMENT----------------------------------------------------------------
HANCOCK'SSEQUENTIAL
STEPPARAMETERCHI-SQUARED.F.PROB.CHI-SQUAREPROB.D.F.PROB.
----------------------------------------------------------1V4,F622.5801.00022.580.0001371.000
2V2,F239.0732.00016.493.0001361.0003V1,F244.6873.0005.614.0181351.000
4V17,F249.7454.0005.058.0251341.000
5V2,F554.6475.0004.902.0271331.0006V8,F159.2016.0004.554.0331321.000
Lagrange Multiplier test
Should we add the relationship? No face validity (unless substantive reasons)Should we associate it only to F6? Significant loading on F1, same problemWe should delete it and run the model again
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Psychometric properties: Validity onvergent validity. An annotated example (Rusbult, 1980; Hatcher, 1994)
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GOODNESSOFFITSUMMARYFORMETHOD=ML
INDEPENDENCEMODELCHI-SQUARE=2167.771ON153DEGREESOFFREEDOM
INDEPENDENCEAIC=1861.77095INDEPENDENCECAIC=1176.23320MODELAIC=-59.12792MODELCAIC=-596.80459
CHI-SQUARE=180.872BASEDON120DEGREESOFFREEDOM
PROBABILITYVALUEFORTHECHI-SQUARESTATISTICIS.00028
THENORMALTHEORYRLSCHI-SQUAREFORTHISMLSOLUTIONIS174.047.
FITINDICES
-----------BENTLER-BONETTNORMEDFITINDEX=.917
BENTLER-BONETTNON-NORMEDFITINDEX=.961COMPARATIVEFITINDEX(CFI)=.970
BOLLEN(IFI)FITINDEX=.970
MCDONALD(MFI)FITINDEX=.881
LISRELGFIFITINDEX=.925
LISRELAGFIFITINDEX=.893ROOTMEAN-SQUARERESIDUAL(RMR)=.197
STANDARDIZEDRMR=.042
ROOTMEAN-SQUAREERROROFAPPROXIMATION(RMSEA)=.046
90%CONFIDENCEINTERVALOFRMSEA(.032,.059)
GoF of revised model 1
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Psychometric properties: Validity Convergent validity. An annotated example (Rusbult, 1980; Hatcher, 1994)
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V1=V1=2.201*F1+1.000E1.129
17.079@V2=V2=2.398*F1+1.000E2
.157
15.301@V3=V3=2.551*F1+1.000E3
.13618.733@
V5=V5=1.596*F2+1.000E5.105
15.162@
V6=V6=1.830*F2+1.000E6.113
16.236@
V7=V7=1.800*F2+1.000E7.109
16.513@V8=V8=.944*F3+1.000E8
.094
9.991@V9=V9=.893*F3+1.000E9
.0949.455@
V10=V10=1.294*F3+1.000E10
.11411.383@
V11=V11=2.143*F4+1.000E11.171
12.516@
V12=V12=2.326*F4+1.000E12.178
13.082@V13=V13=1.093*F4+1.000E13
.157
6.977@
Are loadings significant?
V14=V14=1.773*F5+1.000E14.124
14.333@
V15=V15=1.569*F5+1.000E15
.13611.523@
V16=V16=1.032*F5+1.000E16
.1228.486@
V17=V17=1.368*F6+1.000E17.130
10.493@
V18=V18=1.493*F6+1.000E18
.12711.771@
V19=V19=1.591*F6+1.000E19.141
11.247@
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Psychometric properties: Validity
Convergent validity. An annotated example (Rusbult, 1980; Hatcher, 1994)
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Loadings size
V1=V1=.885*F1+.465E1.784
V2=V2=.824*F1+.566E2.680
V3=V3=.937*F1+.351E3.877
V5=V5=.828*F2+.561E5.685
V6=V6=.866*F2+.500E6.750
V7=V7=.875*F2+.483E7.766
V8=V8=.666*F3+.746E8.444
V9=V9=.634*F3+.773E9.402
V10=V10=.751*F3+.661E10.563
V11=V11=.826*F4+.564E11.682
V12=V12=.864*F4+.503E12.747
V13=V13=.463*F4+.886E13.215
V14=V14=.843*F5+.537E14.711
V15=V15=.707*F5+.707E15.500
V16=V16=.551*F5+.835E16.303
V17=V17=.684*F6+.730E17.467
V18=V18=.760*F6+.650E18.577
V19=V19=.728*F6+.685E19.530
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Psychometric properties: Validity
Discriminant validity Three criteria:
Chi-square difference test (Anderson y Gerbing, 1988) Confidence interval test (Anderson y Gerbing, 1988) Average Variance Extracted test (Fornell y Larcker, 1981)
Must be applied for each pair of factors!!! For time constraint reasons, in this example we will apply them just to the two
factors that exhibit higher correlations (and can more feasibly havediscriminant validity problems)
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IIIF4-F4-.224*I
IF2-F2.071II-3.148@I
II
IF5-F5.635*IIF2-F2.052I
I12.181@III
IF6-F6-.375*IIF2-F2.069I
I-5.424@I
IIIF4-F4-.092*I
IF3-F3.082I
I-1.131III
IF5-F5.516*IIF3-F3.069I
I7.479@I
IIIF6-F6-.424*I
IF3-F3.075II-5.633@I
II
IF5-F5.008*IIF4-F4.079I
I.102III
IF6-F6.255*I
IF4-F4.076II3.340@I
IIIF6-F6-.300*I
IF5-F5.077I
I-3.895@III
Psychometric properties: Validity
Discriminant validity. An annotated example (Rusbult, 1980; Hatcher, 1994)
27
Problematic factor
MAXIMUMLIKELIHOODSOLUTION(NORMALDISTRIBUTIONTHEORY)
COVARIANCESAMONGINDEPENDENTVARIABLES---------------------------------------
STATISTICSSIGNIFICANTATTHE5%LEVELAREMARKEDWITH@.
VF
------IF2-F2.609*I
IF1-F1.047II12.867@I
II
IF3-F3.440*IIF1-F1.066I
I6.624@I
IIIF4-F4-.016*I
IF1-F1.073II-.220I
II
IF5-F5.714*IIF1-F1.044I
I16.107@III
IF6-F6-.223*I
IF1-F1.073II-3.046@I
IIIF3-F3.534*I
IF2-F2.062I
I8.549@I
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Psychometric properties: Validity
Discriminant validity. An annotated example (Rusbult, 1980; Hatcher, 1994) Chi-square difference test (Anderson & Gerbing, 1988)
The CFA for the measurement model is estimated again, but thecovariance between the two problematic factors is fixed to 1 (F1 & F5)
The chi-square of the original measurement model CFA is subtracted fromthe chi-square of this restricted CFA. The same is done with their degreesof freedom.
This difference (should be positive) is distributed as a Chi-square with asmany degrees of freedom as the difference between the two models df.
If this statistic (the chi-square difference) is significant, it will indicate thatrestricting the correlation to be 1, significantly worsens the model fit andis not a reasonable assumption.
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Psychometric properties: Validity
Discriminant validity. An annotated example (Rusbult, 1980; Hatcher, 1994) Chi-square difference test (Anderson & Gerbing, 1988)
29
INDEPENDENCEMODELCHI-SQUARE=2167.771ON153DEGREESOFFREEDOM
INDEPENDENCEAIC=1861.77095INDEPENDENCECAIC=1176.23320
MODELAIC=-59.12792MODELCAIC=-596.80459
CHI-SQUARE=180.872BASEDON120DEGREESOFFREEDOM
PROBABILITYVALUEFORTHECHI-SQUARESTATISTICIS.00028
THENORMALTHEORYRLSCHI-SQUAREFORTHISMLSOLUTIONIS174.047.
INDEPENDENCEMODELCHI-SQUARE=2167.771ON153DEGREESOFFREEDOM
INDEPENDENCEAIC=1861.77095INDEPENDENCECAIC=1176.23320
MODELAIC=9.13766MODELCAIC=-533.01965
CHI-SQUARE=251.138BASEDON121DEGREESOFFREEDOM
PROBABILITYVALUEFORTHECHI-SQUARESTATISTICIS.00000
THENORMALTHEORYRLSCHI-SQUAREFORTHISMLSOLUTIONIS255.781.
Measurement model
CFAwhere/COVF1,F5=1
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Psychometric properties: Validity
Discriminant validity. An annotated example (Rusbult, 1980; Hatcher, 1994) Chi-square difference test (Anderson & Gerbing, 1988)
Chi-square difference: 251,138180,872=70,266 Degrees of freedom difference: 1 Critical value:
p
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Psychometric properties: Validity
Discriminant validity. An annotated example (Rusbult, 1980; Hatcher, 1994) Confidence interval test (Anderson & Gerbing, 1988)
A confidence interval for the correlation estimation is built: correlationestimation 2 SE (standard errors)
If value 1 forms part of the confidence intervaI, discriminant validity cannotbe assumed
Interval: Lower extreme: 0.714 - 20,044=0.626 Upper extreme: 0.714 + 20,044=0.802
Value 1 does not belong to the CI, no threaten to discriminant validity
31
IF4-F4-.016*I
IF1-F1.073I
I-.220I
IIIF5-F5.714*I
IF1-F1.044I
I16.107@I
II
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Psychometric properties: Validity
Discriminant validity. An annotated example (Rusbult, 1980; Hatcher, 1994) AVE test(Fornell y Larcker, 1981)
AVE for each pair of factors is calculated (You did it to check reliability! Sono extra work)
The AVEs of the two evaluated factors are compared to the squaredcorrelation between them
If both AVEs are higher than the squared correlation, no evidence ofdiscriminant validity problems is found
Squared correlation: 0.7142=0.510
32
IF4-F4-.016*I
IF1-F1.073I
I-.220I
II
IF5-F5.714*I
IF1-F1.044II16.107@I
II
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Psychometric properties: Validity Discriminant validity. An annotated example (Rusbult, 1980; Hatcher, 1994) AVE test(Fornell y Larcker, 1981)
Although AVE for F5 is slightly lower than the squared correlation, previousresults would lead us to conclude that no relevant discriminant validityproblems are present
33
AVEF1=
2,340
2,340 + 0, 660= 0, 780
AVEF5=
1,514
1,514 +1, 486
= 0,504
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Psychometric properties: Validity
Nomological validity: Usually it is tested by examining whether the correlations between the
constructs in the measurement model make sense. The construct correlationsare used to assess this.
In my opinion a more sensible (although more exigent) way, is comparing themeasurement model and the structural model fit.
Structural model adds the theoretical value (structural part) to justmeasurement, so it should have a better fit.
If our final structural model (without non-significant relationships and with thenew relationships we could have added on a theory basis) exhibits a betterdegree of fit than the only measurement model, or at least are notdistinguishable, nomological validity can be assumed.
Chi-square difference test is used to evaluate goodness of fit.
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IIIF4-F4-.224*I
IF2-F2.071II-3.148@I
II
IF5-F5.635*IIF2-F2.052I
I12.181@III
IF6-F6-.375*IIF2-F2.069I
I-5.424@I
IIIF4-F4-.092*I
IF3-F3.082I
I-1.131III
IF5-F5.516*IIF3-F3.069I
I7.479@I
IIIF6-F6-.424*I
IF3-F3.075II-5.633@I
II
IF5-F5.008*IIF4-F4.079I
I.102III
IF6-F6.255*I
IF4-F4.076I
IF5-F5.077I
I-3.895@III
Psychometric properties: Validity
Nomological validity. An annotated example (Rusbult, 1980; Hatcher, 1994)
35
MAXIMUMLIKELIHOODSOLUTION(NORMALDISTRIBUTIONTHEORY)
COVARIANCESAMONGINDEPENDENTVARIABLES---------------------------------------
STATISTICSSIGNIFICANTATTHE5%LEVELAREMARKEDWITH@.
VF
------IF2-F2.609*I
IF1-F1.047II12.867@I
II
IF3-F3.440*IIF1-F1.066I
I6.624@I
IIIF4-F4-.016*I
IF1-F1.073II-.220I
II
IF5-F5.714*IIF1-F1.044I
I16.107@III
IF6-F6-.223*I
IF1-F1.073II-3.046@I
IIIF3-F3.534*I
IF2-F2.062I
I8.549@I
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Psychometric properties: Validity Validity. An annotated example (Rusbult, 1980; Hatcher, 1994)
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Psychometric properties: Validity Nomological validity. An annotated example (Rusbult, 1980; Hatcher, 1994)
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Structural model 1
/TITLE/SPECIFICATIONS
VARIABLES=19;CASES=240;
METHOD=ML;
ANALYSIS=COVARIANCE;MATRIX=COR;
/MATRIX
/STANDARDDEVIATIONS
/EQUATIONSV1=*F1+E1;
V2=*F1+E2;
V3=F1+E3;
V5=*F2+E5;
V6=*F2+E6;
V7=F2+E7;
V8=*F3+E8;
V9=*F3+E9;
V10=F3+E10;
V11=*F4+E11;
V12=F4+E12;V13=*F4+E13;
V14=F5+E14;
V15=*F5+E15;
V16=*F5+E16;
V17=*F6+E17;
V18=F6+E18;
V19=*F6+E19;
F1=*F2+*F5+*F6+D1;
F2=*F3+*F4+D2;
/VARIANCESF3TOF6=*;
E1TOE3=*;
E5TOE19=*;
D1TOD2=*;
/COVARIANCES
F3TOF6=*;
/WTEST
/LMTEST
/PRINTFIT=ALL;
/END
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Psychometric properties: Validity
Nomological validity. An annotated example (Rusbult, 1980; Hatcher, 1994)
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INDEPENDENCEMODELCHI-SQUARE=2167.771ON153DEGREESOFFREEDOM
INDEPENDENCEAIC=1861.77095INDEPENDENCECAIC=1176.23320
MODELAIC=-59.12792MODELCAIC=-596.80459
CHI-SQUARE=180.872BASEDON120DEGREESOFFREEDOM
PROBABILITYVALUEFORTHECHI-SQUARESTATISTICIS.00028
THENORMALTHEORYRLSCHI-SQUAREFORTHISMLSOLUTIONIS174.047.
INDEPENDENCEMODELCHI-SQUARE=2167.771ON153DEGREESOFFREEDOM
INDEPENDENCEAIC=1861.77095INDEPENDENCECAIC=1176.23320
MODELAIC=-31.24841MODELCAIC=-586.84764
CHI-SQUARE=216.752BASEDON124DEGREESOFFREEDOM
PROBABILITYVALUEFORTHECHI-SQUARESTATISTICIS.00000
THENORMALTHEORYRLSCHI-SQUAREFORTHISMLSOLUTIONIS214.296.
Measurement model
Structural model 1
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Psychometric properties: Validity
Nomological validity. An annotated example (Rusbult, 1980; Hatcher, 1994) Chi-square structural model 1: 216.75 (df124) Chi-square measurement model: 180.87 (df120) Structural model 1 chi-square greater than measurement model (worst
fit), but is the difference significant?
Chi-square difference: 35.88 Degrees of freedom difference: 4 Critical value p
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Psychometric properties: Validity
Nomological validity. An annotated example (Rusbult, 1980; Hatcher, 1994) What relationships are not necessary (not significant) and are worsening
our fit?
Are there any theory based relationships that could be added? Can Wald and Lagrange help us?
40
MULTIVARIATELAGRANGEMULTIPLIERTESTBYSIMULTANEOUSPROCESSINSTAGE1CUMULATIVEMULTIVARIATESTATISTICSUNIVARIATEINCREMENT
----------------------------------------------------------------
HANCOCK'S
SEQUENTIALSTEPPARAMETERCHI-SQUARED.F.PROB.CHI-SQUAREPROB.D.F.PROB.
----------------------------------------------------------1F2,F534.1811.00034.181.0001241.000
2V1,F244.9242.00010.743.0011231.000
3V2,F551.6743.0006.750.0091221.000
WALDTEST(FORDROPPINGPARAMETERS)MULTIVARIATEWALDTESTBYSIMULTANEOUSPROCESS
CUMULATIVEMULTIVARIATESTATISTICSUNIVARIATEINCREMENT
------------------------------------------------------
STEPPARAMETERCHI-SQUARED.F.PROBABILITYCHI-SQUAREPROBABILITY
-------------------------------------------------------------1F5,F4.0071.935.007.935
2F1,F6.8512.653.845.358
3F4,F32.6843.4431.833.176
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Psychometric properties: Validity Nomological validity. An annotated example (Rusbult, 1980; Hatcher, 1994)
Wald suggests two covariances to be deleted (F5,F4) y (F4,F3), but only oneregression coefficient is not significant F1 to F6.
Lagrange suggests adding a regression coefficient between F2 and F5. Only ifwe can find substantive theory to support this, this step should be done
Model is re-estimated
41
INDEPENDENCEMODELCHI-SQUARE=2167.771ON153DEGREESOFFREEDOM
INDEPENDENCEAIC=1861.77095INDEPENDENCECAIC=1176.23320MODELAIC=-59.12792MODELCAIC=-596.80459
CHI-SQUARE=180.872BASEDON120DEGREESOFFREEDOM
PROBABILITYVALUEFORTHECHI-SQUARESTATISTICIS.00028
THENORMALTHEORYRLSCHI-SQUAREFORTHISMLSOLUTIONIS174.047.
INDEPENDENCEMODELCHI-SQUARE=2167.771ON153DEGREESOFFREEDOM
INDEPENDENCEAIC=1861.77095INDEPENDENCECAIC=1176.23320
MODELAIC=-64.80851MODELCAIC=-620.40774
CHI-SQUARE=183.191BASEDON124DEGREESOFFREEDOM
PROBABILITYVALUEFORTHECHI-SQUARESTATISTICIS.00044
THENORMALTHEORYRLSCHI-SQUAREFORTHISMLSOLUTIONIS176.052.
Measurement model
Structural model 2
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Psychometric properties: Validity Nomological validity. An annotated example (Rusbult, 1980; Hatcher, 1994)
Chi-square structural model 2: 183.19 (df124)
Chi-square measurement model: 180.87 (df
120) Structural model 2 chi-square is greater than the measurement model one
(worst fit), but is the difference significant?
Chi-square difference: 2.32 Degrees of freedom difference: 4 Critical value for p
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Psychometric properties: Validity Model nomologically valid
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Psychometric properties: Validity Example of presentation in a paper (Bign, Alds, Ruiz y Sanz, 2008)
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Appendix
Variable Indicator F actor loading Robust t-value CA CR AVE
P er ce iv ed u se ful nes s US EF UL 2 0 .7 40 * * 19.03 0.87 0.87 0.57US EF UL 3 0 .7 80 * * 20.86US EF UL 4 0 .7 60 * * 19.59US EF UL 5 0 .7 32 * * 18.01US EF UL 6 0 .7 58 * * 19.51
Perceived ease of use EASE1 0.750 * * 16.99 0.74 0.75 0.43EASE3 0.600 * * 12.04EASE4 0.646 * * 14.01
EASE6 0.629 * * 12.88Innovativeness INN1 0.701 * * 8.81 0.78 0 .80 0.67
INN2 0.920 * * 10.67Attitud e to onl in e shopp in g ATT I4 0.743 * * 16.82 0.81 0.81 0.59
ATTI5 0.873 * * 20.29ATTI7 0.678 * * 14.03
I nf or ma ti on de pen den cy DE P1 0 .8 40 * * 18.27 0.73 0.74 0.59DEP3 0.685 * * 13.24
S-B x2 (94 df) 252.31 (p , 0.01); NFI 0.90; NNFI 0.92; CFI 0.94; IFI 0.94;RMSEA 0.06
Notes: *p , 0.05; * *p , 0.01. CA Cronbachs a; CR composite reliability; AVE averagevariance extracted
Table AI.Validation of the finalmeasurement model
reliability and convergentvalidity
Online shoppinginformation
667
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Psychometric properties: Validity Example of presentation in a paper (Bign, Alds, Ruiz y Sanz, 2008)
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1 2 3 4 5
1. Perceived usefulness 0.75 0.60 * * 0.20 * * 0.65 * * 0.62 * *
2. Perceived ease of use [0.51;0.69] 0.66 0.33 * * 0.47 * * 0.55 * *
3. Innovativeness [0.09;0.32] [0.19;0.46] 0.82 0.17 * * 0.084. Attitude to online shopping [0.59;0.71] [0.36;0.58] [0.05;0.29] 0.77 0.44* *
5. Online information dependency [0.53;0.72] [0.43;0.67] [-0.04;0.21] [0.33;0.55] 0.77
Notes: *p, 0.05; * *p, 0.01. Diagonal represents the square root of the average variance extracted;while above the diagonal the shared variance (squared correlations) are represented. Below thediagonal the 95 per cent confidence interval for the estimated factors correlations is provided
Table AII.Validation of the finalmeasurement model
discriminant validity