Consumer BehaviorMeasurement invariance
Issues in assessing Issues in assessing measurement invariance in measurement invariance in
crosscross--national researchnational research
Hans BaumgartnerHans BaumgartnerPenn State UniversityPenn State University
JanJan--Benedict Benedict SteenkampSteenkampTilburg Tilburg UniversityUniversity
Consumer BehaviorMeasurement invariance
Measurement invariance in actual research practice
• often lack of concern for measurement invariance in cross-national research;
• “without evidence of measurement invariance, the conclusions of a study must be weak” (Horn 1991, p. 119);
• discussion of a variety of important issues that arise when measurement invariance is assessed, with special consideration for situations in which comparisons are conducted across many different groups;
Consumer BehaviorMeasurement invariance
DataARG Argentina 397 JPN Japan 449AUT Austria 393 NED Netherlands 504BEL Belgium 508 NOR Norway 549BRA Brazil 398 POL Poland 391CHN China 419 POR Portugal 439CZE Czech Republic 493 ROM Romania 431DEN Denmark 519 RUS Russia 396ESP Spain 546 SUI Switzerland 393FRA France 402 SVK Slovakia 379GBR Great Britain 355 SWE Sweden 419GER Germany 640 THA Thailand 402HUN Hungary 577 TRE Taiwan 396IRL Ireland 552 UKR Ukraine 396ITA Italy 397 USA United States 1181
Consumer BehaviorMeasurement invariance
Satisfaction With Life Scale(Diener et al. 1985)
• scale to measure global life satisfaction as a cognitive-judgmental process;
• five items rated on 5-point agree/disagree scales:– In most ways my life is close to my ideal.– The conditions of my life are excellent.– I am satisfied with my life.– So far I have gotten the important things I want in life.– If I could live my life over, I would change almost
nothing.
Consumer BehaviorMeasurement invariance
Affectometer 2 (Kammann and Flett 1983)
• scale to measure a person’s general happiness or sense of well-being based on the balance of positive and negative feelings in recent experience;
• Below are various feelings that you might use to describe how happy or unhappy you are in your life. Please indicate how often you have felt each of the following feelings over the past month (5-point scale from none of the time to all the time):– Positive Affect (confident, optimistic, satisfied, useful, etc.)– Negative Affect (depressed, hopeless, lonely, discontented,
etc.)
Consumer BehaviorMeasurement invariance
Measurement model for SWB
LS PA NA
I am satisfied with my life.
etc.(5 items)
confidentoptimistic
usefuletc.
(10 items)
depressedhopelesslonely
etc.(10 items)
Consumer BehaviorMeasurement invariance
Measurement modelLet xg be a p x 1 vector of observed variables in country g, ξg an m x 1 vector of latent variables, δg a p x 1 vector of errors of measurement, τg a p x 1 vector of item intercepts, and Λg a p x m matrix of factor loadings. Then
xg = τg + Λgξg + δg
The means part of the model is given byµg = τg + Λgκg
and the covariance part is given byΣg = Λg Φg Λg ' + Θg
Consumer BehaviorMeasurement invariance
Model identification• to identify the covariance part, the latent constructs
have to be assigned a scale in which they are measured; this is done by choosing a marker item and setting its loading to one;
• to identify the means part, the intercepts of the marker items are fixed to zero (which equates the means of the latent constructs to the means of their marker variables, µm
g = κmg);
• although these constraints help with identifying the model, they are not sufficient for meaningful cross-national comparisons;
Consumer BehaviorMeasurement invariance
Forms of measurement invariance• levels of invariance:
imposition of increasingly stringent forms of invariance on the measurement model;
• full vs. partial invariance: extent to which measurement invariance of a given kind is satisfied,
Consumer BehaviorMeasurement invariance
Levels of invariance• configural invariance: the pattern of salient and
nonsalient loadings is the same across different countries;
• metric invariance: the scale metrics are the same across countries;
Λ1 = Λ2 = ... = ΛG
• scalar invariance: in addition to the scale metrics the item intercepts are the same across countries;
τ1 = τ2 = ... = τG
Consumer BehaviorMeasurement invariance
Linking the level of invariance requiredto the research objective
Conducting cross-national comparisons of means
Examining structural relationships with other
constructs cross-nationally
Exploring the basic structure of the construct
cross-nationally
Scalar invariance
Metric invariance
Configural invariance
Consumer BehaviorMeasurement invariance
Full vs. partial invariance• full measurement invariance of a given type (e.g.,
metric, scalar) is frequently not satisfied in practice;• partial measurement invariance as a “compromise”
between full measurement invariance and complete lack of invariance (Byrne et al. 1989);
• issue of the minimal degree of partial measurement invariance necessary for comparisons of cross-national differences in factor means to be meaningful (Steenkamp and Baumgartner 1998);
Consumer BehaviorMeasurement invariance
Partial measurement invariance• for identification purposes, one item per factor has to have
invariant loadings and intercepts (marker item); the marker item has to be chosen carefully;
• at least one other invariance constraint on the loadings/ intercepts is necessary to ascertain whether the marker item satisfies metric/scalar invariance;
• Cheung and Rensvold (1999) proposed the factor-ratio test in which all possible pairs of items are tested for metric invariance and sets of invariant items are identified;
• an alternative is to start with the fully invariant model of a given kind and relax invariance constraints based on significant modification indices, changes in alternative fit indices, and expected parameter changes;
Consumer BehaviorMeasurement invariance
Assessing metric invariance across many different countries
• choose a marker item for each factor– graph the factor loadings– compute variance components– calculate individual item reliabilities
• compare the configural with the full metric invariance model;
• use Bonferroni-adjusted modification indices (or other fit indices) and expected parameter changes to free loadings that are not invariant;
Consumer BehaviorMeasurement invariance
Variance components for loadings
.162
.150
.006
.031
.308
.004
.010
.053
.009
.032
.037
.037
.020
.027
.060
.017
.021
.026
.017
.034
.017
.018
.002
.0031.148.063.352.046.020.123
.035
.039
.022
.017
.074
.026
.050
.033
.030
.052
.011
.016
.004
.011
.015
.013
.009
.011
.010
.013
Item 10Item 9Item 8Item 7Item 6Item 5Item 4Item 3Item 2Item 1
Positive affect
Item 10Item 9Item 8Item 7Item 6Item 5Item 4Item 3Item 2Item 1
Negative affect
Item 5Item 4Item 3Item 2Item 1
Life satisfaction
Consumer BehaviorMeasurement invariance
Individual-item reliabilities
.27
.29
.35
.39
.18
.53
.46
.38
.52
.33
(.04-.45)
(.08-.49)
(.13-.52)
(.11-.60)
(.03-.18)
(.30-.66)
(.23-.60)
(.15-.50)
(.38-.65)
(.21-.44)
.31
.27
.44
.51
.25
.35
.25
.36
.36
.23
(.20-.45)
(.15-.34)
(.28-.59)
(.35-.67)
(.00-.49)
(.17-.56)
(.01-.56)
(.08-.54)
(.19-.53)
(.05-.46)
.38
.47
.64
.54
.53
(.16-.50)
(.31-.68)
(.32-.79)
(.42-.69)
(.20-.71)
Item 10
Item 9
Item 8
Item 7
Item 6
Item 5
Item 4
Item 3
Item 2
Item 1
Positive affect
Item 10
Item 9
Item 8
Item 7
Item 6
Item 5
Item 4
Item 3
Item 2
Item 1
Negative affect
Item 5
Item 4
Item 3
Item 2
Item 1
Life satisfaction
Consumer BehaviorMeasurement invariance
Model comparison tests
.915.912.0948,74438,987Initial scalar invariance
.944.943.0738,55628,242Final scalar invariance
.945.946.0728,18026,590Partial metric invariance
.943.944.0748,21027,462Full metric invariance
.945.950.073
(.060-.092)7,61624,888Configural
invariance
TLICFIRMSEAdfχ2
Factor loadings
Consumer BehaviorMeasurement invariance
Metric invariance: Life satisfaction
5432154321
USAITA
UKRIRL
TREHUN
THAGER
SWEGBR
SVKFRA
SUIESP
RUSDEN
ROMCZE
PORCHN
POLBRA
NORBEL
NEDAUT
JPNARG
Consumer BehaviorMeasurement invariance
Metric invariance: Positive affect109876543211
0987654321
USAITA
UKRIRL
TREHUN
THAGER
SWEGBR
SVKFRA
SUIESP
RUSDEN
ROMCZE
PORCHN
POLBRA
NORBEL
NEDAUT
JPNARG
Consumer BehaviorMeasurement invariance
Metric invariance: Negative affect109876543211
0987654321
USAITA
UKRIRL
TREHUN
THAGER
SWEGBR
SVKFRA
SUIESP
RUSDEN
ROMCZE
PORCHN
POLBRA
NORBEL
NEDAUT
JPNARG
Consumer BehaviorMeasurement invariance
PAPA
NANA
LSLS
papa
nana
lsls
γγ1con
γγ2con
λλpacon
λλnacon
λλlscon
lsls = = λλlscon LS + LS + εε
PAPA
NANA
LSLS
papa
nana
lsls
γγ1fm
γγ2fm
λλpafm
λλnafm
λλlsfm
lsls = = λλlsfm ((λλls
con / / λλlsfm ) LS + ) LS + εε
== λλlsfm LSLS∗∗ + + εε
Consumer BehaviorMeasurement invariance
Comparison of the path coefficients in the configural and full metric invariance solutions
γ1fm = (λls
con / λlsfm) / (λpa
con / λpafm) (γ1
con)
λpacon / λpa
fm
??
??| γ1fm |> | γ1
con |
| γ1fm |< | γ1
con |λls
con / λlsfm
< 1< 1
> 1> 1
< 1< 1 > 1> 1
Consumer BehaviorMeasurement invariance
Some illustrative examples
γ1fm
γ1con
λpacon / λpa
fm
λlscon / λls
fm
GBRJPNSVKSWE
.81.63.73.82
.76.81.62.88
.911.17.851.09
1.00.941.061.04
Consumer BehaviorMeasurement invariance
Evaluating the difference between the path coefficients in the configural and full metric
invariance solutions
γ1con γ1
fm
• best estimate of the structural coefficient in each group;
• however, since the scale metrics may differ, it’s dangerous to compare the effects across groups;
• the scale metrics are specified to be the same and thus the effects are in principle comparable;
• however, the invariance constraints on the scale metrics may not be justified and the estimated effects may be distorted;
Comparison of paths from PA to LS in configural, full metric and partial metric invariance solutions
Comparison of paths from PA to LS in configural, full metric and partial metric invariance solutions
Consumer BehaviorMeasurement invariance
Assessing scalar invariance across many different countries
• compare the final metric invariance model with the full (or initial) scalar invariance model;
• use Bonferroni-adjusted modification indices (or other fit indices) and expected parameter changes to free item intercepts that are not invariant;
Consumer BehaviorMeasurement invariance
Model comparison tests
.915.912.0948,74438,987Initial scalar invariance
.944.943.0738,55628,242Final scalar invariance
.945.946.0728,18026,590Partial metric invariance
.943.944.0748,21027,462Full metric invariance
.945.950.0737,61624,888Configural invariance
TLICFIRMSEAdfχ2
Consumer BehaviorMeasurement invariance
Scalar invariance: Life satisfaction
5432154321
USAITA
UKRIRL
TREHUN
THAGER
SWEGBR
SVKFRA
SUIESP
RUSDEN
ROMCZE
PORCHN
POLBRA
NORBEL
NEDAUT
JPNARG
Consumer BehaviorMeasurement invariance
Scalar invariance: Positive affect109876543211
0987654321
USAITA
UKRIRL
TREHUN
THAGER
SWEGBR
SVKFRA
SUIESP
RUSDEN
ROMCZE
PORCHN
POLBRA
NORBEL
NEDAUT
JPNARG
Consumer BehaviorMeasurement invariance
Scalar invariance: Negative affect109876543211
0987654321
USAITA
UKRIRL
TREHUN
THAGER
SWEGBR
SVKFRA
SUIESP
RUSDEN
ROMCZE
PORCHN
POLBRA
NORBEL
NEDAUT
JPNARG
Item intercepts
Consumer BehaviorMeasurement invariance
LSLS lslsλλls
con
lsls = = ττlscon ++ λλls
con LS + LS + εε
LSLS lslsλλls
fs
lsls = = ττlsfs ++ λλls
fs [ ( [ ( ττlscon –– ττls
fs ) / ) / λλlsfs + (+ ( λλls
con / / λλlsfs ) LS ] + ) LS ] + εε
= = ττlsfs ++λλls
fs LSLS∗∗ + + εε
Consumer BehaviorMeasurement invariance
Comparison of means in the configural and full scalar invariance solutions
κlsfs = ( τls
con - τlsfs ) / λls
fs + (λlscon / λls
fs)(κlscon)
λlscon / λls
fs
??
??
| κlsfs |< | κls
con |
| κlsfs | > | κls
con |
< 0< 0
> 0> 0
< 1< 1 > 1> 1
( τlscon - τls
fs ) / λlsfs
Mean life satisfaction
Mean positive affect
Mean negative affect
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