Factor Analysis:Application using WVS data from selected Arab countries

Irina Vartanova

Institute for Futures Studies, Stockholm

ERF Workshop – May 10, 2015

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Exploitative Factor Analysis: Understanding of Democracy

• V136. Civil rights protect people from state oppression.

• V133. People choose their leaders in free elections.

• V139 Women have the same rights as men.

• V132. Religious authorities ultimately interpret the laws.

• V135. The army takes over when government is incompetent.

• V138. People obey their rulers.

• V137. The state makes people’s incomes equal.

• V131. Governments tax the rich and subsidize the poor.

• V134. People receive state aid for unemployment.

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Motivation

• Measure the underlying concept with smaller measurementerror.

• Dimensionality reduction to reduce the effect ofmulti-collinearity.

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Data

• WVS, 6th wave

• 57 countries (Bahrain excluded)

• Pool sample: 82289 cases

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Understanding of democracy: the World

V139

V133

V136

V134

V131

V137

V138

V135

V132

V132 V135 V138 V137 V131 V134 V136 V133 V139

0.00

0.25

0.50

0.75

1.00value

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Understanding of democracy: the MENA countries

V139

V136

V137

V133

V134

V138

V131

V135

V132

V132 V135 V131 V138 V134 V133 V137 V136 V139

0.25

0.50

0.75

1.00value

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Understanding of democracy: the Western Countries

V137

V131

V134

V136

V139

V133

V135

V132

V138

V138 V132 V135 V133 V139 V136 V134 V131 V137

−0.25

0.00

0.25

0.50

0.75

1.00value

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Number of Factors: Scree plot

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●

●●

●

● ●● ●

0.0

0.5

1.0

1.5

2.0

Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8 Comp.9x

Eig

en v

alue

s

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Full Sample ResultsLoadings:

Factor1 Factor2V136 0.69V133 0.72V139 0.64V132 0.71V135 0.61V138 0.44V137 0.42V131 0.32V134 0.55

Factor1 Factor2SS loadings 1.96 1.41Proportion Var 0.22 0.16Cumulative Var 0.22 0.37

Factor Correlations:Factor1 Factor2

Factor1 1.00 -0.36Factor2 -0.36 1.00

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MENA Region ResultsLoadings:

Factor1 Factor2V136 0.73V133 0.56V139 0.57V132 0.88V135V138V137 0.49V131V134 0.51

Factor1 Factor2SS loadings 1.87 0.98Proportion Var 0.21 0.11Cumulative Var 0.21 0.32

Factor Correlations:Factor1 Factor2

Factor1 1.00 -0.55Factor2 -0.55 1.00

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Why economists do not like survey data

1

We can address the country-item-bias with measurementinvariance concept.

1The figure is reproduced from: Stegmueller, D. (2011). Apples andoranges? The problem of equivalence in comparative research. Political Analysis

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Measuring Latent Variable:Attitudes towards Gender Equality example

• V45. When jobs are scarce, men should have more right to ajob than women.

• V51. On the whole, men make better political leaders thanwomen do.

• V52. A university education is more important for a boy thanfor a girl.

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Data

• WVS, 6th wave

• 12 MENA countries: Algeria, Egypt, Iraq, Jordan, Kuwait,Lebanon, Libya, Morocco, Palestine, Qatar, Yemen

• Bahrain excluded

• Pool sample: 13260 after listwize deletion of missing cases

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Confirmatory Factor Analysis

GendEqAtt y2

y1

y3

e2

e1

e3

y2 = v2 + λ2GendEqAtt + e2

y1 = v1 + 1GendEqAtt + e1

y3 = v3 + λ3GendEqAtt + e3

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CFA results: pooled sample

GendEqAtt y2

y1

y3

1.00

1.14

0.66

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Degrees of Freedom and the Model Identification

• The power or CFA approach that it has clear criteria of modelfit. The estimated variance-covariance matrix is comparedwith the one implied by the theoretical model. How big arediscrepancies is tested with the Pearson χ2.

• However, we need degrees of freedom to test the model fit,meaning that the number of known parameters should belarger than the number of parameters to be estimated.

• One factor with 3 indicators is just identified model and doesnot provide us with fit indices.

• We can obtain additional degrees of freedom by fixing someparameters.

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CFA results: pooled sample with the second item loadingfixed to 1

GendEqAtt y2

y1

y3

1.00

1.00

0.62

Fit: χ2 = 10.160; df = 1; P-value = 0,001;CFI = 0.998; RMSEA = 0.026; P-value RMSEA ≤ 0.05 = 0.995

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Measurement Invariance

• To establish the cross-country comparability, we use theconcept of measurement invariance / equivalence.

• 3 level of invariance - structural, metric and scalar - providedifferent levels of available model comparisons.

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Structural invariance

GendEqAtt y2

y1

y3

y2g = v2g + λ2gGendEqAtt + e2g

y1g = v1g + 1 ∗ GendEqAtt + e1g

y3g = v3g + λ3gGendEqAtt + e3g

• The structural equivalence implies that used items measurethe same concept in all comparing countries.

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Metric Invariance

GendEqAtt y2

y1

y3

y2g = v2g + λ2GendEqAtt + e2g

y1g = v1g + 1 ∗ GendEqAtt + e1g

y3g = v3g + λ3GendEqAtt + e3g

• The metric equivalence implies that the scale of the latentlatent variable has the same metric in groups, that is changein one unit on the scale has the same meaning in differentgroups.

• It is achieved by equality of item loadings across all the groups

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Scalar Invariance

GendEqAtt y2

y1

y3

y2 = v2 + λ2GendEqAtt + e2g

y1 = v1 + 18GendEqAtt + e1g

y3 = v3 + λ3GendEqAtt + e3g

• Scalar equivalence determines correspondence of the absolutevalue at the scale in different groups.

• It is justified by equal items intercepts.

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Partial Invariance

GendEqAtt y2

y1

y3

y2 = v2 + λ2GendEqAtt + e2g

y1 = v1 + 1 ∗ GendEqAtt + e1g

y3g = v3 + λ3gGendEqAtt + e3g

• Partial invariance can be confirmed he at leas two items areinvariant.

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Gender Equality: Test for Invariance

χ2 df p-value CFI RMSEA p-RMSEA

Structural 0 0 1 0Metric 147.576 22 0 0.974 0.061 0.046

Partial in Morocco 97.745 21 0 0.984 0.047 0.648Scalar 737.365 66 0 0.861 0.077 0.000

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Model comparisons

• χ2 difference test, but it is sensitive to large sample sizes.

• Differences in their overall goodness-of-fit indices (Chen,2007).

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Existing alternatives

• Multilevel modelling approach (Stegmueller, 2011)

• ”EPC-interest” (Oberski, 2014)

• Alignment method (Muthen & Asparouhov, 2014)

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Multilevel modelling approach

2

• Non-invariance or systematic country-item-bias can be

captured by allowing the country level latent variable ν(3)k to

directly affect the items’ variance2The figure is reproduced from: Stegmueller, D. (2011). Apples and

oranges? The problem of equivalence in comparative research. Political Analysis26 / 28

EPC-interest approach

• Useful when the relationship between the latent variable andsome covariate vector ”structuralmodel” is of interest.

• Measurement invariance is usually tested by various fitmeasures. However, violations not necessarily seriously affectthe fit measures, and substantial bias in the parameter ofinterest may still remain.

• Expected Parameter Change approach accounts for the effectthat violations of measurement invariance assumption have onthe parameters of interest

• EPC-interest assesses impact on the parameter of interest if aparticular possible direct effect was freed.

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Alignment method approach

• To define the metric system of a latent variable and make theidentifiable one of the two options is used: one of the itemloadings is fixed 1 or variance of the latent variable is fixed to1. Other parameters are estimated base on thevariance-covariance matrix.

• The alignment approach can estimate the parameters byincorporating the assumption that the number of non-invariantmeasurement parameters can be held to minimum.

• For clarification, the authors relate to the analogy of usingrotation in EFA.

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