Department of Data Analysis
Ghent University
Structural Equation Modeling with lavaan Yves Rosseel Department of Data Analysis Ghent University Summer School – Using R for personality research August 23–28, 2014 Bertinoro, Italy
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Department of Data Analysis
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Contents 1
Introduction to SEM 1.1 From regression to structural equation modeling . . . . . . . 1.2 The model-implied covariance matrix (the essence of SEM) . 1.3 Matrix representation in a CFA model . . . . . . . . . . . . 1.4 The implied covariance matrix for the full SEM model . . . 1.5 Model parameters and model matrices . . . . . . . . . . . . 1.6 Model estimation . . . . . . . . . . . . . . . . . . . . . . . 1.7 Model evaluation . . . . . . . . . . . . . . . . . . . . . . . 1.8 Model respecification . . . . . . . . . . . . . . . . . . . . . 1.9 Reporting your results . . . . . . . . . . . . . . . . . . . . . 1.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . .
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3 3 9 13 22 26 34 35 37 38 39
2
Introduction to lavaan
40
3
Further topics 3.1 Meanstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Multiple groups . . . . . . . . . . . . . . . . . . . . . . . . . . .
62 62 66
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3.3 3.4 3.5 3.6 4
Ghent University
Measurement invariance . . . . . . . . . . Missing data . . . . . . . . . . . . . . . . . Nonnormal data and alternative estimators . Handling categorical endogenous variables
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68 73 75 85
Appendices 98 4.1 Rules about variances and covariances . . . . . . . . . . . . . . . 98 4.2 lavaan: a brief user’s guide . . . . . . . . . . . . . . . . . . . . . 105 4.3 Estimator ML, MLM and MLR: robust standard errors and scaled test statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
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1 1.1
Ghent University
Introduction to SEM From regression to structural equation modeling
univariate linear regression 1 x1 x1
β0
β1
x2
β2 β3
x3
x2 y
y x3
β4
x4 x4 yi = β0 + β1 xi1 + β2 xi2 + β3 xi3 + β4 xi4 + i Yves Rosseel
Structural Equation Modeling with lavaan
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multivariate regression
x1 x2
y1
x3
y2
x4
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path analysis • testing models of causal relationships among observed variables • all variables are observed (manifest) • system of regression equations
y1
y5
y2 y3 y4
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y6
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y7
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measurement error • in the social sciences, observed variables are not without measurement error • single indicator measurement model
y
η
• multiple indicator measurement model
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1
y1
2
y2
3
y3
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η
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confirmatory factor analysis (CFA) • factor analysis: representing the relationship between one or more latent variables and their (observed) indicators y1 y2
η1
y3 y4 y5
η2
y6
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structural equation modeling (SEM) • path analysis with latent variables y7
y8
y9
y10
y11
y12
y1 y2
η1
η3
η4
y3 y4 y5
η2
y6 Yves Rosseel
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1.2
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The model-implied covariance matrix (the essence of SEM) • the goal of SEM is to test an a priori specified theory (which often can be depicted as a path diagram) • we may have several alternative models, each one with its own path diagram • each path diagram can be converted to a SEM: – measurement model (relationship latent variables and indicators) – structural equations (regressions among latent/observed variables) • each diagram has ‘model-based’ implications ˆ – for the model-implied covariance matrix: Σ ˆ – for the model-implied mean vector: µ – ... • different diagrams lead to (potentially) different implications; some implications may not fit with your data
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example model-implied covariance matrix (1) • suppose we have three observed variables, y1 , y2 and y3 ; to explain why they are correlated, we may postulate the following model: y1
a
y2
b
y3 • if we use the following values for the model parameters: a = 3 and b = 5, σ 2 (y1 ) = 10, σ 2 (2 ) = 20 and σ 2 (3 ) = 30, then we find (see Appendix): 10 ˆ = Σ 30 110 50 150 280
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example model-implied covariance matrix (2) • but if we change the path diagram (and keep the parameter values fixed), the model-implied covariance matrix will also change: y1
a
y2 b
y3 we find
10 ˆ = Σ 30 110 150 550 2780
• two models are said to be equivalent, if they imply the same covariance matrix (but note that we did not estimate the parameters here) Yves Rosseel
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example model-implied covariance matrix (3) • we can also postulate that the correlations among the three observed variables are explained by a common ‘factor’: y1 1
y2
a
η
b
y3 we find (using σ 2 (1 ) = 10, σ 2 (2 ) = 20, σ 2 (3 ) = 30, σ 2 (η) = 1): 11 ˆ = Σ 4 36 5 20 55
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1.3
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Matrix representation in a CFA model
classic example CFA • well-known dataset; based on Holzinger & Swineford (1939) data • also analyzed by J¨oreskog (1969) • 9 observed ‘indicators’ measuring three ‘latent’ factors: – a ‘visual’ factor measured by x1, x2 and x3 – a ‘textual’ factor measured by x4, x5 and x6 – a ‘speed’ factor measured by x7, x8 and x9 • N=301 • we assume the three factors are correlated
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diagram of the model x1 x2
visual
x3 x4 x5
textual
x6 x7 x8
speed
x9
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how does the data look like? • our dataset contains measures for all observed variables in the model • in many SEM applications, we do not need the full dataset; all we need are summary statistics: – the sample-based variance/covariance matrix (S); if there are p variables, the covariance matrix is of size p × p and contains p(p + 1)/2 non-duplicated elements – the sample size (N) – for some SEM applications, we also need the sample-based p-dimensional mean vector • note that this is true for many statistical techniques (ANOVA, regression, factor analysis, . . . ) • but for more advanced applications, we need the full dataset anyway (e.g. if we wish to correct for non-normality, if we have missing values, . . . ) Yves Rosseel
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data 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ... 301
x1 3.3333333 5.3333333 4.5000000 5.3333333 4.8333333 5.3333333 2.8333333 5.6666667 4.5000000 3.5000000 3.6666667 5.8333333 5.6666667 6.0000000 5.8333333 4.6666667
x2 7.75 5.25 5.25 7.75 4.75 5.00 6.00 6.25 5.75 5.25 5.75 6.00 4.50 5.50 5.75 4.75
x3 0.375 2.125 1.875 3.000 0.875 2.250 1.000 1.875 1.500 0.750 2.000 2.875 4.125 1.750 3.625 2.375
x4 2.3333333 1.6666667 1.0000000 2.6666667 2.6666667 1.0000000 3.3333333 3.6666667 2.6666667 2.6666667 2.0000000 2.6666667 2.6666667 4.6666667 5.0000000 2.6666667
x5 5.75 3.00 1.75 4.50 4.00 3.00 6.00 4.25 5.75 5.00 3.50 4.50 4.00 4.00 5.50 4.25
x6 1.2857143 1.2857143 0.4285714 2.4285714 2.5714286 0.8571429 2.8571429 1.2857143 2.7142857 2.5714286 1.5714286 2.7142857 2.2857143 1.5714286 3.0000000 0.7142857
x7 3.391304 3.782609 3.260870 3.000000 3.695652 4.347826 4.695652 3.391304 4.521739 4.130435 3.739130 3.695652 5.869565 5.130435 4.000000 4.086957
4.3333333 6.00 3.375 3.6666667 5.75 3.1428571 4.086957
x8 5.75 6.25 3.90 5.30 6.30 6.65 6.20 5.15 4.65 4.55 5.70 5.15 5.20 4.70 4.35 3.80
x9 6.361111 7.916667 4.416667 4.861111 5.916667 7.500000 4.861111 3.666667 7.361111 4.361111 4.305556 4.138889 5.861111 4.444444 5.861111 5.138889
6.95 5.166667
• data is complete • the ‘covariance matrix’ contains all information about the interrelations among the observed variables Yves Rosseel
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observed covariance matrix: S • p is the number of observed variables: p = 9 • observed covariance matrix (elements divided by N-1): x1 x2 x3 x4 x5 x6 x7 x8 x9
x1 x2 x3 x4 1.36 0.41 1.38 0.58 0.45 1.28 0.51 0.21 0.21 1.35 0.44 0.21 0.11 1.10 0.46 0.25 0.24 0.90 0.09 -0.10 0.09 0.22 0.26 0.11 0.21 0.13 0.46 0.24 0.37 0.24
x5
x6
1.66 1.01 0.14 0.18 0.30
1.20 0.14 0.17 0.24
x7
x8
x9
1.18 0.54 1.02 0.37 0.46 1.02
• we want to ‘explain’ the observed correlations/covariances by postulating a number of latent variables (factors) and a corresponding factor structure • we will ‘rewrite’ the p(p + 1)/2 = 45 elements in the covariance matrix as a function a smaller number of ‘free parameters’ in the CFA model, summarized in a number of (typically sparse) matrices Yves Rosseel
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the standard CFA model: matrix representation • the classic LISREL representation uses three matrices (for CFA) • the LAMBDA matrix contains the ‘factor structure’: Λ=
x x x 0 0 0 0 0 0
0 0 0 x x x 0 0 0
0 0 0 0 0 0 x x x
• the variances/covariances of the latent variables are summarized in the PSI matrix: Yves Rosseel
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x Ψ= x x
x x
x
• what we can not explain by the set of common factors (the ‘residual part’ of the model) is written in the (typically diagonal) matrix THETA: Θ=
x
x x x x x x x x
• note that we have only 24 parameters (of which 21 are estimable) Yves Rosseel
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the standard CFA model: the model implied covariance matrix • in the standard CFA model, the ‘implied’ covariance matrix is: Σ = ΛΨΛ0 + Θ • all parameters are included in three model matrices • simple matrix multiplication (and addition) gives us the model implied covariance matrix • for identification purposes, some parameters need to be fixed to a constant • estimation problem: choose the ‘free’ parameters, so that the estimated imˆ is ‘as close as possible’ to the observed covariplied covariance matrix (Σ) ance matrix S – generalized (weighted) least-squares estimation (GLS, WLS) – maximum likelihood estimation (ML) – Bayesian approaches Yves Rosseel
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observed covariance matrix x1 x2 x3 x4 x5 x6 x7 x8 x9
x1 x2 x3 x4 x5 x6 x7 x8 x9 1.358 0.407 1.382 0.580 0.451 1.275 0.505 0.209 0.208 1.351 0.441 0.211 0.112 1.098 1.660 0.455 0.248 0.244 0.896 1.015 1.196 0.085 -0.097 0.088 0.220 0.143 0.144 1.183 0.264 0.110 0.212 0.126 0.181 0.165 0.535 1.022 0.458 0.244 0.374 0.243 0.295 0.236 0.373 0.457 1.015
model-implied covariance matrix x1 x2 x3 x4 x5 x6 x7 x8 x9
x1 1.358 0.448 0.590 0.408 0.454 0.378 0.262 0.309 0.284
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x2
x3
x4
x5
x6
x7
x8
1.382 0.327 0.226 0.252 0.209 0.145 0.171 0.157
1.275 0.298 0.331 0.276 0.191 0.226 0.207
1.351 1.090 0.907 0.173 0.205 0.188
1.660 1.010 0.193 0.228 0.209
1.196 0.161 1.183 0.190 0.453 1.022 0.174 0.415 0.490 1.015
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The implied covariance matrix for the full SEM model • in the LISREL representation, we need an additional matrix (B): Σ = Λ(I − B)−1 Ψ(I − B)0−1 Λ0 + Θ where B summarizes the regressions among the latent variables
• we need this extended model for – second-order CFA – MIMIC models – SEM models • in LISREL parlance, this the ‘all-y’ model
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example: Political Democracy • Industrialization and Political Democracy dataset (N=75) • This dataset is used throughout Bollen’s 1989 book (see pages 12, 17, 36 in chapter 2, pages 228 and following in chapter 7, pages 321 and following in chapter 8). • The dataset contains various measures of political democracy and industrialization in developing countries: y1: y2: y3: y4: y5: y6: y7: y8: x1: x2: x3:
Yves Rosseel
Expert ratings of the freedom of the press in 1960 The freedom of political opposition in 1960 The fairness of elections in 1960 The effectiveness of the elected legislature in 1960 Expert ratings of the freedom of the press in 1965 The freedom of political opposition in 1965 The fairness of elections in 1965 The effectiveness of the elected legislature in 1965 The gross national product (GNP) per capita in 1960 The inanimate energy consumption per capita in 1960 The percentage of the labor force in industry in 1960
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model diagram y1
x1
x2
x3
y2 y3
dem60
ind60
y4 y5 y6 dem65 y7 y8
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selection of the output Latent variables: ind60 =˜ x1 x2 x3 dem60 =˜ y1 y2 y3 y4 dem65 =˜ y5 y6 y7 y8 Regressions: dem60 ˜ ind60 dem65 ˜ ind60 dem60 ...
Yves Rosseel
Estimate
Std.err
Z-value
P(>|z|)
Std.lv
Std.all
1.000 2.180 1.819
0.139 0.152
15.742 11.967
0.000 0.000
0.670 1.460 1.218
0.920 0.973 0.872
1.000 1.257 1.058 1.265
0.182 0.151 0.145
6.889 6.987 8.722
0.000 0.000 0.000
2.223 2.794 2.351 2.812
0.850 0.717 0.722 0.846
1.000 1.186 1.280 1.266
0.169 0.160 0.158
7.024 8.002 8.007
0.000 0.000 0.000
2.103 2.493 2.691 2.662
0.808 0.746 0.824 0.828
1.483
0.399
3.715
0.000
0.447
0.447
0.572 0.837
0.221 0.098
2.586 8.514
0.010 0.000
0.182 0.885
0.182 0.885
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Model parameters and model matrices
model parameters > coef(fit) ind60=˜x2 2.180 dem65=˜y7 1.280 y2˜˜y4 1.313 x2˜˜x2 0.120 y5˜˜y5 2.351 dem65˜˜dem65 0.172
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ind60=˜x3 1.819 dem65=˜y8 1.266 y2˜˜y6 2.153 x3˜˜x3 0.467 y6˜˜y6 4.954
dem60=˜y2 1.257 dem60˜ind60 1.483 y3˜˜y7 0.795 y1˜˜y1 1.891 y7˜˜y7 3.431
dem60=˜y3 dem60=˜y4 dem65=˜y6 1.058 1.265 1.186 dem65˜ind60 dem65˜dem60 y1˜˜y5 0.572 0.837 0.624 y4˜˜y8 y6˜˜y8 x1˜˜x1 0.348 1.356 0.082 y2˜˜y2 y3˜˜y3 y4˜˜y4 7.373 5.067 3.148 y8˜˜y8 ind60˜˜ind60 dem60˜˜dem60 3.254 0.448 3.956
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model matrices: free parameters > inspect(fit) $lambda ind60 dem60 dem65 x1 0 0 0 x2 1 0 0 x3 2 0 0 y1 0 0 0 y2 0 3 0 y3 0 4 0 y4 0 5 0 y5 0 0 0 y6 0 0 6 y7 0 0 7 y8 0 0 8 $theta x1 x2 x3 y1 y2 y3 y4 y5 y6 y7 y8 x1 18 x2 0 19 x3 0 0 20 y1 0 0 0 21 y2 0 0 0 0 22 y3 0 0 0 0 0 23 y4 0 0 0 0 13 0 24 y5 0 0 0 12 0 0 0 25 Yves Rosseel
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y6 y7 y8
0 0 0
0 0 0
0 0 0
Ghent University
0 14 0 0 0 0 15 0 0 0 0 16
0 26 0 0 27 0 17 0 28
$psi ind60 dem60 dem65 ind60 29 dem60 0 30 dem65 0 0 31 $beta ind60 dem60 dem65
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ind60 dem60 dem65 0 0 0 9 0 0 10 11 0
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model matrices: estimated values > inspect(fit, "est") $lambda ind60 x1 1.000 x2 2.180 x3 1.819 y1 0.000 y2 0.000 y3 0.000 y4 0.000 y5 0.000 y6 0.000 y7 0.000 y8 0.000
dem60 0.000 0.000 0.000 1.000 1.257 1.058 1.265 0.000 0.000 0.000 0.000
dem65 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 1.186 1.280 1.266
$theta x1 x1 0.082 x2 0.000 x3 0.000 y1 0.000 y2 0.000 y3 0.000 y4 0.000 y5 0.000
x2
x3
y1
y2
0.120 0.000 0.000 0.000 0.000 0.000 0.000
0.467 0.000 0.000 0.000 0.000 0.000
1.891 0.000 0.000 0.000 0.624
7.373 0.000 5.067 1.313 0.000 3.148 0.000 0.000 0.000 2.351
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y3
y4
y5
Structural Equation Modeling with lavaan
y6
y7
y8
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y6 0.000 0.000 0.000 0.000 2.153 0.000 0.000 0.000 4.954 y7 0.000 0.000 0.000 0.000 0.000 0.795 0.000 0.000 0.000 3.431 y8 0.000 0.000 0.000 0.000 0.000 0.000 0.348 0.000 1.356 0.000 3.254 $psi ind60 dem60 dem65 ind60 0.448 dem60 0.000 3.956 dem65 0.000 0.000 0.172 $beta ind60 dem60 dem65 ind60 0.000 0.000 0 dem60 1.483 0.000 0 dem65 0.572 0.837 0
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computing the model-implied covariance matrix (optional) # easy way is: fitted(fit) # manual attach(inspect(fit, "est")) IB
