SEM for those who think they don t care
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- Moris Caldwell
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1 SEM for those who think they don t care Vince Wiggins Vice President, Scientific Development StataCorp LP 2011 London Stata Users Group Meeting V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 1 / 20
2 SEM The short story y = By + Γx + α + ζ Where: y, x, α and ζ are vector y and x may contain both latent and observed variables ζ is a vector of errors Cov(X, ζ) = 0 V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 2 / 20
3 SEM The short story Where: y, x, α and ζ are vector y = By + Γx + α + ζ y and x may contain both latent and observed variables ζ is a vector of errors Cov(X, ζ) = 0 Some interesting things to note: y s can depend on other y s Ignore (mostly) the extensively published rumors that y, x, and/or ζ must be multivariate normal SEM subsumes and extends most linear models. V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 2 / 20
4 SEM I m not going to talk about what most SEMers (SEMians?) use SEM for. V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 3 / 20
5 Path diagrams ε 10 ε 11 ε 12 ε 13 c1 c2 c3 c4 Aptitude C score1 score2 score3 score4 A ε 9 B ε 1 ε 2 ε 3 ε 4 a1 a2 a3 a4 b1 b2 b3 b4 ε 1 ε 2 ε 3 ε 4 ε 5 ε 6 ε 7 ε 8 ε 3 S_e m1 F1 m2 m3 F2 m4 m5 x21 x22 y21 B 1 B y22 ε 4 ε 1 S_e S_e Vi2 S_v Ui ε 1 ε 2 ε 3 ε 4 ε 5 x11 x12 B y11 B 1 y Vi1 S_v ε 2 S_e V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 4 / 20
6 Measurement models/components Aptitude score1 score2 score3 score4 ε 1 ε 2 ε 3 ε 4 V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 5 / 20
7 Multiple factor models (confirmatory or otherwise) F1 F2 m1 m2 m3 m4 m5 ε 1 ε 2 ε 3 ε 4 ε 5 V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 6 / 20
8 Full SEM models ε 10 ε 11 ε 12 ε 13 c1 c2 c3 c4 A C ε 9 B a1 a2 a3 a4 b1 b2 b3 b4 ε 1 ε 2 ε 3 ε 4 ε 5 ε 6 ε 7 ε 8 V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 7 / 20
9 I am also not going to talk about Extensions to linear and multivariate regression Extensions to SURE (including missing values in some y s) MIMIC models Correlations with missing data High-order CFA models Correlated uniqueness models SEM of latent endogenous variables measured by indicators/measurements V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 8 / 20
10 Simultaneous systems and other forms of endogeneity y 1 = β 1 y 2 + β 2 x 1 + β 3 x 2 + ɛ 1 y 2 = β 4 y 1 + β 5 x 1 + β 6 x 3 + ɛ 2 V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 9 / 20
11 Simultaneous systems and other forms of endogeneity y 1 = β 1 y 2 + β 2 x 1 + β 3 x 2 + ɛ 1 y 2 = β 4 y 1 + β 5 x 1 + β 6 x 3 + ɛ 2. reg3 (y1 y2 x1 x2) (y2 y1 x1 x3) V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 9 / 20
12 Simultaneous systems and other forms of endogeneity y 1 = β 1 y 2 + β 2 x 1 + β 3 x 2 + ɛ 1 y 2 = β 4 y 1 + β 5 x 1 + β 6 x 3 + ɛ 2. reg3 (y1 y2 x1 x2) (y2 y1 x1 x3). sem (y1 <- y2 x1 x2) (y2 <- y1 x1 x3), cov(e.y1*e.y1) V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 9 / 20
13 Simultaneous systems and other forms of endogeneity y 1 = β 1 y 2 + β 2 x 1 + β 3 x 2 + ɛ 1 y 2 = β 4 y 1 + β 5 x 1 + β 6 x 3 + ɛ 2. reg3 (y1 y2 x1 x2) (y2 y1 x1 x3). sem (y1 <- y2 x1 x2) (y2 <- y1 x1 x3), cov(e.y1*e.y1) V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 9 / 20
14 Simultaneous systems and other forms of endogeneity y 1 = β 1 y 2 + β 2 x 1 + β 3 x 2 + ɛ 1 y 2 = β 4 y 1 + β 5 x 1 + β 6 x 3 + ɛ 2. reg3 (y1 y2 x1 x2) (y2 y1 x1 x3). sem (y1 <- y2 x1 x2) (y2 <- y1 x1 x3), cov(e.y1*e.y1) SEM extensions control and constrain the structure of the error covariance matrix Obtain SEs, confidence intervals (CIs), etc. that are robust to lack of independence groups of observations option vce(cluster <group>). Handle missing data in the dependent variables, so long as it is missing on observables. Estimate via GMM (generalized method of moments) option method(adf). Estimate direct, indirect, and total effects of all regressors, including the y s estat teffects V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 9 / 20
15 Multilevel random effects models y ijk = βx ijk + µ i + ν ij + ɛ ijk V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 10 / 20
16 Simulating multilevel random effects y ijk = βx ijk + µ i + ν ij + ɛ ijk I = 3, J = 2, K = 2 V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 11 / 20
17 Simulating multilevel random effects y ijk = βx ijk + µ i + ν ij + ɛ ijk I = 3, J = 2, K = 2. set obs 3 V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 11 / 20
18 Simulating multilevel random effects y ijk = βx ijk + µ i + ν ij + ɛ ijk I = 3, J = 2, K = 2. gen i = _n i V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 11 / 20
19 Simulating multilevel random effects y ijk = βx ijk + µ i + ν ij + ɛ ijk I = 3, J = 2, K = 2. gen Ui = rnormal() i Ui 1 µ 1 2 µ 2 3 µ 3 V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 11 / 20
20 Simulating multilevel random effects y ijk = βx ijk + µ i + ν ij + ɛ ijk I = 3, J = 2, K = 2. expand 2 i Ui 1 µ 1 2 µ 2 3 µ 3 1 µ 1 2 µ 2 3 µ 3 V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 11 / 20
21 Simulating multilevel random effects y ijk = βx ijk + µ i + ν ij + ɛ ijk I = 3, J = 2, K = 2. by i, sort: gen j = _n i Ui j 1 µ µ µ µ µ µ 3 2 V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 11 / 20
22 Simulating multilevel random effects y ijk = βx ijk + µ i + ν ij + ɛ ijk I = 3, J = 2, K = 2. gen Vij = rnormal() i Ui j Vij 1 µ 1 1 ν 1 2 µ 2 1 ν 2 3 µ 3 1 ν 3 1 µ 1 2 ν 4 2 µ 2 2 ν 5 3 µ 3 2 ν 6 V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 11 / 20
23 Simulating multilevel random effects y ijk = βx ijk + µ i + ν ij + ɛ ijk I = 3, J = 2, K = 2. expand 2 i Ui j Vij 1 µ 1 1 ν 1 2 µ 2 1 ν 2 3 µ 3 1 ν 3 1 µ 1 2 ν 4 2 µ 2 2 ν 5 3 µ 3 2 ν 6 1 µ 1 1 ν 1 2 µ 2 1 ν 2 3 µ 3 1 ν 3 1 µ 1 2 ν 4 2 µ 2 2 ν 5 3 µ 3 2 ν 6 V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 11 / 20
24 Simulating multilevel random effects y ijk = βx ijk + µ i + ν ij + ɛ ijk I = 3, J = 2, K = 2. by i j, sort: gen k = _n i Ui j Vij k 1 µ 1 1 ν µ 2 1 ν µ 3 1 ν µ 1 2 ν µ 2 2 ν µ 3 2 ν µ 1 1 ν µ 2 1 ν µ 3 1 ν µ 1 2 ν µ 2 2 ν µ 3 2 ν 6 2 V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 11 / 20
25 Simulating multilevel random effects y ijk = βx ijk + µ i + ν ij + ɛ ijk I = 3, J = 2, K = 2. gen Eijk = rnormal() i Ui j Vij k Eijk 1 µ 1 1 ν 1 1 ɛ 1 2 µ 2 1 ν 2 1 ɛ 2 3 µ 3 1 ν 3 1 ɛ 3 1 µ 1 2 ν 4 1 ɛ 4 2 µ 2 2 ν 5 1 ɛ 5 3 µ 3 2 ν 6 1 ɛ 6 1 µ 1 1 ν 1 2 ɛ 7 2 µ 2 1 ν 2 2 ɛ 8 3 µ 3 1 ν 3 2 ɛ 9 1 µ 1 2 ν 4 2 ɛ 10 2 µ 2 2 ν 5 2 ɛ 11 3 µ 3 2 ν 6 2 ɛ 12 V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 11 / 20
26 Simulating multilevel random effects y ijk = βx ijk + µ i + ν ij + ɛ ijk I = 3, J = 2, K = 2. gen x = uniform() i Ui j Vij k Eijk x 1 µ 1 1 ν 1 1 ɛ 1 x 1 2 µ 2 1 ν 2 1 ɛ 2 x 2 3 µ 3 1 ν 3 1 ɛ 3 x 3 1 µ 1 2 ν 4 1 ɛ 4 x 4 2 µ 2 2 ν 5 1 ɛ 5 x 5 3 µ 3 2 ν 6 1 ɛ 6 x 6 1 µ 1 1 ν 1 2 ɛ 7 x 7 2 µ 2 1 ν 2 2 ɛ 8 x 8 3 µ 3 1 ν 3 2 ɛ 9 x 9 1 µ 1 2 ν 4 2 ɛ 10 x 10 2 µ 2 2 ν 5 2 ɛ 11 x 11 3 µ 3 2 ν 6 2 ɛ 12 x 12 V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 11 / 20
27 Simulating multilevel random effects y ijk = βx ijk + µ i + ν ij + ɛ ijk I = 3, J = 2, K = 2. gen y = x + Ui + Vij + Eijk i Ui j Vij k Eijk x y 1 µ 1 1 ν 1 1 ɛ 1 x 1 y 1 2 µ 2 1 ν 2 1 ɛ 2 x 2 y 2 3 µ 3 1 ν 3 1 ɛ 3 x 3 y 3 1 µ 1 2 ν 4 1 ɛ 4 x 4 y 4 2 µ 2 2 ν 5 1 ɛ 5 x 5 y 5 3 µ 3 2 ν 6 1 ɛ 6 x 6 y 6 1 µ 1 1 ν 1 2 ɛ 7 x 7 y 7 2 µ 2 1 ν 2 2 ɛ 8 x 8 y 8 3 µ 3 1 ν 3 2 ɛ 9 x 9 y 9 1 µ 1 2 ν 4 2 ɛ 10 x 10 y 10 2 µ 2 2 ν 5 2 ɛ 11 x 11 y 11 3 µ 3 2 ν 6 2 ɛ 12 x 12 y 12 V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 11 / 20
28 Sorting by groups y ijk = βx ijk + µ i + ν ij + ɛ ijk I = 3, J = 2, K = 2 i Ui j Vij k Eijk x y 1 µ 1 1 ν 1 1 ɛ 1 x 1 y 1 1 µ 1 1 ν 1 2 ɛ 2 x 2 y 2 1 µ 1 2 ν 2 1 ɛ 3 x 3 y 3 1 µ 1 2 ν 2 2 ɛ 4 x 4 y 4 2 µ 2 1 ν 3 1 ɛ 5 x 5 y 5 2 µ 2 1 ν 3 2 ɛ 6 x 6 y 6 2 µ 2 2 ν 4 1 ɛ 7 x 7 y 7 2 µ 2 2 ν 4 2 ɛ 8 x 8 y 8 3 µ 3 1 ν 5 1 ɛ 9 x 9 y 9 3 µ 3 1 ν 5 2 ɛ 10 x 10 y 10 3 µ 3 2 ν 6 1 ɛ 11 x 11 y 11 3 µ 3 2 ν 6 2 ɛ 12 x 12 y 12 V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 12 / 20
29 Reshape 1 y ijk = βx ijk + µ i + ν ij + ɛ ijk I = 3, J = 2, K = 2. egen ij = group(i j). reshape wide eijk y x, i(ij) j(k) k = 1 k = 2 i Ui j Vij Eij1 x1 y1 Eij2 x2 y2 1 µ 1 1 ν 1 ɛ 1 x 1 y 1 ɛ 2 x 2 y 2 1 µ 1 2 ν 2 ɛ 3 x 3 y 3 ɛ 4 x 4 y 4 2 µ 2 1 ν 1 ɛ 5 x 5 y 5 ɛ 6 x 6 y 6 2 µ 2 2 ν 2 ɛ 7 x 7 y 7 ɛ 8 x 8 y 8 3 µ 3 1 ν 1 ɛ 9 x 9 y 9 ɛ 10 x 10 y 10 3 µ 3 2 ν 2 ɛ 11 x 11 y 11 ɛ 12 x 12 y 12 Variable names above are not quite what reshape gives. V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 13 / 20
30 Reshape 2 y ijk = βx ijk + µ i + ν ij + ɛ ijk I = 3, J = 2, K = 2. drop ij. reshape wide eij1 eij2 y1 y1 x1 x2, i(i) j(k) j = 1 j = 2 k = 1 k = 2 k = 1 k = 2 i Ui Vi1 Ei11 x11 y11 Vi1 Ei12 x12 y12 Vi2 Ei21 x21 y21 Vi2 Ei22 x22 y22 1 µ 1 ν 1 ɛ 1 x 1 y 1 ν 1 ɛ 2 x 2 y 2 ν 2 ɛ 3 x 3 y 3 ν 2 ɛ 4 x 4 y 4 2 µ 2 ν 1 ɛ 5 x 5 y 5 ν 1 ɛ 6 x 6 y 6 ν 2 ɛ 7 x 7 y 7 ν 2 ɛ 8 x 8 y 8 3 µ 3 ν 1 ɛ 9 x 9 y 9 ν 1 ɛ 10 x 10 y 10 ν 2 ɛ 11 x 11 y 11 ν 2 ɛ 12 x 12 y 12 V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 14 / 20
31 Reshape 2 y ijk = βx ijk + µ i + ν ij + ɛ ijk I = 3, J = 2, K = 2. drop ij. reshape wide eij1 eij2 y1 y1 x1 x2, i(i) j(k) j = 1 j = 2 k = 1 k = 2 k = 1 k = 2 i Ui Vi1 Ei11 x11 y11 Vi1 Ei12 x12 y12 Vi2 Ei21 x21 y21 Vi2 Ei22 x22 y22 1 µ 1 ν 1 ɛ 1 x 1 y 1 ν 1 ɛ 2 x 2 y 2 ν 2 ɛ 3 x 3 y 3 ν 2 ɛ 4 x 4 y 4 2 µ 2 ν 1 ɛ 5 x 5 y 5 ν 1 ɛ 6 x 6 y 6 ν 2 ɛ 7 x 7 y 7 ν 2 ɛ 8 x 8 y 8 3 µ 3 ν 1 ɛ 9 x 9 y 9 ν 1 ɛ 10 x 10 y 10 ν 2 ɛ 11 x 11 y 11 ν 2 ɛ 12 x 12 y 12 Think of each bounded column set as a linear regression. V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 14 / 20
32 Reshape 2 y ijk = βx ijk + µ i + ν ij + ɛ ijk I = 3, J = 2, K = 2. drop ij. reshape wide eij1 eij2 y1 y1 x1 x2, i(i) j(k) j = 1 j = 2 k = 1 k = 2 k = 1 k = 2 i Ui Vi1 Ei11 x11 y11 Vi1 Ei12 x12 y12 Vi2 Ei21 x21 y21 Vi2 Ei22 x22 y22 1 µ 1 ν 1 ɛ 1 x 1 y 1 ν 1 ɛ 2 x 2 y 2 ν 2 ɛ 3 x 3 y 3 ν 2 ɛ 4 x 4 y 4 2 µ 2 ν 1 ɛ 5 x 5 y 5 ν 1 ɛ 6 x 6 y 6 ν 2 ɛ 7 x 7 y 7 ν 2 ɛ 8 x 8 y 8 3 µ 3 ν 1 ɛ 9 x 9 y 9 ν 1 ɛ 10 x 10 y 10 ν 2 ɛ 11 x 11 y 11 ν 2 ɛ 12 x 12 y 12 Think of each bounded column set as a linear regression. With some creative constraints and a seemingly unrelated regressions structure, this is the estimator for a multilevel random-effects model. V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 14 / 20
33 Path diagram for multilevel RE model ε 3 S_e x21 x22 x11 x12 y21 B 1 B B y22 ε 4 ε 1 y11 y12 S_e S_e B Vi2 S_v Ui Vi1 S_v ε 2 S_e V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 15 / 20
34 Likelihood is identical to xtmixed V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 16 / 20
35 Unbalanced data Different number of observations in some groups? V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 17 / 20
36 Unbalanced data Different number of observations in some groups? No worries? V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 17 / 20
37 Unbalanced data Different number of observations in some groups? No worries? add method(mlmv) Results in the exactly same estimator as xtmixed with unbalanced panels V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 17 / 20
38 Unbalanced y ijk = βx ijk + µ i + ν ij + ɛ ijk I = 3, J = 2, K = 2 k = 1 k = 2 i Ui j Vij Eij1 x1 y1 Eij2 x2 y2 1 µ 1 1 ν 1 ɛ 1 x 1 y 1 ɛ 2 x 2 y 2 1 µ 1 2 ν 2 ɛ 3 x 3 y 3 ɛ 4 x 4 y 4 2 µ 2 1 ν 1 ɛ 5 x 5 y 5 ɛ 6 x 6 y 6 2 µ 2 2 ν 2 ɛ 7 x 7 y 7 ɛ 8 x 8 y 8 3 µ 3 1 ν 1 ɛ 9 x 9 y 9 ɛ 10 x 10 y 10 3 µ 3 2 ν 2 ɛ 11 x 11 y 11 ɛ 12 x 12 y 12 V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 18 / 20
39 I should also mention For the multilevel RE model (and all the other models) SEM supports: robust and cluster-robust SEs estimation by GMM survey data missing data MAR heteroskedastic effects at any level correlated effects at any level V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 19 / 20
40 Uninterested in SEM? So was I. I m interested now. V. Wiggins (StataCorp) SEM for nonbelievers 2011 London UG 20 / 20
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