Robust Structural Equation Modeling with Missing Data and Auxiliary Variables

Abstract

The paper develops a two-stage robust procedure for structural equation modeling (SEM) and an R package rsem to facilitate the use of the procedure by applied researchers. In the first stage, M-estimates of the saturated mean vector and covariance matrix of all variables are obtained. Those corresponding to the substantive variables are then fitted to the structural model in the second stage. A sandwich-type covariance matrix is used to obtain consistent standard errors (SE) of the structural parameter estimates. Rescaled, adjusted as well as corrected and F-statistics are proposed for overall model evaluation. Using R and EQS, the R package rsem combines the two stages and generates all the test statistics and consistent SEs. Following the robust analysis, multiple model fit indices and standardized solutions are provided in the corresponding output of EQS. An example with open/closed book examination data illustrates the proper use of the package. The method is further applied to the analysis of a data set from the National Longitudinal Survey of Youth 1997 cohort, and results show that the developed procedure not only gives a better endorsement of the substantive models but also yields estimates with uniformly smaller standard errors than the normal-distribution-based maximum likelihood.

Appendices

Appendix A. Mathematical Details for Evaluating the Matrix $\hat{\boldsymbol{\Upsilon}}$

This appendix provides the development and formulas for evaluating the $\hat{\boldsymbol{\Upsilon}}$ in (8b) with Huber-type weight. The formulas are programmed in the R package introduced in Section 4.

With the Huber-type weight, w _i3(d _i )=1. Then the estimating equations in (1) and (2) are derived from

$$ g_{i1}(\boldsymbol{\alpha})=w_{i1}(d\boldsymbol{ \nu}_i)'\mathbf {V}_i^{-1}( \mathbf{x}_i-\boldsymbol{\nu}_i), $$

(A.1)

and

$$ g_{i2}(\boldsymbol{\alpha})=\frac{1}{2}\operatorname{tr} \bigl \{ \mathbf{V}_i^{-1}(d\mathbf{V}_i) \mathbf{V}_i^{-1}\bigl[w_{i2}( \mathbf{x}_i-\boldsymbol{\nu}_i) (\mathbf {x}_i-\boldsymbol{\nu}_i)'- \mathbf{V}_i\bigr] \bigr\}, $$

(A.2)

where d is for differentials. It follows from (A.1) and (A.2) that

(A.3)

and

(A.4)

Noting that both w _i1 and w _i2 are function of $d_{i}=[(\mathbf{x}_{i}-\boldsymbol{\nu}_{i})'\mathbf{V}_{i}^{-1}(\mathbf{x}_{i}-\boldsymbol {\nu}_{i})]^{1/2}$, we have

(A.5)

and

(A.6)

Thus, when d _i >ρ _i , we have

(A.7)

and

(A.8)

Notice that, for matrices A, B, C, and D of proper orders, there exists

$$ \operatorname{tr}(\mathbf{A}\mathbf{B}\mathbf{C}\mathbf {D})= \operatorname{vec}'(\mathbf{D}) \bigl(\mathbf{A}\otimes\mathbf {C}'\bigr)\operatorname{vec}\bigl(\mathbf{B}'\bigr)= \operatorname{vec}'\bigl(\mathbf{D}'\bigr) \bigl( \mathbf{C}'\otimes\mathbf{A}\bigr)\operatorname {vec}(\mathbf{B}). $$

(A.9)

Let

$$\mathbf{E}_i=\frac{\partial\boldsymbol{\nu}_i}{\partial \boldsymbol{\nu}'}, \quad \mbox{and} \quad \mathbf{F}_i=\frac{\partial\operatorname{vec}(\mathbf {V}_i)}{\partial\boldsymbol{v}'}. $$

Using (A.9), it follows from (A.3) to (A.8) that, when d _i ≤ρ _i ,

$$\frac{\partial\mathbf{g}_{i1}(\boldsymbol{\alpha})}{\partial \boldsymbol{\nu}'}=-\mathbf{E}_i'\mathbf{V}_i^{-1} \mathbf{E}_i, \qquad \frac{\partial\mathbf{g}_{i1}(\boldsymbol{\alpha})}{\partial \boldsymbol{v}'}=-\mathbf{E}_i' \bigl(\mathbf{V}_i^{-1}\otimes\mathbf{b}_i' \bigr)\mathbf{F}_i, $$

$$\frac{\partial\mathbf{g}_{i2}(\boldsymbol{\alpha})}{\partial \boldsymbol{\nu}'}=-\frac{1}{\kappa_i}\mathbf{F}_i' \bigl(\mathbf{b}_i\otimes\mathbf{V}_i^{-1} \bigr)\mathbf{E}_i, \qquad \frac{\partial\mathbf{g}_{i2}(\boldsymbol{\alpha})}{\partial \boldsymbol{v}'}=- \mathbf{F}_i'\biggl[\frac {1}{\kappa_i}\bigl( \mathbf{H}_i\otimes\mathbf{V}_i^{-1}\bigr)- \mathbf {W}_i\biggr]\mathbf{F}_i; $$

and when d _i >ρ _i ,

Appendix B. R Code for Robust SEM and Its Output

Appendix C. EQS Code for the Model in Equations (12) and (13)

Appendix D. EQS Code for Confirmatory Factor Analysis with Four Variables

Appendix E. EQS Code for the Unconditional Latent Growth Curve Model in Equation (14)

Appendix F. EQS Code for the Conditional Latent Growth Curve Model in Equations (15) and (16)