Multilevel structural equation modeling-based quasi-experimental synthetic cohort design

Abstract

This paper provides a theoretical foundation to examine the effectiveness of post-hoc adjustment approaches such as propensity score matching in reducing the selection bias of synthetic cohort design (SCD) for causal inference and program evaluation. Compared with the Solomon four-group design, the SCD often encounters selection bias due to the imbalance of covariates between the two cohorts. The efficiency of SCD is ensured by the historical equivalence of groups (HEoG) assumption, indicating the comparability between the two cohorts. The multilevel structural equation modeling framework is used to define the HEoG assumption. According to the mathematical proof, HEoG ensures that the use of SCD results in an unbiased estimator of the schooling effect. Practical considerations and suggestions for future research and use of SCD are discussed.

Appendix 1: Variance–covariance decomposition of the extended Solomon four-group design (SFGD) based on two-level SEM framework

1.1 Appendix 1.1: Variance–covariance matrix of SFGD experimental group 1

The following are the variance–covariance matrix of Y and X in the pre- and post-test design—SFGD’s experimental

$$\begin{aligned} \begin{array}{l} {\mathbb{COV}}(Y,X)=\left[ {{\begin{array}{ll} {{\mathbb{V}}(Y)} & \\ {{\mathbb{COV}}(Y,X)} & {\mathbb {V}(X)} \\ \end{array} }} \right] =\left[ {{\begin{array}{llll} {{\mathbb{V}}(Y_0 )} & & & \\ {{\mathbb{COV}}(Y_0 ,Y_1 )} & {{\mathbb{V}}(Y_1 )} & & \\ {{\mathbb{COV}}(Y_0 ,X_0 )} & {{\mathbb{COV}}(Y_1 ,X_0 )} &{{\mathbb{V}}(X_0 )} & \\ {{\mathbb{COV}}(Y_0 ,X_1 )} & {{\mathbb{COV}}(Y_1 ,X_1 )} & {{\mathbb{COV}}(X_0 ,X_1 )} & {{\mathbb{V}}(X_1 )} \\ \end{array} }} \right] \\ =\left[ {{\begin{array}{llll} {{\Lambda _0} (\Pi _0 \Psi _{\xi _0 }^B \Pi '_0 +\Psi _{u_0 }^B ){\Lambda _0} ^{'}+\Theta _{\varepsilon _0 }^B } & & &\\ {{\Lambda _0} (\Pi _0 \Psi _{\xi _0 }^B \Pi '_0 +\Psi _{u_0 }^B ){{\Lambda } }'_1 {\mathcal{V}}'} & {{\Lambda _1} (\Pi _1 {\mathcal{K}} \Psi _{\xi _0 }^B {\mathcal{K}}' \Pi '_1+\Psi _{u_1 }^B ){\Lambda _1} ^{'}+\Theta _{\varepsilon _1 }^B } & & \\ {{\Lambda _0} \Pi _0 \Psi _{\xi _0 }^B {\Gamma }'_0 } & {{\Lambda _1} \Pi _1 {\mathcal{K}} \Psi _{\xi _0 }^W {\Gamma }'_0 } & {\Gamma _0 \Psi _{\xi _0 }^B {\Gamma }'_0 +\Theta _{\zeta _0 }^B } & \\ {{\Lambda _0} \Pi _0 {\mathcal{K}} \Psi _{\xi _0 }^B {\Gamma }'_1 } & {{\Lambda _1} \Pi _1 {\mathcal{K}} \Psi _{\xi _0 }^B {\mathcal{K} }'{\Gamma }'_1 } &{\Gamma _0 \Psi _{\xi _0 }^B {\Gamma }'_1 {\mathcal{K} }'} &{\Gamma _1 {\mathcal{K}} \Psi _{\xi _0 }^B {\mathcal{K} }^{'} {\Gamma }'_1 +\Theta _{\zeta _1 }^B } \\ \end{array} }} \right] \\ +\, \left[ {{\begin{array}{llll} {{\Lambda _0} (\Pi _0 \Psi _{\xi _0 }^W \Pi _0 ^{'}+\Psi _{u_0 }^W ){\Lambda _0} ^{'}+\Theta _{\varepsilon _0 }^W } &&&\\ {{\Lambda _0} (\Pi _0 \Psi _{\xi _0 }^W \Pi'_0+\Psi _{u_0 }^W ){{\Lambda } }'_1 {\mathcal{V}}'} & {{\Lambda _1} (\Pi _1 {\mathcal{K}} \Psi _{\xi _0 }^W {\mathcal{K} }' \Pi _1 ^{'}+\Psi _{u_1 }^W ){\Lambda _1} ^{'}+\Theta _{\varepsilon _1 }^W } &&\\ {{\Lambda _0} \Pi _0 \Psi _{\xi _0 }^W {\Gamma }'_0 } & {{\Lambda _1} \Pi _1 {\mathcal{K}} \Psi _{\xi _0 }^W {\Gamma }'_0 } & {\Gamma _0 \Psi _{\xi _0 }^W {\Gamma }'_0 +\Theta _{\zeta _0 }^W } & \\ {{\Lambda _0} \Pi _0 {\mathcal{K}} \Psi _{\xi _0 }^W {\Gamma }'_1 } & {{\Lambda _1} \Pi _1 {\mathcal{K}} \Psi _{\xi _0 }^W {\mathcal{K}}'{\Gamma }'_1 } &{\Gamma _0 \Psi _{\xi _0 }^W {\Gamma }'_1 {\mathcal{K} }'} & {\Gamma _1 {\mathcal{K}} \Psi _{\xi _0 }^W {\mathcal{K} }' {\Gamma }'_1 +\Theta _{\zeta _1 }^W } \\ \end{array} }} \right] \\ \end{array} \end{aligned}$$

1.2 Appendix 1.2: Variance–covariance matrix of SFGD experimental group 2

$$ \begin{aligned} \begin{array}{ll} {\mathbb{COV}}(Y,X)= & \left[ {{\begin{array}{ll} {\mathbb{V}(Y)} & \\ {\mathbb{COV}(Y,X)} & {\mathbb{V}(X)} \\ \end{array} }} \right] \end{array}\\= & \left[ \begin{array}{llll} {\mathbb{V}(Y_0 )} & & & \\ {\mathbb{COV}(Y_0 ,Y_1 )} & {\mathbb{V}(Y_1 )} & & \\ {\mathbb{COV}(Y_0 ,X_0 )} & {\mathbb{COV}(Y_1 ,X_0 )} & {\mathbb{V}(X_0 )} & \\ {\mathbb{COV}(Y_0 ,X_1 )} & {\mathbb{COV}(Y_1 ,X_1 )} & {\mathbb{COV}(X_0 ,X_1 )} & {\mathbb{V}(X_1 )} \\ \end{array} \right]\\ = & \left[ {{\begin{array}{llll} {{\Lambda _0} (\Pi _0 \Psi _{\xi _0 }^B \Pi _0 ^{'}+\Psi _{u_0 }^B ){\Lambda _0} ^{'}+\Theta _{\varepsilon _0 }^B } & & & \\ {{\Lambda _0} (\Pi _0 \Psi _{\xi _0 }^B \Pi _0 ^{'}+\Psi _{u_0 }^B ){{\Lambda } }'_1 } & {{\Lambda _1} (\Pi _1 {\mathcal {K}} \Psi _{\xi _0 }^B {{\mathcal {K}} }' \Pi _1 ^{'}+\Psi _{u_1 }^B ){\Lambda _1} ^{'}+\Theta _{\varepsilon _1 }^B } & & \\ {{\Lambda _0} \Pi _0 \Psi _{\xi _0 }^B {\Gamma }'_0 } & {{\Lambda _1} \Pi _1 {\mathcal {K}} \Psi _{\xi _0 }^W {\Gamma }'_0 } & {\Gamma _0 \Psi _{\xi _0 }^B {\Gamma }'_0 +\Theta _{\zeta _0 }^B } & \\ {{\Lambda _0} \Pi _0 {\mathcal {K}} \Psi _{\xi _0 }^B {\Gamma }'_1 } & {{\Lambda _1} \Pi _1 {\mathcal {K}} \Psi _{\xi _0 }^B {{\mathcal {K}} }'{\Gamma }'_1 } & {\Gamma _0 \Psi _{\xi _0 }^B {\Gamma }'_1 {{\mathcal {K}} }'} & {\Gamma _1 {\mathcal {K}} \Psi _{\xi _0 }^B {{\mathcal {K}} }' {\Gamma }'_1 +\Theta _{\zeta _1 }^B } \\ \end{array} }} \right] \\&+ \left[ {{\begin{array}{llll} {{\Lambda _0} (\Pi _0 \Psi _{\xi _0 }^W \Pi _0 ^{'}+\Psi _{u_0 }^W ){\Lambda _0} ^{'}+\Theta _{\varepsilon _0 }^W } & & & \\ {{\Lambda _0} (\Pi _0 \Psi _{\xi _0 }^W \Pi _0 ^{'}+\Psi _{u_0 }^W ){{\Lambda } }'_1 } & {{\Lambda _1} (\Pi _1 {\mathcal {K}} \Psi _{\xi _0 }^W {{\mathcal {K}} }' \Pi _1 ^{'}+\Psi _{u_1 }^W ){\Lambda _1} ^{'}+\Theta _{\varepsilon _1 }^W } & & \\ {{\Lambda _0} \Pi _0 \Psi _{\xi _0 }^W {\Gamma }'_0 } & {{\Lambda _1} \Pi _1 {\mathcal {K}} \Psi _{\xi _0 }^W {\Gamma }'_0 } & {\Gamma _0 \Psi _{\xi _0 }^W {\Gamma }'_0 +\Theta _{\zeta _0 }^W } & \\ {{\Lambda _0} \Pi _0 {\mathcal {K}} \Psi _{\xi _0 }^W {\Gamma }'_1 } & {{\Lambda _1} \Pi _1 {\mathcal {K}} \Psi _{\xi _0 }^W {{\mathcal {K}} }'{\Gamma }'_1 } & {\Gamma _0 \Psi _{\xi _0 }^W {\Gamma }'_1 {{\mathcal {K}} }'} & {\Gamma _1 {\mathcal {K}} \Psi _{\xi _0 }^W {{\mathcal {K}} }' {\Gamma }'_1 +\Theta _{\zeta _1 }^W } \\ \end{array} }} \right] \\ \end{aligned}$$

1.3 Appendix 1.3: Variance–covariance matrix of SFGD control group 1

$$\begin{aligned} \mathbb {COV}(Y,X)= & {} \left[ {{\begin{array}{ll} {\mathbb {V}(Y)} &{} \\ {\mathbb {COV}(Y,X)} &{} {\mathbb {V}(X)} \\ \end{array} }} \right] \\= & {} \left[ {{\begin{array}{llll} &{} &{} &{} \\ &{} {\mathbb {V}(Y_1 )} &{} &{} \\ &{} &{} &{} \\ &{} {\mathbb {COV}(Y_1 ,X_1 )} &{} &{} {\mathbb {V}(X_1 )} \\ \end{array} }} \right] \\= & {} \left[ {{\begin{array}{llll} &{} &{} &{} \\ &{} {{\Lambda _1} (\Pi _1 \Psi _{\xi _1 }^B \Pi _1 ^{'}+\Psi _{u_1 }^B ){\Lambda _1} ^{'}+\Theta _{\varepsilon _1 }^B } &{} &{} \\ &{} &{} &{} \\ &{} {{\Lambda _1} \Pi _1 \Psi _{\xi _1 }^B {\Gamma }'_1 } &{} &{} {\Gamma _1 \Psi _{\xi _1 }^B {\Gamma }'_1 +\Theta _{\zeta _1 }^B } \\ \end{array} }} \right] \\\\&+ \left[ {{\begin{array}{llll} &{} &{} &{} \\ &{} {{\Lambda _1} (\Pi _1 \Psi _{\xi _1 }^W \Pi _1 ^{'}+\Psi _{u_1 }^W ){\Lambda _1} ^{'}+\Theta _{\varepsilon _1 }^W } &{} &{} \\ &{} &{} &{} \\ &{} {{\Lambda _1} \Pi _1 \Psi _{\xi _1 }^W {\Gamma }'_1 } &{} &{} {\Gamma _1 \Psi _{\xi _1 }^W {\Gamma }'_1 +\Theta _{\zeta _1 }^W } \\ \end{array} }} \right] \\ \end{aligned}$$

1.4 Appendix 1.4: Variance–covariance matrix of SFGD control group 2

$$\begin{aligned} \mathbb {COV}(Y,X)= & {} \left[ {{\begin{array}{ll} {\mathbb {V}(Y)} &{} \\ {\mathbb {COV}(Y,X)} &{} {\mathbb {V}(X)} \\ \end{array} }} \right] \\= & {} \left[ {{\begin{array}{llll} &{} &{} &{} \\ &{} {\mathbb {V}(Y_1 )} &{} &{} \\ &{} &{} &{} \\ &{} {\mathbb {COV}(Y_1 ,X_1 )} &{} &{} {\mathbb {V}(X_1 )} \\ \end{array} }} \right] \\= & {} \left[ {{\begin{array}{llll} &{} &{} &{} \\ &{} {{\Lambda _1} (\Pi _1 \Psi _{\xi _1 }^B \Pi _1 ^{'}+\Psi _{u_1 }^B ){\Lambda _1} ^{'}+\Theta _{\varepsilon _1 }^B } &{} &{} \\ &{} &{} &{} \\ &{} {{\Lambda _1} \Pi _1 \Psi _{\xi _1 }^B {\Gamma }'_1 } &{} &{} {\Gamma _1 \Psi _{\xi _1 }^B {\Gamma }'_1 +\Theta _{\zeta _1 }^B } \\ \end{array} }} \right] \\&+ \left[ {{\begin{array}{llll} &{} &{} &{} \\ &{} {{\Lambda _1} (\Pi _1 \Psi _{\xi _1 }^W \Pi _1 ^{'}+\Psi _{u_1 }^W ){\Lambda _1} ^{'}+\Theta _{\varepsilon _1 }^W } &{} &{} \\ &{} &{} &{} \\ &{} {{\Lambda _1} \Pi _1 \Psi _{\xi _1 }^W {\Gamma }'_1 } &{} &{} {\Gamma _1 \Psi _{\xi _1 }^W {\Gamma }'_1 +\Theta _{\zeta _1 }^W } \\ \end{array} }} \right] \\ \end{aligned}$$

1.5 Appendix 1.5: Detailed variance–covariance decomposition

This appendix discusses the detailed variance–covariance decomposition of the SFGD’s experimental group 1. The measurement model is defined in Eq. (14), where $\varepsilon \sim N(0,\Theta _{\varepsilon })$. $\varepsilon$ is independent of $\eta$, $\xi$ and $\zeta$. $\zeta \sim N(0,\Theta _{\zeta })$ is independent of $\eta$, $\xi$ and $\varepsilon$. The latent variable $\xi$ is hierarchically structured and includes both within-cluster and between-cluster components. The latent variable $\eta$ of Y is hierarchically structured (Muthén 1994, p. 379). This is because $\eta$ has a functional relationship with $\xi$ in the structural model.

If a is the intercept vector and $\Pi$ the loading matrix, then the structural model (Jöreskog and Sörbom 1996) is written as Eq. (15), where $u\sim N(0,\Theta _u)$ is independent of $\xi$, $\varepsilon$ and $\zeta$. The independence assumption will be used in the computation of the covariance of Y and X. This model is a two-level factor analysis model (Muthén and Muthén 1998–2012). Correspondingly, based on the SEM framework, data variation can be decomposed into within- and between-cluster components (Muthén 1994, p. 380).

1.5.1 Appendix 1.5.1: Decomposition of variation of X

The variation of the latent variable $\xi$ can be decomposed as

$$\begin{aligned} \mathbb {V}(\xi )=\Psi _\xi =\Psi _\xi ^B +\Psi _\xi ^W. \end{aligned}$$

(30)

The variation of X’s residual can be decomposed into between- and within-cluster components. That is,

$$\begin{aligned} \mathbb {V}(\zeta )=\Theta _{\zeta }^B +\Theta _{\zeta }^W. \end{aligned}$$

(31)

The variation of outcome X is decomposed as

$$\begin{aligned} \mathbb {V}(X)=\Phi _X^B +\Phi _X^W, \end{aligned}$$

(32)

with

$$\begin{aligned} \Phi _X^B =\Gamma \Psi _\xi ^B \Gamma ^{'}+\Theta _{\zeta }^B, \end{aligned}$$

(33)

and

$$\begin{aligned} \Phi _X^W =\Gamma \Psi _\xi ^W \Gamma ^{'}+\Theta _{\zeta }^W. \end{aligned}$$

(34)

1.5.2 Appendix 1.5.2: Decomposition of latent variable $\eta$

The variation of $\eta$ can be decomposed using the structural model. First, the residual variance is decomposed as

$$\begin{aligned} {\mathbb {V}(u)=\Theta _u^B +\Theta _u^W}. \end{aligned}$$

(35)

The variation of $\eta$ is decomposed as

$$\begin{aligned} \mathbb {V}(\eta )=\Psi _\eta =\Psi _\eta ^B +\Psi _\eta ^W, \end{aligned}$$

(36)

with

$$\begin{aligned} \Psi _\eta ^B =\Pi \Psi _\xi ^B \Pi ^{'}+{\Psi _u^B}, \end{aligned}$$

(37)

and

$$\begin{aligned} \Psi _\eta ^W =\Pi \Psi _\xi ^W \Pi ^{'}+{\Psi _u^W}. \end{aligned}$$

(38)

1.5.3 Appendix 1.5.3: Decomposition of variation of Y

The variation of Y’s residual can also be decomposed into between- and within-cluster components,

$$\begin{aligned} \mathbb {V}(\varepsilon )=\Theta _{\varepsilon }^B +\Theta _{\varepsilon }^W. \end{aligned}$$

(39)

Now the variation of variable Y is decomposed as

$$\begin{aligned} \mathbb {V}(Y)=\Phi _Y^B +\Phi _Y^W, \end{aligned}$$

(40)

with

$$\begin{aligned} \Phi _Y^B ={\Lambda } \Psi _\eta ^B {\Lambda } ^{'}+\Theta _{\varepsilon }^B, \end{aligned}$$

(41)

and

$$\begin{aligned} \Phi _Y^W ={\Lambda } \Psi _\eta ^W {\Lambda } ^{'}+\Theta _{\varepsilon }^W. \end{aligned}$$

(42)

1.5.4 Appendix 1.5.4: Covariance of Y and X

Based on independence assumptions in the measurement and structural models above, the covariance of Y and X is computed as:

$$\begin{aligned} {\rm COV}[Y,X]= & {} {\rm COV}[{\Lambda }{(a+\Pi \xi +u)}+\varepsilon ,v+\Gamma \xi +\zeta ] \nonumber \\= & {} {\Lambda \Pi \Gamma }Var(\xi )\nonumber \\= & {} {{\Lambda }\Pi \Gamma }\Psi _\xi ^B+{{\Lambda } \Pi \Gamma }\Psi _\xi ^W. \end{aligned}$$

(43)

Thus, the whole data variance–covariance matrix is shown below:

$$\begin{aligned} \mathbb {COV}(Y,X)= & {} \left[ {{\begin{array}{ll} {\mathbb {V}(Y)} &{} \\ {{\rm COV}[Y,X]} &{} {\mathbb {V}(X)} \\ \end{array} }} \right] =\left[ {{\begin{array}{ll} {\Phi _Y^B +\Phi _Y^W } &{} \\ {{{\Lambda } \Pi \Gamma }\Psi _\xi ^B +{{\Lambda } \Pi \Gamma }\Psi _\xi ^W } &{} {\Phi _X^B +\Phi _X^W } \\ \end{array} }} \right] \\= & {} \left[ {{\begin{array}{ll} {\Phi _Y^B } &{} \\ {{{\Lambda } \Pi \Gamma }\Psi _\xi ^B } &{} {\Phi _X^B } \\ \end{array} }} \right] +\left[ {{\begin{array}{ll} {\Phi _Y^W } &{} \\ {{\Lambda \Pi \Gamma }\Psi _\xi ^W } &{} {\Phi _X^W } \\ \end{array} }} \right] \\= & {} \left[ {{\begin{array}{ll} {{\Lambda } \Psi _\eta ^B {\Lambda } ^{'}+\Theta _{\varepsilon }^B } &{} \\ {{\Lambda \Pi \Gamma }\Psi _\xi ^B } &{} {\Gamma \Psi _\xi ^B \Gamma ^{'}+\Theta _{\zeta }^B } \\ \end{array} }} \right] +\left[ {{\begin{array}{ll} {{\Lambda } \Psi _\eta ^W {\Lambda } ^{'}+\Theta _{\varepsilon }^W } &{} \\ {{\Lambda \Pi \Gamma }\Psi _\xi ^W } &{} {\Gamma \Psi _\xi ^W \Gamma ^{'}+\Theta _{\zeta }^W } \\ \end{array} }} \right] \\= & {} \left[ {{\begin{array}{ll} {{{\Lambda } (\Pi }\Psi _\xi ^B {\Pi ^{'}}+\Psi _E^B ){\Lambda } ^{'}+\Theta _{\varepsilon }^B } &{} \\ {{\Lambda \Pi \Gamma }\Psi _\xi ^B } &{} {\Gamma \Psi _\xi ^B \Gamma ^{'}+\Theta _{\zeta }^B } \\ \end{array} }} \right] \\&+ \left[ {{\begin{array}{ll} {{{\Lambda } (\Pi }\Psi _\xi ^W {\Pi ^{'}}+\Psi _E^W ){\Lambda } ^{'}+\Theta _{\varepsilon }^W } &{} \\ {{\Lambda \Pi \Gamma }\Psi _\xi ^W } &{} {\Gamma \Psi _\xi ^W \Gamma ^{'}+\Theta _{\zeta }^W } \\ \end{array} }} \right] \end{aligned}$$

1.5.5 Appendix 1.5.5: Variance–covariance decomposition across times

This is the temporal decomposition of variance–covariance in Appendix 1.1.

Note the measurement model in Eq. (14), $\varepsilon \sim N(0,\Theta _{\varepsilon })$. $\varepsilon$ is independent of $\eta$, $\xi$ and $\zeta$. $\zeta \sim N(0,\Theta _{\zeta })$ is independent of $\eta$, $\xi$ and $\varepsilon$. Rewrite each component in the model into two parts. One part represents the measure collected at time 0 and the other in time 1. For the first equation, $Y=\left( {{\begin{array}{c} {Y_0 }\\ {Y_1 }\\ \end{array} }} \right)$, $\delta =\left( {{\begin{array}{c} {\delta _0 }\\ {\delta _1 }\\ \end{array} }} \right)$, ${\Lambda } =\left( {{\begin{array}{c} {{\begin{array}{c} {{\Lambda _0} } \\ 0 \\ \end{array} }} \quad {{\begin{array}{c} 0 \\ {{\Lambda _1} }\\ \end{array} }} \\ \end{array} }} \right)$, $\eta =\left( {{\begin{array}{c} {\eta _0 } \\ {\eta _1 } \\ \end{array} }} \right)$, $\varepsilon =\left( {{\begin{array}{c} {\varepsilon _0 }\\ {\varepsilon _1 }\\ \end{array} }} \right)$. Variation of variable $Y_t$ is decomposed as

$$\begin{aligned} {\mathbb {V}}(Y_t)=\Phi _{Y_t }^B +\Phi _{Y_t }^W, \end{aligned}$$

(44)

with

$$\begin{aligned} \Phi _{Y_t }^B ={\Lambda }_t \Psi _{\eta _t }^B {\Lambda }_t ^{'}+\Theta _{\varepsilon _{t} }^B, \end{aligned}$$

(45)

and

$$\begin{aligned} \Phi _{Y_t}^W ={\Lambda }_t \Psi _{\eta _t}^W {\Lambda }_t^{'}+\Theta _{\varepsilon _{t} }^W, \end{aligned}$$

(46)

for t = 0, 1.

Similarly, write

$$\begin{aligned} X=\left( {{\begin{array}{c} {X_0 } \\ {X_1 } \\ \end{array} }} \right) , {v}=\left( {{\begin{array}{c} {v_0 } \\ {v_1 } \\ \end{array} }} \right) , \Gamma =\left( {{\begin{array}{c} {{\begin{array}{*{20}c} {\Gamma _0 } \\ 0 \\ \end{array} }} \quad {{\begin{array}{c} 0 \\ {\Gamma _1 } \\ \end{array} }} \\ \end{array} }} \right) , \xi =\left( {{\begin{array}{c} {\xi _0 } \\ {\xi _1 } \\ \end{array} }} \right) , \zeta =\left( {{\begin{array}{c} {\zeta _0 } \\ {\zeta _1 } \\ \end{array} }} \right) . \end{aligned}$$

Correspondingly, the variation of outcome $X_{t}$ is decomposed as

$$\begin{aligned} {\mathbb {V}}(X_t)=\Phi _{X_t }^B +\Phi _{X_t }^W, \end{aligned}$$

(47)

with

$$\begin{aligned} \Phi _{X_t }^B =\Gamma _t \Psi _{\xi _t }^B \Gamma _t^{'}+\Theta _{e_{2i}}^B, \end{aligned}$$

(48)

and

$$\begin{aligned} \Phi _{X_t }^W =\Gamma _t \Psi _{\xi _t }^W \Gamma _t ^{'}+\Theta _{e_{2i} }^W, \end{aligned}$$

(49)

for t = 0,1.

The latent variables $\xi _0$, $\xi _1$, $\eta _0$, and $\eta _1$ are hierarchically structured and include within-cluster and between-cluster components (Muthén 1994, p. 379). This is because $\eta$ has a functional relationship with $\xi$ in the structural model of Eq. (20), where $U\sim N(0,\Theta _U)$ is independent of $\xi$, $\varepsilon$ and $\zeta$. Thus, it results in

$$\begin{aligned} {\mathbb {V}}(\eta _0 )=\Psi _{\eta _0 } =\Psi _{\eta _0 }^B +\Psi _{\eta _0 }^W, \end{aligned}$$

(50)

with

$$\begin{aligned} \Psi _{\eta _0 }^B =\Pi _0 \Psi _{\xi _0 }^B \Pi _0 ^{'}+\Psi _{u_0 }^B, \end{aligned}$$

(51)

and

$$\begin{aligned} \Psi _{\eta _0 }^W =\Pi _0 \Psi _{\xi _0 }^W \Pi _0 ^{'}+\Psi _{u_0 }^W. \end{aligned}$$

(52)

Now, write the variance–covariance matrix as

$$\begin{aligned} {\mathbb {COV}}(Y,X)= & {} \left[ {{\begin{array}{ll} {\mathbb {V}(Y)} &{} \\ {\mathbb {COV}(Y,X)} &{} {\mathbb {V}}(X) \\ \end{array}}} \right] \\= & {} \left[ {{\begin{array}{llll} {\mathbb {V}}(Y_0 ) &{} &{} &{} \\ {\mathbb {COV}(Y_0 ,Y_1 )} &{} {\mathbb {V}}(Y_1 ) &{} &{} \\ {\mathbb {COV}}(Y_0 ,X_0 ) &{} {\mathbb {COV}}(Y_1 ,X_0 ) &{} {\mathbb {V}}(X_0 ) &{} \\ {\mathbb {COV}}(Y_0 ,X_1 ) &{} {\mathbb {COV}}(Y_1 ,X_1 ) &{} {\mathbb {COV}}(X_0 ,X_1 ) &{} {\mathbb {V}}(X_1 ) \\ \end{array} }}\right]\, . \end{aligned}$$

1.5.6 Appendix 1.5.6: Compute ${\mathbb {V}}(Y)$

In order to determine ${\mathbb {COV}}(Y_0, Y_1)$, the structural relationship between $\eta _1$, and $\eta _1$ is displayed by Eq. (17), where slope ${\mathcal {V}}$ represents schooling effect and $\gamma$ represents maturation effect; $\tau$ represents the pre-test effect at time 1 (Solomon 1949).

Thus, ${\mathbb {COV}}(Y_0 ,Y_1 )={\mathbb {COV}}[\delta _1 +{\Lambda _0} \eta _0 +\varepsilon _0,\delta _2 +{\Lambda _1} \eta _1 +\varepsilon _1 )= {\mathbb {COV}}[{\Lambda _0} \eta _0 ,{\Lambda _1} (\tau +\gamma +{\mathcal {V}} \eta _0 )]={\Lambda _0} {\mathbb {V}}(\eta _0 ){{\Lambda } }'_1 {\mathcal {V} }',$ with ${\mathbb {COV}}[\varepsilon _0 ,\varepsilon _1 )=0$.

Plug in those components and write

$$\begin{aligned} {\mathbb {V}}(Y)= & {} {\mathbb {COV}}(Y,Y)\\ {}= & {} \left[ {{\begin{array}{ll} {\mathbb {V}}(Y_0 ) &{} \\ {\mathbb {COV}}(Y_0 ,Y_1 ) &{} {\mathbb {V}}(Y_1 ) \\ \end{array} }} \right] \\= & {} \left[ {{\begin{array}{ll} {\Phi _{Y_0 }^B +\Phi _{Y_0 }^W } &{} \\ {\mathbb {COV}}(Y_0 ,Y_1 ) &{} {\Phi _{Y_1 }^B +\Phi _{Y_1 }^W } \\ \end{array} }} \right] \\= & {} \left[ {{\begin{array}{lll} {{\Lambda _0} \Psi _{\eta _0 }^B {\Lambda _0} ^{'}+\Theta _{\varepsilon _0 }^B } &{} \\ {{\Lambda _0} \Psi _{\eta _0 }^B {{\Lambda } }'_1 {\mathcal {V}}'} &{} {{\Lambda _1} \Psi _{\eta _1 }^B {\Lambda _1} ^{'}+\Theta _{\varepsilon _1 }^B } \\ \end{array} }} \right] \\&+ \left[ {{\begin{array}{ll} {{\Lambda _0} \Psi _{\eta _0 }^W {\Lambda _0} ^{'}+\Theta _{\varepsilon _0 }^W } &{} \\ {{\Lambda _0} \Psi _{\eta _0 }^W {{\Lambda } }'_1 {\mathcal {V}}'} &{} {{\Lambda _1} \Psi _{\eta _1 }^W {\Lambda _1} ^{'}+\Theta _{\varepsilon _1 }^W } \\ \end{array} }} \right] \\= & {} \left[ {{\begin{array}{ll} {{\Lambda _0} (\Pi _0 \Psi _{\xi _0 }^B \Pi _0 ^{'}+\Psi _{u_0 }^B ){\Lambda _0} ^{'}+\Theta _{\varepsilon _0 }^B } &{} \\ {{\Lambda _0} (\Pi _0 \Psi _{\xi _0 }^B \Pi _0 ^{'}+\Psi _{u_0 }^B ){{\Lambda } }'_1 {\mathcal {V} }' } &{} {{\Lambda _1} (\Pi _1 \Psi _{\xi _1 }^B \Pi _1 ^{'}+\Psi _{u_1 }^B ){\Lambda _1} ^{'}+\Theta _{\varepsilon _1 }^B } \\ \end{array} }} \right] \\&+ \left[ {{\begin{array}{ll} {{\Lambda _0} (\Pi _0 \Psi _{\xi _0 }^W \Pi _0 ^{'}+\Psi _{u_0 }^W ){\Lambda _0} ^{'}+\Theta _{\varepsilon _0 }^W } &{} \\ {{\Lambda _0} (\Pi _0 \Psi _{\xi _0 }^W \Pi _0 ^{'}+\Psi _{u_0 }^W ){{\Lambda } }'_1 {{\mathcal {V}} }'} &{} {{\Lambda _1} (\Pi _1 \Psi _{\xi _1 }^W \Pi _1 ^{'}+\Psi _{u_1 }^W ){\Lambda _1} ^{'}+\Theta _{\varepsilon _1 }^W } \\ \end{array} }} \right] \\= & {} \left[ {{\begin{array}{ll} {{\Lambda _0} (\Pi _0 \Psi _{\xi _0 }^B \Pi _0 ^{'}+\Psi _{u_0 }^B ){\Lambda _0} ^{'}+\Theta _{\varepsilon _0 }^B } &{} \\ {{\Lambda _0} (\Pi _0 \Psi _{\xi _0 }^B \Pi _0 ^{'}+\Psi _{u_0 }^B ){{\Lambda } }'_1 {{\mathcal {V}} }' } &{} {{\Lambda _1} (\Pi _1 {\mathcal {K}} \Psi _{\xi _0 }^B {{\mathcal {K}} }'\Pi _1 ^{'}+\Psi _{u_1 }^B ){\Lambda _1} ^{'}+\Theta _{\varepsilon _1 }^B } \\ \end{array} }} \right] \\&+ \left[ {{\begin{array}{ll} {{\Lambda _0} (\Pi _0 \Psi _{\xi _0 }^W \Pi _0 ^{'}+\Psi _{u_0 }^W ){\Lambda _0} ^{'}+\Theta _{\varepsilon _0 }^W } &{} \\ {{\Lambda _0} (\Pi _0 \Psi _{\xi _0 }^W \Pi _0 ^{'}+\Psi _{u_0 }^W ){{\Lambda } }'_1 {{\mathcal {V}} }' } &{} {{\Lambda _1} (\Pi _1 {\mathcal {K}} \Psi _{\xi _0 }^W {{\mathcal {K}} }'\Pi _1 ^{'}+\Psi _{u_1 }^W ){\Lambda _1} ^{'}+\Theta _{\varepsilon _1 }^W } \\ \end{array} }} \right] \\ \end{aligned}$$

1.5.7 Appendix 1.5.7: Compute ${\mathbb {V}}(X)$

In order to determine ${\mathbb {COV}}(X_0 ,X_1 )$, the structural relationship between $\xi _0$, and $\xi _1$ is displayed by Eq. (19).

Thus, ${\mathbb {COV}}(X_0 ,X_1 )={\mathbb {COV}}[v_0 +\Gamma _0 \xi _0 +\zeta _0 ,v_1 +\Gamma _1 \xi _1 +\zeta _1 )={\mathbb {COV}}[\Gamma _0 \xi _0 ,\Gamma _1 (\beta +{\mathcal {K}} \xi _0 )]=\Gamma _0 {\mathbb {V}}(\xi _0 ){\Gamma }'_1 {{{\mathcal {K}}} }',$ with ${\mathbb {COV}}[\zeta _0 ,\zeta _1 )=0$, and

$$\begin{aligned} {\mathbb {V}}(\xi _0 )=\Psi _{\xi _0 }=\Psi _{\xi _0 }^B +\Psi _{\xi _0 }^W. \end{aligned}$$

(53)

Thus,

$$\begin{aligned} {\mathbb {V}}(X)= & {} {\mathbb {COV}}(X,X)\\= & {} \left[ {{\begin{array}{ll} {\mathbb {V}}(X_0 ) &{} \\ {\mathbb {COV}}(X_0 ,X_1 ) &{} {\mathbb {V}}(X_1) \\ \end{array} }} \right] \\= & {} \left[ {{\begin{array}{ll} {\Phi _{X_0 }^B +\Phi _{X_0 }^W } &{} \\ {\mathbb {COV}}(X_0 ,X_1 ) &{} {\Phi _{X_1 }^B +\Phi _{X_1 }^W } \\ \end{array} }} \right] \\= & {} \left[ {{\begin{array}{ll} {\Gamma _0 \Psi _{\xi _0 }^B {\Gamma }'_0 +\Theta _{\zeta _0 }^B } &{} \\ {\Gamma _0 \Psi _{\xi _0 }^B {\Gamma }'_1 {{\mathcal {K}} }'} &{} {\Gamma _1 \Psi _{\xi _1 }^B {\Gamma }'_1 +\Theta _{\zeta _1 }^B } \\ \end{array} }} \right] \\&+\left[ {{\begin{array}{*{20}c} {\Gamma _0 \Psi _{\xi _0 }^W {\Gamma }'_0 +\Theta _{\zeta _0 }^W } &{} \\ {\Gamma _0 \Psi _{\xi _0 }^W {\Gamma }'_1 {{\mathcal {K}} }'} &{} {\Gamma _1 \Psi _{\xi _1 }^W {\Gamma }'_1 +\Theta _{\zeta _1 }^W } \\ \end{array} }} \right] \\&=\left[ {{\begin{array}{ll} {\Gamma _0 \Psi _{\xi _0 }^B {\Gamma }'_0 +\Theta _{\zeta _0 }^B } &{} \\ {\Gamma _0 \Psi _{\xi _0 }^B {\Gamma }'_1 {{\mathcal {K}} }'} &{} {\Gamma _1 {\mathcal {K}} \Psi _{\xi _0 }^B {{\mathcal {K}} }'{\Gamma }'_1 +\Theta _{\zeta _1 }^B } \\ \end{array} }} \right] \\&+\left[ {{\begin{array}{ll} {\Gamma _0 \Psi _{\xi _0 }^W {\Gamma }'_0 +\Theta _{\zeta _0 }^W } &{} \\ {\Gamma _0 \Psi _{\xi _0 }^W {\Gamma }'_1 {{\mathcal {K}} }'} &{} {\Gamma _1 {\mathcal {K}} \Psi _{\xi _0 }^W {{\mathcal {K}} }'{\Gamma }'_1 +\Theta _{\zeta _1 }^W } \\ \end{array} }} \right] . \end{aligned}$$

1.5.8 Appendix 1.5.8: Compute ${\mathbb {COV}}(Y_t,X_{t^{'}})$

Components of ${\mathbb {COV}}(Y_t ,X_{t^{'}})$—for $t,t^{'} = 0,1$—are computed as

$$\begin{aligned} {\mathbb {COV}}(Y_t ,X_{t^{'}} )={\mathbb {COV}}[{\Lambda } _t (a_t +\Pi _t \xi _t +u_t )+\varepsilon _{t} ,v_t^{'} +\Gamma _t^{'} \xi _t^{'} +\zeta _{t^{'}} ]={\Lambda }_t \Pi _t{\mathbb {COV}}(\xi _t ,\xi _t^{'} ){\Gamma }'_{t^{'}}. \end{aligned}$$

(54)

Thus, the four components are computed as

$$\begin{aligned} {\mathbb {COV}}(Y_0 ,X_0 )={\Lambda _0} \Pi _0 (\Psi _{\xi _0 }^B +\Psi _{\xi _0 }^W ){\Gamma }'_0, \end{aligned}$$

(55)

$$\begin{aligned} {\mathbb {COV}}(Y_0 ,X_1 )={\Lambda _0} \Pi _0 {\mathcal {K}} (\Psi _{\xi _0 }^B +\Psi _{\xi _0 }^W ){\Gamma }'_1, \end{aligned}$$

(56)

$$\begin{aligned} {\mathbb {COV}}(Y_1 ,X_0 )={\Lambda _1} \Pi _1 {\mathcal {K}} (\Psi _{\xi _0 }^B +\Psi _{\xi _0 }^W ){\Gamma }'_0, \end{aligned}$$

(57)

and

$$\begin{aligned} {\mathbb {COV}}(Y_1 ,X_1 )={\Lambda _1} \Pi _1 {\mathcal {K}} (\Psi _{\xi _0 }^B +\Psi _{\xi _0 }^W ){{\mathcal {K}} }'{\Gamma }'_1. \end{aligned}$$

(58)

These procedures derive all components displayed in the matrices of Appendix 1.1. Other variance–covariance matrices in Appendices 1.2–1.4 can be derived with similar procedures.