Appendix 1: Variance–covariance decomposition of the extended Solomon four-group design (SFGD) based on two-level SEM framework
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}$$
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}$$
Appendix 1.3: Variance–covariance matrix of SFGD control group 1
$$\begin{aligned} \mathbb {COV}(Y,X)= & {} \left[ {{\begin{array}{ll} {\mathbb {V}(Y)} &{} \\ {\mathbb {COV}(Y,X)} &{} {\mathbb {V}(X)} \\ \end{array} }} \right] \\= & {} \left[ {{\begin{array}{llll} &{} &{} &{} \\ &{} {\mathbb {V}(Y_1 )} &{} &{} \\ &{} &{} &{} \\ &{} {\mathbb {COV}(Y_1 ,X_1 )} &{} &{} {\mathbb {V}(X_1 )} \\ \end{array} }} \right] \\= & {} \left[ {{\begin{array}{llll} &{} &{} &{} \\ &{} {{\Lambda _1} (\Pi _1 \Psi _{\xi _1 }^B \Pi _1 ^{'}+\Psi _{u_1 }^B ){\Lambda _1} ^{'}+\Theta _{\varepsilon _1 }^B } &{} &{} \\ &{} &{} &{} \\ &{} {{\Lambda _1} \Pi _1 \Psi _{\xi _1 }^B {\Gamma }'_1 } &{} &{} {\Gamma _1 \Psi _{\xi _1 }^B {\Gamma }'_1 +\Theta _{\zeta _1 }^B } \\ \end{array} }} \right] \\\\&+ \left[ {{\begin{array}{llll} &{} &{} &{} \\ &{} {{\Lambda _1} (\Pi _1 \Psi _{\xi _1 }^W \Pi _1 ^{'}+\Psi _{u_1 }^W ){\Lambda _1} ^{'}+\Theta _{\varepsilon _1 }^W } &{} &{} \\ &{} &{} &{} \\ &{} {{\Lambda _1} \Pi _1 \Psi _{\xi _1 }^W {\Gamma }'_1 } &{} &{} {\Gamma _1 \Psi _{\xi _1 }^W {\Gamma }'_1 +\Theta _{\zeta _1 }^W } \\ \end{array} }} \right] \\ \end{aligned}$$
Appendix 1.4: Variance–covariance matrix of SFGD control group 2
$$\begin{aligned} \mathbb {COV}(Y,X)= & {} \left[ {{\begin{array}{ll} {\mathbb {V}(Y)} &{} \\ {\mathbb {COV}(Y,X)} &{} {\mathbb {V}(X)} \\ \end{array} }} \right] \\= & {} \left[ {{\begin{array}{llll} &{} &{} &{} \\ &{} {\mathbb {V}(Y_1 )} &{} &{} \\ &{} &{} &{} \\ &{} {\mathbb {COV}(Y_1 ,X_1 )} &{} &{} {\mathbb {V}(X_1 )} \\ \end{array} }} \right] \\= & {} \left[ {{\begin{array}{llll} &{} &{} &{} \\ &{} {{\Lambda _1} (\Pi _1 \Psi _{\xi _1 }^B \Pi _1 ^{'}+\Psi _{u_1 }^B ){\Lambda _1} ^{'}+\Theta _{\varepsilon _1 }^B } &{} &{} \\ &{} &{} &{} \\ &{} {{\Lambda _1} \Pi _1 \Psi _{\xi _1 }^B {\Gamma }'_1 } &{} &{} {\Gamma _1 \Psi _{\xi _1 }^B {\Gamma }'_1 +\Theta _{\zeta _1 }^B } \\ \end{array} }} \right] \\&+ \left[ {{\begin{array}{llll} &{} &{} &{} \\ &{} {{\Lambda _1} (\Pi _1 \Psi _{\xi _1 }^W \Pi _1 ^{'}+\Psi _{u_1 }^W ){\Lambda _1} ^{'}+\Theta _{\varepsilon _1 }^W } &{} &{} \\ &{} &{} &{} \\ &{} {{\Lambda _1} \Pi _1 \Psi _{\xi _1 }^W {\Gamma }'_1 } &{} &{} {\Gamma _1 \Psi _{\xi _1 }^W {\Gamma }'_1 +\Theta _{\zeta _1 }^W } \\ \end{array} }} \right] \\ \end{aligned}$$
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).
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)
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)
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)
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:
$$\begin{aligned} \mathbb {COV}(Y,X)= & {} \left[ {{\begin{array}{ll} {\mathbb {V}(Y)} &{} \\ {{\rm COV}[Y,X]} &{} {\mathbb {V}(X)} \\ \end{array} }} \right] =\left[ {{\begin{array}{ll} {\Phi _Y^B +\Phi _Y^W } &{} \\ {{{\Lambda } \Pi \Gamma }\Psi _\xi ^B +{{\Lambda } \Pi \Gamma }\Psi _\xi ^W } &{} {\Phi _X^B +\Phi _X^W } \\ \end{array} }} \right] \\= & {} \left[ {{\begin{array}{ll} {\Phi _Y^B } &{} \\ {{{\Lambda } \Pi \Gamma }\Psi _\xi ^B } &{} {\Phi _X^B } \\ \end{array} }} \right] +\left[ {{\begin{array}{ll} {\Phi _Y^W } &{} \\ {{\Lambda \Pi \Gamma }\Psi _\xi ^W } &{} {\Phi _X^W } \\ \end{array} }} \right] \\= & {} \left[ {{\begin{array}{ll} {{\Lambda } \Psi _\eta ^B {\Lambda } ^{'}+\Theta _{\varepsilon }^B } &{} \\ {{\Lambda \Pi \Gamma }\Psi _\xi ^B } &{} {\Gamma \Psi _\xi ^B \Gamma ^{'}+\Theta _{\zeta }^B } \\ \end{array} }} \right] +\left[ {{\begin{array}{ll} {{\Lambda } \Psi _\eta ^W {\Lambda } ^{'}+\Theta _{\varepsilon }^W } &{} \\ {{\Lambda \Pi \Gamma }\Psi _\xi ^W } &{} {\Gamma \Psi _\xi ^W \Gamma ^{'}+\Theta _{\zeta }^W } \\ \end{array} }} \right] \\= & {} \left[ {{\begin{array}{ll} {{{\Lambda } (\Pi }\Psi _\xi ^B {\Pi ^{'}}+\Psi _E^B ){\Lambda } ^{'}+\Theta _{\varepsilon }^B } &{} \\ {{\Lambda \Pi \Gamma }\Psi _\xi ^B } &{} {\Gamma \Psi _\xi ^B \Gamma ^{'}+\Theta _{\zeta }^B } \\ \end{array} }} \right] \\&+ \left[ {{\begin{array}{ll} {{{\Lambda } (\Pi }\Psi _\xi ^W {\Pi ^{'}}+\Psi _E^W ){\Lambda } ^{'}+\Theta _{\varepsilon }^W } &{} \\ {{\Lambda \Pi \Gamma }\Psi _\xi ^W } &{} {\Gamma \Psi _\xi ^W \Gamma ^{'}+\Theta _{\zeta }^W } \\ \end{array} }} \right] \end{aligned}$$
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
$$\begin{aligned} {\mathbb {COV}}(Y,X)= & {} \left[ {{\begin{array}{ll} {\mathbb {V}(Y)} &{} \\ {\mathbb {COV}(Y,X)} &{} {\mathbb {V}}(X) \\ \end{array}}} \right] \\= & {} \left[ {{\begin{array}{llll} {\mathbb {V}}(Y_0 ) &{} &{} &{} \\ {\mathbb {COV}(Y_0 ,Y_1 )} &{} {\mathbb {V}}(Y_1 ) &{} &{} \\ {\mathbb {COV}}(Y_0 ,X_0 ) &{} {\mathbb {COV}}(Y_1 ,X_0 ) &{} {\mathbb {V}}(X_0 ) &{} \\ {\mathbb {COV}}(Y_0 ,X_1 ) &{} {\mathbb {COV}}(Y_1 ,X_1 ) &{} {\mathbb {COV}}(X_0 ,X_1 ) &{} {\mathbb {V}}(X_1 ) \\ \end{array} }}\right]\, . \end{aligned}$$
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}$$
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,
$$\begin{aligned} {\mathbb {V}}(X)= & {} {\mathbb {COV}}(X,X)\\= & {} \left[ {{\begin{array}{ll} {\mathbb {V}}(X_0 ) &{} \\ {\mathbb {COV}}(X_0 ,X_1 ) &{} {\mathbb {V}}(X_1) \\ \end{array} }} \right] \\= & {} \left[ {{\begin{array}{ll} {\Phi _{X_0 }^B +\Phi _{X_0 }^W } &{} \\ {\mathbb {COV}}(X_0 ,X_1 ) &{} {\Phi _{X_1 }^B +\Phi _{X_1 }^W } \\ \end{array} }} \right] \\= & {} \left[ {{\begin{array}{ll} {\Gamma _0 \Psi _{\xi _0 }^B {\Gamma }'_0 +\Theta _{\zeta _0 }^B } &{} \\ {\Gamma _0 \Psi _{\xi _0 }^B {\Gamma }'_1 {{\mathcal {K}} }'} &{} {\Gamma _1 \Psi _{\xi _1 }^B {\Gamma }'_1 +\Theta _{\zeta _1 }^B } \\ \end{array} }} \right] \\&+\left[ {{\begin{array}{*{20}c} {\Gamma _0 \Psi _{\xi _0 }^W {\Gamma }'_0 +\Theta _{\zeta _0 }^W } &{} \\ {\Gamma _0 \Psi _{\xi _0 }^W {\Gamma }'_1 {{\mathcal {K}} }'} &{} {\Gamma _1 \Psi _{\xi _1 }^W {\Gamma }'_1 +\Theta _{\zeta _1 }^W } \\ \end{array} }} \right] \\&=\left[ {{\begin{array}{ll} {\Gamma _0 \Psi _{\xi _0 }^B {\Gamma }'_0 +\Theta _{\zeta _0 }^B } &{} \\ {\Gamma _0 \Psi _{\xi _0 }^B {\Gamma }'_1 {{\mathcal {K}} }'} &{} {\Gamma _1 {\mathcal {K}} \Psi _{\xi _0 }^B {{\mathcal {K}} }'{\Gamma }'_1 +\Theta _{\zeta _1 }^B } \\ \end{array} }} \right] \\&+\left[ {{\begin{array}{ll} {\Gamma _0 \Psi _{\xi _0 }^W {\Gamma }'_0 +\Theta _{\zeta _0 }^W } &{} \\ {\Gamma _0 \Psi _{\xi _0 }^W {\Gamma }'_1 {{\mathcal {K}} }'} &{} {\Gamma _1 {\mathcal {K}} \Psi _{\xi _0 }^W {{\mathcal {K}} }'{\Gamma }'_1 +\Theta _{\zeta _1 }^W } \\ \end{array} }} \right] . \end{aligned}$$
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.