Background
I'm working with longitudinal survey data collected at two time points. Pre-treatment data were collected, subjects were randomly assigned to a treatment/control group, then post-treatment variables were collected. My aim is to estimate the following equation, where $\beta_1$ is the parameter of interest: $$\eta_{t_n} = \alpha + \beta_1 (Treat_n) + \beta_2 (\eta_{t-1_n}) + \epsilon_n$$
- $\eta_t$ is a latent variable measured post-treatment by 3 items $[y_{4_n}, y_{5_n}, y_{6_n}]$ for each subject $n \in N$.
- $Treat_n$ is an observed dummy variable indicating the experiment group of subject $n$.
- $\eta_{t-1}$ is a latent variable measured pre-treatment by 3 items $[y_{1_n}, y_{2_n}, y_{3_n}]$ for each $n$.
I am specifying this as a Bayesian structural equation model identified by fixing the first loading of each factor and the latent factor means to 0. I don't want to fix parameters to be invariant across time as $\beta_2$ is a nuisance parameter rather than of substantive interest, but do want to correlate the item residual variances. Assume no missing data and all $y$ are standardized to mean 0 and standard deviation of 1. $Treat$ is not standardized/centered.
Question
While I can model the two latent factors $\eta_t$ and $eta_{t-1}$ as bivariate normal with estimated variances and covariances to obtain the covariance between the factors, how can I incorporate the exogenous observed variable $Treat_n$ so that it is correlated with $\eta_{t_n}$ but not $\eta_{t-1_n}$ and I can estimate the equation above (with the intercept)?
An example without $Treat$ is below, where we'll assume the $\eta$ values are bivariate normal. I've left "??" in the parameters since I'm not sure how what goes there.
Note:
- Observed variables assumed uncorrelated conditional on the latent variable, thus the zeroes on the residual error covariance matrix.
- Model is identified by setting $\lambda_1$ to 1 and $\xi$ to 0.
- Assumes $y$ have been standardized to mean 0 and standard deviation 1. $$ \begin{pmatrix} y_{1_{n}} \\ y_{2_{n}} \\ y_{3_{n}} \\ y_{4_{n}} \\ y_{5_{n}} \\ y_{6_{n}} \\ \end{pmatrix} = \begin{pmatrix} \tau_1 \\ \tau_2 \\ \tau_3 \\ \tau_4 \\ \tau_5 \\ \tau_6 \\ \end{pmatrix} + \begin{pmatrix} 1 \\ \lambda_2 \\ \lambda_3 \\ 0 \\ 0 \\ 0 \\ \end{pmatrix} \begin{pmatrix} \eta_{t-1_{n}} \end{pmatrix} + \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \\ \lambda_5 \\ \lambda_6 \\ \end{pmatrix} \begin{pmatrix} \eta_{1_{n}} \end{pmatrix} + \begin{pmatrix} \epsilon_{1_n}\\ \epsilon_{2_n} \\ \epsilon_{3_n} \\ \epsilon_{4_n} \\ \epsilon_{5_n} \\ \epsilon_{6_n} \end{pmatrix} \\ $$
$$ \begin{pmatrix} \epsilon_{1_n}\\ \epsilon_{2_n} \\ \epsilon_{3_n} \\ \epsilon_{4_n} \\ \epsilon_{5_n} \\ \epsilon_{6_n} \end{pmatrix} \sim MVN(\begin{pmatrix} 0 \\0 \\0 \\0 \\0 \\0 \end{pmatrix}, \begin{pmatrix} 1 & 0 & 0 & \sigma_{1,4} & 0 & 0 \\ 0 & 1 & 0 & 0 & \sigma_{2,5} & 0 \\ 0 & 0 & 1 & 0 & 0 & \sigma_{3,6} \\ \sigma_{1,4} & 0 & 0 & 1 & 0 & 0 \\ 0 & \sigma_{2,5} & 0 & 0 & 1 & 0 \\ 0 & 0 & \sigma_{3,6} & 0 & 0 & 1 \\ \end{pmatrix}) $$
$$ \begin{pmatrix} \eta_{t_n} \\ \eta_{t-1_{n}} \end{pmatrix} = MVN(\begin{pmatrix} 0 \\ 0 \end{pmatrix}, \begin{pmatrix} ?? & ?? \\ ?? & ?? \end{pmatrix}) $$
