y_{31}, y_{32})^T$ (group 1) and $\mathbf{y}_2 = (y_{11}, y_{13}, y_{21}, y_{23}, y_{31}, y_{33})^T$ (group 2). Note that neither group contains both methods 2 and 3. As above, the CFA model per group is $$\label{eq:cfa-group}
    \boldsymbol{\Sigma}_g(\boldsymbol{\theta}) = \Lambda_g \Phi_g \Lambda_g^T + \Psi_g,$$ where subscripts $g$ indicate group specific vectors. To facilitate further analysis, we redefine the vector of (co)variances as $\boldsymbol{\sigma}(\boldsymbol{\theta}) := (\boldsymbol{\sigma}_1(\boldsymbol{\theta})^T, \boldsymbol{\sigma}_2(\boldsymbol{\theta})^T)^T$, thus deleting the unobserved moments, and will do the same for observed moments $\mathbf{s}$.

As in @revilla_split-ballot_2013, we then specify the “correlated-trait uncorrelated-method” (CTUM) model with equal method loadings, $$\begin{aligned}
\label{eq:lambda-1}
\Lambda_1 =    \left(
\begin{array}{ccccc}
 \lambda_{11} & 0 & 0 & 1 & 0 \\
 \lambda_{12} & 0 & 0 & 0 & 1 \\
 0 & \lambda_{21} & 0 & 1 & 0 \\
 0 & \lambda_{22} & 0 & 0 & 1 \\
 0 & 0 & \lambda_{31} & 1 & 0 \\
 0 & 0 & \lambda_{32} & 0 & 1 \\
\end{array} 
\right)&\;& 
    \Lambda_2 = \left(
\begin{array}{ccccc}
 \lambda_{11} & 0 & 0 & 1 & 0 \\
 \lambda_{13} & 0 & 0 & 0 & 1 \\
 0 & \lambda_{21} & 0 & 1 & 0 \\
 0 & \lambda_{23} & 0 & 