for the reduced-group split-ballot multitrait multimethod (SBMTMM) model.

Rank-deficiencies in reduced-group split-ballot multitrait-multimethod models
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The split-ballot MTMM design [@saris_new_2004] is a two-group randomized design in which different elements of $\mathbf{y}$ are observed for two groups of subjects. This yields a planned missing data design. In the reduced-group SBMTMM design, not all possible combinations of methods are observed. For example, in the European Social Survey (ESS), group 1 receives a questionnaire with versions 1 and 2 of a questionnaire (method 1 and 2), whereas group 2 receives a questionnaire with versions 1 and 3 (methods 1 and 3). Therefore, all covariances between measures obtained with methods 2 and 3 are completely missing. @revilla_split-ballot_2013 noted that this particular design yields serious problems with nonconvergence, instability, and inadmissible estimates.

Here, we will follow the ESS design discussed by @revilla_split-ballot_2013: three traits, three methods, and two groups in which methods 1 and 2 (group 1) and methods 1 and 3 (group 2) have been used. Let $\mathbf{y}_g$ be the vector of observed variables for group $g$, and let $y_{tm}$ indicate the measure of the $t$-th trait obtained with the $m$-th method. The observed variable vectors are then $\mathbf{y}_1 = (y_{11}, y_{12}, y_{21}, y_{22}, 