A note on applying the BCH method under linear equality and inequality constraints

Researchers often wish to relate estimated scores on latent variables to exogenous covariates not previously used in analyses. The BCH method corrects for asymptotic bias in estimates due to these scores’ uncertainty and has been shown to be relatively robust. When applying the BCH approach however, two problems arise. First, negative cell proportions can be obtained. Second, the approach cannot deal with situations where marginals need to be fixed to specific values, such as edit restrictions. The BCH approach can handle these problems when placed in a framework of quadratic loss functions and linear equality and inequality constraints. This research note gives the explicit form for equality constraints and demonstrates how solutions for inequality constraints may be obtained using numerical methods.