  We investigate the geometry of a family of log-linear statistical models
called quasi-independence models. The toric fiber product is useful for
understanding the geometry of parameter inference in these models because the
maximum likelihood degree is multiplicative under the TFP. We define the
coordinate toric fiber product, or cTFP, and give necessary and sufficient
conditions under which a quasi-independence model is a cTFP of lower-order
models. We show that the vanishing ideal of every 2-way quasi-independence
model with ML-degree 1 can be realized as an iterated toric fiber product of
linear ideals. We also classify which Lawrence lifts of 2-way
quasi-independence models are cTFPs and give a necessary condition under which
a $k$-way model has ML-degree 1 using its facial submodels.
