We will discuss hierarchical models and certain toric ideals as a way of studying these objects in algebraic statistics. Some algebraic properties of these ideals will be described and a formula for the Krull dimension of the corresponding toric rings will be presented.  One goal is to study the invariance properties of families of ideals arising from hierarchical models with varying parameters. We will present classes of examples where we have information about an equivariant Hilbert series. This is joint work with Uwe Nagel.