Identifiability

A statistical model is identifiable when distinct parameter values determine distinct probability distributions. In a nonidentifiable parameterization, data can at most estimate combinations of parameters that remain unchanged along each equivalent parameter set.

In a shared zero-inflated Gamma-Poisson count model, write $m_j=(1-\pi ){\mu }_j$ and $\kappa =(\tau +\pi )/(1-\pi )$. Its first two moments are $\mathrm{Var}(Y_j)=m_j+\kappa m_j^2$ and $\mathrm{Cov}(Y_j,Y_k)=\kappa m_j{m}_k$. These moments do not separately identify $\pi$, $\tau$, and the component-mean intercept: taking ${\pi }^{\prime }=0$, ${\tau }^{\prime }=\kappa$ and ${\mu }_j^{\prime }=m_j$ gives the same first two moments. The full count distribution can carry information absent from the moments. A consistent estimator of a marginal mean ratio therefore need not consistently estimate a structural-zero fraction from a misspecified moment parameterization.

In a normal linear model with an intercept and factor effects, set the effect of one factor level to zero. The intercept then denotes that reference level's mean, and the remaining effects are contrasts to it. The constraint removes redundant statistical parameters without altering the fitted mean space.