Abstract
Most statistical software packages implement numerical strategies for computation of maximum likelihood estimates in random effects models. Little is known, however, about the algebraic complexity of this problem. For the one-way layout with random effects and unbalanced group sizes, we give formulas for the algebraic degree of the likelihood equations as well as the equations for restricted maximum likelihood estimation. In particular, the latter approach is shown to be algebraically less complex. The formulas are obtained by studying a univariate rational equation whose solutions correspond to the solutions of the likelihood equations. Applying techniques from computational algebra, we also show that balanced two-way layouts with or without interaction have likelihood equations of degree four. Our work suggests that algebraic methods allow one to reliably find global optima of likelihood functions of linear mixed models with a small number of variance components.
Highlights
Linear models with fixed and random effects are widely used for dependent observations
Since linear mixed models have rational likelihood equations, this involves careful clearing of denominators and applying symbolic and specialized numerical techniques to determine all solutions of the resulting polynomial system
We consider balanced two-way layouts. These are known to have restricted maximum likelihood (REML) degree equal to one, and we show that the maximum likelihood (ML) degree is four, which means that ML estimates are available in closed form in the sense of Cardano’s formula
Summary
Linear models with fixed and random effects are widely used for dependent observations. The layout is balanced, that is, all batch sizes are equal, here ni = 5 In this case, the likelihood equations are well-known to be equivalent to a linear equation system, and the ML estimators are rational functions of the observations Yij. In terminology we will use later on, balanced one-way layouts have ML degree one. By carefully clearing terms from the numerator and the denominator appearing, our proof produces a polynomial in θ whose roots correspond to the solutions of the rational equation The degree of this polynomial is the ML/REML degree; recall Example 1. These are known to have REML degree equal to one, and we show that the ML degree is four, which means that ML estimates are available in closed form in the sense of Cardano’s formula.
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