Random Effects Anova That Is Correct for Non-normal Errors

Consider a fixed effects ANOVi\ model. We propose to estimate cell means /lij by MLE's ~ij instead of cell averages ~j, and we propose to estimate error variance by a MLEQ2 rather than the usual mean square for error. Even so, much of the usual ANOVi\ methodology carries over when applied to the ~ij 's, and the efficiency is often increased. When this is generalized to common random effects and mixed effects models, the resulting procedures are asymptotically correct. Fixed, random, mixed. 1. THE We first consider the model I Yij =J.l.i +aeEij for 1Si Sl and 1Sj SJi (let 11. == ':rJi and Ai 1 In) (1.1) = I _ I =J.l. + (Xi + ae Eij where J.l. == J.l. == L Ai J.l.i, <Xi == J.l.i J.l. and (X == L Ai <Xi =0 ; 1 1 here the Eij 's are assumed to be independent rv's from a "regular" density f that has been standardized to have mean 0 and variance 1. Let eYe, iiI..... iii denote the MLE's of the parameters. Let g(x)== (logj(x»)'. The information matrix n for these parameters is easily shown to be (1.2) qoo AlqOI 'AzqQ1 AlqOI A1qQ1 Alq 11 0 0 11. 0 n=(1.3) a; 0 Alq01 0 0 Alq11 with qoo==-E(I+E2g'(E)+2eg(E»). qOI ==-E(g(E)+Eg'(E») and qu ==-Eg'(E). Invert this matrix to find the asymptotic distribution of these estimators, and then perform a linear transformation to the rv's A A A and (Xi ==J.l.i ..... J.l. (1.4) to find that as min(Jl' ... .JI) -+00 we have [ ""(cr,-crJ1 d oo dOl 0 ~1 dOl d11 d ll ..r; (,i-J.l.) [1 1 ;.#Ie N , 0; 0 d ll dZl Al -lJ • "In (a~ -al) :::: l oj 0 d ll -d22 -dZl (1.5)