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Saturated Model

Saturated Model. Why, for a 3-d table, can’t we estimate a 3-way interaction? ln ( e ijk ) = u + u 1( i ) + u 2(j) + u 3(k) + u 12( ij ) + u 13( ik ) + u 23( jk ) + u 123( ijk ) the “ saturated model ”. Analogous to ANOVA w/o >1 replications per cell -- we’ve run out of df :

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Saturated Model

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  1. Saturated Model Why, for a 3-d table, can’t we estimate a 3-way interaction? ln(eijk) = u + u1(i) + u2(j) + u3(k) + u12(ij) + u13(ik) + u23(jk) + u123(ijk) the “saturated model”. Analogous to ANOVA w/o >1 replications per cell -- we’ve run out of df: For the 2x2x3 ex, fitting [ABC] would leave: 12 - 1 - 1 - 1 - 2 - 1 - 2 - 2 - 2 = 0 df 1(i)2(j)3(k)12(ij)13(ik)23(jk)123(ijk) In general for 3-d table, [ABC] leaves: IJK - 1 - (I-1) - (J-1) - (K-1) - (I-1)(J-1) - (I-1)(K-1) - (J-1)(K-1) - (I-1)(J-1)(K-1) =IJK - 1 - I+1 - J+1 - K+1 - IJ+I+J - 1 - IK+I+K - 1 - JK+J+K - 1 - IJK+IJ­+IK - I+JK - J - K+1 = 0 df.

  2. Saturated Model The saturated model will always (trivially) fit the data perfectly, because there are as many parameters to estimate as there are data points (i.e., 0 df). G2 would = 0 because all eijk = oijk, etc. In general, including the n-way interaction in hierarchial models for n-dimensional tables results in the saturated model.

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