Fixed e®ects Completely Randomized Design model In this model, yij = ¹ + ¿i + "ij; the t values of ¿i, ¿1;¿2; :::; ¿t; are the only e®ects of interest, and "ij is the only random term, "ij has a normal distribution with mean 0 and variance 2 2 σe ("ij » N(0; σe ) ). Random e®ects Completely Randomized Design model In this model, yij = ¹ + ¿i + "ij; 2 The ¿i terms are now random, ¿i » N(0 ; σt ) , and the ¿i and "ij are independent. Now the ¿i `s are viewed as a random sample from a population of ¿i `s, 2 2 and the ANOVA H0 is H0: σ¿ = 0 vs. Ha : σ¿ > 0 . How do you know if an e®ect is ¯xed or random? 1. How were the levels for the factor in question chosen? 2. Is it desired to generalize the results to levels that weren't used? Implications for random e®ects 1. The denominator of the F statistic may change. 2. Generally you are not interested in doing multiple comparisons. There 2 2 may be interest in estimating σ¿ , σe , etc. 2 2 3. Estimating σt may still be of interest even if we reject H0 : σ¿¯ = 0 Random e®ects model for Randomized Block Design The model is: yij = ¹ + ¿i + ¯j + "ij; 2 2 2 Where ¿i » N(0; σ¿ ); ¯j » N(0; σ¯);"ij » N(0; σe ); and ¿i , ¯j , and "ij are mutually independent. For the RB design (no replication) we assume 2 that σ¿¯ = 0. 1 Generalized Randomized Block Design with random e®ects 2 1. Test H0 : σ¿¯ = 0 using F = MSAB/MSE 2 2. If the P value for H0 : σ¿¯ = 0 is large ( > .15 or > .25), you may choose to pool terms, creating MSE(pooled) = (SSAB +SSE)/(dfAB + dfE), also denoted by MSRES. Mixed E®ects Models A mixed e®ect model includes both ¯xed and random e®ects. The lotions for allergies experiment is an example of a mixed model. 2 2 2 In the two factor random e®ect model, E(MSA) = σe +n σ¿¯ + bn σ¿ ; 2 2 and E(MSAB) = σe +n σ¿¯ . 2 When H0 : σ¿ = 0 is true, then E(MSA) = E(MSAB) and the sample F value is close to 1. 2 However, E(MSE) = σe so even if the null hypothesis for factor A is true 2 (σ¿ = 0) the sample F value will tend to be greater than 1 if we use F = MSA/MSE . Instead we use F = MSA/MSAB. 2.