1. Every achievement originates from the seed of determination. 1Random Effect

2. One-Way Anova (Fixed Effect) Basically, the same assumptions as in pooled t tests: MODEL: Y ij =  +  i +  ij 1.) the  ij are indep. random variables 2.) Each  ij is Normally Distributed E(  ij ) = 0 for all i, j 3.)  2 (  ij ) = constant for all i, j Normality plot Residual plot Run order plot 2Random Effect

3. This assumes a “fixed model”: Inherent interest in the specific levels of the factors under study - there’s no direct interest in extrapolating to other levels - inference will be limited to levels that appear in the experiment. Experimenter selects the levels If a “random model”: Levels in experiment randomly selected from a population of such levels, and inference is to be made about the entire population of levels. Then, besides assumptions 1 to 3, there is another assumption: 4) a) the  i are independent random variables which are normally distributed with constant variance b) the  i and  ij are independent 3Random Effect

4. Random Effect 1-Way Anova Model See Section 17.2. and Minitab Unit 3 4Random Effect

5. Example: Y = LIFETIME (HOURS) BRAND 3 replications per level 1 2 3 4 5 6 7 8 1.8 4.2 8.6 7.0 4.2 4.2 7.8 9.0 5.0 5.4 4.6 5.0 7.8 4.2 7.0 7.4 1.0 4.2 4.2 9.0 6.6 5.4 9.8 5.8 2.6 4.6 5.8 7.0 6.2 4.6 8.2 7.4 5.8 Now, suppose the brands of test are randomly selected from a large pool of brands. 5Random Effect

6. Minitab: Stat>>General Linear Model, random factors, results, tick “In addition, coefficients for all terms” and “display mean squares and variance components” Analysis of Variance for lifetime, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P brand 7 69.120 69.120 9.874 3.38 0.021 Error 16 46.720 46.720 2.920 Total 23 115.840 S = 1.70880 R-Sq = 59.67% R-Sq(adj) = 42.02% Expected Mean Squares, using Adjusted SS Expected Mean Square Source for Each Term 1 brand (2) + 3.0000 (1) 2 Error (2) Error Terms for Tests, using Adjusted SS Synthesis Source Error DF Error MS of Error MS 1 brand 16.00 2.920 (2) 6Random Effect Variance Components, using Adjusted SS: Estimated Source Value brand 2.318 Error 2.920