1 RANDOM FACTORS 2 Two Types of Factors Fixed Factor levels of factor are fixed across replications of the experiment e g If experimenter is interested in specific form of smiling therapy vs no therapy and would use same EXACT forms of therapy across replications therapy would be a fixed factor Random Factor levels of factor vary randomly across replications of the experiment e g Assume there are multiple forms of smiling therapy and researcher is interested in generalizing results to all forms of smiling therapy If researcher randomly selects one form of smiling therapy would be a random factor 3 Models Associated with Factors Fixed effects Model All factors in the model are fixed Random effects Model All factors in the model are random Mixed effects Model Some factors in the model are fixed and others are random All models can be expressed with same nomenclature Yijk j k jk ijk 4 Consequence of a Random Factor in a Factorial Design Error term for the main effect of other factor needs to be adjusted Yijk j k jk ijk If is a random factor the error term for would not be MSW In a one factor model the analysis of random fixed factors is identical However there are conceptual differences 5 Conceptual Differences in Fixed vs Random One Factor Model H0 H1 are phrased differently Expected MSeffect is different 6 One Factor Fixed Effect Model Yijk j ijk j A given score is a function of the effect of Factor A j and random error ij The same treatment is used across replications of an experiment Consequently variation in random error MSW is the only parameter that produces changes across replications 2 7 H0 H1 One Factor Fixed Effect Model H 0 1 2 3 H1 1 2 3 or or H 0 2j 0 H1 2j 0 Null indicates that there is no effect Alternative indicates that there is an effect 8 E MSeffect One Factor Fixed Effect Model F MS factor MSerror compares variation b w w in groups Variation b w groups arises from error treatment effect Variation w in groups arises only from error When H0 is false MSeffect 2 n 2j Error Treatment Effect F Error When H0 is true MSeffect 2 0 random error effect a 1 2 n 2j a 1 2 1 random error Error 0 2 0 F 2 Error 1 One Factor Random Effect Model Yijk j ijk j 9 A given score is a function of the effect of Factor A j and random error ij The levels of Factor A are selected randomly across replications of an experiment Consequently variation in random error MSW and variation in the effect can both produce changes across replications So two sources of random variation and 2 2 2 2 H0 H1 One Factor Random Effect Model H 0 2 0 H1 2 0 10 Null indicates that all of the j s are equal i e all means are the same and there is no effect Alternative indicates that j s are not all equal i e all means are not the same and there is an effect E MSeffect One Factor Random Effect Model F When H0 is false MSeffect MS factor MSerror 2 n 2 F error variation in the effect Error Treatment Effect 2 2 Error 2 1 11 When H0 is true MSeffect F 2 0 error Error 0 2 0 2 Error 1 Analysis of 1 Factor Fixed vs Random Model Models differ in regard to H0 E MSeffect However both models have in common fact that the numerator denominator of the F ratio differ in regard to only 1 parameter F Error Treatment Effect Error 12 Formulas for the F ratio are algebraically equivalent for Fixed and Random One factor models In SAS analyze a one factor random effects model as you have been doing for a one factor fixed effects model Factorial Design with a Random Factor Including a random factor changes E MSmain effect for the other factor E MSmain effect error Main effect interaction with random factor Numerator denominator of F ratio will differ by more than 1 parameter 13 Adjustments need to be made to the denominator of F Example Superintendent of a school district is interested in testing the relative effectiveness of 3 methods of teaching math Conceptual Problem Solving Memorization Three teachers are randomly selected from the 9 teachers in the district Each teacher teaches 3 classes using one method for each class 14 3 method x 3 teacher mixed factorial with teacher as random factor Example Assume we know the effectiveness of all 9 teachers with each of the methods Population Effectiveness Means for Method and Teachers Teacher Method A B C D E F G H I Mean Conceptual 9 8 7 1 1 1 2 3 4 4 Problem solving 2 3 4 9 8 7 1 1 1 4 Memorization 1 1 1 2 3 4 9 8 7 4 Mean 4 4 4 4 4 4 4 4 4 15 On average the methods are equally effective On average all teachers are equally effective Method x Teacher Some teachers are more effective with one method than the other methods Example When conducting the experiment the superintendent randomly selects 3 of the 9 teachers The results of the experiment will differ depending on which teachers are selected Method Teacher A B I Conceptual Problem solving 9 2 8 3 4 1 Mea n 7 2 16 Memorization Mean 1 4 1 4 7 4 3 With teachers A B I it appears as if the conceptual approach is most effective on average Consequence of a Random Factor Interaction b w fixed random factor in the population has no effect on the main effect of the random factor but can change the main effect of the fixed factor If teacher is analyzed as fixed effect Fmethod Error Method Method x Teacher Error 17 The test of the method main effect is not accurate because it includes variation due to the main effect variation due to the interaction Solution Fmethod Error Method Method x Teacher Error Adjust error term so numerator denominator differ only in regard to the desired effect Fmethod Error Method Method x Teacher Error Method x Teacher Such an adjustment is made when teacher is treated as a random factor Notice that error term is MSMethod x Teacher 18 FMethod x Teacher Error Method x Teacher Error E MS Error Terms for Fixed Mixed Random Models Effect A B AxB Fixed Effects Model Mixed Effects Random Effects A B both fixed A fixed B random A B both random Expected MS 2 2 bn 2 j a 1 an k2 b 1 n 2jk 2 a 1 b 1 Error Term MSW Expected MS MSW 2 2 MSW 2 2 2 MSAB MSW 2 2 MSW 2 2 MSW 2 n 2j a 1 2 Error Error Term Expected MS Term MSAB MSAB 2 2 2 19 Testing A Mixed Random Factorial in SAS Problem 10 p 450 Maxwell Delaney 2000 Psychologist is interested in the relative effectiveness of three therapies rational emotive …
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