ModelExpected MSAAxB1RANDOM FACTORS2Two Types of Factors-Fixed Factor-levels of factor are fixed across replications of the experimente.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 experimente.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.3Models 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:ijkjkkjijkY )(4Consequence of a Random Factor in aFactorial Design-Error term for the main effect of “other factor” needs to be adjustedijkjkkjijkY )(-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 differences5Conceptual Differences in Fixed vsRandom One-Factor Model-H0 & H1 are phrased differently-Expected MSeffect is different6One-Factor Fixed Effect ModelijkjijkY(-=-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 (2 = MSW) is the only parameter that produces changes across replications.7H0 & H1: One-Factor Fixed Effect Model3210:H or 0:20 jH3211:H or 0:21 jH-Null indicates that there is no effect-Alternative indicates that there is an effect8E(MSeffect): One-Factor Fixed Effect ModelerrorfactorMSMSF 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 =122anj = random error + effect2221anErrorEffectTreatmentErrorFj > 1-When H0 is true: MSeffect =02 = random error 2200ErrorErrorF = 1One-Factor Random Effect ModelijkjijkY(-=-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 (2 = MSW) and variation in the effect (2) can both produce changes across replications.-So, two sources of random variation: 2 and 2H0 & H1: One-Factor Random Effect Model0:20H0:21H10-Null indicates that all of the -j’s are equal (i.e., all means arethe 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 ModelerrorfactorMSMSF -When H0 is false: MSeffect =22n = error + variation in the effect222ErrorEffectTreatmentErrorF > 111-When H0 is true: MSeffect =02 = error 2200ErrorErrorF = 1Analysis 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 parameterErrorEffectTreatmentErrorF12-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 modelFactorial Design with a Random Factor-Including a random factor changes E(MSmain effect) for the other factorE(MSmain effect) = error + Main effect + interaction with random factor-Numerator & denominator of F-ratio will differ by more than 1 parameter13-Adjustments need to be made to the denominator of FExample-Superintendent of a school district is interested in testing therelative 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 class14-3(method) x 3(teacher) mixed factorial with teacher as random factorExample-Assume we know the effectiveness of all 9 teachers with each of the methodsPopulation Effectiveness Means for Method and TeachersTeacherMethod A B C D E F G H I MeanConceptual 9 8 7 1 1 1 2 3 4 4Problem solving 2 3 4 9 8 7 1 1 1 4Memorization 1 1 1 2 3 4 9 8 7 4Mean 4 4 4 4 4 4 4 4 415-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 whichteachers are selectedTeacherMethod A B I MeanConceptual 9 8 4 7Problem solving 2 3 1 216Memorization 1 1 7 3Mean 4 4 4-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 factorIf teacher is analyzed as fixed effect,ErrorTeacherxMethodMethodErrorFmethod17-The test of the method main effect is not accurate because it includes variation due to the main effect & variation due to the interaction.SolutionErrorTeacherxMethodMethodErrorFmethod-Adjust error term so numerator & denominator differ only inregard to the desired effectTeacherxMethodErrorTeacherxMethodMethodErrorFmethod-Such an adjustment is made when teacher is treated as a random factor-Notice that error term is MSMethod x Teacher18ErrorTeacherxMethodErrorFTeacherxMethodE(MS) & Error Terms for Fixed, Mixed, & Random ModelsModelFixed Effects Mixed Effects Random Effects(A & B both fixed) (A fixed & B random) (A & B both random)Effect
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