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# class03_21 - 1.017/1.010 Class 21 Multifactor Analysis of...

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1.017/1.010 Class 21 Multifactor Analysis of Variance Multifactor Models We often wish to consider several factors contributing to variability ratherr than just one. Extend concepts of single factor ANOVA to multiple factors. Focus on the two-factor case. Suppose there I treatments for Factor A and J treatments for factor B ., giving IJ random variables described by CDFs F xij ( x ij ). The different F xij ( x ij ) are assumed identical (except for their means) and normally distributed (check this, as in single factor case). A random sample [ x ij 1 , x ij 2 , . .., x ijK ] of size K is obtained for treatment combination ( i , j ). Two-factor model describing x ijk : x ijk = µ ij + e ijk = + a i + b j + c ij + e ijk ij = E [ x ijk ] = + a i + b j + c ij = unknown mean of x ijk (for all k ) µ = unknown grand mean (average of i 's ). a i = unknown main effects of Factor A b j = unknown main effects of Factor B c ij = unknown interactions between Factors A and B e ijk = random residual for treatment i , replicate j E [ e ijk ] = 0, Var [ e ijk ] = σ 2 , for all i , j , k Note that c ij can only be distinguished from e ijk if number of replicates K >1. Constraints: Objective is to estimate/test values of a i 's, b j 's, and c ij 's, which are distributional parameters for the F xij ( x ij )'s. Formulating the Problem as a Hypothesis Test Formulate three sum-of-squares hypotheses that insure that all a i ' s, all b i 's, or all c ij 's are zero: 1

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Derive test statistics based on sums-of-squares of data. Sums-of-Squares Computations Define treatment and grand sample means: Test statistics are computed from sums-of-squares: Corresponding mean-sums-of-squares are: 2
Expected values of these mean-sums-of-squares show depends on main effects and interactions: Test Statistic Use ratios as test statistics for the three hypotheses: When H0

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class03_21 - 1.017/1.010 Class 21 Multifactor Analysis of...

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