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Analysis of Variance

Analysis of Variance. Analysis of variance is a General Linear Models procedure to ascertain “treatments” effects on the “mean” of the response variable.

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Analysis of Variance

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  1. Analysis of Variance • Analysis of variance is a General Linear Models procedure to ascertain “treatments” effects on the “mean” of the response variable. • Multiple regression is also a General Linear Models procedure to ascertain the impact of independent (predictor) variables on the mean of the dependent (response) variable.

  2. ANOVA vs. Regression • In ANOVA independent variables are qualitative • In traditional multiple regression independent variables are quantitative • Indicator variables introduce qualitative variable to regression models • Covariates introduce quantitative variable to Anova models

  3. General Linear Models Traditional definitions of both regression and analysis of variance are subsumed by the General Linear Model in which the left hand (response) variable is quantitative and the right hand variables (predictor) can be a mix of qualitative and quantitative. The purpose of regression is still somewhat different from the purpose of ANOVA.

  4. Purpose of Regression • Prediction, yhat: • Interpolation, importance of R • Extrapolation Generally unacceptable except with time series • Estimation, beta hat Unreliable with multicollinearity or high influence points

  5. Purpose of ANOVA • The purpose of Analysis of Variance is to determine whether or not different treatments have different mean responses. • Classic ANOVA model without interaction Y = m + a + b + e Ho: a = 0 for all levels of factor A Ho: b = 0 for all levels of factor B

  6. Purpose of ANOVA cont’d • Classic ANOVA model with interaction Y = m + a + b + (ab) + e Ho: ab = 0 for all factor level combinations • General Linear Model Y = b0 + b1A1 + b2A2 + b3B1 + b4B2 +b5(A1*B1) + b6(A1*B2) + b7(A2*B1) +b8(A2*B2) + e

  7. Purpose of ANOVA cont’d Ho: b1 = b2 = 0 Ho: b3 = b4 = 0 Ho: b5 = b6 = b7 = b8 = 0. • ANOVA for GLM: Sources of variation df SS Model (a-1)+(b-1)+(a-1)(b-1) TSS-SSE Error n - ab SSE Total n - 1 TSS

  8. There is an effect! Now what? • Multiple comparisons • All pairwise • LSD • Tukey • Bonferroni • Scheffe • Planned • Control vs. treatment average • All orthogonal

  9. Interactions and Main Effects • Interactions • Repeat tests required for testing • Graphs can indicate presence of interactions and their relative importance even without tests of hypotheses • Main Effects • Tests must be interpreted in light of interaction • Plots effectively convey results of comparisons

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