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Two-Way ANOVA

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  1. Two-Way ANOVA Lab 5 instruction

  2. ANOVA: analysis of variance • a collection of statistical methods to compare several groups according to their means on a quantitative response variable • Two-Way ANOVA two factors are used consider “main effect” and “interaction effect”

  3. Example • Response: Blood Alcohol Content (BAC) • Factors: • Gender(GEN) • Alcohol Consumption (ALC) • Factor levels: • GEN: 1=Male; 0=Female • ALC: 1=1 drink; 2=2 drinks; 4=4 drinks • Main effect of GEN, main effect of ALC, and their interaction

  4. Explore the difference in average response across both factors. • Table of marginal means (contingency table) • In SPSS: Analyze  Compare Means  Means /Or General linear models  descriptive stat

  5. e.g. Table of marginal means (contingency table) • Response: in Dependent list • Factors: in independent list of two seperate layers

  6. e.g. Table of marginal means (contingency table) • Interpret

  7. Explore the difference in average response across both factors. • Plots of marginal means (profile plot) • In SPSS: Graphs legacy dialogs  Line  multiple /Or GLM  univariate  (plots)

  8. e.g. Plot of marginal means (profile plot) • Profile plot with lines representing gender

  9. e.g. Plot of marginal means (profile plot) • Interpretation: • 6 points • Effect of gender • Effect of drink consumption • Interaction (parallel or crossed)

  10. Explore the difference in average response across both factors. • Clustered boxplot • In SPSS: Graphs  legacy dialogs  Boxplot Clustered

  11. e.g. Clustered boxplot • Clustered boxplot with clusters defined by gender

  12. e.g. Clustered boxplot

  13. General Linear Model (GLM)-Assumptions • Independent samples • Normality: • Boxplot • QQ plot • Equal standard deviation (variance): • Boxplot • Summary statistics (rule of thumb: the ratio of the largest s.d. over the smallest s.d. is less than 2)

  14. General Linear Model (GLM) • Univariate GLM is the general linear model now often used to implement such essential statistical procedures as regression and ANOVA. In particular, the procedure can be used to carry out two-way analysis of variance. • Procedure: Analyze  General Linear Model  Univariate (defaulted: model with Interactions)

  15. GLM: Non-additive model output

  16. GLM: Non-additive model output • In columns “SS” and “df” • Corrected total = corrected model + error • Corrected model=GEN + ALC + GEN*ALC • The column “F” contains the value of the f statistic for each effect. The value of F is computed as follows:

  17. GLM: Non-additive model output Hypothesis test : no interaction between GEN and ALC

  18. Equivalent to the regression models: Since the interaction term is not significant, we tend to modify the model. the model without the interaction term: additive model

  19. GLM: Define additive model

  20. GLM: Additive model output & equivalent regression model

  21. GLM: Additive model output & equivalent regression model

  22. GLM: Additive model output & equivalent regression model

  23. GLM - Options • Model: define the model, Include intercept or not • Contrasts: The linear combination of the level means • Profile Plots: the plot of estimated means • Post Hoc • Save • Options

  24. GLM - Options • Profile Plots: different from the one obtained with line chart • This plots estimated means form the model

  25. Equivalent regression models and tests • Define dummy variables • Write down estimated model equation • Use extra-sum-of-squares F-test for testing interaction term • Interpret meaning of estimated coefficient, t-test on coefficients and their 95% CI

  26. Equivalent regression models and tests • estimated regression model with interaction term (full non-additive model)

  27. Equivalent regression models and tests • estimated regression model with NO interaction term (additive model)

  28. Equivalent regression models and tests • Construct extra-of-sum F-test use the above two ANOVA tables, • In the regression model with NO interaction term, interpret meaning of , and its CI. • Is “GEN” effective in predicting BAC? (t-test on )

  29. Note on Q3 (b) • Linear combination of mean difference in BAC for male V.S. female • Substitute ’s by corresponding ’s

  30. Note on Q3 (c) and Q7 (c) • Rate of increase in mean BAC for an increase in drinks consumption • Thus, in Q3 (c) • rate (1to2) = , rate (2to4) = • In Q7 (c) • rate (1to2) = , rate (2to4) =