2. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. Chapter 14 Comparing Groups: Analysis of Variance Methods Section 14.3 Two-Way ANOVA

3. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 3 Type of ANOVA One-way ANOVA is a bivariate method:  It has a quantitative response variable  It has one categorical explanatory variable Two-way ANOVA is a multivariate method:  It has a quantitative response variable  It has two categorical explanatory variables

4. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 4 Example: Amounts of Fertilizer and Manure A recent study at Iowa State University:  A field was portioned into 20 equal-size plots.  Each plot was planted with the same amount of corn seed.  The goal was to study how the yield of corn later harvested depended on the levels of use of nitrogen-based fertilizer and manure.  Each factor (fertilizer and manure) was measured in a binary manner.

5. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 5 There are four treatments you can compare with this experiment found by cross-classifying the two binary factors: fertilizer level and manure level. Table 14.7 Four Groups for Comparing Mean Corn Yield These result from the two-way cross classification of fertilizer level with manure level. Example: Amounts of Fertilizer and Manure

6. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 6 Inference about Effects in Two-Way ANOVA In two-way ANOVA, a null hypothesis states that the population means are the same in each category of one factor, at each fixed level of the other factor. We could test: : Mean corn yield is equal for plots at the low and high levels of fertilizer, for each fixed level of manure. Example: Amounts of Fertilizer and Manure

7. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 7 We could also test: : Mean corn yield is equal for plots at the low and high levels of manure, for each fixed level of fertilizer. The effect of individual factors tested with the two null hypotheses (the previous two pages) are called the main effects. Example: Amounts of Fertilizer and Manure

8. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 8 Assumptions for the Two-way ANOVA F-test  The population distribution for each group is normal.  The population standard deviations are identical.  The data result from a random sample or randomized experiment.

9. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 9 SUMMARY: F-test Statistics in Two-Way ANOVA For testing the main effect for a factor, the test statistic is the ratio of mean squares:  The MS for the factor is a variance estimate based on between-groups variation for that factor.  The MS error is a within-groups variance estimate that is always unbiased.

10. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 10  When the null hypothesis of equal population means for the factor is true, the F-test statistic values tend to fluctuate around 1.  When it is false, they tend to be larger.  The P-value is the right-tail probability above the observed F-value. SUMMARY: F-test Statistics in Two-Way ANOVA

11. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 11 Example: Corn Yield Data and sample statistics for each group: Table 14.9 Corn Yield by Fertilizer Level and Manure Level

12. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 12 Output from Two-way ANOVA: Table 14.10 Two-Way ANOVA for Corn Yield Data in Table 14.9 Example: Corn Yield

13. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 13 First consider the hypothesis: : Mean corn yield is equal for plots at the low and high levels of fertilizer, for each fixed level of manure. From the output, you can obtain the F-test statistic of 6.33 with its corresponding P-value of 0.022. The small P-value indicates strong evidence that the mean corn yield depends on fertilizer level. Example: Corn Yield

14. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 14 Next consider the hypothesis: : Mean corn yield is equal for plots at the low and high levels of manure, for each fixed level of fertilizer. From the output, you can obtain the F-test statistic of 6.88 with its corresponding P-value of 0.018. The small P-value indicates strong evidence that the mean corn yield depends on manure level. Example: Corn Yield

15. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 15 Exploring Interaction between Factors in Two-Way ANOVA No interaction between two factors means that the effect of either factor on the response variable is the same at each category of the other factor.

16. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 16 Figure 14.5 Mean Corn Yield, by Fertilizer and Manure Levels, Showing No Interaction. Exploring Interaction between Factors in Two-Way ANOVA

17. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 17 A graph showing interaction: Figure 14.6 Mean Corn Yield, by Fertilizer and Manure Levels, Displaying Interaction. Exploring Interaction between Factors in Two-Way ANOVA

18. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 18 Testing for Interaction In conducting a two-way ANOVA, before testing the main effects, it is customary to test a third null hypothesis stating that their is no interaction between the factors in their effects on the response.

19. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 19 The test statistic providing the sample evidence of interaction is: When is false, the F-statistic tends to be large. Testing for Interaction

20. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 20 Example: Corn Yield Data ANOVA table for a model that allows interaction: Table 14.14 Two-Way ANOVA of Mean Corn Yield by Fertilizer Level and Manure Level, Allowing Interaction

21. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 21 The test statistic for : no interaction is: F = (MS for interaction)/(MS error) = 3.04 / 2.78 = 1.10 ANOVA table reports corresponding P-value of 0.311  There is not much evidence of interaction.  We would not reject at the usual significance levels, such as 0.05. Example: Corn Yield Data

22. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 22 Check Interaction Before Main Effects In practice, in two-way ANOVA, you should first test the hypothesis of no interaction. It is not meaningful to test the main effects hypotheses when there is interaction.

23. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 23 If the evidence of interaction is not strong (that is, if the P-value is not small), then test the main effects hypotheses and/or construct confidence intervals for those effects. Check Interaction Before Main Effects

24. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 24 If important evidence of interaction exists, plot and compare the cell means for a factor separately at each category of the other factor. Check Interaction Before Main Effects

25. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 25 Why Not Instead Perform Two Separate One-Way ANOVAs? When you have two factors, you could perform two separate One-Way ANOVAs rather than a Two-Way ANOVA but  you learn more with a Two-Way ANOVA -it indicates whether there is interaction.  more cost effective to study the variables together rather than running two separate experiments.  the residual variability tends to decrease so we get better predictions, larger test statistics and hence greater power for rejecting false null hypotheses.

26. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 26 Factorial ANOVA The methods of two-way ANOVA can be extended to the analysis of several factors. A multifactor ANOVA with observations from all combinations of the factors is called factorial ANOVA, e.g., with three factors - three-way ANOVA considers main effects for all three factors as well as possible interactions.

27. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 27 Use Regression With Categorical and Quantitative Predictors In practice, when you have several predictors, both categorical and quantitative, it is sensible to build a multiple regression model containing both types of predictors.