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Chapter 14 Comparing Groups: Analysis of Variance Methods

Chapter 14 Comparing Groups: Analysis of Variance Methods. Section 14.3 Two-Way ANOVA. 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:

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Chapter 14 Comparing Groups: Analysis of Variance Methods

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  1. Chapter 14Comparing Groups: Analysisof Variance Methods Section 14.3 Two-Way ANOVA

  2. 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

  3. 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.

  4. Example: Amounts of Fertilizer and Manure • What are the four treatments that you can compare with this experiment by cross-classifying the two binary factors? Table 14.7 Four Groups for Comparing Mean Corn Yield These result from the two-way cross classification of fertilizer level with manure level.

  5. Example: Amounts of Fertilizer and Manure • 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.

  6. Example: Amounts of Fertilizer and Manure • 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.

  7. Example: Amounts of Fertilizer and Manure • 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.

  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. 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. SUMMARY: F-test Statistics in Two-Way ANOVA • 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.

  11. Example: Corn Yield • Data and sample statistics for each group: Table 14.9 Corn Yield by Fertilizer Level and Manure Level

  12. Example: Corn Yield • Output from Two-way ANOVA: Table 14.10 Two-Way ANOVA for Corn Yield Data in Table 14.9

  13. Example: Corn Yield • 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.

  14. Example: Corn Yield • 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.

  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. Exploring Interaction between Factors in Two-Way ANOVA Figure 14.5 Mean Corn Yield, by Fertilizer and Manure Levels, Showing No Interaction.

  17. Exploring Interaction between Factors in Two-Way ANOVA • A graph showing interaction: Figure 14.6 Mean Corn Yield, by Fertilizer and Manure Levels, Displaying Interaction.

  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. Testing for Interaction • The test statistic providing the sample evidence of interaction is: • When is false, the F-statistic tends to be large.

  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. Example: Corn Yield Data • 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.

  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. Check Interaction Before Main Effects • 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.

  24. Check Interaction Before Main Effects • If important evidence of interaction exists, plot and compare the cell means for a factor separately at each category of the other factor.

  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. 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. 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.

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