The Analysis of Variance

1. 10 The Analysis of Variance

2. 10.1 Single-Factor ANOVA

3. Single-Factor ANOVA • Single-factor ANOVA focuses on a comparison of more than two population or treatment means. Let • l = the number of populations or treatments being compared • 1 = the mean of population 1 or the true average response when treatment 1 is applied • . • . • . • I = the mean of population I or the true average response when treatment I is applied

4. Single-Factor ANOVA • The relevant hypotheses are • H0: 1 = 2 = ··· = I • versus • Ha: at least two the of the i’s are different • If I =4, H0 is true only if all four i’s are identical. Ha would be true, for example, if 1 = 23 = 4, if 1 = 3 = 42, or if all four i’s differ from one another.

5. The Idea of ANOVA • The sample means for the three samples are the same for each set. • The variation among sample meansfor (a) is identical to (b). • The variation among the individuals within the three samples is much less for (b). • CONCLUSION: the samples in (b) contain a larger amount of variation among the sample means relative to the amount of variation within the samples, so ANOVA will find more significant differences among the means in (b) • assuming equal sample sizes here for (a) and (b). • Note: larger samples will find more significant differences.

6. Comparing Several Means Do SUVs, trucks and midsize cars have same gas mileage? • Response variable: gas mileage (mpg) • Groups: vehicle classification • 31 midsize cars • 31 SUVs • 14 standard-size pickup trucks Data from the Environmental Protection Agency’s Model Year 2003 Fuel Economy Guide, www.fueleconomy.gov.

7. Comparing Several Means Means: Midsize: 27.903 SUV: 22.677 Pickup: 21.286 • Mean gas mileage for SUVs and pickups appears less than for midsize cars. • Are these differences statistically significant?

8. Comparing Several Means Means: Midsize: 27.903 SUV: 22.677 Pickup: 21.286 Null hypothesis: The true means (for gas mileage) are the same for all groups (the three vehicle classifications). We could look at separate t tests to compare each pair of means to see if they are different: 27.903 vs. 22.677, 27.903 vs. 21.286, & 22.677 vs. 21.286 H0: μ1 = μ2H0: μ1 = μ3H0: μ2 = μ3 However, this gives rise to the problem of multiplecomparisons!

9. The One-Way ANOVA Model Random sampling always produces chance variations. Any “factor effect” would thus show up in our data as the factor-driven differences plus chance variations (“error”): Data = fit + residual The one-way ANOVA model analyzes situations where chance variations are normally distributed N(0,σ) such that:

10. The ANOVA F Test To determine statistical significance, we need a test statistic that we can calculate: The ANOVA F Statistic The analysis of variance F statistic for testing the equality of several means has this form: Difference in means small relative to overall variability Difference in means large relative to overall variability  F tends to be large  F tends to be small Larger F-values typically yield more significant results. How large depends on the degrees of freedom (I− 1 and N− I).

11. The ANOVA F Test • The measures of variation in the numerator and denominator are mean squares: • Numerator: Mean Square for Treatments (MSTr) • Denominator: Mean Square for Error (MSE)

12. Notation • The individual sample means will be denoted by X1, X2, . . ., XI. • That is, • for i=1,…,I • Similarly, the average of all N observations, called the grand mean, is

13. Notation • Additionally, let , denote the sample variances: • for i=1,…,I

14. The ANOVA Table • The computations are often summarized in a tabular format, called an ANOVA table in below Table. • Tables produced by statistical software customarily include a P-value column to the right of f. • An ANOVA Table

15. F Distributions and the F Test • Both v1 and v2 are positive integers. Figure 10.3 pictures an F density curve and the corresponding upper-tail critical value Appendix Table A.9 gives these critical values for  = .10, .05, .01, and .001. • Values of v1 are identified with different columns of the table, and the rows are labeled with various values of v2. • An F density curve and critical value • Figure 10.3

16. Nematodes and plant growth Hypotheses: All mi are the same (H0) versus not All mi are the same (Ha) Do nematodes affect plant growth? A botanist prepares 16 identical planting pots and adds different numbers of nematodes into the pots. Seedling growth (in mm) is recorded two weeks later.

17. Output for the one-way ANOVA numerator denominator Here, the calculated F-value (12.08) is larger than Fcritical (3.49) for a=0.05. Thus, the test is significant at a 5%  Not all mean seedling lengths are the same; the number of nematodes is an influential factor.

18. Using F-table The F distribution is asymmetrical and has two distinct degrees of freedom. This was discovered by Fisher, hence the label “F.” Once again, what we do is calculate the value of F for our sample data and then look up the corresponding area under the curve in F-Table.

19. Fcritical for a 5% is 3.49 F = 12.08 > 10.80 Thus p< 0.001