Application of ANOVA

Explanation of One Way Anova

1. Business Statistics Presentation Presented by:- Siddharth Nahata Rohit Patidar Deepali Agarwal Rajat Srivastava Prachi Mandhani Sumant Singh Application of ANOVA

2. 2 STATITICAL DATA ANALYSIS COMMON TYPES OF ANALYSIS? Examine Strength and Direction of Relationships Bivariate (e.g., Pearson Correlation—r) Between one variable and another: r xy or Y = a + b 1 x 1 Multivariate (e.g., Multiple Regression Analysis) Between one dep. var . and each of several indep . variables, while holding all other indep . variables constant : Y = a + b 1 x 1 + b 2 x 2 + b 3 x 3 + . . . + b k x k Compare Groups Compare Proportions (e.g., Chi-Square Test—  2 ) H : P 1 = P 2 = P 3 = … = P k Compare Means (e.g., Analysis of Variance) H : µ 1 = µ 2 = µ 3 = …= µ k

3. ONE-WAY ANOVA To compare the mean values of a certain characteristic among two or more groups . To see whether two or more groups are equal (or different) on a given metric characteristic. 3 ANOVA was developed in 1919 by Sir Ronald Fisher, a British statistician and geneticist/evolutionary biologist When Do You Use ANOVA? Sir Ronald Fisher (1890-1962)

4. ONE-WAY ANOVA 4 H : There are no differences among the mean values of the groups being compared (i.e., the group means are all equal )– H : µ 1 = µ 2 = µ 3 = …= µ k H 1 (Conclusion if H rejected)? Not all group means are equal (i.e., at least one group mean is different from the rest ). H in ANOVA?

5. ONE-WAY ANOVA Scenario 1 . When comparing 2 groups, a one-step test : 2 Groups: A B Step 1: Check to see if the two groups are different or not, and if so, how. Scenario 2 . When comparing > 3 groups, if H is rejected , it is a two-step test : > 3 Groups: A B C Step 1: Overall test that examines if all groups are equal or not. And, if not all are equal (H rejected) , then: Step 2: Pair-wise ( post-hoc ) comparison tests to see where (i.e., among which groups ) the differences exit, and how . 5 So, the number of steps involved in ANOVA depend on if we are comparing 2 groups or > 2 groups:

6. ANOVA TABLE 6 Typical solution presented in statistics classes require… Constructing an ANOVA TABLE Test Statistic

7. ONE-WAY ANOVA Sample Data: A random sample of 9 banks, 10 retailers, and 10 utilities. Table 1. Earnings Per Share (EPS) of Sample Firms in the Three Industries Banking Retailing Utility 6.42 3.52 3.55 2.83 4.21 2.13 8.94 4.36 3.24 6.80 2.67 6.47 5.70 3.49 3.06 4.65 4.68 1.80 6.20 3.30 5.29 2.71 2.68 2.96 8.34 7.25 2.90 ----- 0.16 1.73 n B = 9 n R = 10 n U = 10 n = 29 H : There were no differences in average EPS of Banks, Utilities, and Retailers. First logical thing you do? _ _ _ = x B = 5.84 x R = 3.63 x U = 3.31 X = 4.21 7 EXAMPLE: Whether or not average earnings per share (EPS) for commercial banks , retailing operations, & utility companies (variable Industry) was the same last year.

8. EPS in various sectors 8

9. ONE-WAY ANOVA 9 Why is it called ANOVA ? Differences in EPS (Dep. Var.) among all 29 firms has two components -- differences among the groups and differences within the groups. That is, There are some differences in EPS among the three groups of firms (Banks vs. Retailers vs. Utilities), and There are also some differences/variations in EPS of the firms within each of these groups (among banks themselves, among retailers themselves, and among utilities themselves). ANOVA will partition/analyze the variance of the dependent variable (i.e., the differences in EPS) and traces it to its two components/sources --i.e., to differences between groups vs. differences within groups.

10. ONE-WAY ANOVA Table 1. Earnings Per Share (EPS) of Sample Firms in the Three Industries Banking Retailing Utility 6.42 3.52 3.55 2.83 4.21 2.13 8.94 4.36 3.24 6.80 2.67 6.47 5.70 3.49 3.06 4.65 4.68 1.80 6.20 3.30 5.29 2.71 2.68 2.96 8.34 7.25 2.90 ----- 0.16 1.73 n B = 9 n R = 10 n U = 10 n = 29 _ _ _ = x B = 5.84 x R = 3.63 x U = 3.31 X = 4.21 Total WITHIN Group Variance (or Mean Square WITHIN )? 10

11. Mean Square WITHIN Groups (MSW): 11 Called “ Degrees of Freedom ” = (n B -1)+(n R -1)+(n U -1) Let ’ s see what we just did: The generic mathematical formula for MSW: ONE-WAY ANOVA

12. ONE-WAY ANOVA 12 Table 1. Earnings Per Share (EPS) of Sample Firms in the Three Industries Banking Retailing Utility 6.42 3.52 3.55 2.83 4.21 2.13 8.94 4.36 3.24 6.80 2.67 6.47 5.70 3.49 3.06 4.65 4.68 1.80 6.20 3.30 5.29 2.71 2.68 2.96 8.34 7.25 2.90 ----- 0.16 1.73 n B = 9 n R = 10 n U = 10 n = 29 _ _ _ = x B = 5.84 x R = 3.63 x U = 3.31 x = 4.21 Let ’ s now compute the BETWEEN Group Variance ( Mean Square BETWEEN--MSB )?

13. Mean Square BETWEEN Groups (MSB): 13 Called Degrees of Freedom Let ’ s see what we just did: Mathematical formula for MSB: Weighted by respective group sizes ONE-WAY ANOVA

14. ONE-WAY ANOVA 14 Mean Square Between Groups = MSB = 17.698 MSB represents the portion of the total differences /variations in EPS (the dependent variable) that is attributable to (or explained by) differences BETWEEN groups (e.g., industries) That is, the part of differences in companies ’ EPS that result from whether they are banks, retailers, or utilities .

15. ONE-WAY ANOVA 15 Mean Square Within Groups (MS Residual/Error ) = MSW = 3.35 MSW represents: The differences in EPS (the dependent variable) that are due to all other factors that are not examined and not controlled for in the study (e.g., diversification level, firm size, etc.) Plus . . . The natural variability of EPS (the dependent variable) among members within each of the comparison groups (Note that even banks with the same size and same level of diversification would have different EPS levels).

16. ONE-WAY ANOVA 16 Now, let ’ s compare MSB & MSW: MSB = 17.6 and MSW = 3.35. QUESTION: Based on the logic of ANOVA, when would we consider two (or more) groups as different/unequal ? When MSB is significantly larger than MSW . QUESTION: What would be a reasonable index (a single number) that will show how large MSB is compared to MSW ? (i.e., a single number that will show if MSB is larger than, equal to, or smaller than MSW)?

17. Compare BETWEEN and WITHIN Group Variances/Mean Squares--Compute the F-Ratio: Ratio of MSB and MSW (Call it F-Ratio): What can we infer when F-ratio is close to 1 ? MSB and MSW are likely to be equal and, thus, there is a strong likelihood that NO difference exists among the comparison groups. How about when F-ratio is significantly larger than 1 ? The more F-ratio exceeds 1, the larger MSB is compared to MSW and, thus, the stronger would be the likelihood/evidence that group difference(s) exist. Results of the above computations are usually summarized in an ANOVA TABLE such as the one that follows: 17

18. ANOVA TABLE 18

19. ONE-WAY ANOVA For our sample companies , EPS difference across the three industries (MSB) is more than 5 times the EPS difference among firms within the industries (MSW) QUESTION: What is our null Hypothesis? QUESTION: Is the above F-ratio of 5.28 large enough to warrant rejecting the null? ANSWER: It would be if the chance of being wrong (in rejecting the null) does not exceed 5%. So, look up the F-value in the table of F-distribution (under appropriate degrees of freedom) to find out what the - level will be if, given this F-value, we decide to reject the null. Degrees of Freedom: v 1 = k – 1 = 2 v 2 = n – k = 26 19 Interpretation and Conclusion: QUESTION: What does the F = 5.28 mean, intuitively?

20. 20 F = 3.37 is significant at  = 0.05 (If F=3.37 and we reject H , 5% chance of being wrong) 11

21. 21 F = 4.27 is significant at  = 0.025. That is, if F=4.27 and we reject H , we would face 5% chance of being wrong. But, our F = 5.28 > 4.27 So, what can we say about our -level ? Will it be larger or smaller than 0.025?

22. ONE-WAY ANOVA 22 The odds of being wrong, if we decide to reject the null, would be less than 2.5% (i.e.,  < 0.025) . Would rejecting the null be a safe bet? Conclusion? Reject the null and conclude that the average EPS is NOT EQUAL FOR ALL GROUPS (industries) being compared . Our F = 5.28 > 4.27

23. 23 Thank You