Application of ANOVA
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This document presents an analysis of statistical data comparing common types of analysis such as examining relationships between variables and comparing groups. It then provides an example comparing the average earnings per share of banks, retailers, and utilities using analysis of variance (ANOVA). The ANOVA calculates between-group and within-group variances and uses an F-ratio test to determine if there are statistically significant differences between the industry groups' means. The analysis finds an F-ratio of 5.28 which is greater than the critical value of 4.27, allowing the researchers to reject the null hypothesis that the group means are equal and conclude that average earnings per share differs between industries.
Application of ANOVA
- 1. Presentation Presented by:- 1. Siddharth Nahata 2. Rohit Patidar 3. Deepali Agarwal 4. Rajat Srivastava 5. Prachi Mandhani 6. Sumant Singh
- 2. 2 STATITICAL DATA ANALYSIS COMMON TYPES OF ANALYSIS? 1. Examine Strength and Direction of Relationships a. Bivariate (e.g., Pearson Correlation—r) Between one variable and another: rxy or Y = a + b1 x1 b. 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 + b1 x1 + b2 x2 + b3 x3 + . . . + bk xk 2. Compare Groups a. Compare Proportions (e.g., Chi-Square Test—2) H0: P1 = P2 = P3 = … = Pk b. Compare Means (e.g., Analysis of Variance) H0: µ1 = µ2 = µ3 = …= µk
- 3. • 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. 4 H0: There are no differences among the mean values of the groups being compared (i.e., the group means are all equal)– H0: µ1 = µ2 = µ3 = …= µk H1 (Conclusion if H0 rejected)? Not all group means are equal (i.e., at least one group mean is different from the rest). H0 in ANOVA?
- 5. • 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 H0 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 (H0 rejected), then: Step 2: Pair-wise (post-hoc) comparison tests to see where (i.e., 5 So, the number of steps involved in ANOVA depend on if we are comparing 2 groups or > 2 groups:
- 6. Sum of Squares df Mean Squares F-Ratio SSB (Between Groups Sum Of Squares) K – 1 MSB = SSB / K-1 F = MSB / MSW corresponding SSW (Within Groups Sum of Squares) N – K MSW = SSW / N-K SST (Total Sum of Squares) N – 1 Kn xxxxxx kkiii 222 )(...)()( MSW 2211 1 22 22 2 11 )(...)()( K kknnn xxxxxx MSB 6 Typical solution presented in statistics classes require… • Constructing an ANOVA TABLE Test Statistic
- 7. • 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 nB = 9 nR = 10 nU = 10 n = 29 H0: There were no differences in average EPS of Banks, Utilities, and Retailers. First logical thing you do? _ _ _ = xB = 5.84 xR = 3.63 xU = 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. 8 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 1 2 3 4 5 6 7 8 9 10 Banking Retailing Utility
- 9. 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, a. There are some differences in EPS among the three groups of firms (Banks vs. Retailers vs. Utilities), and b. 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. • 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 nB = 9 nR = 10 nU = 10 n = 29 _ _ _ = xB = 5.84 xR = 3.63 xU = 3.31 X = 4.21 Total WITHIN Group Variance (or Mean Square WITHIN)? 10)310109( )31.373.1()31.355.3()63.316.0(...)63.352.3()84.534.8(...5.84)-(6.42 MSW 222222
- 11. 11 K)-(N WithinSS GroupsofNumber-SizeSampleTotal MeansGroupRespectiveTheirFromnsObservatioAllofDeviationsSquaredofSum MSW K xxxxxx n UUiRRiBBi 222 )()()( MSW 350.3 26 112.87 MSW Called “Degrees of Freedom”= (nB-1)+(nR-1)+(nU-1) )310109( )31.373.1()31.355.3()63.316.0(...)63.352.3()84.534.8(...5.84)-(6.42 MSW 222222 Let’s see what we just did: The generic mathematical formula for MSW: ONE-WAY ANOVA
- 12. 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 nB = 9 nR = 10 nU = 10 n = 29 _ _ _ = xB = 5.84 xR = 3.63 xU = 3.31 x = 4.21 Let’s now compute the BETWEEN Group Variance (MeanSquare BETWEEN--MSB)? 698.17 2 397.35 13 )21.431.3(10)21.463.3(10)21.484.5(9 222 MSB
- 13. 698.17 2 397.35 13 )21.431.3(10)21.463.3(10)21.484.5(9 222 MSB 13 1-K BetweenSS 1-GroupsofNumber MeanGrandthefromMeansGroupofDeviationsSquaredofSum MSB Called Degrees of Freedom 1 )()()( 222 K xxnxxnxxn MSB uuRRBB Let’s see what we just did: Mathematical formula for MSB: Weighted by respective group sizes ONE-WAY ANOVA
- 14. 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. 15 Mean Square Within Groups (MS Residual/Error) = MSW = 3.35 MSW represents: a. 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 . . . b. 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. 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. • 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: 17 282.5 350.3 698.17 MSW MSB F
- 18. Source Sum of Squares df Mean Squares F Between Groups 35.397 K – 1 = 2 35.39 / 2 = 17.698 17.698 / 3.35 = 5.282 Within Groups 87.112 N – K = 26 87.11 / 26 = 3.350 Total 122.509 N – 1 = 28 698.17 2 397.35 13 )21.431.3(10)21.463.3(10)21.484.5(9 222 MSB 18 350.3 26 112.87 MSW )310109( )31.373.1()31.355.3()63.316.0(...)63.352.3()84.534.8(...5.84)-(6.42 MSW 222222
- 19. 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: v1 = k – 1 = 2 v2 = n – k = 26 19 Interpretation and Conclusion: QUESTION: What does the F = 5.28 mean, intuitively?
- 20. 20F = 3.37 is significant at = 0.05 (If F=3.37 and we reject H0, 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 H0, 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. 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