1. 1 Review of ANOVA & Inferences About The Pearson Correlation Coefficient Heibatollah Baghi, and Mastee Badii

2. 2 Review of ANOVA (1) SourceSSDFMSFcFc FαFα -------------------- -------------- Between70235.09.133.88 Within46123.83 ---------------------- ------- Total11614

3. 3 Review of ANOVA (2) Source SSDFMSFcFc FαFα -------------- ------------------------- Explained 70 2359.133.88 Unexplained 46 123.83 ------- --------------- ------- Total 116 14

4. 4 Review of ANOVA (3) S.V.SSDFMS F c F α ------------------------------------------ Systematic Effect 702359.133.88 Random Effect 46123.83 ---------------------------------- Total 11614

5. 5 Practical Significance or Effect Size in ANOVA Statistical significance does not provide information about the effect size in ANOVA. The index of effect size is η 2 (eta-squared) η 2 = SS B / SS T or η 2 = 70/116 =.60 60 % of the variability in stress scores is explained by different treatments.

6. 6 Practical Significance or Effect Size in ANOVA, Continued SourceSSDFMS F c F α η 2 -------------------------- ------------ ---- Between70235.09.133.88.60 Within46123.83 ---------------------------------- Total11614

7. 7 Sample Size in ANOVA To estimate the minimum sample size needed in ANOVA, you need to do the power analysis. Given the: α =.05, effect size =.10, and a power ( 1- beta) of.80, 30 subjects per group would be needed. (Refer to Table 7-7, page 178).

8. 8 Inferences About The Pearson Correlation Coefficient Refer to Session 5 GPA and SAT Example

9. 9 STUDENTSY(GPA)X(SAT) A1.6400-0.97-145.80141.43 B2.0350-0.57-195.80111.61 C2.2500-0.37-45.8016.95 D2.84000.23-145.80-33.53 E2.84500.23-95.80-22.03 F2.65500.034.200.13 G3.25500.634.202.65 H2.0600-0.5754.20-30.89 I2.4650-0.17104.20-17.71 J3.46500.83104.2086.49 K2.87000.23154.2035.47 L3.07500.43204.2087.81 Sum30.806550.0378.33 Mean2.57545.80 S.D.0.54128.73

10. 10 Calculation of Covariance & Correlation

11. 11 Population of visual acuity and neck size “scores” ρ=0 Sample 1 Etc Sample 2Sample 3 r = -0.8r = +.15r = +.02 Relative Frequency r: 0µr0µr The development of a sampling distribution of sample v:

12. 12 Steps in Test of Hypothesis 1.Determine the appropriate test 2.Establish the level of significance:α 3.Determine whether to use a one tail or two tail test 4.Calculate the test statistic 5.Determine the degree of freedom 6.Compare computed test statistic against a tabled/critical value Same as Before

13. 13 1. Determine the Appropriate Test Check assumptions: Both independent and dependent variable (X,Y) are measured on an interval or ratio level. Pearson’s r is suitable for detecting linear relationships between two variables and not appropriate as an index of curvilinear relationships. The variables are bivariate normal (scores for variable X are normally distributed for each value of variable Y, and vice versa) Scores must be homoscedastic (for each value of X, the variability of the Y scores must be about the same) Pearson’s r is robust with respect to the last two specially when sample size is large

14. 14 2. Establish Level of Significance α is a predetermined value The convention α =.05 α =.01 α =.001

15. 15 3. Determine Whether to Use a One or Two Tailed Test H 0 : ρ XY = 0 H a : ρ XY ≠ 0 H a : ρ XY > or < 0 Two Tailed Test if no direction is specified One Tailed Test if direction is specified

16. 16 4. Calculating Test Statistics

17. 17 5. Determine Degrees of Freedom For Pearson’s r df = N – 2

18. 18 6. Compare the Computed Test Statistic Against a Tabled Value α =.05 Identify the Region (s) of Rejection. Look up t α corresponding to degrees of freedom

19. 19 Formulate the Statistical Hypotheses. H o : ρ XY = 0 H a : ρ XY ≠ 0 α = 0.05 Collect a sample of data, n = 12 Example of Correlations Between SAT and GPA scores

20. 20 Data

21. 21 Calculation of Difference of Y and mean of Y

22. 22 Calculation of Difference of X and Mean of X

23. 23 Calculation of Product of Differences

24. 24 Covariance & Correlation

25. 25 Calculate t-statistics

26. 26 Identify the Region (s) of Rejection. t α = 2.228 Make Statistical Decision and Form Conclusion. t c < t α Fail to reject H o p-value = 0.095 > α = 0.05 Fail to reject H o Or use Table B-6: r c = 0.50 < r α =.576 Fail to reject H o Check Significance

27. 27 Practical Significance in Pearson r Judge the practical significance or the magnitude of r within the context of what you would expect to find, based on reason and prior studies. The magnitude of r is expressed in terms of r 2 or the coefficient of determination. In our example, r 2 is.50 2 =.25 (The proportion of variance that is shared by the two variables).

28. 28 Intuitions about Percent of Variance Explained

29. 29 Sample Size in Pearson r To estimate the minimum sample size needed in r, you need to do the power analysis. For example, Given the: α =.05, effect size (population r or ρ) = 0.20, and a power of.80, 197 subjects would be needed. (Refer to Table 9-1). Note: [ρ =.10 (small), ρ=.30 (medium), ρ =.50 (large)]

30. 30 Magnitude of Correlations ρ =.10 (small) ρ =.30 (medium) ρ =.50 (large)

31. 31 Factors Influencing the Pearson r Linearity. To the extent that a bivariate distribution departs from normality, correlation will be lower. Outliers. Discrepant data points affect the magnitude of the correlation. Restriction of Range. Restricted variation in either Y or X will result in a lower correlation. Unreliable Measures will results in a lower correlation.

32. 32 Take Home Lesson How to calculate correlation and test if it is different from a constant