Review of ANOVA & Inferences About The Pearson Correlation Coefficient

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

2. Review of ANOVA (1)

3. Review of ANOVA (2)

4. Review of ANOVA (3) S.V. SS DF MS Fc Fα -------------- ------ ------ ------ ----- ----- Systematic Effect 70 2 35 9.13 3.88 Random Effect 46 12 3.83 ------- ---------- ----- ----- ------- Total 116 14

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 η2 (eta-squared) • η2 = SSB / SST or η2 = 70/116 = .60 • 60 % of the variability in stress scores is explained by different treatments.

6. Practical Significance or Effect Size in ANOVA, Continued Source SS DF MS Fc Fαη2 --------- ------ ------ ----- ------- ----- ---- Between 70 2 35.0 9.13 3.88 .60 Within 46 12 3.83 ------- ------ ---- ----- ----- ------- Total 116 14

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

10. Calculation of Covariance & Correlation

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

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

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

15. 3. Determine Whether to Use a One or Two Tailed Test • H0 : ρXY = 0 • Ha : ρXY ≠ 0 • Ha : ρXY > or < 0 Two Tailed Test if no direction is specified One Tailed Test if direction is specified

16. 4. Calculating Test Statistics

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

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

19. Example of Correlations Between SAT and GPA scores • Formulate the Statistical Hypotheses. • Ho : ρXY = 0 Ha : ρXY ≠ 0 • α = 0.05 • Collect a sample of data, n = 12

20. Data

21. Calculation of Difference of Y and mean of Y

22. Calculation of Difference of X and Mean of X

23. Calculation of Product of Differences

24. Covariance & Correlation

25. Calculate t-statistics

26. Check Significance • Identify the Region (s) of Rejection. • tα = 2.228 • Make Statistical Decision and Form Conclusion. • tc < tα Fail to reject Ho • p-value = 0.095 > α = 0.05 Fail to reject Ho • Or use Table B-6: rc = 0.50 < rα =.576 Fail to reject Ho

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 r2 or the coefficient of determination. • In our example, r2 is .50 2 = .25 (The proportion of variance that is shared by the two variables).

28. Intuitions about Percent of Variance Explained

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

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