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

2 Review of ANOVA (1) SourceSSDFMSFcFc FαFα Between Within Total11614

3 Review of ANOVA (2) Source SSDFMSFcFc FαFα Explained Unexplained Total

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

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 = % of the variability in stress scores is explained by different treatments.

6 Practical Significance or Effect Size in ANOVA, Continued SourceSSDFMS F c F α η Between Within Total11614

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 5 GPA and SAT Example

9 STUDENTSY(GPA)X(SAT) A B C D E F G H I J K L Sum Mean S.D

10 Calculation of Covariance & Correlation

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 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 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 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 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 t α corresponding to degrees of freedom

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 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 Identify the Region (s) of Rejection. t α = Make Statistical Decision and Form Conclusion. t c < t α Fail to reject H o p-value = > 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 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 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 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 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