Understanding Correlation Coefficient in Inferential Statistics

1. S519: Evaluation of Information Systems Social Statistics Inferential Statistics Chapter 13: correlation coefficient

2. This week • Testing correlation coefficient • The interpretation • PEARSON • Using Excel to calculate correlation coefficient

3. Which test to use • Figure 13.1 (p261) • The relationship between variables, and not the difference between groups, is being examined. • Only two variables are being used • The appropriate test statistic to use is the t test for the correlation coefficient

4. Example

5. Correlation coefficient • CORREL() and PEARSON() • Same value • There is no difference • Spearman’s rank correlation coefficient • Kendall's tau

6. T test for the significance of the correlation coefficient • Step1: A statement of the null and research hypotheses • Null hypothesis: there is no relationship between the quality of the marriage and the quality of the relationship between parents and children • Research hypothesis: (two-tailed, nondirectional) there is a relationship between the two variables

7. T test for the significance of the correlation coefficient • Step2: setting the level of risk (or the level of significance or Type I error) associated with the null hypothesis • 0.05 or 0.01 • What does it mean? • on any test of the null hypothesis, there is a 5% (1%) chance you will reject it when the null is true when there is no group difference at all. • Why not 0.0001? • So rigorous in your rejection of false null hypothesis that you may miss a true one; such stringent Type I error rate allows for little leeway

8. T test for the significance of the correlation coefficient • Step 3 and 4: select the appropriate test statistics • The relationship between variables, and not the difference between groups, is being examined. • Only two variables are being used • The appropriate test statistic to use is the t test for the correlation coefficient

9. T test for the significance of the correlation coefficient • Step5: determination of the value needed for rejection of the null hypothesis using the appropriate table of critical values for the particular statistic. • Table B4 • compute the correlation coefficient (r=0.393) • Compute df=n-2 (df=27) • If obtained value>the critical value reject null hypothesis • If obtained value<the critical value accept null hypothesis

10. T test for the significance of the correlation coefficient • Step6: compare the obtained value with the critical value • obtained value: 0.393 • critical value: 0.349

11. T test for the significance of the correlation coefficient • Step 7 and 8: make decisions • What could be your decision? And why, how to interpret? • obtained value: 0.393 > critical value: 0.349 (level of significance: 0.05) • Coefficient of determination is 0.154, indicating that 15.4% of the variance is accounted for and 84.6% of the variance is not. • There is a 5% chance that the two variables are not related at all

12. Causes and associations • Two variables are related to each other One causes another • having a great marriage cannot ensure that the parent-child relationship will be of a high quality as well; • The two variables maybe correlated because they share some traits that might make a person a good husband or wife and also a good parent; • It’s possible that someone can be a good husband or wife but have a terrible relationship with his/her children.

13. A critique • a correlation can be taken as evidence for a possible causal relationship, but cannot indicate what the causal relationship, if any, might be. • These examples indicate that the correlation coefficient, as a summary statistic, cannot replace the individual examination of the data.

14. Exercise: S-P267-Q1