1. B AD 6243: Applied Univariate Statistics Correlation Professor Laku Chidambaram Price College of Business University of Oklahoma

2. BAD 6243: Applied Univariate Statistics 2 Correlation Is a measure of association between two (usually) interval/ratio level variables The correlation coefficient, r, refers to the strength (or magnitude) of the association in the sample; it serves as an estimate for the population parameter,  When computing r, three outcomes are possible: –positive correlation (both scores move together in the same direction); –negative correlation (one score moves in one direction, while the other moves in the opposite direction) –No correlation (no systematic movement)

3. BAD 6243: Applied Univariate Statistics 3 Scatter Plots of Correlations Variable 1 Variable 2 Variable 1 Variable 2 Variable 1 Variable 2 Variable 1 Variable 2

4. BAD 6243: Applied Univariate Statistics 4 An Example

5. BAD 6243: Applied Univariate Statistics 5 Results of the Analysis

6. BAD 6243: Applied Univariate Statistics 6 A Graphical Representation

7. BAD 6243: Applied Univariate Statistics 7 Significance of Correlation Is r significantly different from zero? –H 0 :  xy = 0 –H a :  xy  0 One-tailed vs. two-tailed tests Significance level is affected by sample size, so even “small” correlation coefficients can be significant, if sample sizes are large However, despite statistically significant correlations, their practical relevance may be difficult to determine

8. BAD 6243: Applied Univariate Statistics 8 Coefficient of Determination R 2 refers to the shared variance between variables and is obtained by squaring the correlation coefficient, r Variance of y Variance of x

9. BAD 6243: Applied Univariate Statistics 9 Some Cautionary Notes Problem of outliers Correlation is not causation It only summarizes linear relationships The “third variable” problem Restriction in range of values Use of extreme groups

10. BAD 6243: Applied Univariate Statistics 10 Impact of Outliers Pearson’s Corr Coefficient, r = -0.473 (p =.198) Kendall’s tau = 0.515 (p =.066) Spearman’s rho = 0.429 (p =.249)

11. BAD 6243: Applied Univariate Statistics 11 What to Use? Pearson’s Product Moment Correlation: –Interval/Ratio level variables –Interval/Ratio and dichotomous variables Spearman’s Rho: –Non-normal distributions –Ordinal variables Kendall’s Tau: –Non-normal distributions –Ordinal and dichotomous variables –Small sample with numerous tied ranks

12. BAD 6243: Applied Univariate Statistics 12 Partial Correlation A partial correlation refers to the correlation between two variables, with the influence of a third variable removed from both In other words, in our example you become aware that wage rates are not influenced by overall work experience per se, but relevant work experience (say, in a similar job) So, we examine if this proposition is true by looking at the partial correlation

13. Results of Analysis