Statistical Relationship

1. Statistical Relationship (Lesson – 02E) As One Set of Data Move in One Direction What Do the Other Set of Data Do? Dr. C. Ertuna

2. Statistical Relationship The price for TRC stock and S&P 500 index are given on the left. Is there any relationship between those two? To answer this question we need to measure the “Statistical Relationship” between those two. Data: St-CE-Ch02-x1-Examples-Slide 60 Dr. C. Ertuna

3. Statistical Relationship The descriptive statistics that measures the degree of relation between 2 variables are called correlation coefficients. Three measures for statistical relationship are: Dr. C. Ertuna

4. Statistical Relationship (Cont.) • Pearson Correlation coefficient (ρ, r) measures the strength of linear relationship between two variables (X and Y) assuming normal distribution. {significance!} Dr. C. Ertuna

5. Statistical Relationship (Cont.) • Correlation coefficient will range from -1 to +1 • A correlation of 0 indicates that there is no linear relationship between two variables • Even a high correlation could be observed just by chance; to be sure we need to run a statistical test. • Correlation between two variables does not mean causal relationship between them • Correlation Matrix provides pair-wise correlation between more than two variables. Dr. C. Ertuna

6. Statistical Relationship (Cont.) Square of Pearson’s r (r2)can be interpreted as explained variance if there is a Dependent Variable (DV) Independent Variable (IV) relationship exists. • For example if r = 0.933 than r2 = 0.87049 that means IV explains 87.05% of the variations in the DV. Dr. C. Ertuna

7. Correlation Test Assumptions Parametric Correlation Test • Pearson’s r: • Interval data • Normality • Equal Variance (not needed if n > 30) • Linearity Dr. C. Ertuna

8. Statistical Relationship (Cont.) • Kendall's tau-b A distribution-free (nonparametric) measure of association for ordinal (or ranked) variables that take ties into account. The sign of the coefficient indicates the direction of the relationship, and its absolute value indicates the strength, with larger absolute values indicating stronger relationships. Possible values range from -1 to 1. Dr. C. Ertuna

9. Statistical Relationship (Cont.) • Spearman’s rho Commonly used distribution-free (nonparametric) measure of correlation between two ordinal variables. For all of the cases, the values of each of the variables are ranked from smallest to largest, and the Pearson correlation coefficient is computed on the ranks. Dr. C. Ertuna

10. Correlation Test Assumptions Non-Parametric Correlation Test • Kandell’s tau-b & Spearman’s rho: • Ordinal data • Monotonicity Dr. C. Ertuna

11. 2-Test Chi-square test is suitable for analyzing nominal and ordinal data. (Interval and ratio data should be grouped first) Chi-square test is used for - Goodness-of-fit (1-Way classification; 1-DV, 1-IV) - Test for independence (2-Way classification; 1-DV, 2+IV) Categorical data in Rows Ordinal data in Columns Dr. C. Ertuna

12. 2-Test Assumptions • Categorical data • Any cell’s raw frequency > 5 • Random Sampling Dr. C. Ertuna

13. 2-Test PHStat2 / Multiple-Sample Tests / / Chi-Square Test Significance Level: to be entered Number of Raws: to be entered Number of Columns: to be entered If p_value < 0,05 There is a relationship Dr. C. Ertuna

14. 2-Test Strength of the Relationship is measured by Where N = total number of observations k = min( #rows, #columns) Dr. C. Ertuna

15. 2-Test • Cramer’s V has a value between 0 and 1 • Where 0 means independence or no relationship and 1 means perfect relation ship. Dr. C. Ertuna

16. Interpreting the Association Although there is no theoretical guideline on how to interpret the value of association, here are some guidelines: • Interpret the squared value of the association • 1.00 – 0.80 High (strong) association • 0.80 – 0.60 Moderately high association • 0.60 – 0.40 Moderate association • 0.40 – 0.20 Weak association • 0.20 – 0.00 Very weak association Dr. C. Ertuna

17. Interpreting of Cramer’s V Although there is no theoretical guideline on how to interpret the value of association, here are some guidelines for Cramer’s V: • 1.00 – 0.40 Worrisomely High (strong) association • 0.40 – 0.35 Very High (strong) association • 0.35 – 0.30 High (strong) association • 0.30 – 0.25 Moderately high association • 0.25 – 0.20 Moderate association • 0.20 – 0.10 Weak association • 0.10 – 0.00 Very weak association/Not acceptable Dr. C. Ertuna

18. SPSS – Nominal Association Cathegories should be coded first: • Data / Weight Cases / (variable that stands for frequencies) • Analyze / Discriptives / Crosstabs / Dr. C. Ertuna

19. Example: Statistical Relationship The price for TRC stock and S&P 500 index are given on the left. 1. Compute the correlation between S&P500 and TRC 2. Explain the meaning of the result. Data: St-CE-Ch02-x1-Examples-Slide 60 Dr. C. Ertuna

20. Example: Statistical Relationship • Analyze/Correlate/ Bivariate • Select the variables & move to the right pane • Select Pearson, Kendall’s tau_b, Spearman. (2-tailed* ; 1-tailed) • Ok Dr. C. Ertuna

21. Example: Statistical Relationship Dr. C. Ertuna

22. Example: Statistical Relationship (cont.) Data: St-CE-Ch02-x1-Examples-Slide 60 Dr. C. Ertuna

23. Example: Statistical Relationship (cont.) The correlation between Tracway stock price and S&P 500 index is 0.93. • Explain the meaning of the result. Dr. C. Ertuna

24. Meaning: Statistical Relationship (cont.) Pearson correlation coefficient of 0.93 indicates thatthere is a • strong (most of the time holding) , • positive (when one changes, the other one changes in the same direction) , • linear relationship between S&P 500 index and Tracway stock price (assuming both normally distributed). Dr. C. Ertuna

25. Next Lesson (Lesson - 03A) Random Variables & Probability Distribution Dr. C. Ertuna