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# Correlation & Regression

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### Correlation & Regression

1. 1. Correlation & Regression<br />Grant M. Heller<br />Stats 5030<br />
2. 2. Perfect Correlation<br />
3. 3. Perfect Correlation<br />
4. 4. Perfect Correlation<br />
5. 5. Perfect Correlations<br />
6. 6. Y = ???<br />
7. 7. Review of Concepts<br />Mean<br />Standard Deviation<br />Sum of Scores (SS)<br />variance<br />z-scores (Standard Scores)<br />
8. 8. Mean = sum of all scores divided by number of scores<br />Xi = {2, 4, 5, 7, 8}<br />Mean<br />
9. 9. Standard Deviation<br />
10. 10. Z-scores (Standard Scores)<br />Calculated by subtracting the mean from each observation and dividing the difference by the standard deviation.<br />
11. 11. Types of Relationships<br />Positive Relationships<br />Negative Relationship<br />No Relationship<br />Strong vs. Weak Relationship<br />Image from: http://member.tripod.com/~BDaugherty/KeySkills/lineGraphs.html#SCATTER<br />
12. 12. Correlation: strength of relationship<br />Pearson’s correlation coeffecient (r)<br />Any value between -1.00 and +1.00<br />Sign (+/-) indicates direction of relationship (whether positive or negative)<br />Absolute value of r indicates the strength of the linear relationship.<br />Effect size (Cohen, 1988)<br />r ≤ .10 : small (weak) relationship<br />r of .30 : medium (moderate) relationship<br />r ≥ .50 : large (strong) relationship <br />Animation from: http://www.ats.ucla.edu/stat/sas/teach/corrrelation/corr.htm<br />
13. 13. Correlation Coefficient<br />Z Score Formula:<br />To calculate r, w need to:<br />1)Calculate means and standard deviations for variables X and Y<br />2) Calculate Standard Scores (z scores) for all variable X and Y observations<br />3) Multiply the z scores for X & Y (Zx & Zy) for each pair of observations.<br />4) Sum the product of Zx * Zy<br />5) Divide the results by the number of paired observations<br />
14. 14. Calculation of r from example problemusing z-score method<br />
15. 15. Calculation of r: computational formula<br />
16. 16. Steps for calculation of r<br />1) determine the number of paired observations (n).<br />2) sum all scores for X and for Y separately<br />3) find the product of each pair of X & Y scores (multiply)<br />4) sum the products of X & Y scores – save this #<br />5) square each X score and sum them up <br />do the same for each Y score, and save these 2 numbers.<br />
17. 17. Calculation of r: an example<br />
18. 18. Calculation of r: an example<br />
19. 19. Regression: Line of best fit<br />
20. 20. Least Squares Regression Equation<br />Least Square Regression Equation<br />Finding Values for b and a<br />Solving for b<br />Solving for a<br />
21. 21. Least Squares Regression: Example<br />
22. 22. Finding the Standard Error of Estimate<br />Standard Error of Estimate (definition formula)<br />Standard Error of Estimate (computation formula)<br />
23. 23. Standard Error of Estimate: example<br />
24. 24. Assumptions & Limitations<br />Assumptions<br />Linearity<br />Homoscedasticity<br />Adequate range<br />Limitations<br />Question of direction of effect<br />Does X influence Y or does Y influence X?<br />
25. 25. References<br />Caldwell, S. (2004). Statistics Unplugged (2nd ed.). Belmont, CA: Thompson.<br />Cohen, B. H. (2008). Explaining Psychological Statistics (3rd ed.). Hoboken, NJ: Wiley.<br />Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum.  <br />Introduction to SAS. UCLA: Academic Technology Services, Statistical Consulting Group. from http://www.ats.ucla.edu/stat/sas/notes2/ (accessed October 4, 2010). <br />Green, S. B., & Salkind, N. J. (2003). Using SPSS for Windows and Macintosh: Analyzing and Understanding Data (3rd ed.). Upper Saddle River, NJ: Prentice Hall.<br />Witte, R. S., & Witte, J. S. Cohen, J. (1988). Statistics (8th ed.). Hoboken, NJ: Wiley.<br />