2. USES OF CORRELATION Test reliability Test validity Predict future scores (regression) Test hypotheses about relationships between variables

3. Doing Correlational Research Any of the descriptive methods can be used (observation, survey, archival, physical traces). Must have pairs of scores

4. Doing Correlational Research The pairs of scores should be independent. Both variables should be normally distributed. If there is a relationship between variables, it should be linear.

5. Correlation and Cause A correlation by itself does not show that one variable causes the other. A correlation is consistent with a causal relationship.

6. The Third Variable Problem A correlation between X and Y could be caused by a third variable influencing both X and Y. example: The use birth control is correlated with the number of electrical appliances in the household.

7. The Directionality Problem A correlation between X and Y could be a result of X causing Y or Y causing X example: Amount of TV watching and the level of aggression are correlated.

8. The Cross-Lagged Panel Design used to determine direction of cause Measure both variables at two different points in time Cause cannot work backwards in time

9. Time 1Time 2variable Xvariable Y

10. The Correlation Coefficient Strength of relationship – 0 means no relationship at all – -1 or +1 means perfectly related Direction of relationship – positive: variable X increases as variable Y increases – negative: variable X decreases as variable Y increases

11. The Scatterplot X Y low high o o o o o o o o o o o o o o o o o o o o o o o o o o o positive r

12. X Y low high o o o o o o o o o o o o o o o o o o o o o o o o o o o negative r

13. X Y low high o o o o o o o o o o o o o o o o o o o o o o o o o o o zero r

14. X Y low high o o o o o o o o o o o o o o o o o o o o o o o o o o o Non-linear relationship

15. Correlation Coefficients X dataY dataCoefficient interval/ratiointerval/ratioPearson r ordinalordinalSpearman rho dichotomousinterval/ratioPoint Biserial dichotomous dichotomousPhi dichotomous: having only two values

16. More on Dichotomous Variables With dichotomous variables, whether r is negative or positive depends on how the numbers were assigned

17. More on Dichotomous Variables If the correlation between gender and GPA is positive, it could mean that – females have higher GPAs, if males were 1’s and females were 2’s – males have higher GPAs, if females were 1’s and males were 2’s

18. Pearson r formula

19. Computation of Pearson r Example: Compute the correlation between scores on Exam1 and Exam2. StudentExam1Exam2 19786 28295 37479 48995 59390

20. STEP 1:Convert the x scores to z-scores. Exam1x-  (x-  ) 2 z x 9710100+1.22 82-525-.61 74-13169-1.59 8924+.24 93636+.73  =87  =334  x = 334/5 = 8.17

21. STEP 2:Convert the y scores to z-scores. Exam2y-  (y-  ) 2 z y 86-39-.50 95636+1.00 79-10100-1.66 95636+1.00 9011+.17  =89  =182  y = 182/5 = 6.03

22. STEP 3: Multiply the z-scores. z x z y +1.22-.50-.61 -.61+1.00-.61 -1.59-1.66+2.64 +.24+1.00+.24 +.73+.17+.12

23. STEP 4: Add up the z x z y products. z x z y -.61 +2.64 +.24 +.12  = 1.78

24. STEP 5: Divide  z x z y by N.

25. Coefficient of Determination Measures proportion of explained variance in Y based on X. Square r to get r 2. Example: r =.36r 2 =.13 We can explain 13% of the differences in Exam 2 scores by knowing Exam 1 scores.

26. What Could a Low r Mean? Lack of a relationship. Unreliable measurement. Non-linear relationship. Restricted range : full range of scores not measured on both variables