1. Design and Data Analysis in Psychology I Salvador Chacón Moscoso Susana Sanduvete Chaves School of Psychology Dpt. Experimental Psychology 1

2. Lesson 11 Relationship between two quantitative variables 2

3. INTRODUCTION  When assumptions are accepted (parametric tests): Simple linear regression (it is going to be studied next academic year in the subject Design and Data Analysis in Psychology II). Pearson correlation.  When assumptions are not accepted (non- parametric tests): Spearman correlation. 3

4. PEARSON CORRELATION: DEFINITION r XY Coefficient useful to measure covariation between variables: in which way changes in a variable are associated to the changes in other variable. Quantitative variables (interval or ratio scale). Linear relationship EXCLUSIVELY. Values: -1 ≤ r XY ≤ +1. Interpretation: +1: perfect positive correlation (direct association). -1: perfect negative correlation (inverse association). 0: no correlation. 4

5. 5 Perfect positive correlation: r xy = +1 (difficult to find in psychology)

6. 6 Positive correlation: 0 < r xy < +1

7. 7 Perfect negative correlation: r xy = -1 (difficult to find in psychology)

8. 8 Negative correlation: -1 < r xy < 0

9. 9 No correlation

10. Formulas 10 Raw scores Deviation scores Standard scores

11. Example X: 2 4 6 8 10 12 14 16 18 20 Y:1 6 8 10 12 10 12 13 10 22 1. Calculate r xy in raw scores. 2. Calculate r xy in deviation scores. 3. Calculate r xy in standard scores. 11

12. Example: scatter plot 12

13. Example : calculation of r xy in raw scores XYXYX2X2 Y2Y2 21241 46241636 68483664 8108064100 1012120100144 1210120144100 1412168196144 1613208256169 1810180324100 2022440400484 110104139015401342 13

14. Example : calculation of r xy in raw scores 14

15. Example : calculation of r xy in deviation scores XYxyxyx2x2 y2y2 21-9-9.484.68188.36 46-7-4.430.84919.36 68-5-2.412255.76 810-3-0.41.290.16 10121.6-1.612.56 12101-0.4 10.16 141231.64.892.56 161352.613256.76 18107-0.4-2.8490.16 2022911.6104.481134.56 11010400246330260.4 15

16. Example : calculation of r xy in deviation scores 16

17. Example : calculation of r xy in standard scores XYZxZyZxZy 21-1.567-1.8422.886 46-1.218-0.8621.051 68-0.870-0.4700.409 810-0.522-0.0780.041 1012-0.1740.314-0.055 12100.174-0.078-0.014 14120.5220.3140.164 16130.8700.5100.443 18101.218-0.078-0.096 20221.5672.2733.561 110104008.391 17

18. Example : calculation of r xy in standard scores 18

19. Significance  Does the correlation coefficient show a real relationship between X and Y, or is that relationship due to hazard?  Null hypothesis  H 0 : r xy = 0. The correlation coefficient is drawn from a population whose correlation is zero (ρ XY = 0).  Alternative hypothesis  H 1 :. The correlation coefficient is not drawn from a population whose correlation is different to zero (ρ XY ). 19

20. Significance  Formula:  Interpretation:  Null hypothesis is rejected. The correlation is not drawn from a population whose score ρ xy = 0. Significant relationship between variables exists.  Null hypothesis is accepted. The correlation is drawn from a population whose score ρ xy = 0. Significant relationship between variables does not exist.  Exercise: conclude about the significance of the example. 20

21. Significance: example Conclusions: we reject the null hypothesis with a maximum risk to fail of 0.05. The correlation is not drawn from a population whose score ρ xy = 0. Relationship between variables exists. 21

22. Other questions to be considered  Correlation does not imply causality.  Statistical significance depends on sample size (higher N, likelier to obtain significance).  Other possible interpretation is given by the coefficient of determination, or proportion of variability in Y that is ‘explained’ by X.  The proportion of Y variability that left unexplained by X is called coefficient of non-determination:  Exercise: calculate the coefficient of determination and the coefficient of non-determination and interpret the results. 22

23. Coefficient of determination: example 70.4% of variability in Y is explained by X. 29.6% of variability in Y is not explained. 23

24. Which is the final conclusion? Significant effect Non-significant effect High effect size ( ≥ 0.67) The effect probably exists The non- significance can be due to low statistical power Low effect size ( ≤ 0.18) The statistical significance can be due to an excessive high statistical power The effect probably does not exist 24

25. Which is the final conclusion? Significant effect Non-significant effect High effect size ( ≥ 0.67) The effect probably exists The non- significance can be due to low statistical power Low effect size ( ≤ 0.18) The statistical significance can be due to an excessive high statistical power The effect probably does not exist 25