Design and Data Analysis in Psychology I

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

2. Relationship between two quantitative variables Lesson 11

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.

4. PEARSON CORRELATION: DEFINITION • rXY • 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 ≤ rXY ≤ +1. • Interpretation: +1: perfect positive correlation (direct association). -1: perfect negative correlation (inverse association). 0: no correlation.

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

6. Positive correlation: 0 < rxy < +1

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

8. Negative correlation: -1 < rxy < 0

9. No correlation

10. Formulas 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 • Calculate rxy in raw scores. • Calculate rxy in deviation scores. • Calculate rxy in standard scores.

12. Example: scatter plot

13. Example :calculation of rxy in raw scores

14. Example :calculation of rxy in raw scores

15. Example :calculation of rxy in deviation scores

16. Example :calculation of rxy in deviation scores

17. Example :calculation of rxy in standard scores

18. Example :calculation of rxy in standard scores

19. Significance • Does the correlation coefficient show a real relationship between X and Y, or is that relationship due to hazard? • Null hypothesis  H0: rxy = 0. The correlation coefficient is drawn from a population whose correlation is zero (ρXY = 0). • Alternative hypothesis  H1: . The correlation coefficient is not drawn from a population whose correlation is different to zero (ρXY ).

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

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 ρxy = 0. Relationship between variables exists.

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.

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

24. Which is the final conclusion?

25. Which is the final conclusion?