1. C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Designing Experiments In designing experiments we: Manipulate the independent variable and hold all other conditions constant. Any observed changes in the dependent variable can be attributed to the manipulation of the independent variable.

2. C82MCP Diploma Statistics School of Psychology University of Nottingham 2 Correlational Studies Sometimes the manipulation of independent variables is only part of what we do Sometimes we are more interested in finding out whether two variables are related to each other. For Example: Are IQ and reading ability associated with each other In other words, Do high IQ scores and high reading ability scores occur together and low IQ scores and reading ability scores occur together

3. C82MCP Diploma Statistics School of Psychology University of Nottingham 3 Correlational Studies When we ask about the relationship between two variables we have a correlational design The null hypothesis associated with correlational designs is that there is no relationship between variables The alternative hypothesis associated with correlational designs is that there is a relationship between variables

4. C82MCP Diploma Statistics School of Psychology University of Nottingham 4 Covariance - A Measure of Association Covariance is defined as the sum of the product of two variables The value of the covariance is closely related to the relationship between X and Y The larger the absolute value of the covariance the stronger the relationship between X and Y. The sign of the covariance tells us about the kind of association

5. C82MCP Diploma Statistics School of Psychology University of Nottingham 5 Covariance - A Measure of Association A Problem With Covariance When there is an association between two variables the value of the covariance depends on: The degree or size of the association The ranges of the two variables

6. C82MCP Diploma Statistics School of Psychology University of Nottingham 6 Pearson's Product-Moment Coefficient of Correlation If we divide the covariance of two variables by the standard deviation of each variable we can account for the differences between the ranges Pearson's Product-Moment Coefficient of Correlation, r, does precisely this or

7. C82MCP Diploma Statistics School of Psychology University of Nottingham 7 Pearson's Product-Moment Coefficient of Correlation The absolute value of this coefficient always lies between zero and one regardless of the different ranges of the variables The sign of the coefficient tells us what kind of association we have There are tables of values that estimate the probability of getting a particular value for Pearson's product-moment coefficient. Using these we can test the null hypothesis of no association between two variables.

8. C82MCP Diploma Statistics School of Psychology University of Nottingham 8 An Example of a Positive Correlation Suppose a researcher measured the reading age and IQ of six 8 year old children The results of the measurement are placed on a scatter graph.

9. C82MCP Diploma Statistics School of Psychology University of Nottingham 9 An Example of a Positive Correlation Given these data: The critical value of r for N=6 is 0.811 For a correlation to be significant r observed  r critical For these data there is no evidence of a significant association between IQ and Reading Age.

10. C82MCP Diploma Statistics School of Psychology University of Nottingham 10 An Example of a Negative Correlation Suppose a researcher measured the reading age and the emotional and behavioural difficulties of six 8 year old children The results of the measurement are placed on a scatter graph.

11. C82MCP Diploma Statistics School of Psychology University of Nottingham 11 Given these data: The critical value of r for N=6 is 0.811 For a correlation to be significant r observed  r critical For these data there is no evidence of a significant association between EBD and Reading Age. An Example of a Negative Correlation

12. C82MCP Diploma Statistics School of Psychology University of Nottingham 12 Spearman's Rank Correlation Coefficient Spearman Rank Coefficient of Correlation is an adaptation of Pearson's product-moment coefficient of correlation for ranks. For each variable we rank the scores associated with that variable. We calculate the covariance and the standard deviations for the ranks. Dividing the covariance by the two standard deviations we get Spearman's Rank Coefficient of Correlation.

13. C82MCP Diploma Statistics School of Psychology University of Nottingham 13 An Example of Spearman's Rank Correlation (Rho) For a correlation to be significant rho observed  rho critical For these data there is no evidence of a significant association between IQ and Reading Age.

14. C82MCP Diploma Statistics School of Psychology University of Nottingham 14 Summary of Bivariate Correlations There are two regularly used correlation coefficient Pearson's Product Moment Coefficient A parametric correlation coefficient Spearman's Rho A non-parametric correlation coefficient Correlation coefficient indicate The degree of a relationship between two variables The direction of the relationship between two variables

15. C82MCP Diploma Statistics School of Psychology University of Nottingham 15 Multiple Correlations We may be interested in looking at the pattern of correlations between several variables Such patterns correlations are usually summarised in a correlation table. *p<0.05

16. C82MCP Diploma Statistics School of Psychology University of Nottingham 16 There are many different models of the relationships between these variables. For example Reading Ability and Verbal Ability may be caused by IQ Reading Ability and Verbal Ability may cause IQ Correlations and Causal Models Reading Ability Verbal Ability IQ Reading Ability Verbal Ability IQ

17. C82MCP Diploma Statistics School of Psychology University of Nottingham 17 Correlations and Causal Models It is difficult to know, on the basis of the correlations, which model is the most appropriate. We do not know which variables are directly or indirectly causing relationships between the variables Even worse some of the correlations may be spurious, i.e. not ‘true’ correlations at all.

18. C82MCP Diploma Statistics School of Psychology University of Nottingham 18 Partial Correlations A technique known as partial correlations has been developed that allows us to look at the relationship between two variables in the presence of a third (or many other) variables. The partial correlation examines the degree of a relationship when other variables are controlled for. Given the zero order correlations, r xy, r yz and r xz the partial correlation is defined as:

19. C82MCP Diploma Statistics School of Psychology University of Nottingham 19 Partial Correlations Looking at the partial correlation between reading ability and verbal ability controlling for IQ we find:

20. C82MCP Diploma Statistics School of Psychology University of Nottingham 20 Comparing Zero Order & Partial Correlations The zero order correlation between reading ability and verbal ability is 0.621 and the partial correlation when we have taken into account of IQ is equal to 0.495. Looking at the relationship between reading ability and IQ, the zero order correlation is 0.736 and the partial correlation when we account for verbal ability is 0.661. The zero order correlation for verbal ability and IQ is 0.433. The partial correlation is -0.045.

21. C82MCP Diploma Statistics School of Psychology University of Nottingham 21 Causal Models and Correlations Given this pattern of results, one plausible explanation seems to be that IQ and verbal ability "cause" reading ability. However, correlation cannot itself inform us about causes. We must appeal to theoretical considerations to try and understand the causal relationships between different variables. Reading Ability Verbal Ability IQ

22. C82MCP Diploma Statistics School of Psychology University of Nottingham 22 Causal Models and Correlations Given the pattern of zero order and partial correlations shown above, there are alternative accounts which are equally plausible. There are three other models as far as the statistical methods are concerned Reading Ability Verbal Ability IQ Reading Ability Verbal Ability IQ Reading Ability Verbal Ability IQ