1. Lecture 1: Correlations and multiple regression Aims & Objectives -Should know about a variety of correlational techniques -Multiple correlations and the Bonferroni correction -Partial correlations -3 type of multiple regression -Simultaneous -Stepwise -Hierarchical

2. Questions & techniques What is the association between a set of variables This takes a number of multi-variate forms –Associations between a number of variables (multiple-correlations) –Associations between 1 variable (DV) and many variables (IVs) – MODEL BUILDING regression and partial correlations –Associations between 1 set of variables and another set of variables canonical correlations

3. Correlations Vary between –1 and 1 +1 Low High Low High

4. Types of correlation Pearsons (Interval and ratio data) Spearmans (Ordinal data) Phi (both true dichotomies) Tau (rating) Biserial (Interval & dichotomised) Point-biserial (interval & true dichotomy)

5. Factors affecting correlations Outliers Homoscedecence Restriction of range Multi-collinearity Singularity

6. Outliers Outlier or influential point Cooks distance of 1 or greater

7. Homoscedasticity When the variability of scores (errors) in one continuous variable is the same in a second variable At group level data this is Termed homogeneity of variance

8. Heteroscedasicity One variable is skew or the relationship is non-linear

9. Singularity & Multicollinearity Singularity: –when variables are redundant, one variable is a combination of two or more other variables. Multi-collinearity: –when variables are highly correlated (.90+). For example two measures of IQ Problems –Logical: Dont want to measure the same thing twice. –Statistical: Singularity prevents matrix inversion (division) as determinants = zero, for multi-collinearity determinant zero to many decimal places Screening –Bivariate correlations –Examine SMC: large = problems –Tolerance (1 – SMC) Solutions: –Composite score –Remove 1 variable

10. IQ: Multi-collinearity & Singularity IQ1 VerbalSpatial Memory Maths Total IQ is singular with its own sub-scales (total is a function of combining subscales One total IQ test (MD5) is multicolinear with another (MAT) IQ2 Multicolinear Singular

11. Multiple correlations StressESContDep Stress1 ES.32*1 Cont.24*.121 Dep.23*.62*.43*1

12. Partial correlations DV IV1 IV2 a c b d Neuroticism Stress Depression Partial r Neuroticism (N) = once the overlap of stress with N and the Stress with Depression is removed Semi-partial r for N = once overlap of Stress with N is removed [N] [S] [Dep]

13. Bonferroni correction With multiple r matrix [R] or many (k) IVs in regression analysis then the possibility of chance effects increases Correct the level (0.05/N) Correct for the number of effects expected by chance = * N (0.05 * N)

14. Multiple regression Y X B A (intercept) (slope)

15. Regression assumptions N:IVs ratio –Assume medium effect size for Multiple Correlations N > 50 + 8m (m = N of IVs) For simple linear regression N > 104 + m –(8/f 2 ) + (m – 1). Where f 2 = ES =.10,.15 –or f 2 =.35 f 2 = R 2 /(1 – R 2 ) for a more accurate estimate – Stepwise 40:1 Outlier = Cook distance Singularity-Multi-collinearity = SMCs Normality = residual plots

16. Types of regression Simultaneous (Standard) –No theory and enter all IV in one block Stepwise –No theory. Allows the computer to choose on statistical ground the best sub-set of IVs to fit the equations. Capitalises on chance effects Hierarchical (sequential)– –Theory driven. A-priori sequence of entry.

17. Types of regression: An example Simultaneous Age Gender Stress N Control Stepwise Age Control Hierarchical Step 1 Age Gender Step 2 Stress N Control

18. Venn Diagrams ab c d e f g Depression Age Sex Neuroticism Stress

19. Standard Regression a c e g Depression Age Sex Neuroticism Stress

20. Hierarchical ab c d e f g Depression Age Sex Neuroticism Stress Step 1 Step 2

21. Stepwise ab c d e f g Depression Age Sex Neuroticism Stress

22. Stepwise ab c d e f g Depression Age Sex Neuroticism Stress

23. Statistical terms B = un-standardized Beta Beta = standardized (-1 to +1) T-test = Is the beta significant? R 2 0-1 (amount of variance accounted for) R 2 = Change in from one block to the next F = is the change in R significant? F = Is the equation significant?