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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

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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

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

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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)

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

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Outliers Outlier or influential point Cooks distance of 1 or greater

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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

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

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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

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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

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

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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]

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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)

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

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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

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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.

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

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

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Standard Regression a c e g Depression Age Sex Neuroticism Stress

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

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

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

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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?

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