Correlation

Correlation

1. Variance and Covariance

   • How far numbers are spread
   • How 2 variables change together

2. Hypothesis

   • H0 : ρ = 0
     • Normal
     • T-Dist 2df
   • H0 : ρ = ρ 0 Where ρ0 ≠ 0
     • Not Normal Fisher
       • (1+r) Z=(1/2)ln------- (1-r)
     • Bivariate Normal
       • Mean
       • Standard Error
     • Test
       • Z and Z0
         • (Z-Z0) Sqrt (n-3)
   • H0 : ρ xy = ρ x'y
   • H0 : ρ1 = ρ2

3. Dichotomies

   • Biserial Correlation
     • rb when x is quantitative, y is artificial dichotomy
     • X¯ p be the mean of X for the pass group, and let p be the proportion of people who passed the exam
     • X¯ q be the mean of X for the fail group, and let q be the proportion of people who failed the exam
     • sX be the sample standard deviation of all the X scores
     • I h be the height of the normal distribution at the point where the pass/fail cut-off was set for Y .
     • X¯ p − X¯ q rb=------------------------ × pq h sX 
   • Tetrachoric Correlation
     • Both X and Y are artificial dichotomies
     • equal split
       • rtet = sin  π 2  b + c − a − d n 
     • Not equal
       • rtet = cos π 1 + p bc/ad !
   • phi correlation
     • bc − ad φ=------------------------------------- sqrt (a + b)(a + c)(b + d)(c + d)
       • b and c are concordant
       • a and d are discordant
       • φ > 0 if bc large
       • φ< 0 if ad large
       • φ= 0 if ad=bc
     • test = χ² =nφ²
   • Point Biserial correlation
   • true Dichotomy - no underlying continuum between categories EG- male /Female
     • Only linear correlation
   • Artificial - has been derived from a continuous variable EG exam results: pass or fail, but score is continuous

4. ·

   • Effect Size
     • r measures effect size
     • r = ±0.1 Small effect r 2 = 0.01 (1% explained) r = ±0.3 Medium effect r 2 = 0.09 (9% explained) r = ±0.5 Large effect r 2 = 0.25 (25% explained)
   • Factors that influence r
     • Correlations provide info for other multivariate analysis
     • Artefacts that influence the values of sample correlation will also affect the results of these other multivariate analyses
   • Non-Linearity
     • apply transformation replace X by logX
     • rank values perform ANOVA
   • Distribution Shapes
     • Regression towards the Mean
       • Can predict the score of Y from X
       • If X is ONE SD from mean(X), Y will be r SD from mean (Y)
       • r is symmetric index of the relation of Y and X
       • Zy=rZx
     • Regression Equation
       • r= +1 distance of Y from mean (Y) = distance of X of mean(X)
       • r < 1 y will be closer to mean (Y) than X to mean (X)
   • Statistical power
     • 1-β
     • chance of a sample correlation r large enough to correctly reject the hypothesis
     • Depends on:
       • true effect size
       • Sample size 'n'
       • α-level used as criterion
     • can find the minimum sample size required to obtain adequate statistical power for different hypothesized values of true correlation
       • find adequate statistical power
       • evaluate assumptions adequately
       • avoid undue influence from outliers
   • Unreliability
     • X and Y are subject to error = Correlation decreases
     • Test - Retest
       • Reliability coefficient for X, measure same group twice, compute correlation
     • Attenuation
       • rxy= ρxy( sqrt(rxxryy))
         • ryy<1 rxx<1
         • Smaller than true correlation
         • rearrange to get ρ(hat)xy

5. ·

   • Pearsons
     • -Most Common -Represented by 'r' -Measures Degree of correlation -LINEAR Relationship
     • Measures correlation of a Sample
     • |r| >0.7 Very Strong 0.5-0.7 Strong 0.3-0.5 Moderate 0.2-0.3 Weak 0-0.2 Non
     • (Sxy) r=--------- (SxSy)
       • Covariance
       • Standard Deviations of X and Y
       • Test against the product-moment correlation coefficent
     • Bonferroni Correction When applying to multiple pairs
     • Causality - if independent variable had been manipulated and outcome variable measured in the context of a well-controlled experimental design
     • Reasons of Correlation
       • X is a cause of Y Y is a cause of X
       • X may cause Z, Z causes Y Z mediates the effect of X on Y
       • X may be confounded with Z, Z predicts or causes Y
       • X and Y may measure same underlying variable
       • X and Y both caused by Z
       • Sampling Error
     • Conditions
       • Independent
       • Quantitative
       • Normal
       • Linear Correlation
       • Variances Should be approximately equal
   • Partial correlation
   • Spearmans

6. ·

   • Samples that Combine Groups
     • Interaction Effect
       • Lack of correlation in one group
       • Obscure a strong correlation
     • Spurious Correlation
       • 2 groups with no correlation
       • may be grouped together to result in a correlation
   • Biased Sample
     • Accidentally Bias Intentionally selective
       • Restricted range of observations
         • Cannot assume for whole population
           • Need samples from entire range