Correlation• The coefficient of correlation is a numerical measure of the strength of the linear relationship between 2 variables.• X, Y must be measurable quantitatively.
Correlation• Two variables X, Y have a positive correlation if large values of X tend to be associated with large values of Y; similarly, for small values.• Two variables X, Y have a negative correlation if large values of X tend to be associated with small values of Y, and vice-versa.
What Are Correlation Tests?• Provides a quantitative perspective on the strength of the relationship if one is found, denoted by an “r” value.• Values of r are always between -1 & 1; i.e., between 0 and 1 in absolute value.• r = 0 means no correlation; r = +-1 means perfect correlation; both rare.• Tests can be run for both parametric and non-parametric data sets.
Correlation RelationshipsNegative Relationship No Relationship Positive Relationship
Parametric vs. Non-Parametric• Parametric data: – Assumes normal distribution, homogenous variance, and data sets are typically ratio or interval. – Can draw more conclusions. • Pearson Correlation• Non-Parametric data: – No assumption on distribution or variance relationship, and data sets are typically ordinal or nominal. – More simple and less affected by outliers. • Spearman Correlation
Pearson Product Moment Correlation• More simply known as Pearson Correlation, designated by the Greek letter rho (ρ) but reported as a “r” value.• Depicts linear relationships.• Data sets must meet assumptions of parametric data.• Like Spearman’s, Hypothesis testing states; – Ho : ρ = 0 – Ha: ρ ≠ 0
Correlation CoefficientPearson Product-moment Correlation Coefficient 1 X i −X (Yi − ) Y rxy = *∑ * (n − ) 1 Sx Sy Cov xy rxy = Sx * S y http://www.drvkumar.com/mr9/ 9
Testing the Significance of the Correlation Coefficient• Null hypothesis: Ho : p = 0• Alternative hypothesis: Ha : p ≠ 0• Test statistic 6 −2 t = .70 2 = 1.96 1 − 0.70 Example: n = 6 and r = .70 At α = .05 , n-2 = 4 degrees of freedom, Critical value of t = 2.78 Since 1.96<2.78, we fail to reject the null hypothesis. http://www.drvkumar.com/mr9/ 11