Correlation Analysis

• A correlation deals with the relationship between two quantitative variables.
• A correlation analysis involves measuring the strength of a particular relationship between variables. This measure is in the form of a single number called a correlation coefficient.
• The linear correlation coefficient, denoted by r, measures the strength and the direction of a linear relationship between two variables. This coefficient is sometimes called Pearson product moment correlation since it was developed by the English mathematician and biostatistician, Karl Pearson.
• A bivariate data contains two sets of related data, (X, Y).
• The Pearson Product-Moment Correlation Coefficient, r, is a widely used statistical measure of the strength of a linear relationship between two variables.
• A positive linear correlation means that as the values of X increases, the value of Y also increases. Likewise, as X decreases, Y also decreases. The variables X and Y have a strong positive linear correlation if the value of r is close to 1. Thus, r = 1 indicates a perfect positive correlation.
• A negative linear correlation means that as the values of X increases, the value of Y decreases. The variables X and Y have a strong negative linear correlation if the value of r is close to −1. Thus, r = −1 indicates a perfect negative correlation.
• The variables X and Y have a weak positive or negative linear correlation if the value of r is close to 0. Likewise, r = 0 implies that X and Y has no linear correlation.
• .

  A) Perfect positive correlation

  B) High positive correlation

  C) Low positive correlation

  D) No correlation

  E) Low negative correlation

  F) High negative correlation

  G) Perfect negative association

  
• Pearson's r Product Moment Correlation Chart

  A) Perfect correlation

  B) Very strong correlation

  C) Strong correlation

  D) Moderate correlation

  E) Weak correlation

  F) Very weak correlation

  G) No correlation

  
• The coefficient of determination, denoted by r^2, is the percentage of the total variation in the Y values (the extent to which the Y’s are different) that is accounted for, or explained by its linear relationship with X.
• The coefficient of determination (R^2) is a number between 0 and 1

  that measures how well a statistical model predicts an outcome. You

  can interpret the R^2 as the proportion of variation in the dependent

  variable that is predicted by the statistical model.
• The Spearman’s rank correlation coefficient or Spearman’s rho, named after Charles Spearman, is another formula for correlation coefficient and is denoted by the ρ or r sub s.
• Spearman's Rank Correlation Coefficient or Spearman's Rho


  It is a nonparametric version of the Pearson product-moment

  correlation which measures the strength and direction of

  association between two ranked variables.
• The value of ρ will always be between 1.0 (a perfect positive correlation) and -1.0 (a perfect negative correlation). A ρ of 0 indicates no association between ranks.