Correlation and Covariance R. F. Riesenfeld (Based on web slides by James H. Steiger)

Goals ⇨ Introduce concepts of  Covariance  Correlation ⇨ Develop computational formulas 2R F Riesenfeld Sp 2010CS5961 Comp Stat

Covariance ⇨ Variables may change in relation to each other ⇨ measures how much the movement in one variable predicts the movement in a corresponding variable ⇨ Covariance measures how much the movement in one variable predicts the movement in a corresponding variable 3R F Riesenfeld Sp 2010CS5961 Comp Stat

Smoking and Lung Capacity ⇨ Example: investigate relationship between and ⇨ Example: investigate relationship between cigarette smoking and lung capacity ⇨ Data: sample group response data on smoking habits, measured lung capacities, respectively ⇨ Data: sample group response data on smoking habits, and measured lung capacities, respectively R F Riesenfeld Sp 2010CS5961 Comp Stat4

Smoking v Lung Capacity Data 5 N Cigarettes ( X )Lung Capacity ( Y ) R F Riesenfeld Sp 2010CS5961 Comp Stat

Smoking and Lung Capacity 6

Smoking v Lung Capacity ⇨ Observe that as smoking exposure goes up, corresponding lung capacity goes down ⇨ Variables inversely ⇨ Variables covary inversely ⇨ and quantify relationship ⇨ Covariance and Correlation quantify relationship 7R F Riesenfeld Sp 2010CS5961 Comp Stat

Covariance ⇨ Variables that inversely, like smoking and lung capacity, tend to appear on opposite sides of the group means ⇨ Variables that covary inversely, like smoking and lung capacity, tend to appear on opposite sides of the group means  When smoking is above its group mean, lung capacity tends to be below its group mean. ⇨ Average measures extent to which variables covary, the degree of linkage between them ⇨ Average product of deviation measures extent to which variables covary, the degree of linkage between them 8R F Riesenfeld Sp 2010CS5961 Comp Stat

The Sample Covariance ⇨ Similar to variance, for theoretical reasons, average is typically computed using ( N -1 ), not N. Thus, 9R F Riesenfeld Sp 2010CS5961 Comp Stat

Cigs ( X )Lung Cap ( Y ) Calculating Covariance R F Riesenfeld Sp 2010CS5961 Comp Stat10

Calculating Covariance Cigs (X ) Cap (Y ) ∑= R F Riesenfeld Sp 2010CS5961 Comp Stat11

Evaluation yields, Evaluation yields, 12 Covariance Calculation (2) R F Riesenfeld Sp 2010CS5961 Comp Stat

Covariance under Affine Transformation 13R F Riesenfeld Sp 2010CS5961 Comp Stat

14 CovarianceunderAffineTransf Covariance under Affine Transf (2) R F Riesenfeld Sp 2010CS5961 Comp Stat

(Pearson) Correlation Coefficient r xy ⇨ Like covariance, but uses Z-values instead of deviations. Hence, invariant under linear transformation of the raw data. 15R F Riesenfeld Sp 2010CS5961 Comp Stat

Alternative (common) Expression 16R F Riesenfeld Sp 2010CS5961 Comp Stat

Computational Formula 1 17R F Riesenfeld Sp 2010CS5961 Comp Stat

Computational Formula 2 18R F Riesenfeld Sp 2010CS5961 Comp Stat

Table for Calculating r xy 19 Cigs ( X ) X 2 XYY 2 Cap (Y ) ∑= R F Riesenfeld Sp 2010CS5961 Comp Stat

Computing from Table Computing r xy from Table 20R F Riesenfeld Sp 2010CS5961 Comp Stat

Computing Correlation 21R F Riesenfeld Sp 2010CS5961 Comp Stat

Conclusion ⇨ r xy = implies almost certainty smoker will have diminish lung capacity ⇨ Greater smoking exposure implies greater likelihood of lung damage 22R F Riesenfeld Sp 2010CS5961 Comp Stat

End Covariance & Correlation 23 Notes R F Riesenfeld Sp 2010CS5961 Comp Stat