Math 4030 – 12a Correlation

Correlation Two random variables, X and Y, both continuous numerical; Correlation exists when the value of one variable go “consistently” up or down with the change of the other variable. Correlation coefficient: r  [-1,1]

x y x2 y2 xy x1 y1 … xn yn xi yi xi2 yi2 xiyi Calculation: or

Meaning of r values: r = 0.5 r = 0.01 b = 0.8 b = 0.9 r = - 0.95

r vs. b: r and b have the same sign; b is the slope of the linear relationship; r is the strength of the linear relationship; r  [-1,1], b  (-, +).

Testing about the normal population correlation coefficient : Distribution of sample statistic r? Fisher Z transformation: r  (-1, 1)  Fisher-  (- , ) If joint distribution of (X,Y) is approximately bivariate normal, then

Test statistic for H0:  = 0

Confidence interval for : Confidence interval for Fisher-Z score: Solve the two boundary value for  using relationship

Strength vs. significance of the correlation: the significance, given by P-value, depends on the statistical evidence. When small, the correlation (despite of the strength) exists. the strength, given by the r value, is meaningful only it is supported by statistical significance.

Correlation does not mean causation! Final Remark: Correlation does not mean causation!