Math 4030 – 13a Correlation & Regression

Correlation (Sec. 11.6):  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]

Calculation: or xyx2x2 y2y2 xy x1x1 y1y1 …… xnxn ynyn xixi yiyi xi2xi2 yi2yi2 xiyixiyi

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

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  (- , +  ).

Correlation Coefficient and the Efficiency of the (Linear) Regression Model

Decomposition of Variability

Coefficient of determination: Proportion of total variability explained by the linear regression:

Coefficient of Determination Correlation Coefficient

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 H 0 :  =  0 Test statistic for H 0 :  = 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 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. the strength, given by the r value, is meaningful only it is supported by statistical significance.