Notes

Outline

Statistics 9 Correlation and Regression Sherril M. Stone, Ph.D. Department of Family Medicine OSU-College of Osteopathic Medicine

Basic Terms Correlation – statistical technique that describes degree of relationship between 2 variables Regression – technique that uses data to write an equation for a straight line. The equation then is used to make some type of prediction of the data. Draw a line of best fit (straight line) Make prediction of 2nd score when 1st score is known Linear regression - from the phenomenon of regression toward the mean, a consequence of the laws of probability when correlation is less than perfect (as correlation gets lower, prediction gets closer to z = 0).

Line of Best Fit Slope-intercept formula for a straight line is Y = mX + b Y and X = variables (scores) on the Y and X axes m = slope of the line (a constant) b = intercept (intersection) of line with Y axis Positive – highest point on line is Right of lowest point Negative – highest point on line is Left of lowest point Equal – horizontal lines Large (+ or -) – vertical lines

Line of Best Fit Regression Equation Formula – finds the line of best fit Least squares method (LSM)–Pearson’s mathematics to create a straight line. LSM produces a value for the slope and the intercept. With slope and intercept – write an equation for a straight line (this is one that best fits your data) Ŷ = a + bX Ŷ = predicted value of Y from X a = point where line intersects Y axis a, b = regression b = slope of the regression line           coefficients X = X score (data) used to predict Y

Correlation Coefficient Correlation coefficient (r)  Pearson product-moment correlation – measures the degree  and direction of linear relationship between 2 variables Coefficient of determination (r2) – proportion of shared variance (s2) between 2 variables Effect size for r –   Small  .10                    Med    .30                                  Large  .50

Correlation Coefficient Formula r Blanched formula        åXY       – (X)(Y)     r     =         N (SDX)(SDY) Raw score formula         NåXY – (åX)( åY) r = Ö  (NåX2 – (åX)2       (NåY2 – (åY)2

Regression Coefficient Formulas           SDY       b =  r    SDX - OR -        NåXY – (åX) (åY) b =                N åX2 – (åX)2

Steps to Calculate Regression Calculate mean and SD Calculate r and r2 Calculate b Calculate a  Predict Y

Example 1        X X2 Y   Y2            XY 5 3 7 4 1 1 6 3 9 5 9      5 10 7 4 3 3      2 2 2 ∑X           ∑Y N = SDX r = SDY r2 = b = a = Ŷ = a + bX

Example 2        X X2 Y   Y2            XY 21 8 14 9 16 10 11 15 15 11 10 16 9 14   8                     21 ∑X           ∑Y N = SDX r = SDY r2 = b = a = Ŷ = a + bX

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