Ch 8: Correlation & Regression Wed, Feb 25 th, 2004

Relationships among 2 variables wUse scatter diagram (scatterplot) to graph relationships. –Can indicate nature of relationship –Plot independent var on X axis; dependent on Y axis –Draw the best-fitting straight line through all points (the one that comes closest to all points)

Direction of relationship wLook on scatterplot to determine: –Pos relationship – high values on 1 variable go w/high values on 2 nd var –Neg relationship – high values on 1 st variable go w/low values on 2 nd var; OR · Low values on 1 st var go w/high values on 2 nd –No association – no pattern (0 relationship)

Pearson’s Correlation wMeasures the strength & direction of relationship betw 2 variables –Direction – pos or neg association –Strength – correlation can range from –1.0 to 1.0 with 0 as midpoint. · 0 indicates no relationship; +1.0 indicates perfect pos relationship; -1.0 indicates perfect neg relationship

(cont.) wConceptually: correlation is Covariability of x and y Variability of x and y separately r = Syx / Sx Sy (book notation) Where Syx = [  (x-xbar) (y-ybar)] / N-1 Sx =  (x-xbar) 2 / N-1 Sy =  (y-ybar) 2 / N-1 Example?

Regression wFocuses on finding the best fitting line or regression equation to describe the scatterplot relationship –Can then use reg equation to predict y given x wEquation for a line is y = bX + a Where y is Dep variable (being predicted), X is Indep variable b is slope of the line (change in y given 1 unit change in x) a is y intercept – where line crosses y axis (when x=0)

(cont.) wHow do we find the best-fitting line? –Want to minimize error betw predicted and actual values = Least Squares Solution –Error = y – yhat (where y is actual observed y value; yhat is predicted y) –Using Least Squares, formulas for b and a are…

Finding b and a b = Syx / S 2 x Syx is covariance of x and y; S 2 x is variance of x a = ybar – b (xbar) After finding a and b, put together reg equation: Yhat = a + b(x) Then use it to predict y values given x Example:

Assessing Accuracy wIdeally, reg equation/line will predict new scores, but there will always be some error –Base reg equation off of 1data set, get a different data set, they will differ –R-squared (R 2 ) gives you index of accuracy (coefficient of determination) –Gives % of variance in y explained by x (want it to be as close to 100% as possible)

SPSS handout wLab 12 – –For # 2, 8, 10, only do correlations for height & income; height & weight –Skip #11 and 12 wLab 13: start at “Getting SPSS to compute the Least Squares regression equation” –Do ONLY #3 – and only need to do height & income; height & weight