Correlation and Linear Regression INCM 9102

Correlation  Correlation coefficients assess strength of linear relationship between two quantitative variables. The correlation measure ranges from -1 to +1. A negative correlation means that X and Y are inversely related. A positive correlation means that X and Y are directly related. zero correlation means that X and Y are not linearly related. A correlation of +1 indicates X and Y are directly related and that all the points fall on the same straight line. A correlation of -1 indicates X and Y are inversely related and that all the points fall on the same straight line  Plot Scatter Diagram of Each Predictor variable and Dependent Variable Look of Departures from Linearity Look for extreme data points (Outliers)  Examine Partial Correlation Can’t determine causality, but isolate confounding variables

Correlation For example, lets take two variables and evaluate their correlation…open the stats98 dataset in Excel… What would you expect the correlation of the Verbal SAT scores and the Math SAT scores to be? Why? What would you expect the correlation of the Math SAT scores and the percent taking the test to be? Why?

Correlation What would you expect the correlation of the Verbal SAT scores and the Math SAT scores to be? Why?

Correlation What would you expect the correlation of the Math SAT scores and the Percent of HS students that took the test? Why?

Correlation Lets pull up the UCDAVIS2 dataset in Excel…plot Ideal Height versus Actual Height…what would you expect the correlation value to be? Can you explain someone’s Ideal Height using their Actual Height?

Regression

From the previous slide, the “regression line” has been imposed onto the relationship between ideal height and height. The equation of this line takes the general form of y=mx+b, where: Y is the dependent variable (ideal height) M is the slope of the line X is the independent variable (actual height) B is the Y-intercept. When we discussion regression models, we transform this equation to be: Y = b o + b 1 x 1 + …b n x n Where b o is the y-intercept and b 1 is the slope of the line. The “slope” is also the effect of a one unit change of x on y.

Regression From the previous slide, the model equation is presented in the form of the equation of a line: y=.8174x From this, we would say: 1.For every 1 inch of change in someone’s actual height, there is a.8174 inch change in their ideal height. 2.Everyone “starts” with inches. 3.If someone has an actual height of 68 inches, their ideal height is inches. That R2 value of.7372 is interpreted as “73.72% of the change in ideal height can be explained by a linear model with actual height as the only predictor”.