Multiple Regression Scott Hudson January 24, 2011

Bivariate Regression Review Formula Y (dependent variable) = Constant (Intercept) + (Coefficient * X Independent Variable) Example (Compliment of Dr. Cumella) Income at Age 40 = -24,000 + (30,000 * College GPA)

Review Continued Pearson’s r – correlation between two variables – may be anywhere from -1 to plus 1: 0 means no correlation R squared – comes out as a decimal. Move the decimal point 2 spaces to the right to form a percentage. This tells you to what percent the variance of one is explained by the variance in the other-this is not the statistical significance

SPSS Scan the data to look for “Sig”; SIG or Significance A number of errors on the last assignment attributable to using the wrong score from the SPSS tables

Multiple Regression Multiple Regression is the same as a Bivariate Regression analysis, except that we are adding more predictor variables There is essentially an unlimited number to theoretical predictor variables we could add but ultimately we only want to include those predictor variables which are significant

Multiple Regression Example

Equation Y = a + b1*X1 + b2*X bp*Xp Y'i = b0 + b1X1i + b2X2i This is the same formula as we used for Simple Regression but we just added more predictor variables. We are still solving for the Dependent variable “Y”.

Assumptions Normal distribution Linear Variables measured without error Homoscedasticity

SPSS Example Data

Significance Testing SPSS will provide us with a P value which we can use to determine if a predictor variable is significant or not. If it is not statistically significant, we will not include it in our final multiple regression as it is just attributable to chance.

Types of Multiple Linear Regressions Forward Backward Stepwise

Multiple Regression Example Estimated Income at age 40 based on College GPA Thanks to Dr. Cumella for allowing use of this example Estimated Income at Age 40 = -$ $30,000 (GPA) Example: Income = ,000 (4.0) (remember to do the multiplication first) Should equal $96,000

Adding an additional Predictor Variable In this equations we will add an additional variable which has been found to influence income at age 40 (Parent’s income when parent was age 40) This is not a huge variable but it does have a multiplier of.10 (parent’s income at age 40)

Continued Let’s assume parent’s income at age 40 was $100,000 Income = -24, ,000 (4.0) +.10 (100,000) = $106,000

One more example GPA of 3.0 Parent income of $40,000 Income = -24, ,000 (3.0) +.10(40,000) = -24, , $70,000