Chapter 11 REGRESSION

Multiple Regression  Uses  Explanation  Prediction

Multiple Regression  Based On: Correlations Characteristics of a straight line

Regression vs. Multiple Regression  One independent variable vs. more than one independent variable  One dependent variable

Multiple Regression TType of Data Required Independent variables CCategorical - can be coded for entry CContinuous - meet assumptions Dependent variable CContinuous - should be normally distributed

AAssumptions Sample representative of population Variables should have normal distribution Homoscedasticity Linear relationship between variables

 Power analysis If sample size = number of variables, Rsquared will equal Generally need subjects per independent variable Less than 10 subjects per independent variable leads to serious error

Relationship of correlation to regression  Perfect correlation?  No correlation?  Imperfect correlation?

Regression Equation  Formula for a straight line  Predicted score = constant plus regression weight times score  Y’ = a + bX  Y’ = a + b1X1 + b2X2 = B3X3

Regression Equation  Predicted score Y’  Constant a value of Y when X = 0 point where regression line intercepts the Y axis  regression coefficient/s b or beta rate of change in Y with a unit change in X measure of slope of regression line

Regression Equation  Constant or a is based on means of variables involved  Regression coeffients, b or beta, based on correlation between two variables

Regression coefficients  The b-weights are based on raw scores  Beta-weights are based on standardized scores and are partial correlation coefficients

Regression line  Least squares  “Line of best fit”  Deviations around this line sum to zero Deviations are the differences between the actual and predicted scores

Computer Example  What is the multiple correlation between a group of independent variables entered in three blocks and the dependent variable, total positive psychological attitudes?  Block 1: Age and education  Block 2: Smoking hx and exercise  Block 3: Sat. with wt. and Health

SPSS - Multiple Regression  ANALYZE Regression  Linear Statistics  Confidence intervals  R squared change  Descriptives  Part and partial correlations Options  Exclude cases pairwise

Dummy coding  Uses 1s and 0s  a = mean of dependent variable for group assigned 0s throughout  b - tests the difference between the group assigned 1 on the variable and the group assigned 0s throughout

Dummy coding  Vector 1 Republicans = 1 Democrats = 0 Independents = 0  Vector 2 Republicans = 0 Democrats = 1 Independents = 0

SPSS - creating dummy variables  TRANSFORM Compute  New variable with new value  IF create conditional expression

Effect Coding  Uses 1s, 0s, and -1s  a = grand mean of dependent variable  b tests the difference between the mean of the group assigned 1 and the grand mean  the b-weights add up to zero

Effect Coding  Vector 1 Republicans 1 Democrats 0 Independents -1  Vector 2 Republicans 0 Democrats 1 Independents -1

MULTIPLE REGRESSION  Selecting Variables for the Equation Standard (ENTER) Hierarchical (Setwise) Stepwise  Forward  Backward  Stepwise

Mediator and Moderator Variables

Mediator Variable  A variable seen as “between” the independent and dependent variable.  Tested with multiple regression/path analysis

Moderator Variable  Affects the association between an independent and dependent variable.  Test for an interaction between the moderator and another independent variable using hierarchical multiple regression.

Example from the literature