# Advanced Data Analysis: Multiple Regression - PowerPoint PPT Presentation

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Advanced Data Analysis: Multiple Regression. Advanced Data Analysis: Multiple Regression. What is regression analysis?

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## Advanced Data Analysis: Multiple Regression

### Advanced Data Analysis: Multiple Regression

• What is regression analysis?

• Statistical technique that allows researchers to investigate the relationship between a dependent variable (Y) and one (X1) or several independent variables (X1, X2, etc.)

• Provides a mathematical statement of the relationship

• Allows the simultaneous relationship between Y and several Xs

• Variables must be of interval or ratio scale (?)

### Advanced Data Analysis: Multiple Regression

• Correlation versus regression

• Correlation -- closeness of the relationship between two variables (Y and X)

• Regression -- derivation of a linear equation that explains the relationship between two or more variables (Y, X1, X2, etc.)

• General equation: Y = a + biXi + e

• a is an intercept term; b represents the change in Y that is explained (or predicted) by a one unit change in X; e is an error term

### Advanced Data Analysis: Multiple Regression

• The regression equation (example):

• Y = 32 + .55X + e

• When X = 0, Y = 32

• For each increase in X, Y increases by .55

• When X = 1, Y = 32.55

• When X = 2, Y = 33.10

### Advanced Data Analysis: Multiple Regression

• Coefficient of determination (r2 orR2)

• r2 = 1 - [unexplained variation (in Y by X) / total variation in Y] or

• r2 = explained variation (in Y by X) / total variation in Y

• R2 – proportion of variation in Y explained by all X’s

• In a perfect world r2 (R2)= 1

• Should be > 0 -- will test this!

### Advanced Data Analysis: Multiple Regression

• How do researchers derive the mathematical relationship?

• Estimate the “best” linear equation (Ordinary Least Squares algorithm)

### Advanced Data Analysis: Multiple Regression

• Interpretation of Results

• Overall Model Evaluation

• Ho: R2 = 0; Ha: R2 > 0

• F-test

• Are individual b coefficients significant?

• Ho: bi= 0; Ha: bi n.e. 0

• t-test

• EXAMPLE

### Advanced Data Analysis: Multiple Regression

• Multicollinearity -- two or more X variables are significantly correlated

• Reduces the overall predictive (or explanatory) power of each variable (lowers b-values)

• Check correlations of Ivs (VIP)

• Non-Linear Relationship -- relationship between X and Y cannot be explained with a straight line

• Check non-linear relationship (transform X to X2)

### Advanced Data Analysis: Multiple Regression

• X variables are nominal or interval scaled

• Use dummy or effects coding

• One X – code 0 or 1

• Multiple levels – need two variables

• Interpret results in same way

• Use effects coding

• One X – code -1 or 1

• Intercept term (a) is mean value of Y

• X (X1, X2) variables may “interact”

• Create INTERACTION = X1 * X2