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Advanced Data Analysis: Multiple RegressionPowerPoint Presentation

Advanced Data Analysis: Multiple Regression

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Advanced Data Analysis: Multiple Regression

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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 (?)

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

- 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

- 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

- 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!

- How do researchers derive the mathematical relationship?
- Estimate the “best” linear equation (Ordinary Least Squares algorithm)

- Interpretation of Results
- Overall Model Evaluation
- Ho: R2 = 0; Ha: R2 > 0
- F-test

- Ho: R2 = 0; Ha: R2 > 0
- Are individual b coefficients significant?
- Ho: bi= 0; Ha: bi n.e. 0
- t-test

- Ho: bi= 0; Ha: bi n.e. 0

- Overall Model Evaluation
- EXAMPLE

- 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)

- Reduces the overall predictive (or explanatory) power of each variable (lowers b-values)
- Non-Linear Relationship -- relationship between X and Y cannot be explained with a straight line
- Check non-linear relationship (transform X to X2)

- 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

- One X – code 0 or 1
- Use effects coding
- One X – code -1 or 1
- Intercept term (a) is mean value of Y

- One X – code -1 or 1

- Use dummy or effects coding
- X (X1, X2) variables may “interact”
- Create INTERACTION = X1 * X2