Advanced data analysis multiple regression
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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

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Advanced data analysis multiple regression

Advanced Data Analysis: Multiple Regression


Advanced data analysis multiple regression1

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 regression2

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 regression3

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 regression4

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 regression5

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 regression6

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 regression7

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 regression8

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


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