MULTIPLE REGRESSION
After completing this chapter, you should be able to:
n understand model building using multiple regression analysis
n apply multiple regression analysis to business decision-making situations
n analyze and interpret the computer output for a multiple regression model
n test the significance of the independent variables in a multiple regression model
n use variable transformations to model nonlinear relationships
n recognize potential problems in multiple regression analysis and take the steps to correct the problems.
n incorporate qualitative variables into the regression model by using dummy variables.
Multiple Regression Assumptions
n The errors are normally distributed
n The mean of the errors is zero
n Errors have a constant variance
n The model errors are independent
Model Specification
n Decide what you want to do and select the dependent variable
n Determine the potential independent variables for your model
n Gather sample data (observations) for all variables
The Correlation Matrix
n Correlation between the dependent variable and selected independent variables can be found using Excel:
n Tools / Data Analysis… / Correlation
n Can check for statistical significance of correlation with a t test
Example
n A distributor of frozen desert pies wants to evaluate factors thought to influence demand
n Dependent variable: Pie sales (units per week)
n Independent variables: Price (in $)
Advertising ($100’s)
n Data is collected for 15 weeks
Pie Sales Model
Sales = b0 + b1 (Price)
+ b2 (Advertising)
Interpretation of Estimated Coefficients
n Slope (bi)
n Estimates that the average value of y changes by bi units for each 1 unit increase in Xi holding all other variables constant
n Example: if b1 = -20, then sales (y) is expected to decrease by an estimated 20 pies per week for each $1 increase in selling price (x1), net of the effects of changes due to advertising (x2)
n y-intercept (b0)
n The estimated average value of y when all xi = 0 (assuming all xi = 0 is within the range of observed values)
Pie Sales Correlation Matrix
n Price vs. Sales : r = -0.44327
n There is a negative association between
price and sales
n Advertising vs. Sales : r = 0.55632
n There is a positive association between
advertising and sales
Scatter Diagrams
n Computer software is generally used to generate the coefficients and measures of goodness of fit for multiple regression
n Excel:
n Tools / Data Analysis... / Regression
Multiple Regression Output
The Multiple Regression Equation
Using The Model to Make Predictions
Input values
Multiple Coefficient of Determination
n Reports the proportion of total variation in y explained by all x variables taken together
Multiple Coefficient of Determination
Adjusted R2
n R2 never decreases when a new x variable is added to the model
n This can be a disadvantage when comparing models
n What is the net effect of adding a new variable?
n We lose a degree of freedom when a new x variable is added
n Did the new x variable add enough explanatory power to offset the loss of one degree of freedom?
n Shows the proportion of variation in y explained by all x variables adjusted for the number of x variables used
(where n = sample size, k = number of independent variables)
n Penalize excessive use of unimportant independent variables
n Smaller than R2
n Useful in comparing among models
Multiple Coefficient of Determination
Is the Model Significant?
n F-Test for Overall Significance of the Model
n Shows if there is a linear relationship between all of the x variables considered together and y
n Use F test statistic
n Hypotheses:
n H0: β1 = β2 = … = βk = 0 (no linear relationship)
n HA: at least one βi ≠ 0 (at least one independent
variable affects y)
F-Test for Overall Significance
n Test statistic:
where F has (numerator) D1 = k and
(denominator) D2 = (n – k - 1)
degrees of freedom
H0: β1 = β2 = 0
HA: β1 and β2 not both zero
a = .05
df1= 2 df2 = 12
Are Individual Variables Significant?
n Use t-tests of individual variable slopes
n Shows if there is a linear relationship between the variable xi and y
n Hypotheses:
n H0: βi = 0 (no linear relationship)
n HA: βi ≠ 0 (linear relationship does exist
between xi and y)
H0: βi = 0 (no linear relationship)
HA: βi ≠ 0 (linear relationship does exist
between xi and y)
t Test Statistic:
(df = n – k – 1)
Inferences about the Slope: t Test Example
H0: βi = 0
HA: βi ¹ 0
Confidence Interval Estimate for the Slope
Standard Deviation of the Regression Model
n The estimate of the standard deviation of the regression model is:
Standard Deviation of the Regression Model
n The standard deviation of the regression model is 47.46
n A rough prediction range for pie sales in a given week is
n Pie sales in the sample were in the 300 to 500 per week range, so this range is probably too large to be acceptable. The analyst may want to look for additional variables that can explain more of the variation in weekly sales
OUTLIERS
If an observation exceeds UP=Q3+1.5*IQR or if an observation is
smaller than LO=Q1-1.5*IQR where Q1 and Q3 are quartiles and
IQR=Q3-Q1
What to do if there are outliers?
Sometimes it is appropriate to delete the entire observation containing the oulier. This will generally increase the R2 and F test statistic values
Multicollinearity
n Multicollinearity: High correlation exists between two independent variables
n This means the two variables contribute redundant information to the multiple regression model
n Including two highly correlated independent variables can adversely affect the regression results
n No new information provided
n Can lead to unstable coefficients (large standard error and low t-values)
n Coefficient signs may not match prior expectations
Some Indications of Severe Multicollinearity
n Incorrect signs on the coefficients
n Large change in the value of a previous coefficient when a new variable is added to the model
n A previously significant variable becomes insignificant when a new independent variable is added
n The estimate of the standard deviation of the model increases when a variable is added to the model
Output for the pie sales example:
n Since there are only two explanatory variables, only one VIF is reported
n VIF is < 5
n There is no evidence of collinearity between Price and Advertising
Qualitative (Dummy) Variables
n Categorical explanatory variable (dummy variable) with two or more levels:
n yes or no, on or off, male or female
n coded as 0 or 1
n Regression intercepts are different if the variable is significant
n Assumes equal slopes for other variables
n The number of dummy variables needed is (number of levels - 1)
Dummy-Variable Model Example (with 2 Levels)
Interpretation of the Dummy Variable Coefficient
Dummy-Variable Models (more than 2 Levels)
n The number of dummy variables is one less than the number of levels
n Example:
y = house price ; x1 = square feet
n The style of the house is also thought to matter:
Style = ranch, split level, condo
Dummy-Variable Models (more than 2 Levels)
Interpreting the Dummy Variable Coefficients (with 3 Levels)
Nonlinear Relationships
n The relationship between the dependent variable and an independent variable may not be linear
n Useful when scatter diagram indicates non-linear relationship
n Example: Quadratic model
n The second independent variable is the square of the first variable
Polynomial Regression Model
n where:
β0 = Population regression constant
βi = Population regression coefficient for variable xj : j = 1, 2, …k
p = Order of the polynomial
ei = Model error
Linear vs. Nonlinear Fit
Quadratic Regression Model
Testing for Significance: Quadratic Model
n Test for Overall Relationship
n F test statistic =
n Testing the Quadratic Effect
n Compare quadratic model
with the linear model
n Hypotheses
n (No 2nd order polynomial term)
n (2nd order polynomial term is needed)
Higher Order Models
Interaction Effects
n Hypothesizes interaction between pairs of x variables
n Response to one x variable varies at different levels of another x variable
n Contains two-way cross product terms
Effect of Interaction
n Without interaction term, effect of x1 on y is measured by β1
n With interaction term, effect of x1 on y is measured by β1 + β3 x2
n Effect changes as x2 increases
Interaction Example
n Hypothesize interaction between pairs of independent variables
n Hypotheses:
n H0: β3 = 0 (no interaction between x1 and x2)
n HA: β3 ≠ 0 (x1 interacts with x2)
Model Building
n Goal is to develop a model with the best set of independent variables
n Easier to interpret if unimportant variables are removed
n Lower probability of collinearity
n Stepwise regression procedure
n Provide evaluation of alternative models as variables are added
n Best-subset approach
n Try all combinations and select the best using the highest adjusted R2 and lowest sε
n Idea: develop the least squares regression equation in steps, either through forward selection, backward elimination, or through standard stepwise regression
n The coefficient of partial determination is the measure of the marginal contribution of each independent variable, given that other independent variables are in the model
Best Subsets Regression
n Idea: estimate all possible regression equations using all possible combinations of independent variables
n Choose the best fit by looking for the highest adjusted R2 and lowest standard error sε
Aptness of the Model
n Diagnostic checks on the model include verifying the assumptions of multiple regression:
n Each xi is linearly related to y
n Errors have constant variance
n Errors are independent
n Error are normally distributed
Residual Analysis
The Normality Assumption
n Errors are assumed to be normally distributed
n Standardized residuals can be calculated by computer
n Examine a histogram or a normal probability plot of the standardized residuals to check for normality
Chapter Summary
n Developed the multiple regression model
n Tested the significance of the multiple regression model
n Developed adjusted R2
n Tested individual regression coefficients
n Used dummy variables
n Examined interaction in a multiple regression model
n Described nonlinear regression models
n Described multicollinearity
n Discussed model building
n Stepwise regression
n Best subsets regression
n Examined residual plots to check model assumptions