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Quantitative Business Analysis for Decision Making. Multiple Linear Regression Analysis. Outlines. Multiple Regression Model Estimation Testing Significance of Predictors Multicollinearity Selection of Predictors Diagnostic Plots. Multiple Regression Model.

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quantitative business analysis for decision making

Quantitative Business Analysis for Decision Making

Multiple Linear



  • Multiple Regression Model
  • Estimation
  • Testing Significance of Predictors
  • Multicollinearity
  • Selection of Predictors
  • Diagnostic Plots


multiple regression model
Multiple Regression Model

Multiple linear regression model:

are slope coefficients of

X1, X2 ,… ,Xk.

quantifies the amount of change in

response Y for a unit change in Xi when

all other predictors are held fixed.


multiple regression model con t
Multiple Regression Model (con’t)

In the model,

is the mean of Y.

  • Contributes to the variation in Y values from their mean , and
  • is assumed normally distributed with mean 0 and standard deviation



A random sample of n units is taken. Then for

each unit k+1 measurements are made:

Y, X1 , X2 , …., Xk


estimated model
Estimated Model

Estimated multiple regression model is:

Expressions for bi are cumbersome to

write. is an estimate of


standard error
Standard Error

Sample standard deviation around the mean (estimated regression model) is:

It is an estimate of

Standard error of (for specified values of predictors) is denoted by


testing significance of a predictor
Testing Significance of a Predictor

For comparing with a reference ,test

statistic is:

and for estimating by a confidence




coefficient of determination
Coefficient of Determination

Coefficient of determination R2 quantifies the % of

variation in the Y-distribution that is accounted by the

predictors in the model. If

  • R2 = 80%, then 20% variation in the Y-distribution is due to factors other than those in the model.
  • R2 increases as predictors are added in the model but at the cost of complicating it.


testing the model for significance
Testing the Model for Significance

Null hypothesis = predictors in the relationship have no predictive power to explain the variation in Y-distribution

Test statistic: F = . It has

F- distribution with k and (n-k-1) degrees of

freedoms for the numerator and denominator.


multicollinearity and selection of predictors
Multicollinearity and Selection of Predictors
  • Multicollinearity - occurs when predictors are highly

correlated among themselves. In its presence R2 may be high,

but individual coefficients are less reliable.

  • Screening process (e.g. stepwise regression) can eliminate

multicollinearity by selecting only those predictors that are not

strongly correlated among themselves.


diagnostic plots
Diagnostic Plots
  • Residuals are used to diagnose the validity of the model assumptions.
  • A scatter plot of the residuals against the predicted values can serve as a diagnostic tool.
  • A diagnostic plot can identify outliers, unequal

variability, and need for transformation to achieve

homogeneity etc.


indicator variables
Indicator Variables
  • Indicator variables (also called dummy variables) are

numerical codes that are used to represent qualitative


  • For example, 0 for men and 1 for women.
  • For a qualitative variable with c categories, (c-1) indicator variables need to be defined.