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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 l.jpg

Quantitative Business Analysis for Decision Making

Multiple Linear



Outlines l.jpg

  • Multiple Regression Model

  • Estimation

  • Testing Significance of Predictors

  • Multicollinearity

  • Selection of Predictors

  • Diagnostic Plots


Multiple regression model l.jpg
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 l.jpg
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


Sampling l.jpg

A random sample of n units is taken. Then for

each unit k+1 measurements are made:

Y, X1 , X2 , …., Xk


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Estimated Model

Estimated multiple regression model is:

Expressions for bi are cumbersome to

write. is an estimate of


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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


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Testing Significance of a Predictor

For comparing with a reference ,test

statistic is:

and for estimating by a confidence




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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.


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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.


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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.


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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.


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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.