Multi Regression

8/9/2019 Multi Regression

1/11

Chapter

Forecasting with Multiple Regression Method

Contents

Introduction

Example Model assumption

Residual analysis

Other issues: non-linear relationship too many variables

mixed variables

p. 1


2/11

Introduction to multiregression

Multiple linear regression (multiregression) attempts to modelthe relationship between two or more explanatory variables anda response variable simultaneously by fitting a linear equation toobserved data.

p. 2


3/11



What is the different between simple linear regression andmultiregression?

p. 2


4/11



What is the different between simple linear regression andmultiregression?

yi = 0 + 1xi + i

yi = 0 + 1xi1 + 2xi2 + 3xi3 + .... + pxip + i

for i = 1, 2,...,n

p. 2


5/11

Exampl

Household spending in grocery stores

The manager of the marketing division of a grocery store chainwants to conduct a study in a particular US city, where thecompany wants to open a store, to understand the relationshipbetween the number of dollars a household spends in grocerystores each month. A group of 27 grocery shoppers were

selected by simple random sampling from a study populationand are requested to provide the needed information. Thevariables are

amount: Monthly amount spend by household in grocerystore (in US$)

income: Monthly income for the household (in US$)

children: Number of children in the household

adults: Number of adults in the household

p. 3


6/11

Assumption

1. Independence: the response variables Yi are independent.

2. Normality: the response variables Yi are normallydistributed.

3. Homoscedasticity: the response variables Yi all have the

same variance, 2.

4. Linearity: The true relationship between the mean of theresponse variable E[Y] and the exploratory variablesx1, x2,...,xp is a straight line.

p. 4


7/11

Residual Analysi

It is quite difficult to check for validity of model assumptions onthe response variables. Thus, it is much convenient to expressthe assumptions in term of random errors

i = yi (0 + 1xi1 + 2xi2 + 3xi3 + .... + pxip)

Thus, the following assumptions are that the random errors i

are independent, normally distributed, constant variance andhave zero mean.

If these assumptions are satisfied, the random errors i are

independent and identically distributed random variables withdistributionsi N(0,

2), i = 1, 2,...,n

p. 5


8/11

How residual analysis can be done

Similar to simple linear regression, we can perform the residualanalysis following these indicators:

Model adequacy check: R2

, adjusted R2

, results fromF-test and t-test.

Model goodness-of-fit: normality plot, residual plot, partialresidual plot, multicollinearity check etc.

Take note:"Even though most assumptions of multiple regression cannotbe tested explicitly, gross violations can be detected and shouldbe dealt with appropriately. In particular outliers (i.e., extremecases) can seriously bias the results by "pulling" or "pushing"the regression line in a particular direction."

p. 6


9/11

Some problems in modelling proces

What if some of the exploratory variables do not have linearrelationship to the response variable?

p. 7


10/11



What if the effect of constructing a model using a data setwhich the number of variables is larger than the number ofobjects?

p. 7


11/11



What if the effect of constructing a model using a data setwhich the number of variables is larger than the number ofobjects?

Can we construct a single model for different types ofvariables?

p. 7

Multi Regression

Documents

Transcript of Multi Regression