Regression multiple Dan Fisher Marriott School of Management Brigham Young University November 2005 linear

regressionmultiple

Dan FisherMarriott School of Management

Brigham Young University

November 2005

linear

• definitions

• uses of multiple linear regression

• selection of variables

• formula

• application exercise

• multiple regression analysis

• summary

what will be covered:

multipleregression

linear

multipleregression

linear

definitions

Multiple linear regression is a method of determining the relationship between a continuous process output (Y) and several factors (Xs).

Multiple linear regression is a quantitative method of forecasting involving the use of more than one variable to predict some criterion.

multipleregression

definitions

linear

Multiple regression differs from simple regression in that it studies the relationship between a single dependent variable and two or more independent variables.

uses of multiple linear regression

multipleregression

linear

• definitions



• formula



• summary

multipleregression

linear

uses of multiple regression

There are two major uses for multiple regression:

• Forecasting

• Causal analysis

multipleregression

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forecasting

An estimate of the future level of some variable.

We can use forecasting to determine future supply, demand, pricing, sales, or some other variable of interest.

multipleregression

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causal analysis

Independent variables are regarded as causes of the dependent variable

The goal is to determine whether a particular independent variable really affects the dependent variable and how strong that relationship is.

multipleregression

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selection of variables

• definitions



• formula



• summary

multipleregression

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In multiple linear regression we use two primary types of variables:

• dependent variables and

• independent variables

multipleregression

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From the data collected we determine that the main items of interest are as follows:

1. Number of vehicles unloaded

2. Number of different containers (pallets, etc.) handled

3. Total quantity of parts handled

Example: Measuring Work in a Warehouse

In our warehouse example, the dependent variable is standard hours worked.

multipleregression

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dependent variable

The dependent variable is the factor that we are trying to measure by using multiple linear regression.

multipleregression

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independent variables

In multiple regression analysis we will have two or more independent variables.

In our warehouse example, the independent variables are:

1. Quantity of vehicles unloaded

2. Quantity of containers handled

3. Quantity of parts handled

multipleregression

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formula for multiple regression

• definitions



• formula for multiple regression



• summary

multipleregression

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i

k

ii xbay

1

ˆˆˆ

ii

i

xb

a

kitindependenix

tindependenk

ydependenty

able with variassociatedt coefficien slope estimatedˆ

line for the termintercept estimatedˆ

1... where variable,th

variables ofnumber

, variablefor forecast

The multiple regression forecast model is defined as follows:

where:

formula for multiple regression

• computer software programs are available that can perform complex calculations for us

• manual calculations are time-consuming and prone to error

multipleregression

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computer analysis

multipleregression

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application exercise

• definitions






• summary

multipleregression

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application exercise CarMiles Per

GallonHorsepower

Engine Displaceme

nt

1 27 130 112

2 28 81 90

3 30 93 135

4 28 113 97

5 31 90 114

6 33 63 81

7 40 55 61

8 30 102 97

9 33 92 91

10 32 81 90

11 29 103 113

12 31 90 97

13 33 74 98

14 35 73 73

15 29 102 97

16 32 78 89

17 28 100 109

Using a software program capable of regression analysis, such as Microsoft Excel, input the data to the right and analyze the data to determine the relationship between the independent and dependent variables.

multipleregression

linear

• definitions






• summary

multiple regression analysis

multipleregression

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multiple regression analysisdependent variable

independent variables

dependent variableindependent variables

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R2

Coefficients

R

R2, the coefficient of multiple determination• strength of the relationship between the dependent

and independent variables

• R2 = .73

multipleregression

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multiple regression analysisMeasures of effectiveness

R, the correlation coefficient • The square root of R2

• R = .86

multipleregression

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multiple regression analysisMeasures of effectiveness

multipleregression

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The equation for the gas mileage example would be in the form below and when plotted would form a plane similar to the one here.

iixbxbbyi 22110ˆ

22

11

0

lefor variabt coefficien regression the

xlefor variabt coefficien regression the

constant regressionor intercept y the

n observatiofor y of valuepredicted theˆ

xb

b

b

iyi

where:

multipleregression

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the regression coefficients and predicting the dependent variable y

05797.0

10932.0

429.46

2

1

0

b

b

bii

xxy 211 05797.10932.429.46ˆ

ix

ix

iy

i

i

i

car for nt displaceme engine the

car for horsepower the

car for gallon per miles predicted the

2

1

Regression coefficients: Regression equation:

where:

MPG 7.29ˆ

)100(05797.)100(10932.429.46ˆ

1

1

y

ytherefore:

multipleregression

linear

• definitions






• summary

summary

multipleregression

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summary

Multiple regression is a statistical tool that management can use in order to create forecasts and perform causal analysis.

Multiple regression differs from simple regression in that it studies the relationship between a single dependent variable and two or more independent variables.

It is important to select independent variables carefully so that we study the intended relations.

multipleregression

linear

summary

Software programs are available that can perform the complex regression calculations for us.

The R2 value is an indicator of how well the model fits the data. R tells us how tightly the variables are correlated to one another.

By plugging values into the regression equation we end up with a prediction.

multipleregression

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readings list:

Bowerman, O’Connell, Koehler. Forecasting, Time Series, and Regression 4E. Duxbury, 2005.

Golberg, M.A.; Cho, H.A. Introduction to Regression Analysis. WIT Press, 2004.

Lewis-Beck, Michael. Applied Regression. Sage Publications, 1980.

Allison, Paul D. Multiple Regression: A Primer. Pine Forge Press, 1999.

Aguinis, Herman. Regression Analysis for Categorical Moderators. Guilford Press, 2004.

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Regression multiple Dan Fisher Marriott School of Management Brigham Young University November 2005 linear