multiple regression 2

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Business Research Methods

Multiple Regression

PIYOOSH BAJORIA



Multiple Regression

• Multiple Regression Model

• Least Squares Method

• Multiple Coefficient of Determination

• Model Assumptions• Testing for Significance

• Using the Estimated Regression Equation

for Estimation and Prediction• Qualitative Independent Variables

PIYOOSH BAJORIA



The Multiple Regression Model

• The Multiple Regression Model

y = 0 + 1 x1 + 2 x2 + . . . + p x p +

• The Estimated Multiple Regression Equation

y = b0

+ b1 x

1+ b

2 x

2+ . . . + b

p x

p

^

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The Least Squares Method

• Least Squares Criterion

• Computation of Coefficients’ Values

The formulas for the regression coefficients b0, b1, b2, . .

. b p involve the use of matrix algebra. We will rely on

computer software packages to perform the calculations.

• A Note on Interpretation of Coefficients

bi represents an estimate of the change in y

corresponding to a one-unit change in xi when all other independent variables are held constant & is known as

partial regression coefficient

min ( iy y i )2

min ( iy y i

)

2

^

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The Multiple Coefficient of Determination

• Relationship Among SST, SSR, SSE

SST = SSR + SSE

• Multiple Coefficient of Determination

R 2 = SSR/SST

• Adjusted Multiple Coefficient of Determination

( ) ( ) ( )y y y y y yi i i i 2 2 2( ) ( ) ( )y y y y y yi i i i 2 2 2

^^

R Rn

n pa2 21 1

1

1

( )R Rn

n pa2 21 1

1

1

( )

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The Multiple Coefficient of Determination

• Multiple Coefficient of Determination R 2:It is the

square of multiple correlation coefficient R & itmeasures strength of association in multiple

regression

• Adjusted Multiple Coefficient of Determination:

R 2 is adjusted for the number of independent

variables & the sample size to account for

diminishing returns. After the first few variables

the additional variables do not make much

contribution

^^

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Model Assumptions• Assumptions About the Error Term

– The error is a random variable with mean of zero.

– The variance of , denoted by 2, is the same for all

values of the independent variables.

– The values of are independent.

– The error is a normally distributed random variable

reflecting the deviation between the y value and the

expected value of y given by 0 + 1 x1 + 2 x2 + . . . + p x p

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Testing for Significance: F Test

• Hypotheses

H 0: 1 = 2 = . . . = p = 0 H a: One or more of the parameters

is not equal to zero.

• Test Statistic F = MSR/MSE

• Rejection Rule

Reject H 0 if F > F

where F is based on an F distribution with p d.f.

in the numerator and n - p - 1 d.f. in the

denominator. PIYOOSH BAJORIA



Testing for Significance: t Test

• Hypotheses

H 0: i = 0

H a: i = 0

• Test Statistic

• Rejection RuleReject H 0 if t < -t or t > t

where t is based on a t distribution with

n - p - 1 degrees of freedom.

t b

s

i

bi

t b

s

i

bi

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Testing for Significance: Multicollinearity

• The term multicollinearity refers to thecorrelation among the independent variables.

• When the independent variables are highly

correlated (say, |r | > .7), it is not possible todetermine the separate effect of any

particular independent variable on the

dependent variable.

• Every attempt should be made to avoid

including independent variables that are

hi hl correlated.PIYOOSH BAJORIA



Using the Estimated Regression Equation

for Estimation and Prediction

• The procedures for estimating the mean value of y

and predicting an individual value of y in multiple

regression are similar to those in simple regression.

• We substitute the given values of x1, x2, . . . , x p into

the estimated regression equation and use the

corresponding value of y as the point estimate. .

• Software packages for multiple regression will often

provide the interval estimates for an individual value

of y

^

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Example: Programmer Salary Survey

A software firm collected data for a sample of

20computer programmers. A suggestion was made

that regression analysis could be used to determine

if salary was related to the years of experience and

the score on the firm’s programmer aptitude test.The years of experience, score on the aptitude

test, and corresponding annual salary (Rs10000) for

a sample of 20 programmers is shown on the nextslide.

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Exper. Score Salary Exper. Score Salary4 78 24 9 88 38

7 100 43 2 73 26.6

1 86 23.7 10 75 36.2

5 82 34.3 5 81 31.6

8 86 35.8 6 74 29

10 84 38 8 87 34

0 75 22.2 4 79 30.1

1 80 23.1 6 94 33.9

6 83 30 3 70 28.2

6 91 33 3 89 30PIYOOSH BAJORIA

E l P S l S




• Multiple Regression Model

Suppose we believe that salary ( y) is related to theyears of experience ( x1) and the score on the

programmer aptitude test ( x2) by the following

regression model:

y = 0 + 1 x1 + 2 x2 +

where

y = annual salary (Rs.0000)

x1 = years of experience

x2

= score on programmer aptitude testPIYOOSH BAJORIA




• Estimated Regression Equationb0, b1, b2 are the least squares estimates of

0, 1, 2

Thus y = b0 + b1 x1 + b2 x2

^

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• Computer Output

Predictor Coef Stdev t-ratio p

Constant 3.174 6.156 .52 .613

Exper 1.4039 .1986 7.07 .000

Score .25089 .07735 3.24 .005

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INTERPRETATION



INTERPRETATIONPartial regression coefficients are 1.40 for

experience and .251 for score

Increase in salary per unit increase in experienceis 1.40 units when test score is constant

Increase in salary per unit increase in test score

is .251 units when experience is constantThe regression equation is

Salary = 3.17 + 1.40 Exper + 0.251 Score

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The p ratio for

Experience is .000. This is less than .05.Hence experience is a significant variablein explaining salary

The p ratio for

Score is .005 which is also less than .05.

Hence score is also a significant variable inexplaining salary

PIYOOSH BAJORIA




• Computer Output (continued)

Analysis of Variance SOURCE DF SS MS F P

Regression 2 500.33 250.16 42.76 0.000

Error 17 99.46 5.85Total 19 599.79

R-sq = 83.4% R-sq(adj) = 81.5%

Interpretation : The p ratio of overall model is.000.This is less than .05. Hence overall regression

model is significant at 5% level of significance

Since R-Sqare is 81.5 %,the regression model explains

81.5% variation in salaryPIYOOSH BAJORIA




• F Test

– Hypotheses H 0: 1 = 2 = 0

H a: One or both of the parameters

is not equal to zero.

– Rejection Rule

For = .05 and d.f. = 2, 17: F .05 = 3.59

Reject H 0 if F > 3.59.

– Test Statistic

F = MSR/MSE = 250.16/5.85 = 42.76

– Conclusion

We can reject H 0.PIYOOSH BAJORIA



Qualitative Independent Variables

• In many situations we must work with qualitativeindependent variables such as gender (male, female),method of payment (cash, check, credit card), etc.

• For example, x2 might represent gender where x2 = 0indicates male and x2 = 1 indicates female.

• In this case, x2 is called a dummy or indicator variable.• If a qualitative variable has k levels, k - 1 dummy

variables are required, with each dummy variable beingcoded as 0 or 1.

• For example, a variable with levels A, B, and C would berepresented by x1 and x2 values of (0, 0),

(1, 0), and (0,1), respectively.

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Example: Programmer Salary Survey (B)

As an extension of the problem involving thecomputer programmer salary survey, suppose that

management also believes that the annual salary is

related to whether or not the individual has a graduate

degree in computer science or information systems.

The years of experience, the score on the programmer

aptitude test, whether or not the individual has a

relevant graduate degree, and the annual salary (Rs.0000)

for each of the sampled 20 programmers are shown on

the next slide.PIYOOSH BAJORIA




Exp. Score Degr. Salary Exp. Score Degr. Salary4 78 No 24 9 88 Yes 38

7 100 Yes 43 2 73 No 26.6

1 86 No 23.7 10 75 Yes 36.2

5 82 Yes 34.3 5 81 No 31.6

8 86 Yes 35.8 6 74 No 29

10 84 Yes 38 8 87 Yes 34

0 75 No 22.2 4 79 No 30.11 80 No 23.1 6 94 Yes 33.9

6 83 No 30 3 70 No 28.2

6 91 Yes 33 3 89 No 30PIYOOSH BAJORIA




• Multiple Regression Equation

E( y ) = 0 + 1 x1 + 2 x2 + 3 x3 • Estimated Regression Equation

y = b0 + b1 x1 + b2 x2 + b3 x3 where

y = annual salary (Rs.0000) x1 = years of experience

x2 = score on programmer aptitude test

x3 = 0 if individual does not have a grad. degree

1 if individual does have a grad. degree

Note: x3 is referred to as a dummy variable.

^

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E l P S l S (B)




• Computer Output

The regression is Salary = 7.95 + 1.15 Exp + 0.197 Score + 2.28 Deg

Predictor Coef Stdev t-ratio p

Constant 7.945 7.381 1.08 .298

Exp 1.1476 .2976 3.86 .001

Score .19694 .0899 2.19 .044

Deg 2.280 1.987 1.15 .268

SE=S =√MSE= √5.74= 2.396 R-sq = 84.7% R-sq(adj) = 81.8%

PIYOOSH BAJORIA



Example: Programmer Salary

Survey (B)• Computer Output (continued)

Analysis of Variance

SOURCE DF SS MS F P

Regression 3 507.90 169.30 29.48 0.000

Error 16 91.89 5.74

Total 19 599.79

Since p value is < .05 , we reject hypothesis 1= 2= 3=0Regression model is significant & explains = 81.8% of total

variation

PIYOOSH BAJORIA



Partial CorrelationA partial correlation coefficient measures the

association between two variables after controlling for,

or adjusting for, the effects of one or more additional

variables.

• Partial correlations have an order associated with them. Theorder indicates how many variables are being adjusted or controlled.

• The simple correlation coefficient, r , has a zero-order, as it doesnot control for any additional variables while measuring theassociation between two variables.

r x y . z = r x y - ( r x z ) ( r y z ) 1 - r x z 2 1 - r y z 2

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Nonmetric Correlation

• If the nonmetric variables are ordinal and numeric,

Spearman's rho, , and Kendall's tau, , are two measuresof nonmetric correlation, which can be used to examinethe correlation between them.

• Both these measures use rankings rather than the absolutevalues of the variables, and the basic concepts underlying

them are quite similar. Both vary from -1.0 to +1.0 .• In the absence of ties, Spearman's yields a closer

approximation to the Pearson product moment correlationcoefficient, , than Kendall's . In these cases, theabsolute magnitude of tends to be smaller than Pearson's

.• On the other hand, when the data contain a large number of tied ranks, Kendall's seems more appropriate.

s

s

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Regression with Dummy Variables

Product Usage Original Dummy Variable Code

Category VariableCode D1 D2 D3

Nonusers............... 1 1 0 0

Light Users........... 2 0 1 0

Medium Users....... 3 0 0 1

Heavy Users.......... 4 0 0 0

i = a + b1 D1 + b2 D2 + b3 D3

• In this case, "heavy users" has been selected as a reference categoryand has not been directly included in the regression equation.

• The coefficient b1 is the difference in predictedi

for nonusers, ascompared to heavy users.

Y

Y

PIYOOSH BAJORIA

multiple regression 2

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