1. Multiple Regression Analysis
Multiple regression uses many independent
variables to predict or explain the variation
in a dependent variables
The basic multiple regression model is a
first-order model, containing each predictor
but no nonlinear terms such as squared values.
In this model, each slope should be interpreted
as a partial slope, the predicted effect of a one-
change in a variable, holding all other variables
constant.

2. Estimating Equation Describing Relationship among Three Variables
Multiple Regression Analysis
Estimating Equation Describing Relationship
among Three Variables
Estimating Equation Describing Relationship
among n Variables

3. Multiple Regression Analysis – Example File: PPT_MultRegr
The department is interested to know whether the amount of field audits and computer hours spent on tracking have yielded any results. Further the department has introduced the reward system for tracking the culprits. The data on actual unpaid taxes for ten cases is considered for analysis. Initially the regression of Actual Unpaid Taxes(Y) on Field Audits(X1) and Computer Hours(X2) was carried out and as a next step the Reward to Informants(X3) was also considered as a variable and analyzed. The analysis yielded the following SPSS outputs.

4. Regression Statistics
Multiple Regression and Correlation Analysis – Excel Output
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations 10
ANOVA df SS MS F
Significance F
Regression 2
Residual 7
Total 9
29.6
Coefficients
t Stat
P-value
Lower 95%
Upper 95%
Lower 95.0%
Upper 95.0%
Intercept
X Variable 1
X Variable 2

5. Multiple Regression and Correlation Analysis – SPSS Output

6. Regression Statistics
Multiple Regression and Correlation Analysis – Excel Output for 3 Independent Variables
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations 10
ANOVA df SS MS F
Significance F
Regression 3
E-06
Residual 6
Total 9
29.6
Coefficients
t Stat
P-value
Lower 95%
Upper 95%
Intercept
8.288E-05
X Variable 1
X Variable 2
8.29E-06
X Variable 3
7.341E-05

7. Multiple Regression and Correlation Analysis –
SPSS Output for 3 Independent Variables

8. Using two independent variables :
Multiple Regression and Correlation Analysis
Using two independent variables :
Using three independent variables:

9. Coefficient of Determination
From the output, R2 = 0.983
98.3% of the variation in actual unpaid taxes is explained by the three independent variables. 1.7% remains unexplained.

10. Making inferences about Population Parameters
Multiple Regression and Correlation Analysis
Making inferences about Population Parameters
1. Inferences about an individual slope
or whether a variable is significant
2. Regression as a whole

11. Inferences about the Regression as a Whole
Multiple Regression and Correlation Analysis
Inferences about the Regression as a Whole

12. Value of the test statistic:
Multiple Regression and Correlation Analysis
Test Statistic
Value of the test statistic:

13. Multiple Regression and Correlation Analysis

14. Test of whether a variable is significant.
Multiple Regression and Correlation Analysis
Test of whether a variable is significant.
Test whether reward to informants is a significant
explanatory variable.

15. Test statistic, with n-2 degrees of freedom:
Multiple Regression and Correlation Analysis
Test statistic, with n-2 degrees of freedom:
Rejection Region

16. Value of the test statistic:
Multiple Regression and Correlation Analysis
Value of the test statistic:
Conclusion:
The standardized regression coefficient
is which is outside the acceptance
region. Therefore we will reject the null
hypothesis. The reward to informants is
a significant explanatory variable.

17. Interpreting the Coefficients
b0 = This is the intercept, the value of y when all the variables take the value zero. Since the data range of all the independent variables do not cover the value zero, do not interpret the intercept.
b1 = In this model, for each additional field audit, the actual unpaid taxes increases on average by .597% (assuming the other variables are held constant).

18. Estimating the Coefficients and Assessing the Model, Another Example
Where to locate a new motor inn?
La Quinta Motor Inns is planning an expansion.
Management wishes to predict which sites are likely to be profitable.
Several areas where predictors of profitability can be identified are:
Competition
Market awareness
Demand generators
Demographics
Physical quality

19. Estimating the Coefficients and Assessing the Model, Example
Operating Margin
Profitability
Market
awareness
Physical
Competition
Customers
Community
Rooms
Nearest
Office
space
College
enrollment
Income
Disttwn
Number of
hotels/motels
rooms within
3 miles from
the site.
Distance to
the nearest
La Quinta inn.
Median
household
income.
Distance to
downtown.

20. Estimating the Coefficients and Assessing the Model, Example
Data were collected from randomly selected 100 inns that belong to La Quinta, and ran for the following suggested model (Multiple Regr_Margin.sav):
Margin = b0 + b1Rooms + b2Nearest + b3Office + b4College + b5Income + b6Disttwn

21. Regression Analysis, SPSS Output
This is the sample regression equation
(sometimes called the prediction equation)
Margin = Number Nearest Office Space Enrollment Income Distance

22. Coefficient of Determination
From the printout, R2 = 0.525
52.5% of the variation in operating margin is explained by the six independent variables. 47.5% remains unexplained.

23. Testing the Validity of the Model
We pose the question:
Is there at least one independent variable linearly related to the dependent variable?
To answer the question we test the hypothesis
H0: B0 = B1 = B2 = … = Bk
H1: At least one Bi is not equal to zero.
If at least one Bi is not equal to zero, the model has some validity.

24. Testing the Validity of the La Quinta Inns Regression Model
The hypotheses are tested by an ANOVA procedure ( the SPSS output)
MSR/MSE
k =
n–k–1 =
n-1 =
SSR
MSR=SSR/k
MSE=SSE/(n-k-1)
SSE

25. Testing the Validity of the La Quinta Inns Regression Model
Conclusion: There is sufficient evidence to reject
the null hypothesis in favor of the alternative hypothesis.
At least one of the bi is not equal to zero. Thus, at least one independent variable is linearly related to y.
This linear regression model is valid
Fa,k,n-k-1 = F0.05,6, =2.17
F = > 2.17
Also, the p-value (Significance F) =
Reject the null hypothesis.

26. Interpreting the Coefficients
b0 = This is the intercept, the value of y when all the variables take the value zero. Since the data range of all the independent variables do not cover the value zero, do not interpret the intercept.
b1 = – In this model, for each additional room within 3 mile of the La Quinta inn, the operating margin decreases on average by .008% (assuming the other variables are held constant).

27. Interpreting the Coefficients
b2 = In this model, for each additional mile that the nearest competitor is to a La Quinta inn, the operating margin increases on average by 1.646% when the other variables are held constant.
b3 = For each additional 1000 sq-ft of office space, the operating margin will increase on average by .02% when the other variables are held constant.
b4 = For each additional thousand students the operating margin increases on average by .212% when the other variables are held constant.

28. Interpreting the Coefficients
b5 = For additional $1000 increase in median household income, the operating margin increases on average by .413%, when the other variables remain constant.
b6 = For each additional mile to the downtown center, the operating margin decreases on average by .225% when the other variables are held constant.

29. Testing the Coefficients
The hypothesis for each bi is
SPSS printout
H0: bi = 0
H1: bi ¹ 0
Test statistic
d.f. = n - k -1

30. La Quinta Inns, Predictions
Predict the average operating margin of an inn at a site with the following characteristics:
3815 rooms within 3 miles,
Closet competitor .9 miles away,
476,000 sq-ft of office space,
24,500 college students,
$35,000 median household income,
11.2 miles distance to downtown center.
MARGIN = (3815) +1.65(.9) (476)
+0.212(24.5) (35) (11.2) = 37.1%