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© 2000 Prentice-Hall, Inc. Chap. 10 - 2
The Multiple Regression Model
The relationship between one dependent & two or more independent variables is a linear function
Population Y-intercept
Population slopes
Dependent (Response) variable for sample
Independent (Explanatory) variables for sample model
Random Error
0 1 1 2 2i i i p pi iY b b X b X b X e 1 2i i i p pi iY X X X
© 2000 Prentice-Hall, Inc. Chap. 10 - 3
Oil (Gal) Temp Insulation275.30 40 3363.80 27 3164.30 40 1040.80 73 694.30 64 6
230.90 34 6366.70 9 6300.60 8 10237.80 23 10121.40 63 331.40 65 10
203.50 41 6441.10 21 3323.00 38 352.50 58 10
Multiple Regression Model: Example
(0F)Develop a model for estimating heating oil used for a single family home in the month of January based on average temperature and amount of insulation in inches.
© 2000 Prentice-Hall, Inc. Chap. 10 - 4
Sample Multiple Regression Model: Example
CoefficientsIntercept 562.1510092X Variable 1 -5.436580588X Variable 2 -20.01232067
Excel Output
For each degree increase in temperature, the estimated average amount of heating oil used is decreased by 5.437 gallons, holding insulation constant.
For each increase in one inch of insulation, the estimated average use of heating oil is decreased by 20.012 gallons, holding temperature constant.
0 1 1 2 2i i i p piY b b X b X b X
1 2ˆ 562.151 5.437 20.012i i iY X X
© 2000 Prentice-Hall, Inc. Chap. 10 - 5
Slope (bi)
The average Y changes by bi each time Xi is
increased or decreased by 1 unit holding all other variables constant. For example: If b1 = -2, then fuel oil usage (Y) is
expected to decrease by an estimated 2 gallons for each 1 degree increase in temperature (X1) given
the inches of insulation (X2).
Interpretation of Estimated Coefficients
© 2000 Prentice-Hall, Inc. Chap. 10 - 6
Intercept (b0)
The intercept (b0) is the
estimated average value of Y when all Xi = 0.
Interpretation of Estimated Coefficients
© 2000 Prentice-Hall, Inc. Chap. 10 - 7
Using The Model to Make Predictions
Predict the amount of heating oil used for a home if the average temperature is 300 and the insulation is 6 inches.
The predicted heating oil used is 278.97 gallons
1 2ˆ 562.151 5.437 20.012i i iY X X
562.151 5.437 30 20.012 6
278.969
© 2000 Prentice-Hall, Inc. Chap. 10 - 8
Developing the Model
Checking for problems.
Being sure the model passes all tests for model quality.
© 2000 Prentice-Hall, Inc. Chap. 10 - 9
Identifying Problems
Do all the residual tests listed for simple regression.
Check for multicolinearity.
© 2000 Prentice-Hall, Inc. Chap. 10 - 10
Multicolinearity
• This occurs when there is a high correlation between the explanatory variables.
• This leads to unstable coefficients .• The VIF used to measure colinearity
(values exceeding 5 are not good and exceeding 10 are a big problem):
,R
VIFj
j 21
1
2jR = Coefficient of Multiple
Determination of Xj
with all the others
© 2000 Prentice-Hall, Inc. Chap. 10 - 12
Coefficient of Multiple Determination
Regression StatisticsMultiple R 0.982654757R Square 0.965610371Adjusted R Square 0.959878766Standard Error 26.01378323Observations 15
Excel Output
r2
Adjusted r2
The r2 is adjusted downward to reflect small sample sizes.
© 2000 Prentice-Hall, Inc. Chap. 10 - 14
Testing for Overall Significance
•Shows if there is a linear relationship between all of the X variables taken together and Y
•Hypothesis:
H0: 1 = 2 = … = p = 0 (No linear relationships)
H1: At least one i 0 (At least one independent variable effects Y)
© 2000 Prentice-Hall, Inc. Chap. 10 - 15
ANOVAdf SS MS F Significance F
Regression 2 228014.6 114007.3 168.4712 1.65411E-09Residual 12 8120.603 676.7169Total 14 236135.2
Test for Overall SignificanceExcel Output: Example
p = 2, the number of explanatory variables n - 1
MSRMSE
p value
= F Test Statistic
© 2000 Prentice-Hall, Inc. Chap. 10 - 16
F0 3.89
H0: 1 = 2 = … = p = 0
H1: At least one I 0 = .05
df = 2 and 12
Critical value(s):
Test Statistic:
Decision:
Conclusion:
Reject at = 0.05
There is evidence that at least one independentvariable affects Y.
= 0.05
F
Test for Overall Significance
168.47(Excel Output)
© 2000 Prentice-Hall, Inc. Chap. 10 - 17
Test for Significance:Individual Variables
•Shows if there is a linear relationship between each
variable Xi and Y.
•Hypotheses:
H0: i = 0 (No linear relationship)
H1: i 0 (Linear relationship between Xi and Y)
© 2000 Prentice-Hall, Inc. Chap. 10 - 18
Coefficients Standard Error t StatIntercept 562.151009 21.09310433 26.65094X Variable 1 -5.4365806 0.336216167 -16.1699X Variable 2 -20.012321 2.342505227 -8.54313
T Test StatisticExcel Output: Example
t Test Statistic for X1 (Temperature)
t Test Statistic for X2 (Insulation)
k
k
b
btS
© 2000 Prentice-Hall, Inc. Chap. 10 - 19
H0: 1 = 0
h1: 1 0
df = n-2 = 12 critical value(s):
Test Statistic:
Decision:
Conclusion:
Reject H0 at = 0.05
There is evidence of a significant effect of temperature on oil
consumption.t0 2.1788-2.1788
.025
Reject H0 Reject H0
.025
Does temperature have a significant effect on monthly consumption of heating oil? Test at = 0.05.
t Test : Example Solution
t Test Statistic = -16.1699
© 2000 Prentice-Hall, Inc. Chap. 10 - 20
Confidence Interval Estimate For The Slope
Provide the 95% confidence interval for the population slope 1 (the effect of temperature on oil consumption).
111 bpn Stb Coefficients Lower 95% Upper 95%
Intercept 562.151009 516.1930837 608.108935X Variable 1 -5.4365806 -6.169132673 -4.7040285X Variable 2 -20.012321 -25.11620102 -14.90844
-6.169 1 -4.704The estimated average consumption of oil is reduced by between 4.7 gallons to 6.17 gallons per each increase of 10 F.
© 2000 Prentice-Hall, Inc. Chap. 10 - 22
Dummy-variable Models
• Create a categorical variable (dummy variable) with 2 levels: For example, yes and no or male and female. The date is coded as 0 or 1.
• The coding makes the intercepts different.
• This analysis assumes equal slopes.
• The regression model has same form: ipipiii XXXY 22110
© 2000 Prentice-Hall, Inc. Chap. 10 - 23
Dummy-variable Models Assumption
Given:
Y = Assessed Value of House
X1 = Square footage of House
X2 = Desirability of Neighborhood =
Desirable (X2 = 1)
Undesirable (X2 = 0)
iii Xb)bb()(bXbbY 11202110 1
0 if undesirable 1 if desirable
iii Xbb)(bXbbY 1102110 0
iii XbXbbY 22110
Same slopes
© 2000 Prentice-Hall, Inc. Chap. 10 - 24
Dummy-variable Models Assumption
X1 (Square footage)
Y (Assessed Value)
Desirable Location
Undesirableb0 + b2
b0
Same slopes
Intercepts different
© 2000 Prentice-Hall, Inc. Chap. 10 - 25
Interpretation of the Dummy Variable Coefficient
For example:
0 1 1 2 2i i iY b b X b X
1X
1 220 5 6i iX X
: GPA2X
0 Female
1 Male
Y: Annual salary of college graduate in thousand $
This 6 is interpreted as given the same GPA, the male college graduate is making an estimated 6 thousand dollars more than female on average.
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