Document related concepts
no text concepts found
Transcript
```Statistics for
6th Edition
Chapter 13
Multiple Regression
Chap 13-1
Chapter Goals
After completing this chapter, you should be able to:
 Apply multiple regression analysis to business decisionmaking situations
 Analyze and interpret the computer output for a multiple
regression model
 Perform a hypothesis test for all regression coefficients
or for a subset of coefficients
 Fit and interpret nonlinear regression models
 Incorporate qualitative variables into the regression
model by using dummy variables
 Discuss model specification and analyze residuals
Chap 13-2
The Multiple Regression
Model
Idea: Examine the linear relationship between
1 dependent (Y) & 2 or more independent variables (Xi)
Multiple Regression Model with k Independent Variables:
Y-intercept
Population slopes
Random Error
Y  β0  β1X1  β2 X2    βk Xk  ε
Chap 13-3
Multiple Regression Equation
The coefficients of the multiple regression model are
estimated using sample data
Multiple regression equation with k independent variables:
Estimated
(or predicted)
value of y
Estimated
intercept
Estimated slope coefficients
yˆ i  b0  b1x1i  b2 x 2i    bk xki
In this chapter we will always use a computer to obtain the
regression slope coefficients and other regression
summary measures.
Chap 13-4
Multiple Regression Equation
(continued)
Two variable model
y
yˆ  b0  b1x1  b2 x 2
x2
x1
Chap 13-5
Standard Multiple Regression
Assumptions

The values xi and the error terms εi are
independent

The error terms are random variables with
mean 0 and a constant variance, 2.
E[ε i ]  0 and E[ε i2 ]  σ 2
for (i  1, ,n)
(The constant variance property is called
homoscedasticity)
Chap 13-6
Standard Multiple Regression
Assumptions
(continued)

The random error terms, εi , are not correlated
with one another, so that
E[ε iε j ]  0

for all i  j
It is not possible to find a set of numbers, c0,
c1, . . . , ck, such that
c 0  c1x1i  c 2 x 2i    cK xKi  0
(This is the property of no linear relation for
the Xj’s)
Chap 13-7
Example:
2 Independent Variables
 A distributor of frozen desert pies wants to
evaluate factors thought to influence demand
 Dependent variable:
Pie sales (units per week)
 Independent variables: Price (in \$)
 Data are collected for 15 weeks
Chap 13-8
Pie Sales Example
Week
Pie
Sales
Price
(\$)
(\$100s)
1
350
5.50
3.3
2
460
7.50
3.3
3
350
8.00
3.0
4
430
8.00
4.5
5
350
6.80
3.0
6
380
7.50
4.0
7
430
4.50
3.0
8
470
6.40
3.7
9
450
7.00
3.5
10
490
5.00
4.0
11
340
7.20
3.5
12
300
7.90
3.2
13
440
5.90
4.0
14
450
5.00
3.5
15
300
7.00
2.7
Multiple regression equation:
Sales = b0 + b1 (Price)
Chap 13-9
Estimating a Multiple Linear
Regression Equation
 Excel will be used to generate the coefficients
and measures of goodness of fit for multiple
regression
 Excel:
 Tools / Data Analysis... / Regression
 PHStat:
 PHStat / Regression / Multiple Regression…
Chap 13-10
Multiple Regression Output
Regression Statistics
Multiple R
0.72213
R Square
0.52148
0.44172
Standard Error
47.46341
Observations
ANOVA
Regression
Sales  306.526 - 24.975(Pri ce)  74.131(Advertising)
15
df
SS
MS
F
2
29460.027
14730.013
Residual
12
27033.306
2252.776
Total
14
56493.333
Coefficients
Standard Error
Intercept
306.52619
114.25389
2.68285
0.01993
57.58835
555.46404
Price
-24.97509
10.83213
-2.30565
0.03979
-48.57626
-1.37392
74.13096
25.96732
2.85478
0.01449
17.55303
130.70888
t Stat
6.53861
Significance F
P-value
0.01201
Lower 95%
Upper 95%
Chap 13-11
The Multiple Regression Equation
Sales  306.526 - 24.975(Pri ce)  74.131(Advertising)
where
Sales is in number of pies per week
Price is in \$
b1 = -24.975: sales
will decrease, on
average, by 24.975
pies per week for
each \$1 increase in
selling price, net of
the effects of changes
b2 = 74.131: sales will
increase, on average,
by 74.131 pies per
week for each \$100
increase in
effects of changes
due to price
Chap 13-12
Coefficient of Determination, R2
 Reports the proportion of total variation in y
explained by all x variables taken together
SSR regression sum of squares
R 

SST
total sum of squares
2
 This is the ratio of the explained variability to
total sample variability
Chap 13-13
Coefficient of Determination, R2
(continued)
Regression Statistics
Multiple R
0.72213
R Square
0.52148
0.44172
Standard Error
Regression
52.1% of the variation in pie sales
is explained by the variation in
47.46341
Observations
ANOVA
SSR 29460.0
R 

 .52148
SST 56493.3
2
15
df
SS
MS
F
2
29460.027
14730.013
Residual
12
27033.306
2252.776
Total
14
56493.333
Coefficients
Standard Error
Intercept
306.52619
114.25389
2.68285
0.01993
57.58835
555.46404
Price
-24.97509
10.83213
-2.30565
0.03979
-48.57626
-1.37392
74.13096
25.96732
2.85478
0.01449
17.55303
130.70888
t Stat
6.53861
Significance F
P-value
0.01201
Lower 95%
Upper 95%
Chap 13-14
Estimation of Error Variance
 Consider the population regression model
Yi  β0  β1x1i  β2 x 2i    βK xKi  ε i
 The unbiased estimate of the variance of the errors is
n
s2e 
2
e
 i
SSE
n  K 1 n  K 1
i1

where ei  y i  yˆ i
 The square root of the variance, se , is called the
standard error of the estimate
Chap 13-15
Standard Error, se
Regression Statistics
Multiple R
0.72213
R Square
0.52148
0.44172
Standard Error
Regression
The magnitude of this
value can be compared to
the average y value
47.46341
Observations
ANOVA
se  47.463
15
df
SS
MS
F
2
29460.027
14730.013
Residual
12
27033.306
2252.776
Total
14
56493.333
Coefficients
Standard Error
Intercept
306.52619
114.25389
2.68285
0.01993
57.58835
555.46404
Price
-24.97509
10.83213
-2.30565
0.03979
-48.57626
-1.37392
74.13096
25.96732
2.85478
0.01449
17.55303
130.70888
t Stat
6.53861
Significance F
P-value
0.01201
Lower 95%
Upper 95%
Chap 13-16
Determination, R 2
 R2 never decreases when a new X variable is
added to the model, even if the new variable is
not an important predictor variable
 This can be a disadvantage when comparing
models
 What is the net effect of adding a new variable?
 We lose a degree of freedom when a new X
 Did the new X variable add enough
explanatory power to offset the loss of one
degree of freedom?
Chap 13-17
Determination, R 2
(continued)
 Used to correct for the fact that adding non-relevant
independent variables will still reduce the error sum of
squares
SSE / (n  K  1)
2
R  1
SST / (n  1)
(where n = sample size, K = number of independent variables)
 Adjusted R2 provides a better comparison between
multiple regression models with different numbers of
independent variables
 Penalize excessive use of unimportant independent
variables
 Smaller than R2
Chap 13-18
R
Regression Statistics
Multiple R
0.72213
R Square
0.52148
0.44172
Standard Error
47.46341
Observations
ANOVA
Regression
15
df
2
R  .44172
2
44.2% of the variation in pie sales is
explained by the variation in price and
advertising, taking into account the sample
size and number of independent variables
SS
MS
F
2
29460.027
14730.013
Residual
12
27033.306
2252.776
Total
14
56493.333
Coefficients
Standard Error
Intercept
306.52619
114.25389
2.68285
0.01993
57.58835
555.46404
Price
-24.97509
10.83213
-2.30565
0.03979
-48.57626
-1.37392
74.13096
25.96732
2.85478
0.01449
17.55303
130.70888
t Stat
6.53861
Significance F
P-value
0.01201
Lower 95%
Upper 95%
Chap 13-19
Coefficient of Multiple
Correlation
 The coefficient of multiple correlation is the correlation
between the predicted value and the observed value of
the dependent variable
R  r(yˆ , y)  R2
 Is the square root of the multiple coefficient of
determination
 Used as another measure of the strength of the linear
relationship between the dependent variable and the
independent variables
 Comparable to the correlation between Y and X in
simple regression
Chap 13-20
Evaluating Individual
Regression Coefficients
 Use t-tests for individual coefficients
 Shows if a specific independent variable is
conditionally important
 Hypotheses:
 H0: βj = 0 (no linear relationship)
 H1: βj ≠ 0 (linear relationship does exist
between xj and y)
Chap 13-21
Evaluating Individual
Regression Coefficients
(continued)
H0: βj = 0 (no linear relationship)
H1: βj ≠ 0 (linear relationship does exist
between xi and y)
Test Statistic:
t
bj  0
(df = n – k – 1)
Sb j
Chap 13-22
Evaluating Individual
Regression Coefficients
(continued)
Regression Statistics
Multiple R
0.72213
R Square
0.52148
0.44172
Standard Error
47.46341
Observations
ANOVA
Regression
15
df
t-value for Price is t = -2.306, with
p-value .0398
t-value for Advertising is t = 2.855,
with p-value .0145
SS
MS
F
2
29460.027
14730.013
Residual
12
27033.306
2252.776
Total
14
56493.333
Coefficients
Standard Error
Intercept
306.52619
114.25389
2.68285
0.01993
57.58835
555.46404
Price
-24.97509
10.83213
-2.30565
0.03979
-48.57626
-1.37392
74.13096
25.96732
2.85478
0.01449
17.55303
130.70888
t Stat
6.53861
Significance F
P-value
0.01201
Lower 95%
Upper 95%
Chap 13-23
Example: Evaluating Individual
Regression Coefficients
From Excel output:
H0: βj = 0
H1: βj  0
Price
d.f. = 15-2-1 = 12
 = .05
Coefficients
Standard Error
t Stat
P-value
-24.97509
10.83213
-2.30565
0.03979
74.13096
25.96732
2.85478
0.01449
The test statistic for each variable falls
in the rejection region (p-values < .05)
t12, .025 = 2.1788
Decision:
/2=.025
/2=.025
Reject H0 for each variable
Conclusion:
Reject H0
Do not reject H0
-tα/2
-2.1788
0
Reject H0
tα/2
2.1788
There is evidence that both
pie sales at  = .05
Chap 13-24
Confidence Interval Estimate
for the Slope
Confidence interval limits for the population slope βj
b j  tnK 1,α/2Sb j
Coefficients
Standard Error
Intercept
306.52619
114.25389
Price
-24.97509
10.83213
74.13096
25.96732
where t has
(n – K – 1) d.f.
Here, t has
(15 – 2 – 1) = 12 d.f.
Example: Form a 95% confidence interval for the effect of
changes in price (x1) on pie sales:
-24.975 ± (2.1788)(10.832)
So the interval is
-48.576 < β1 < -1.374
Chap 13-25
Confidence Interval Estimate
for the Slope
(continued)
Confidence interval for the population slope βi
Coefficients
Standard Error
…
Intercept
306.52619
114.25389
…
57.58835
555.46404
Price
-24.97509
10.83213
…
-48.57626
-1.37392
74.13096
25.96732
…
17.55303
130.70888
Lower 95%
Upper 95%
Example: Excel output also reports these interval endpoints:
Weekly sales are estimated to be reduced by between 1.37 to
48.58 pies for each increase of \$1 in the selling price
Chap 13-26
Test on All Coefficients
 F-Test for Overall Significance of the Model
 Shows if there is a linear relationship between all
of the X variables considered together and Y
 Use F test statistic
 Hypotheses:
H0: β1 = β2 = … = βk = 0 (no linear relationship)
H1: at least one βi ≠ 0 (at least one independent
variable affects Y)
Chap 13-27
F-Test for Overall Significance
 Test statistic:
MSR
SSR/K
F 2 
se
SSE/(n  K  1)
where F has k (numerator) and
(n – K – 1) (denominator)
degrees of freedom
 The decision rule is
Reject H0 if F  Fk,nK 1,α
Chap 13-28
F-Test for Overall Significance
(continued)
Regression Statistics
Multiple R
0.72213
R Square
0.52148
0.44172
Standard Error
47.46341
Observations
ANOVA
Regression
15
df
MSR 14730.0
F

 6.5386
MSE
2252.8
With 2 and 12 degrees
of freedom
SS
MS
P-value for
the F-Test
F
2
29460.027
14730.013
Residual
12
27033.306
2252.776
Total
14
56493.333
Coefficients
Standard Error
Intercept
306.52619
114.25389
2.68285
0.01993
57.58835
555.46404
Price
-24.97509
10.83213
-2.30565
0.03979
-48.57626
-1.37392
74.13096
25.96732
2.85478
0.01449
17.55303
130.70888
t Stat
6.53861
Significance F
P-value
0.01201
Lower 95%
Upper 95%
Chap 13-29
F-Test for Overall Significance
(continued)
H0: β1 = β2 = 0
H1: β1 and β2 not both zero
 = .05
df1= 2
df2 = 12
Critical
Value:
 = .05
Do not
reject H0
Reject H0
MSR
F
 6.5386
MSE
Decision:
Since F test statistic is in
the rejection region (pvalue < .05), reject H0
F = 3.885
0
Test Statistic:
Conclusion:
F
F.05 = 3.885
There is evidence that at least one
independent variable affects Y
Chap 13-30
Tests on a Subset of
Regression Coefficients
 Consider a multiple regression model involving
variables xj and zj , and the null hypothesis that the z
variable coefficients are all zero:
yi  β0  β1x1i    βK xKi  α1z1i  αr zri  εi
H0 : α1  α2    αr  0
H1 : at least one of α j  0 (j  1,..., r)
Chap 13-31
Tests on a Subset of
Regression Coefficients
(continued)
 Goal: compare the error sum of squares for the
complete model with the error sum of squares for the
restricted model
 First run a regression for the complete model and obtain SSE
 Next run a restricted regression that excludes the z variables
(the number of variables excluded is r) and obtain the
restricted error sum of squares SSE(r)
 Compute the F statistic and apply the decision rule for a
significance level 
( SSE(r)  SSE ) / r
Reject H0 if F 
 Fr,nK r 1,α
2
se
Chap 13-32
Prediction
 Given a population regression model
yi  β0  β1x1i  β2 x 2i    βK xKi  εi (i  1,2,,n)
 then given a new observation of a data point
(x1,n+1, x 2,n+1, . . . , x K,n+1)
the best linear unbiased forecast of y^n+1 is
yˆ n1  b0  b1x1,n1  b2 x 2,n1    bK xK,n1
 It is risky to forecast for new X values outside the range of the data used
to estimate the model coefficients, because we do not have data to
support that the linear model extends beyond the observed range.
Chap 13-33
Using The Equation to Make
Predictions
Predict sales for a week in which the selling
price is \$5.50 and advertising is \$350:
Sales  306.526 - 24.975(Pri ce)  74.131(Adv ertising)
 306.526 - 24.975 (5.50)  74.131 (3.5)
 428.62
Predicted sales
is 428.62 pies
in \$100’s, so \$350
means that X2 = 3.5
Chap 13-34
Predictions in PHStat
 PHStat | regression | multiple regression …
Check the
“confidence and
prediction interval
estimates” box
Chap 13-35
Predictions in PHStat
(continued)
Input values
<
Predicted y value
<
Confidence interval for the
mean y value, given
these x’s
<
Prediction interval for an
individual y value, given
these x’s
Chap 13-36
Residuals in Multiple Regression
Two variable model
y
yˆ  b0  b1x1  b2 x 2
<
Residual =
ei = (yi – yi)
yi
Sample
observation
<
yi
x2i
x2
x1i
x1
Chap 13-37
Nonlinear Regression Models
 The relationship between the dependent
variable and an independent variable may
not be linear
 Can review the scatter diagram to check for
non-linear relationships
Y  β0  β1X1  β2 X12  ε
 The second independent variable is the square
of the first variable
Chap 13-38
Model form:
Yi  β0  β1X1i  β 2 X  ε i
2
1i
 where:
β0 = Y intercept
β1 = regression coefficient for linear effect of X on Y
β2 = regression coefficient for quadratic effect on Y
εi = random error in Y for observation i
Chap 13-39
Linear vs. Nonlinear Fit
Y
Y
X
X
Linear fit does not give
random residuals
residuals
residuals
X
X

Nonlinear fit gives
random residuals
Chap 13-40
Yi  β0  β1X1i  β 2 X  ε i
2
1i
Quadratic models may be considered when the scatter
diagram takes on one of the following shapes:
Y
Y
β1 < 0
β2 > 0
X1
Y
β1 > 0
β2 > 0
X1
Y
β1 < 0
β2 < 0
X1
β1 > 0
β2 < 0
X1
β1 = the coefficient of the linear term
β2 = the coefficient of the squared term
Chap 13-41
Testing for Significance:
 Compare the linear regression estimate
yˆ  b0  b1x1
yˆ  b0  b1x1  b2 x12
 Hypotheses
 H0: β2 = 0 (The quadratic term does not improve the model)
 H1: β2  0 (The quadratic term improves the model)
Chap 13-42
Testing for Significance:
(continued)
Hypotheses
 H0: β2 = 0
(The quadratic term does not improve the model)
 H1: β2  0
(The quadratic term improves the model)
 The test statistic is
b2  β2
t
sb 2
d.f.  n  3
where:
b2 = squared term slope
coefficient
β2 = hypothesized slope (zero)
Sb = standard error of the slope
2
Chap 13-43
Testing for Significance:
(continued)
Compare R2 from simple regression to
 If R2 from the quadratic model is larger than
R2 from the simple model, then the
quadratic model is a better model
Chap 13-44
3
1
7
2
8
3
15
5
22
7
33
8
40
10
54
12
67
13
70
14
78
15
85
15
87
16
99
17
 Purity increases as filter time
increases:
Purity vs. Time
100
80
Purity
Purity
Filter
Time
60
40
20
0
0
5
10
15
20
Time
Chap 13-45
(continued)
 Simple regression results:
y^ = -11.283 + 5.985 Time
Coefficients
Standard
Error
-11.28267
3.46805
-3.25332
0.00691
5.98520
0.30966
19.32819
2.078E-10
Intercept
Time
t Stat
t statistic, F statistic, and
R2 are all high, but the
residuals are not random:
P-value
Regression Statistics
0.96888
0.96628
Standard Error
6.15997
373.57904
Time Residual Plot
Significance F
10
2.0778E-10
Residuals
R Square
F
5
0
-5 0
5
10
15
20
-10
Time
Chap 13-46
(continued)
y^ = 1.539 + 1.565 Time + 0.245 (Time)2
Coefficients
Standard
Error
Intercept
1.53870
2.24465
0.68550
0.50722
Time
1.56496
0.60179
2.60052
0.02467
Time-squared
0.24516
0.03258
7.52406
1.165E-05
Time Residual Plot
P-value
10
Residuals
t Stat
5
0
-5
R Square
0.99494
0.99402
Standard Error
2.59513
F
1080.7330
5
2.368E-13
The quadratic term is significant and
improves the model: R2 is higher and se is
lower, residuals are now random
10
15
20
Time
Significance F
Time-squared Residual Plot
10
Residuals
Regression Statistics
0
5
0
-5
0
100
200
300
400
Time-squared
Chap 13-47
The Log Transformation
The Multiplicative Model:
 Original multiplicative model
Y  β0 X1β1 Xβ22 ε
 Transformed multiplicative model
log(Y)  log(β0 )  β1log(X1 )  β2log(X 2 )  log(ε)
Chap 13-48
Interpretation of coefficients
For the multiplicative model:
log Yi  log β0  β1 log X1i  log ε i
 When both dependent and independent
variables are logged:
 The coefficient of the independent variable Xk can
be interpreted as
a 1 percent change in Xk leads to an estimated bk
percentage change in the average value of Y
 bk is the elasticity of Y with respect to a change in Xk
Chap 13-49
Dummy Variables
 A dummy variable is a categorical independent
variable with two levels:
 yes or no, on or off, male or female
 recorded as 0 or 1
 Regression intercepts are different if the variable
is significant
 Assumes equal slopes for other variables
 If more than two levels, the number of dummy
variables needed is (number of levels - 1)
Chap 13-50
Dummy Variable Example
yˆ  b0  b1x1  b2 x 2
Let:
y = Pie Sales
x1 = Price
x2 = Holiday (X2 = 1 if a holiday occurred during the week)
(X2 = 0 if there was no holiday that week)
Chap 13-51
Dummy Variable Example
(continued)
yˆ  b0  b1x1  b 2 (1)  (b 0  b 2 )  b1x1
Holiday
yˆ  b0  b1x1  b 2 (0) 
No Holiday
y (sales)
b0 + b2
b0
 b1 x 1
b0
Different
intercept
Same
slope
If H0: β2 = 0 is
rejected, then
“Holiday” has a
significant effect
on pie sales
x1 (Price)
Chap 13-52
Interpreting the
Dummy Variable Coefficient
Example:
Sales  300 - 30(Price)  15(Holiday)
Sales: number of pies sold per week
Price: pie price in \$
1 If a holiday occurred during the week
Holiday:
0 If no holiday occurred
b2 = 15: on average, sales were 15 pies greater in
weeks with a holiday than in weeks without a
holiday, given the same price
Chap 13-53
Interaction Between
Explanatory Variables
 Hypothesizes interaction between pairs of x
variables
 Response to one x variable may vary at different
levels of another x variable
 Contains two-way cross product terms

yˆ  b0  b1x1  b 2 x 2  b3 x 3
 b0  b1x1  b 2 x 2  b3 (x1x 2 )
Chap 13-54
Effect of Interaction
 Given: Y  β  β X  (β  β X )X
0
2 2
1
3 2
1
 β0  β1X1  β2 X 2  β3 X1X 2
 Without interaction term, effect of X1 on Y is
measured by β1
 With interaction term, effect of X1 on Y is
measured by β1 + β3 X2
 Effect changes as X2 changes
Chap 13-55
Interaction Example
Suppose x2 is a dummy variable and the estimated
regression equation is yˆ  1 2x1  3x2  4x1x2
y
12
x2 = 1:
^y = 1 + 2x + 3(1) + 4x (1) = 4 + 6x
1
1
1
8
4
x2 = 0:
^y = 1 + 2x + 3(0) + 4x (0) = 1 + 2x
1
1
1
0
0
0.5
1
1.5
x1
Slopes are different if the effect of x1 on y depends on x2 value
Chap 13-56
Significance of Interaction Term
 The coefficient b3 is an estimate of the difference
in the coefficient of x1 when x2 = 1 compared to
when x2 = 0
 The t statistic for b3 can be used to test the
hypothesis
H0 : β3  0 | β1  0, β 2  0
H1 : β3  0 | β1  0, β 2  0
 If we reject the null hypothesis we conclude that there is
a difference in the slope coefficient for the two
subgroups
Chap 13-57
Multiple Regression Assumptions
Errors (residuals) from the regression model:
<
ei = (yi – yi)
Assumptions:
 The errors are normally distributed
 Errors have a constant variance
 The model errors are independent
Chap 13-58
Analysis of Residuals
in Multiple Regression
 These residual plots are used in multiple
regression:
<
 Residuals vs. yi
 Residuals vs. x1i
 Residuals vs. x2i
 Residuals vs. time (if time series data)
Use the residual plots to check for
violations of regression assumptions
Chap 13-59
Chapter Summary






Developed the multiple regression model
Tested the significance of the multiple regression model
Discussed adjusted R2 ( R2 )
Tested individual regression coefficients
Tested portions of the regression model
Used quadratic terms and log transformations in
regression models
 Used dummy variables
 Evaluated interaction effects
 Discussed using residual plots to check model
assumptions