Transcript Chapter 11: Multiple Regression (1)

11-1
COMPLETE
BUSINESS
STATISTICS
by
AMIR D. ACZEL
&
JAYAVEL SOUNDERPANDIAN
6th edition.
11-2
Chapter 11
Multiple Regression
11-3
11 Multiple Regression (1)
• Using Statistics
• The k-Variable Multiple Regression Model
• The F Test of a Multiple Regression Model
• How Good is the Regression
• Tests of the Significance of Individual Regression
Parameters
• Testing the Validity of the Regression Model
• Using the Multiple Regression Model for
Prediction
11-4
11 Multiple Regression (2)
• Qualitative Independent Variables
• Polynomial Regression
• Nonlinear Models and Transformations
• Multicollinearity
• Residual Autocorrelation and the Durbin-Watson
Test
• Partial F Tests and Variable Selection Methods
• Multiple Regression Using the Solver
• The Matrix Approach to Multiple Regression
Analysis
11-5
11 LEARNING OBJECTIVES (1)
After studying this chapter you should be able to:
• Determine whether multiple regression would be
applicable to a given instance
• Formulate a multiple regression model
• Carryout a multiple regression using a spreadsheet
template
• Test the validity of a multiple regression by analyzing
residuals
• Carryout hypothesis tests about the regression
coefficients
• Compute a prediction interval for the dependent
variable
11-6
11 LEARNING OBJECTIVES (2)
After studying this chapter you should be able to:
• Use indicator variables in a multiple regression
• Carryout a polynomial regression
• Conduct a Durbin-Watson test for autocorrelation
in residuals
• Conduct a partial F-test
• Determine which independent variables are to be
included in a multiple regression model
• Solve multiple regression problems using the
Solver macro
11-7
11-1 Using Statistics
y
y
Lines
B
A
Slope: 1
C
A
x1
Intercept: 0
x
Any two points (A and B), or
an intercept and slope (0 and
1), define a line on a twodimensional surface.
Planes
B
x2
Any three points (A, B, and C), or an
intercept and coefficients of x1 and x2
(0 , 1 , and 2), define a plane in a
three-dimensional surface.
11-8
11-2 The k-Variable Multiple
Regression Model
The population regression model of a
dependent variable, Y, on a set of k
independent variables, X1, X2,. . . , Xk is
given by:
x2
y
2
Y= 0 + 1X1 + 2X2 + . . . + kXk +
where 0 is the Y-intercept of the
regression surface and each i , i = 1,2,...,k
is the slope of the regression surface sometimes called the response surface with respect to Xi.
1
0
x1
y   0   1 x1   2 x 2  
Model assumptions:
1. ~N(0,2), independent of other errors.
2. The variables Xi are uncorrelated with the error term.
11-9
Simple and Multiple Least-Squares
Regression
y
Y
x1
y  b0  b1x
X
In a simple regression model,
the least-squares estimators
minimize the sum of squared
errors from the estimated
regression line.
x2
y  b0  b1x1  b2 x2
In a multiple regression model,
the least-squares estimators
minimize the sum of squared
errors from the estimated
regression plane.
11-10
The Estimated Regression
Relationship
The estimated regression relationship:
Y  b0  b1 X 1  b2 X 2 bk X k
where Y is the predicted value of Y, the value lying on the
estimated regression surface. The terms bi, for i = 0, 1, ....,k are
the least-squares estimates of the population regression
parameters i.
The actual, observed value of Y is the predicted value plus an
error:
yj = b0+ b1 x1j+ b2 x2j+. . . + bk xkj+e, j = 1, …, n.
11-11
Least-Squares Estimation:
The 2-Variable Normal Equations
Minimizing the sum of squared errors with respect to the
estimated coefficients b0, b1, and b2 yields the following
normal equations which can be solved for b0, b1, and b2.
 y  nb  b  x  b  x
0
1
1
2
2
x y b x b x b x x
2
1
0
1
1
1
2
1
x y b x b x x b x
2
0
2
1
1
2
2
2
2
2
11-12
Example 11-1
Y
72
76
78
70
68
80
82
65
62
90
--743
X1
12
11
15
10
11
16
14
8
8
18
--123
X2
5
8
6
5
3
9
12
4
3
10
--65
X1X2
60
88
90
50
33
144
168
32
24
180
--869
X12
144
121
225
100
121
256
196
64
64
324
---1615
X22
25
64
36
25
9
81
144
16
9
100
--509
X1Y
864
836
1170
700
748
1280
1148
520
496
1620
---9382
X2Y
360
608
468
350
204
720
984
260
186
900
---5040
Normal Equations:
743 = 10b0+123b1+65b2
9382 = 123b0+1615b1+869b2
5040 = 65b0+869b1+509b2
b0 = 47.164942
b1 = 1.5990404
b2 = 1.1487479
Estimated regression equation:
Y  47164942
.
 15990404
.
X 1  11487479
.
X2
11-13
Example 11-1: Using the Template
Regression results for Alka-Seltzer sales
11-14
Decomposition of the Total Deviation
in a Multiple Regression Model

y

Total deviation: Y  Y
y
Y  Y: Error Deviation
Y  Y : Regression Deviation
x1
x2
Total Deviation = Regression Deviation + Error Deviation
SST
=
SSR
+ SSE
11-15
11-3 The F Test of a Multiple
Regression Model
A statistical test for the existence of a linear relationship between Y and any or
all of the independent variables X1, x2, ..., Xk:
H0: 1 = 2 = ...= k= 0
H1: Not all the i (i=1,2,...,k) are equal to 0
Source of
Variation
Sum of
Squares
Regression SSR
Error
Total
SSE
SST
Degrees of
Freedom Mean Square
k
n - (k+1)
n-1
SSR
MSR 
MSE 
k
SSE
( n  ( k  1))
MST 
SST
( n  1)
F Ratio
11-16
Using the Template: Analysis of
Variance Table (Example 11-1)
F Distribution with 2 and 7 Degrees of Freedom
f(F)
Test statistic 86.34
=0.01
F
0
F0.01=9.55
The test statistic, F = 86.34, is greater
than the critical point of F(2, 7) for any
common level of significance
(p-value 0), so the null hypothesis is
rejected, and we might conclude that
the dependent variable is related to
one or more of the independent
variables.
11-17
11-4 How Good is the Regression
The mean square error is an unbiased
estimator of the variance of the population
2
errors,  , denoted by  :
y
MSE 
SSE
( n  ( k  1))

 ( y  y) 2
( n  ( k  1))
x1
Standard error of estimate:
x2
Errors: y - y
s=
MSE
2
The multiple coefficient of determination, R , measures the proportion of
the variation in the dependent variable that is explained by the combination
of the independent variables in the multiple regression model:
SSR
SSE
R =
=1SST
SST
2
11-18
Decomposition of the Sum of Squares and
the Adjusted Coefficient of Determination
SST
SSR
R
2
SSE
=
SSR
SST
= 1-
SSE
SST
The adjusted multiple coefficient of determinat ion, R 2, is the coefficien t of
determinat ion with t he SSE and SST divided by their respective degrees of freedom :
SSE
R 2 =1- (n -(k +1))
SST
(n -1)
Example 11-1:
s = 1.911
R-sq = 96.1%
R-sq(adj) = 95.0%
11-19
Measures of Performance in Multiple
Regression and the ANOVA Table
Source of
Variation
Sum of
Squares
Degrees of
Freedom Mean Square
Regression SSR
(k)
MSR 
R
2
Error
SSE
(n-(k+1))
=(n-k-1)
Total
SST
(n-1)
SSR
=
SSE
= 1-
SST
SST
F 
R
MSE 
F Ratio
F 
SSR
k
MSR
MSE
SSE
( n  ( k  1))
MST 
SST
( n  1)
SSE
2
( n  ( k  1))
2
(1  R )
(k )
R
2
=1-
(n - (k + 1))
SST
(n - 1)
=
MSE
MST
11-20
11-5 Tests of the Significance of
Individual Regression Parameters
Hypothesis tests about individual regression slope parameters:
(1)
H0: 1= 0
H1: 1  0
(2)
H0: 2 = 0
H1: 2  0
.
.
.
(k)
H0: k = 0
H1: k  0
Test statistic for test i: t
b 0

s(b )
i
( n  ( k 1 )
i
11-21
Regression Results for Individual
Parameters (Interpret the Table)
Variable
Constant
Coefficient
Estimate
Standard
Error
t-Statistic
53.12
5.43
9.783
X1
2.03
0.22
9.227
X2
5.60
1.30
4.308
X3
10.35
6.88
1.504
X4
3.45
2.70
1.259
X5
-4.25
0.38
11.184
n=150
t0.025=1.96
*
*
*
*
11-22
Example 11-1: Using the Template
Regression results for Alka-Seltzer sales
11-23
Using the Template: Example 11-2
Regression results for Exports to Singapore
11-24
11-6 Testing the Validity of the
Regression Model: Residual Plots
Residuals vs M1 (Example 11-2)
It appears that the residuals are randomly distributed with no pattern and
with equal variance as M1 increases
11-25
11-6 Testing the Validity of the
Regression Model: Residual Plots
Residuals vs Price (Example 11-2)
It appears that the residuals are increasing as the Price increases. The
variance of the residuals is not constant.
11-26
Normal Probability Plot for the
Residuals: Example 11-2
Linear trend indicates residuals are normally distributed
11-27
Investigating the Validity of the Regression:
Outliers and Influential Observations
y
Regression line
without outlier
. .
.
.
.
..
. .. ..
.. .
.
y
Regression
line with
outlier
* Outlier
x
Outliers
Point with a large
value of xi
.
.
.
.
.. .. .. .
. .. .
*
Regression line
when all data are
included
No relationship in
this cluster
x
Influential Observations
11-28
Possible Relation in the Region between the
Available Cluster of Data and the Far Point
Point with a large value of xi
y
*
Some of the possible data between the
original cluster and the far point
.
.
.
.
.. .. .. .
. .. .
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
More appropriate curvilinear relationship
(seen when the in between data are known).
x
11-29
Outliers and Influential Observations:
Example 11-2
Unusual Observations
Obs.
M1
EXPORTS
Residual
St.Resid
1
5.10
2.6000
0.1288
-0.0420
2
4.90
2.6000
0.1234
-0.0438
25
6.20
5.5000
0.0676
0.9051
26
6.30
3.7000
0.0651
-0.9311
50
8.30
4.3000
0.0648
-0.8317
67
8.20
5.6000
0.0668
0.6526
Fit
Stdev.Fit
2.6420
-0.14 X
2.6438
-0.14 X
4.5949
2.80R
4.6311
-2.87R
5.1317
-2.57R
4.9474
2.02R
R denotes an obs. with a large st. resid.
X denotes an obs. whose X value gives it large influence.
11-30
11-7 Using the Multiple Regression
Model for Prediction
Sales
Estimated Regression Plane for Example 11-1
89.76
Advertising
18.00
63.42
8.00
Promotions
12
3
11-31
Prediction in Multiple Regression
A (1- ) 100% prediction interval for a value of Y given values of X :
i
yˆ  t 
s2( yˆ)  MSE
( ,(n(k 1)))
2
A (1-α) 100% prediction interval for the conditional mean of Y given
values of X :
i
yˆ  t 
s[Eˆ (Y )]
( ,(n(k 1)))
2
11-32
11-8 Qualitative (or Categorical)
Independent Variables (in Regression)
An indicator (dummy, binary) variable of qualitative level A:
1 if level A is obtained
Xh  
0 if level A is not obtained
MOVIE EARN
1
28
2
35
3
50
4
20
5
75
6
60
7
15
8
45
9
50
10
34
11
48
12
82
13
24
14
50
15
58
16
63
17
30
18
37
19
45
20
72
COST
4.2
6.0
5.5
3.3
12.5
9.6
2.5
10.8
8.4
6.6
10.7
11.0
3.5
6.9
7.8
10.1
5.0
7.5
6.4
10.0
PROM
1.0
3.0
6.0
1.0
11.0
8.0
0.5
5.0
3.0
2.0
1.0
15.0
4.0
10.0
9.0
10.0
1.0
5.0
8.0
12.0
BOOK
0
1
1
0
1
1
0
0
1
0
1
1
0
0
1
0
1
0
1
1
EXAMPLE 11-3
11-33
Picturing Qualitative Variables in
Regression
y
Y
Line for X2=1
b3
b0+b2
Line for X2=0
x1
b0
X1
A regression with one
quantitative variable (X1) and
one qualitative variable (X2):
y  b  b x  b x
0
1
1
2
2
x2
A multiple regression with two
quantitative variables (X1 and X2)
and one qualitative variable (X3):
y  b  b x  b x  b x
0
1
1
2
2
3
3
11-34
Picturing Qualitative Variables in Regression:
Three Categories and Two Dummy Variables
Y
Line for X = 0 and X3 = 1
Line for X2 = 1 and X3 = 0
b0+b3
Line for X2 = 0 and X3 = 0
A qualitative
variable with r
levels or categories
is represented with
(r-1) 0/1 (dummy)
variables.
b0+b2
b0
X1
A regression with one quantitative variable (X1) and two
qualitative variables (X2 and X2):
y  b  b x  b x  b x
0
1
1
2
2
3
3
Category
Adventure
Drama
Romance
X2
0
0
1
X3
0
1
0
11-35
Using Qualitative Variables in
Regression: Example 11-4
Salary = 8547
+
949 Education
+
Experience
- 3256 Gender
(SE)
(32.6)
(45.1)
(78.5)
(212.4)
(t)
(262.2)
(21.0)
(16.0)
(-15.3)
1 if Female
Gender  
0 if Male
1258
On average, female salaries are
$3256 below male salaries
11-36
Interactions between Quantitative and
Qualitative Variables: Shifting Slopes
Line for X2=0
Y
Line for X2=1
Slope = b1
b0
Slope = b1+b3
b0+b2
X1
A regression with interaction between a quantitative
variable (X1) and a qualitative variable (X2 ):
y  b  b x  b x  b x x
0
1
1
2
2
3
1
2
11-37
11-9 Polynomial Regression
One-variable polynomial regression model:
Y= 0+1 X + 2X2 + 3X3 +. . . + mXm +
where m is the degree of the polynomial - the highest power of X appearing in
the equation. The degree of the polynomial is the order of the model.
Y
Y
y  b  b X
0
y  b  b X
0
1
1
y  b  b X  b X
(b  0)
0
1
2
y  b  b X  b X  b X
2
2
0
2
X1
1
2
3
X1
3
11-38
Polynomial Regression: Example 11-5
11-39
Polynomial Regression: Other
Variables and Cross-Product Terms
Variable Estimate Standard Error T-statistic
X1
2.34
0.92
2.54
X2
3.11
1.05
2.96
2
X1
4.22
1.00
4.22
X22
3.57
2.12
1.68
X1X2
2.77
2.30
1.20
11-40
11-10 Nonlinear Models and
Transformations
The multiplicative model:
Y   X X X 
The logarithmic transformation:
1
0
1
2
2
3
3
log Y  log    log X   log X   log X  log 
0
1
1
2
2
3
3
11-41
Transformations: Exponential Model
The exponential model:
Y   e 
The logarithmic transformation:
1X
0
log Y  log 
0
  X  log 
1
1
11-42
Plots of Transformed Variables
Sim ple Regression of Sales on Advertising
Regression of Sales on Log(Advertising)
25
20
Y = 6 .59 2 71 + 1.19 176 X
R- Sq uared = 0 .8 9 5
10
SALES
SALES
30
15
Y = 3.6 6 8 2 5 + 6 .78 4 X
R- Sq uared = 0 .978
5
0
5
10
15
0
1
ADVERT
2
3
LOGADV
Regression of Log(Sales) on Log(Advertising)
Residual Plots: Sales vs Log(Advertising)
1.5
3.5
2.5
Y = 1.70 0 8 2 + 0 .5 53 13 6 X
R- Sq uared = 0 .9 47
RESIDS
LOGSALE
0.5
-0.5
-1.5
1.5
0
1
2
LOGADV
3
2
12
Y-HAT
22
11-43
Variance Stabilizing Transformations
• Square root transformation: Y  
Y
Useful when the variance of the regression errors is
approximately proportional to the conditional mean of Y
• Logarithmic transformation: Y   log(Y )
Useful when the variance of regression errors is approximately
proportional to the square of the conditional mean of Y
• Reciprocal transformation: Y   Y1
Useful when the variance of the regression errors is
approximately proportional to the fourth power of the
conditional mean of Y
11-44
Regression with Dependent Indicator
Variables
The logistic function:
e (  X )
E (Y X ) 
1  e (  X )
0
1
0
1
Transformation to linearize the logistic function:
 p 
p   log

 1  p
y
Logistic Function
1
0
x
11-45
11-11: Multicollinearity
x2
x2
x1
Orthogonal X variables provide
information from independent
sources. No multicollinearity.
x2
x1
Perfectly collinear X variables
provide identical information
content. No regression.
x2
x1
Some degree of collinearity.
Problems with regression depend
on the degree of collinearity.
x1
A high degree of negative
collinearity also causes problems
with regression.
11-46
Effects of Multicollinearity
•
•
•
•
•
•
Variances of regression coefficients are inflated.
Magnitudes of regression coefficients may be different
from what are expected.
Signs of regression coefficients may not be as expected.
Adding or removing variables produces large changes in
coefficients.
Removing a data point may cause large changes in
coefficient estimates or signs.
In some cases, the F ratio may be significant while the t
ratios are not.
11-47
Detecting the Existence of Multicollinearity:
Correlation Matrix of Independent Variables and
Variance Inflation Factors
11-48
Variance Inflation Factor
The variance inflation factor associated with X h :
1
VIF ( X h ) 
1  Rh2
where R 2h is the R 2 value obtained for the regression of X on
the other independent variables.
Relationship between VIF and Rh2
VIF100
50
0
0.0
0.5
1.0
Rh2
11-49
Variance Inflation Factor (VIF)
Observation: The VIF (Variance Inflation Factor) values
for both variables Lend and Price are both greater than
5. This would indicate that some degree of
multicollinearity exists with respect to these two
variables.
11-50
Solutions to the Multicollinearity
Problem
• Drop a collinear variable from the
regression
• Change in sampling plan to include
elements outside the multicollinearity range
• Transformations of variables
• Ridge regression
11-51
11-12 Residual Autocorrelation and
the Durbin-Watson Test
An autocorrelation is a correlation of the values of a variable
with values of the same variable lagged one or more periods
back. Consequences of autocorrelation include inaccurate
estimates of variances and inaccurate predictions.
Lagged Residuals
i
1
2
3
4
5
6
7
8
9
10
1.0
0.0
-1.0
2.0
3.0
-2.0
1.0
1.5
1.0
-2.5
i
*
1.0
0.0
-1.0
2.0
3.0
-2.0
1.0
1.5
1.0
i-1
i-2
*
*
*
*
1.0 *
0.0 1.0
-1.0 0.0
2.0 -1.0
3.0 2.0
-2.0 3.0
1.0 -2.0
1.5 1.0
i-3
*
*
*
*
1.0
0.0
-1.0
2.0
3.0
-2.0
The Durbin-Watson test (first-order
autocorrelation):
i-4
H0: 1 = 0
H1:  0
The Durbin-Watson test statistic:
n
2
 ( ei  ei 1 )
d  i2 n
2
 ei
i 1
11-52
Critical Points of the Durbin-Watson Statistic: =0.05,
n= Sample Size, k = Number of Independent Variables
n
15
16
17
18
.
.
.
65
70
75
80
85
90
95
100
k=1
dL dU
k=2
dL dU
k=3
dL dU
k=4
dL dU
k=5
dL dU
1.08
1.10
1.13
1.16
0.95
0.98
1.02
1.05
0.82
0.86
0.90
0.93
0.69
0.74
0.78
0.82
0.56
0.62
0.67
0.71
1.57
1.58
1.60
1.61
1.62
1.63
1.64
1.65
1.36
1.37
1.38
1.39
.
.
.
1.63
1.64
1.65
1.66
1.67
1.68
1.69
1.69
1.54
1.55
1.57
1.59
1.60
1.61
1.62
1.63
1.54
1.54
1.54
1.53
.
.
.
1.66
1.67
1.68
1.69
1.70
1.70
1.71
1.72
1.50
1.52
1.54
1.56
1.57
1.59
1.60
1.61
1.75
1.73
1.71
1.69
.
.
.
1.70
1.70
1.71
1.72
1.72
1.73
1.73
1.74
1.47
1.49
1.51
1.53
1.55
1.57
1.58
1.59
1.97
1.93
1.90
1.87
.
.
.
1.73
1.74
1.74
1.74
1.75
1.75
1.75
1.76
1.44
1.46
1.49
1.51
1.52
1.54
1.56
1.57
2.21
2.15
2.10
2.06
.
.
.
1.77
1.77
1.77
1.77
1.77
1.78
1.78
1.78
11-53
Using the Durbin-Watson Statistic
Positive
Autocorrelation
0
dL
Test is
Inconclusive
dU
No
Autocorrelation
Test is
Inconclusive
4-dU
Negative
Autocorrelation
4-dL
4
For n = 67, k = 4: dU1.73 4-dU2.27
dL1.47 4- dL2.53 < 2.58
H0 is rejected, and we conclude there is negative first-order
autocorrelation.
11-54
11-13 Partial F Tests and Variable
Selection Methods
Full model:
Y = 0 + 1 X 1 +  2 X 2 + 3 X 3 + 4 X 4 + 
Reduced model:
Y = 0 + 1 X 1 +  2 X 2 + 
Partial F test:
H0: 3 = 4 = 0
H1: 3 and 4 not both 0
Partial F statistic:
(SSE
F

(r, (n  (k  1))
 SSE ) / r
R
F
MSE
F
where SSER is the sum of squared errors of the reduced model, SSEF is the sum of squared
errors of the full model; MSEF is the mean square error of the full model [MSEF =
SSEF/(n-(k+1))]; r is the number of variables dropped from the full model.
11-55
Variable Selection Methods
• All possible regressions
Run regressions with all possible combinations of
independent variables and select best model
A p-value of 0.001 indicates
that we should reject the null
hypothesis H0: the slopes for
Lend and Exch. are zero.
11-56
Variable Selection Methods
• Stepwise procedures
Forward selection
•
Add one variable at a time to the model, on the basis of its F statistic
Backward elimination
•
Remove one variable at a time, on the basis of its F statistic
Stepwise regression
•
Adds variables to the model and subtracts variables from the model, on
the basis of the F statistic
11-57
Stepwise Regression
Compute F statistic for each variable not in the model
Is there at least one variable with p-value > Pin?
No
Stop
Yes
Enter most significant (smallest p-value) variable into model
Calculate partial F for all variables in the model
Is there a variable with p-value > Pout?
No
Remove
variable
11-58
Stepwise Regression: Using the
Computer (MINITAB)
MTB > STEPWISE 'EXPORTS' PREDICTORS
'EXCHANGE'
'M1’
'LEND'
'PRICE’
Stepwise Regression
F-to-Enter:
4.00
Response is EXPORTS
Step
Constant
M1
T-Ratio
on
F-to-Remove:
4 predictors, with N =
67
1
0.9348
0.520
9.89
PRICE
T-Ratio
S
R-Sq
4.00
2
-3.4230
0.361
9.21
0.0370
9.05
0.495
60.08
0.331
82.48
11-59
Using the Computer: MINITAB
MTB > REGRESS
'EXPORTS’
4
'M1’
'LEND’
'PRICE'
'EXCHANGE';
SUBC> vif;
SUBC> dw.
Regression Analysis
The regression equation is
EXPORTS = - 4.02 + 0.368 M1 + 0.0047 LEND + 0.0365 PRICE +
0.27 EXCHANGE
Predictor
t-ratio
Constant
-1.45
M1
5.77
LEND
0.10
PRICE
3.91
EXCHANGE
0.23
Coef
Stdev
p
-4.015
VIF
2.766
0.152
0.36846
0.06385
0.000
3.2
0.00470
0.924
0.036511
0.000
0.268
0.820
0.04922
5.4
0.009326
6.3
1.175
1.4
s = 0.3358
81.4%
Analysis of Variance
R-sq = 82.5%
R-sq(adj) =
11-60
Using the Computer: SAS (continued)
Parameter Estimates
Parameter
Standard
T for H0:
Variable
Prob > |T|
INTERCEP
0.1517
M1
1
DF
Estimate
-4.015461
Error
2.76640057
Parameter=0
-1.452
1
0.368456
0.06384841
5.771
1
0.004702
0.04922186
0.096
1
0.036511
0.00932601
3.915
0.0001
LEND
0.9242
PRICE
EXCHANGE
0.8205
0.0002
1
Variable
INTERCEP
M1
LEND
PRICE
EXCHANGE
0.267896
DF
1
1
1.17544016
Variance
Inflation
0.00000000
1
3.20719533
1
5.35391367
1
6.28873181
1.38570639
0.228