Regression Equation

7/31/2019 Regression Equation

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xbya

xxn

yxyxn

S

S

xVar

yxCovb

n

i

n

i ii

n

ii

n

ii

n

iii

x

xy

1

2

1

2

111

2)(

),(

Now for any fixed value of x, each observation in our sample satisfies the relation

iyixiy

Wherei is a random error representing the vertical deviation of the point from the population

regression line (parametric regression equation). From previous assumptions oni

y , i must

necessarily be a value of a random variable having a mean of zero and the variance2 . In terms of the

sample regression line, we can also write:

iii eyy An essential part of regression analysis involves the construction of confidence intervals for and and

testing hypothesis concerning these regression coefficients. The hypothesis for testing the coefficients

are =0 and =0. However the unknown variance 2

must be estimated from the data. An unbiased

estimate of 2

with n-2 degree of freedom, denoted by2

eS , is given by the formula:

2

)(

22

222

n

yy

n

e

n

SSES iiie

In usual the sample variance formula we use to take one degree of freedom, provide an unbiased

estimate of the population variance, since only is replaced by the sample mean in our calculations.

Here, it is necessary to take 2 degree of freedom in the formula for2eS because 2 degree of freedom are

lost by replacing and by a and b in our calculation of the syi ' .The simple formula for thecalculation of SSE are as follows:

)1(

)(

)1(

)(

))(1(

1 1

22

21 1

22

2

222

nn

yyn

andSnn

xxn

S

Where

SbSnSSE

n

i

n

i

ii

y

n

i

n

i

ii

x

xy


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Test for Linearity of Regression Equation

OR

Validity of the Regression Model

We define the regression to be linear when all the means of y corresponding to each fall on a straight

line. One can always prefer a linear regression model over non linear model. We can test the linearity of

the regression equation by using the ANOVA test. If the linearity will be confirm than we can say that

regression model is valid and then we develop the model.

Calculation of ANOVA:

Values of x

50 55 65 70

Values of

corresponding to

each X

74.893 79.378 88.348 92.833

74.893 79.378 88.348 92.833

79.378 88.348 92.833

79.378

Sum 149.786 317.512 265.044 278.499 1010.841

Square of sum 22435.85 100813.9 70248.32 77561.69 1021800

Square of sum/ni 11217.92 25203.47 23416.11 25853.9 85149.96

Regression sum of square = (11217.92+25203.47+23416.11+25853.9)-85149.96

Regression sum of square = 541.69

Residual sum of square = SSE = 186.557 and 2eS = 18.656

ANOVAb

Model Sum of Squares df Mean Square F Sig.

1 Regression 541.693 1 541.693 29.036 .000a

Residual 186.557 10 18.656

Total 728.250 11

a. Predictors: (Constant), TestScore

b. Dependent Variable: CheScore

Here the significance value of 0.000 < 0.05 which means that the Ho of ANOVA is significant

means that the Ho reject.


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Inference: Ho = all means are equal; which is rejected means Regression line is not

horizontal that implies that line with some slope and slope shows the correlation between the

predictor and the estimator.

Inferences Concerning the regression coefficients:Confidence interval for

A (1-)100% confidence interval for the parameter in the regression line is

)1()1(

1

2

21

2

2

nnS

xSt

annS

xSt

ax

n

iie

x

n

iie

Note that the symbol is being used here in two totally unrelated ways, first as the level ofsignificance and then as the intercept of the regression line.

Confidence interval for

A (1-)100% confidence interval for the parameter in the regression line is

)1()1(

22

nS

Stb

nS

Stb

x

e

x

e

Note that the symbol is being used here in two totally unrelated ways, first as the level ofsignificance and then as the intercept of the regression line.

Predictions

The equation may be used to predict the mean responseoxy

at x = xo, where, xo

is not necessarily one of the pre-chosen values, or it may be used to predict a single value of

the variable when . We would expect the error of the prediction to be higher in the

case of a single predicted value than in the case where a mean is predicted. This, then, will

affect the width of our confidence intervals for the values being predicted.

Predictions foroxy

A (1-) 100% confidence interval for the meanoxY

is given by:

2

2

22

2

2)1(

)(1

)1(

)(1

x

oeoxY

x

oeo

Sn

xx

nSty

Sn

xx

nSty

o


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Predictions for oy

A (1-) 100% confidence interval for the single value oy when is given by:

2

2

22

2

2)1(

)(11)1(

)(11x

oeoo

x

oeoSnxx

nStyy

Snxx

nSty

What is the necessary condition for Simple linear regression?

The necessary condition for simple linear regression is that the test must be run between two

scale variable.

The variable must be correlated with each other.

How to run the test?

For the understanding we take the example from the book wall pole page no. 347.

In this example the two variables are IQ test score and Chemistry test score. Both are scale

measurement and theoretically they have correlation with each other.


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Interpretation of output:

Descriptive Statistics

Mean Std. Deviation N

Chemistry test score 84.2500 8.13662 12

IQ test score 60.4167 7.82140 12

The descriptive of the variables

Correlations

Chemistry test

score IQ test score

Pearson Correlation Chemistry test score 1.000 .862

IQ test score .862 1.000

Sig. (1-tailed) Chemistry test score . .000

IQ test score .000 .

N Chemistry test score 12 12

IQ test score 12 12

The independent and dependent variable having correlation with each other therefore test can

be run.The significant value 0.000 which means that the test is significant. The hypothesis for the test

is there is no correlation between the two study variables which is rejected and test become

significant.

Model Summaryb

Model R R Square

Adjusted R

Square

Std. Error of the

Estimate

1 .862a

.744 .718 4.31923

a. Predictors: (Constant), IQ test score

b. Dependent Variable: Chemistry test score

(18.656)^0.5 = 4.319 (sum of the square of error term or residual

R-square = 0.744 this means 74.4% variation is explained by the predictors of the model.


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ANOVAb


1 Regression 541.693 (no. of predictors)1 541.693 29.036 .000a

Residual 186.557 10 18.656

Total 728.250 (N-no. of predictors)11

a. Predictors: (Constant), IQ test score

b. Dependent Variable: Chemistry test score

The value of F-statistic is 29.036 which is very high and the p-value (or the sig value) is 0.000which is less than 0.05 ( level of significance) this implies that the test of ANOVA is significant

and the model is valid from the given predictors. (See page # 365 for study)

Coefficientsa

Model

Unstandardized

Coefficients

Standardized

Coefficients

t Sig.

95.0% Confidence

Interval for B

Collinearity

Statistics

B Std. Error Beta

Lower

Bound

Upper

Bound Tolerance VIF

1 (Constant) 30.043 10.137 2.964 .014 7.458 52.629

IQ test

score

.897 .167 .862 5.389 .000 .526 1.268 1.000 1.000

a. Dependent Variable: Chemistry test score

1. t-values are calculated by taking the ratio between and the standard error. (e.g.30.043/10.137 = 2.964)

2. as far as standard error is increases the t-value is decreases and as t-value decreases the

significant value (p-value) will increases and if p-value become more than the level of

significance which is usually 0.05 than the predictor become insignificant or less

important for the model

3. Here the significance value (the p-value) is less than 0.14 and 0.000; both are less than

0.05 which means that the constant term as well as the coefficient of x both is significant

for model.4. Standardized Coefficients of Beta: it can be calculated by taking standardized values of

all the predictors and then run the test of regression analysis. In this context, whatever

be the value of be calculated that will be the Standardized Coefficients of Beta.

5. If there are more than one predictors, Standardized Coefficients of Beta will rank the

importance of the predictors. The bigger value will be more important predictor as

compare to the one which has the smaller value.


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6. It shows that 95% confidence Interval which fall between 7.458 and 52.629. It means

researcher is 95% confident that minimum value for 30.043 may be 7.458 and maximum

may be 52.629. (See page no. 358 to 360 of wall Pole for further study).

7. As in this model which is simple linear regression model where only one predictor

therefore the explanation of tolerance and VIF cannot be explain well. It will discuss

when we discuss multiple regression model.

Variable which is save during run of the test is RES_1, which shows the residual value we can

check by taking the difference between predicted value (PRE_1) and actual (chemistryScore)


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The variables 7, 8 and 9, 10 show the confidence interval of 95% for the predicted value of y at

specific value of x on the basis of mean of sample and on the individual basis. See page 361 to

363 of Wall Pole; for further understanding. Here this is important to understand that the range

of individual is more than mean. Because, it depends upon sample and its mean

Final Regression model

xY 897.0043.30 The predictor explaining 74.4% (R-Square = 0744) of the model.


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Exponential regression equation

Or

Log TransformationIf a set of data appears to be best represented by a nonlinear regression curve, we must then try to

determine the form of the curve and estimate the parameters. Non linear regression curve means

the mean values of ys corresponding to each value of x are not fall on a straight line which

shows that the curve is nonlinear. In that situation we mostly apply exponential curve of the

form:

xxy

Where and are parameters to be estimated from the data. Denoting these estimates by c and

d, respectively, we can estimate xy by y from the sample regression curve.

xcdy

Taking log base 10 on both sides

xdcy )(logloglog

And each pair of observations in th sample satisfies the relation

iii

iii

ebxay

exdcy

log

)(logloglog

Where a = log c and b = log d. Therefore, it is possible to find a and b by the formulas discussed

above and then find c and d by taking the anti-log of the values.

Note: the log transformation is usually a good transformation technique for addressing the

nonlinearity of xy .


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Multiple regression equationWhat is Multiple Regression Equation?

Multiple regression equation is a linear regression model with one independent and multiplenumbers of dependent variables. It is a dependence technique.

rrxxxY xxxxr .........332211....,2,1

Why is Multiple Regression Analysis?

Multiple regression analysis is a statistical technique that can be used to analyze the relationship

between a single dependent (criterion) variable and several independent (predictors) variables.

The objective of multiple regression analysis is to use the independent variables whose values

are known to predict the single dependent value selected by the researcher.

When Multiple Regression Analysis? It will carry out when all the variables are scale in measurement.

Some time, the variable with ordinal measurement may also use in MLR but without changing

the measurement.

In SLR can be effective with a sample size of 20, but in multiple regression requires a minimum

sample of 50 and preferably 100 observations for most research situations.

The minimum ratio of observations to variables is 5:1, but the preferred ratio is 15:1 or 20:1,

which should increase when stepwise estimation is used.

As the structure of y and error term are same therefore we study the structure of error

term instead ofy , because it is easy to study. For the study of MLR, the following assumptions about the error term should not violate:

Error term must hold the Normality Variables are Identical All predictors are Independent or lack of multi-collinearity or Independency Error term structure holds the linearity


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Important terms to understand the discussion


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How to perform Multiple Regression Analysis (MLR) on SPSS?

There are two steps involve in the analysis.

1. To run the test in SPSS

2. To understand and interpret the output of the test.

How to run the test in SPSS?

For running the test we consider the file ofcar_sales.sav from the sample files of system file.


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In the beginning remove all the variables from variable no. 15 till end ( these are transformed or

standardized variables which is not useful during the initial level of MLR)

Make a correction in the measurement of the variable type from ordinal to nominal.


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Out of 11 scale variables sales in thousand is dependent variable, while all other scale

variables (10 in number) are independents. In other words we can say that there are 10 predictors

which estimate the car sales.

Here we are using enter method.


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Do continue and ok


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How to understand the output of the test?

Descriptive Statistics

Mean Std. Deviation N

Sales in thousands 59.11232 75.058933 117

4-year resale value 18.03154 11.605632 117

Price in thousands 25.96949 14.149699 117

Engine size 3.049 1.0552 117

Horsepower 181.28 58.592 117

Wheelbase 107.326 8.0506 117

Width 71.190 3.5302 117

Length 187.718 13.8499 117

Curb weight 3.32405 .597177 117

Fuel capacity 17.813 3.7946 117

Fuel efficiency 24.12 4.404 117

Descriptive of all the variables


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Correlations

Sales in

thousands

4-year

resale

value

Price in

thousands

Engine

size Horsepower Wheelbase Width Length

Curb

weight

Fuel

capacity

Fuel

efficiency

Pearso

n

Correlat

ion

Sales in

thousands

1.000 -.275 -.252 .038 -.153 .407 .178 .272 .067 .138 -.067

4-year resale

value

-.275 1.000 .955 .527 .773 -.054 .178 .025 .363 .325 -.399

Price in

thousands

-.252 .955 1.000 .649 .853 .067 .301 .183 .511 .406 -.480

Engine size .038 .527 .649 1.000 .862 .410 .672 .537 .743 .617 -.725

Horsepower -.153 .773 .853 .862 1.000 .226 .507 .401 .599 .480 -.596

Wheelbase .407 -.054 .067 .410 .226 1.000 .676 .854 .676 .659 -.471

Width .178 .178 .301 .672 .507 .676 1.00

0

.743 .736 .672 -.600

Length .272 .025 .183 .537 .401 .854 .743 1.000 .684 .563 -.466

Curb weight .067 .363 .511 .743 .599 .676 .736 .684 1.000 .848 -.819

Fuel capacity .138 .325 .406 .617 .480 .659 .672 .563 .848 1.000 -.809

Fuel efficiency -.067 -.399 -.480 -.725 -.596 -.471 -.600 -.466 -.819 -.809 1.000

Sig. (1-

tailed)

Sales in

thousands

. .001 .003 .342 .050 .000 .028 .001 .236 .069 .237

4-year resale

value

.001 . .000 .000 .000 .283 .027 .393 .000 .000 .000

Price in

thousands

.003 .000 . .000 .000 .236 .000 .024 .000 .000 .000

Engine size .342 .000 .000 . .000 .000 .000 .000 .000 .000 .000

Horsepower .050 .000 .000 .000 . .007 .000 .000 .000 .000 .000

Wheelbase .000 .283 .236 .000 .007 . .000 .000 .000 .000 .000

Width .028 .027 .000 .000 .000 .000 . .000 .000 .000 .000

Length .001 .393 .024 .000 .000 .000 .000 . .000 .000 .000

Curb weight .236 .000 .000 .000 .000 .000 .000 .000 . .000 .000

Fuel capacity .069 .000 .000 .000 .000 .000 .000 .000 .000 . .000

Fuel efficiency .237 .000 .000 .000 .000 .000 .000 .000 .000 .000 .


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The first column shows the correlation between the dependent variable sales in thousands and

other independent variables. So high correlation in this column and low level of significant (p-

value) shows that those variables are important for the model otherwise the variable is

insignificant and less important for the model. On the other hand the high correlation in the other

columns shows that the independent variables are highly correlated with each other. This shows

the property ofmulticollinearity is exist between the independent variables. Multicollineartity

violate the assumption of the independencyamong predictors. The advantage which one can

observe that, when we remove the variable which has the multicollineartity than the R-square of

the model will increase which is also a good sign for the model. One of the solutions of this

problem is to make the factors or components by factor analysis before the regression analysis.

Model Summaryb

Model R R Square

Adjusted R

Square

Std. Error of the

Estimate

1 .565a

.319 .255 64.798014

a. Predictors: (Constant), Fuel efficiency, 4-year resale value, Length,

Width, Engine size, Fuel capacity, Wheelbase, Curb weight,

Horsepower, Price in thousands

b. Dependent Variable: Sales in thousands

R-Square (Coefficient of determination) is equals to 0.319 or 31.9%. It means that all the 10

predictors explain the 31.9% of the variation in the estimators.

ANOVAb


1 Regression 208454.878 10 20845.488 4.965 .000a

Residual 445070.963 106 4198.783

Total 653525.841 116

a. Predictors: (Constant), Fuel efficiency, 4-year resale value, Length, Width, Engine size, Fuel

capacity, Wheelbase, Curb weight, Horsepower, Price in thousands


The test of ANOVA is significant which show that the model will be valid.


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Coefficientsa

Model

Unstandardized

Coefficients

Standardized

Coefficients

t Sig.

95.0% Confidence

Interval for B

Collinearity

Statistics

B Std. Error Beta

Lower

Bound

Upper

Bound Tolerance VIF

1 (Constant) -225.116 191.901 -1.173 .243 -605.579 155.347

4-year resale

value

.011 2.253 .002 .005 .996 -4.456 4.478 .053 18.890

Price in

thousands

-.015 2.146 -.003 -.007 .994 -4.269 4.239 .039 25.464

Engine size 37.640 15.588 .529 2.415 .017 6.735 68.545 .134 7.474

Horsepower -.612 .342 -.477 -1.788 .077 -1.290 .067 .090 11.095

Wheelbase 6.391 1.787 .685 3.576 .001 2.848 9.934 .175 5.718

Width -.375 3.120 -.018 -.120 .905 -6.561 5.811 .298 3.352

Length -.437 1.077 -.081 -.406 .686 -2.573 1.698 .163 6.149

Curb weight -69.476 29.602 -.553 -2.347 .021 -128.165 -10.787 .116 8.633

Fuel capacity -.157 3.697 -.008 -.043 .966 -7.487 7.172 .184 5.437

Fuel efficiency -2.608 2.931 -.153 -.890 .376 -8.418 3.203 .217 4.602

a. Dependent Variable: Sales in thousands

All such predictors which are insignificant should remove one by one from the model start from

the highest significant valued variable.

NOTE:

One should advise to study the residual structure and apply the appropriate transformation before

the removal of the variable. The reason of this practice is just to find out the true significant

variables otherwise it is quite possible that during the process of enter and remove one can

remove the significant variable.

Study of the residual Structure:

Usually we perform the analysis of residual structure by drawing the graph (scatter plot) between

unstandardized predicted value and standardized residual value (the variables unstandardizedpredicted value and standardized residual value generates when researcher runs the testconditionally when we give the option of save during the running the test)


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If the predicted values are very large than residuals will also be very large, and as a result

analysis of residual structure becomes difficult. Therefore we use standardized or studentized

residual values. The Standardized residual (taking on y-axis or dependent) and un-standardized

predicted value (taking on x-axis or independent)


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Do OK

The graph shows that the variation between the two variables initially less but later on it will

increase. Compare with the following set of graph.


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Here the graph which we form shows the resemblance with the graph of C. as shown below

Graph-1


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It means that the graph has the heteroscedasticity. Heteroscedasticity means that the variance

pattern between the values of y for the different predictors is not same. So transformation

requires lifting the assumption violation.

Method of Transformations:

When Data Transformation require?

The data transformation require when the study of residual structure tells us that it is not

following or violating the assumptions of the model. The assumptions are listed below:

Linearity of the phenomenon measured (through scatter plot of the error term). Constant variance of the error terms (Homoscedasticity). Independence of the error term (multicollineartity). All predictors are

Independent. Normality of the error term distribution.

Why Data Transformation require?Data transformation gives us two benefits:

1. Two correct the violations of the statistical assumption mentioned above for themultivariate technique.

2. To improve the relationship between the study variable and the predictors.3. Transformation shows the correct picture of the significant variables. Sometime we may

find such few variables which are insignificant but after transformation it deduces that thevariable was actually significant.

How to do Data Transformation?Criterion for the transformation

FigureViolation of assumption

Situation of

ViolationRemedy

a Null Plot All the

assumptions of

the model are

met.

No remedy is required.

b Non Normality Flat pattern Inverse transformation

(1/y)

Negatively

Skewed

Squared or Cubed (y2 or

y3)

Positively

Skewed

Square root or log

transformation (y or lny)

C Heteroscedasticity Cone shape

distribution

Inverse transformation

(1/y)


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opens rightward

Opens leftward Square root

transformation (y)D,H Heteroscedasticity A diamond shape

pattern shows

that the highvariance in the

middle while at

the end the

variance are less.

It means it is

nonlinearity is

also exist.

Usually log

transformation is

advisable. It is oftenhappened that when we

address one of the

violation the other will

also adjusted

simultaneously.

E Time base dependence

F Event base dependence

G NormalNumerous procedures are available for achieving linearity between two variables but most

simple nonlinear relationships can be placed in one of four categories in the figure below. If the

relationship looks like figure a, then either variable can be squared to achieve linearity. Whenmultiple transformation possibilities are shown, start with the top method in each figure and then

move downward until linearity is achieved.


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How to perform transformation:

Stepwise transformation are shown below.


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The dependent variable sales has been transformed as variable name TransformedSales

Now see the effect of transformation shown below.

Before seeing the effect of transformation, run the regression test again by taking the dependent

variable transformedSales. After that draw the scatter plot of Standardized residual and

Unstandardized predicted value.


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Result of transformation

Before Transformation After transformation

Model Summaryb

Mod

el R

R

Square

Adjusted R

Square

Std. Error

of the

Estimate

1 .565a

.319 .255 64.798014

a. Predictors: (Constant), Fuel efficiency, 4-year

resale value, Length, Width, Engine size, Fuel

capacity, Wheelbase, Curb weight, Horsepower,

Price in thousands


Model Summaryb

Mode

l R

R

Square

Adjusted R

Square

Std. Error of

the Estimate

1 .634a

.402 .345 1.08253

a. Predictors: (Constant), Fuel efficiency, 4-year

resale value, Length, Width, Engine size, Fuel

capacity, Wheelbase, Curb weight, Horsepower,

Price in thousands

b. Dependent Variable: TransformedSales


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Conclusion:

1. R-Square improved from 31.9% to 40.2%

2. The scatter plot also show that this is looks like the null plot.

Enter and Remove the Predictors (Before transformation)

Here we perform the removal of insignificant variable from the model one by one we perform

this exercise without the transformation. The removal of the variable will take place in

descending order on the basis of significant values. The variable which has the highest sig value

will remove first continue this procedure one by one.

Note: It is quite possible that the researcher can think about any particular variable as an

important factor but statistical may not support his/her idea. It does not matter, if researcher

thinks to include that variable to be a part of the model so it can be.

0. Without removing any variable (Consider all predictors)

Model Summary

Model R R Square

Adjusted R

Square

Std. Error of the

Estimate

1 .565a

.319 .255 64.798014





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Model

Unstandardized

Coefficients

Standardized

Coefficients

t Sig.

95.0% Confidence Interval

for B

B Std. Error Beta Lower Bound Upper Bound

1 (Constant) -214.894 152.450 -1.410 .161 -516.258 86.471

Price in

thousands

-.441 .731 -.092 -.604 .547 -1.886 1.004

Engine size 34.469 11.271 .525 3.058 .003 12.189 56.750

Horsepower -.563 .250 -.464 -2.254 .026 -1.056 -.069

Wheelbase 4.729 1.317 .529 3.590 .000 2.125 7.333

Width -.222 2.428 -.011 -.092 .927 -5.022 4.577

Length -.129 .748 -.025 -.173 .863 -1.608 1.349

Curb weight -50.079 19.946 -.462 -2.511 .013 -89.508 -10.649

Fuel capacity .460 2.719 .026 .169 .866 -4.916 5.835

Fuel efficiency -1.172 2.266 -.073 -.517 .606 -5.653 3.308


After the removal of 4-year resale value the significance value of the variables are addressed

few are increase and few are decrease. The variable horse power which was initially

insignificant, but now become significant.

2. Remove variable width

Model Summaryb

Model R R Square

Adjusted R

Square

Std. Error of the

Estimate

1 .560a

.314 .275 58.690836

a. Predictors: (Constant), Fuel efficiency, Length, Price in thousands,

Fuel capacity, Engine size, Wheelbase, Curb weight, Horsepower


This time there is no change in the value of R-square.

Coefficientsa

Model

Unstandardized

Coefficients

Standardized

Coefficients t Sig.


for B


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1 (Constant) -224.403 111.287 -2.016 .046 -444.383 -4.423

Price in

thousands

-.433 .722 -.090 -.599 .550 -1.860 .995

Engine size 34.277 11.036 .522 3.106 .002 12.463 56.091

Horsepower -.565 .248 -.466 -2.283 .024 -1.054 -.076

Wheelbase 4.712 1.299 .527 3.628 .000 2.144 7.279

Length -.141 .734 -.028 -.192 .848 -1.592 1.310

Curb weight -50.374 19.616 -.465 -2.568 .011 -89.148 -11.600

Fuel capacity .447 2.706 .026 .165 .869 -4.902 5.796

Fuel efficiency -1.181 2.257 -.074 -.523 .602 -5.642 3.280


After this run only 4 variables are now left as insignificant while the constant becomes

significant itself.

3. Remove variable Fuel Capacity

Model Summaryb

Model R R Square

Adjusted R

Square

Std. Error of the

Estimate

1 .560a .313 .280 58.492268


Engine size, Wheelbase, Curb weight, Horsepower


Again R-square is dropped slightly i.e. only 0.001

Coefficientsa

Model

Unstandardized

Coefficients

Standardized

Coefficients

t Sig.


for B


1 (Constant) -221.857 109.840 -2.020 .045 -438.963 -4.750

Price in

thousands

-.423 .718 -.089 -.590 .556 -1.842 .995

Engine size 34.347 10.990 .523 3.125 .002 12.624 56.070


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Horsepower -.569 .245 -.469 -2.320 .022 -1.054 -.084

Wheelbase 4.767 1.250 .534 3.815 .000 2.297 7.238

Length -.151 .729 -.030 -.207 .836 -1.592 1.290

Curb weight -48.971 17.621 -.452 -2.779 .006 -83.801 -14.142

Fuel efficiency -1.310 2.111 -.082 -.620 .536 -5.482 2.863


4. Remove variable Length

Model Summaryb

Model R R Square

Adjusted R

Square

Std. Error of the

Estimate

1 .560

a

.313 .285 58.298917

a. Predictors: (Constant), Fuel efficiency, Price in thousands,

Wheelbase, Engine size, Curb weight, Horsepower


No change in the value of R-square

Coefficientsa

Model

Unstandardized

Coefficients

Standardized

Coefficients

t Sig.


for B


1 (Constant) -223.549 109.174 -2.048 .042 -439.327 -7.771

Price in

thousands

-.384 .690 -.080 -.556 .579 -1.747 .979

Engine size 34.172 10.922 .520 3.129 .002 12.586 55.759

Horsepower -.582 .237 -.480 -2.455 .015 -1.050 -.113

Wheelbase 4.585 .884 .513 5.189 .000 2.839 6.331

Curb weight -49.726 17.184 -.459 -2.894 .004 -83.689 -15.762

Fuel efficiency -1.421 2.035 -.089 -.698 .486 -5.443 2.602


5. Remove variable Price in thousand

Model Summaryb


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Model R R Square

Adjusted R

Square

Std. Error of the

Estimate

1 .560a

.313 .290 57.965284

a. Predictors: (Constant), Fuel efficiency, Wheelbase, Horsepower,

Curb weight, Engine size


No change in the value of R-square

Coefficientsa

Model

Unstandardized

Coefficients

Standardized

Coefficients

t Sig.

95.0% Confidence Interval for

B


1 (Constant) -224.038 108.178 -2.071 .040 -437.823 -10.253

Engine size 36.113 10.310 .549 3.503 .001 15.737 56.489

Horsepower -.682 .156 -.563 -4.382 .000 -.990 -.374

Wheelbase 4.746 .830 .531 5.715 .000 3.105 6.386

Curb weight -53.227 15.946 -.491 -3.338 .001 -84.739 -21.715

Fuel efficiency -1.541 2.006 -.096 -.768 .444 -5.506 2.424


6. Remove variable Fuel efficiency

Model Summaryb

Model R R Square

Adjusted R

Square

Std. Error of the

Estimate

1 .556a

.309 .291 57.595036

a. Predictors: (Constant), Curb weight, Horsepower, Wheelbase,

Engine size


This time R-square dropped by 0.004 unit but it is still predictors explain the estimator by 30.9%.

Coefficientsa

Model

Unstandardized

Coefficients

Standardized

Coefficients

t Sig.

95.0% Confidence Interval for

B



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1 (Constant) -288.879 73.316 -3.940 .000 -433.745 -144.012

Engine size 36.856 9.985 .561 3.691 .000 17.127 56.585

Horsepower -.669 .154 -.552 -4.354 .000 -.973 -.365

Wheelbase 4.747 .820 .531 5.787 .000 3.126 6.367

Curb weight -46.288 13.208 -.427 -3.505 .001 -72.386 -20.190


Now only significant variables are left.

Enter and Remove the Predictors (After Transformation)

0. First output with considering all the variables.

Model Summaryb

Model R R Square

Adjusted R

Square

Std. Error of the

Estimate

1 .634a

.402 .345 1.08253




b. Dependent Variable: LogSale

Coefficientsa

Model

Unstandardized

Coefficients

Standardized

Coefficients

t Sig.

95.0% Confidence

Interval for B

Collinearity

Statistics

B Std. Error Beta

Lower

Bound

Upper

Bound Tolerance VIF

1 (Constant) -1.097 3.206 -.342 .733 -7.453 5.259

4-year resale

value

-.012 .038 -.101 -.311 .757 -.086 .063 .053 18.890

Price in

thousands

-.036 .036 -.385 -1.015 .312 -.107 .035 .039 25.464

Engine size .310 .260 .244 1.190 .237 -.206 .826 .134 7.474

Horsepower -.003 .006 -.118 -.470 .639 -.014 .009 .090 11.095

Wheelbase .093 .030 .559 3.111 .002 .034 .152 .175 5.718

Width -.026 .052 -.068 -.492 .624 -.129 .078 .298 3.352


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Length -.018 .018 -.188 -1.008 .316 -.054 .018 .163 6.149

Curb weight .262 .495 .117 .530 .597 -.718 1.242 .116 8.633

Fuel capacity -.059 .062 -.166 -.949 .345 -.181 .064 .184 5.437

Fuel efficiency .026 .049 .087 .538 .592 -.071 .123 .217 4.602

a. Dependent Variable: LogSale

1. Remove variable 4-year resale value

The variable 4-year resale value having the heighest sig value i.e. 0.757 so remove it first.

Model Summaryb

Model R R Square

Adjusted R

Square

Std. Error of the

Estimate

1 .678a

.459 .425 1.01072


Width, Fuel capacity, Engine size, Wheelbase, Curb weight,

Horsepower


Coefficientsa

Model

Unstandardized

Coefficients

Standardized

Coefficients

t Sig.

95.0% Confidence

Interval for B

Collinearity

Statistics

B Std. Error Beta

Lower

Bound

Upper

Bound Tolerance VIF

1 (Constant) -.414 2.616 -.158 .874 -5.586 4.758

Price in

thousands

-.057 .013 -.616 -4.542 .000 -.082 -.032 .207 4.835

Engine size .408 .193 .321 2.110 .037 .026 .791 .164 6.095

Horsepower -.004 .004 -.151 -.828 .409 -.012 .005 .114 8.760

Wheelbase .061 .023 .355 2.711 .008 .017 .106 .222 4.499

Width -.049 .042 -.127 -1.175 .242 -.131 .033 .325 3.080

Length -.003 .013 -.028 -.217 .828 -.028 .023 .226 4.418

Curb weight .424 .342 .202 1.238 .218 -.253 1.100 .142 7.019

Fuel capacity -.026 .047 -.077 -.560 .576 -.118 .066 .200 4.991

Fuel efficiency .047 .039 .153 1.218 .225 -.030 .124 .241 4.144


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Coefficientsa,b

Model

Unstandardized

Coefficients

Standardized

Coefficients

t Sig.

95.0% Confidence

Interval for B

Collinearity

Statistics

B Std. Error Beta

Lower

Bound

Upper

Bound Tolerance VIF

1 Price in

thousands

-.057 .012 -.498 -4.603 .000 -.082 -.033 .045 22.048

Engine size .413 .190 .376 2.174 .031 .038 .789 .018 56.259

Horsepower -.004 .004 -.194 -.836 .405 -.012 .005 .010 101.939

Wheelbase .061 .022 1.848 2.720 .007 .017 .105 .001 870.319

Width -.053 .030 -1.072 -1.757 .081 -.114 .007 .001 702.034

Length -.003 .013 -.141 -.209 .834 -.028 .023 .001 859.164

Curb weight .424 .341 .410 1.241 .216 -.251 1.098 .005 205.739

Fuel capacity -.027 .046 -.138 -.574 .567 -.118 .065 .009 109.013

Fuel efficiency .044 .034 .304 1.299 .196 -.023 .112 .010 102.955


b. Linear Regression through the Origin

3. Remove variable Length

Model Summary

c,d

Model R R Squareb

Adjusted R

Square

Std. Error of the

Estimate

1 .961a

.924 .920 1.00392

a. Predictors: Fuel efficiency, Price in thousands, Engine size, Fuel

capacity, Horsepower, Curb weight, Wheelbase, Width

b. For regression through the origin (the no-intercept model), R Square

measures the proportion of the variability in the dependent variable about

the origin explained by regression. This CANNOT be compared to R

Square for models which include an intercept.

c. Dependent Variable: LogSale

d. Linear Regression through the Origin

Coefficientsa,b


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Model

Unstandardized

Coefficients

Standardized

Coefficients

t Sig.

95.0% Confidence

Interval for B

Collinearity

Statistics

B Std. Error Beta

Lower

Bound

Upper

Bound Tolerance VIF

1 Price in

thousands

-.057 .012 -.492 -4.703 .000 -.080 -.033 .048 20.811

Engine size .411 .189 .374 2.173 .031 .037 .785 .018 56.089

Horsepower -.004 .004 -.205 -.904 .367 -.012 .004 .010 97.349

Wheelbase .058 .017 1.755 3.432 .001 .025 .091 .002 495.998

Width -.055 .030 -1.096 -1.833 .069 -.114 .004 .001 678.380

Curb weight .410 .334 .397 1.228 .221 -.250 1.070 .005 198.513

Fuel capacity -.026 .046 -.133 -.558 .578 -.117 .065 .009 108.047

Fuel efficiency .043 .033 .294 1.288 .200 -.023 .109 .010 99.044



4. Remove variable Fuel Capacity

Model Summaryc,d

Model R R Squareb

Adjusted RSquare

Std. Error of theEstimate

1 .961a

.924 .920 1.00154

a. Predictors: Fuel efficiency, Price in thousands, Engine size, Curb

weight, Horsepower, Wheelbase, Width







Coefficientsa,b

Model

Unstandardized

Coefficients

Standardized

Coefficients t Sig.

95.0% Confidence

Interval for B

Collinearity

Statistics


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B Std. Error Beta

Lower

Bound

Upper

Bound Tolerance VIF

1 Price in

thousands

-.057 .012 -.499 -4.814 .000 -.081 -.034 .049 20.521

Engine size .410 .189 .372 2.172 .031 .037 .783 .018 56.081

Horsepower -.003 .004 -.188 -.841 .402 -.012 .005 .010 95.713

Wheelbase .055 .016 1.680 3.413 .001 .023 .087 .002 461.716

Width -.057 .030 -1.135 -1.916 .057 -.115 .002 .001 669.012

Curb weight .334 .304 .323 1.098 .274 -.267 .934 .006 164.991

Fuel efficiency .050 .031 .343 1.632 .105 -.011 .111 .012 84.344



5. Remove variable Horse Power

Model Summaryc,d

Model R R Squareb

Adjusted R

Square

Std. Error of the

Estimate

1 .961a

.924 .920 1.00053

a. Predictors: Fuel efficiency, Price in thousands, Engine size, Curbweight, Wheelbase, Width







Coefficients

a,b

Model

Unstandardized

Coefficients

Standardized

Coefficients

t Sig.

95.0% Confidence

Interval for B

Collinearity

Statistics

B Std. Error Beta

Lower

Bound

Upper

Bound Tolerance VIF


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1 Price in

thousands

-.065 .008 -.566 -8.443 .000 -.080 -.050 .117 8.584

Engine size .308 .145 .280 2.128 .035 .022 .595 .030 33.125

Wheelbase .055 .016 1.662 3.383 .001 .023 .087 .002 460.831

Width -.062 .029 -1.235 -2.132 .035 -.119 -.004 .002 641.711

Curb weight .403 .292 .390 1.381 .169 -.174 .980 .007 152.731

Fuel efficiency .053 .031 .362 1.732 .085 -.007 .113 .012 83.408



6. Remove variable Curb weight

Model Summaryc,d

Model R R Squareb

Adjusted R

Square

Std. Error of the

Estimate

1 .961a

.923 .920 1.00022

a. Predictors: Fuel efficiency, Price in thousands, Engine size,

Wheelbase, Width







Coefficientsa,b

Model

Unstandardized

Coefficients

Standardized

Coefficients

t Sig.

95.0% Confidence

Interval for B

Collinearity

Statistics

B Std. Error Beta

Lower

Bound

Upper

Bound Tolerance VIF

1 Price in

thousands

-.062 .007 -.542 -8.407 .000 -.077 -.047 .125 7.975

Engine size .332 .143 .303 2.317 .022 .049 .615 .030 32.794

Wheelbase .063 .015 1.930 4.265 .000 .034 .093 .003 393.006

Width -.049 .027 -.989 -1.793 .075 -.103 .005 .002 583.801

Fuel efficiency .027 .024 .186 1.125 .262 -.021 .075 .019 52.739


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7. Remove variable Fuel efficiency

8. Remove variable WidthModel Summary

c,d

Model R R Squareb

Adjusted R

Square

Std. Error of the

Estimate

1 .960a

.922 .921 .99940

a. Predictors: Wheelbase, Price in thousands, Engine size

Model Summaryc,d

Model R R Squareb

Adjusted R

Square

Std. Error of the

Estimate

1 .961a

.923 .921 .99628

a. Predictors: Width, Price in thousands, Engine size, Wheelbase







Coefficientsa,b

Model

Unstandardized

Coefficients

Standardized

Coefficients

t Sig.

95.0% Confidence

Interval for B

Collinearity

Statistics

B Std. Error Beta

Lower

Bound

Upper

Bound Tolerance VIF

1 Price in

thousands

-.062 .007 -.541 -8.480 .000 -.077 -.048 .125 8.005

Engine size .223 .106 .203 2.094 .038 .013 .432 .054 18.427

Wheelbase .061 .015 1.849 4.161 .000 .032 .090 .003 388.695

Width -.031 .022 -.630 -1.398 .164 -.076 .013 .003 399.409




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Coefficientsa,b

Model

Unstandardized

Coefficients

Standardized

Coefficients

t Sig.

95.0% Confidence

Interval for B

Collinearity

Statistics

B Std. Error Beta

Lower

Bound

Upper

Bound Tolerance VIF

1 Price in

thousands

-.064 .007 -.559 -8.913 .000 -.078 -.050 .130 7.684

Engine size .221 .107 .202 2.078 .039 .011 .432 .054 18.426

Wheelbase .041 .003 1.237 16.255 .000 .036 .046 .088 11.328



Conclusion

After all these exercise we conclude that there are three predictors; (1) price in thousand, (2)

engine size and (3) wheel base are the significant predictors but the VIF of engine size is veryhigh that is 18.426 so it shows the effect of multicollinearity exist here although it is significant

coefficient. So it depends upon the researcher and theory, whether to include or not to include

this variable in the regression model. On the other hand, when the test was run before the

transformation we find the other four variables with constant term was (1) engine size (2) horse

power (3) wheel base and (4) curb weight. After transformation we get theoretically more logical

predictors as compare to the result of before transformation while the value of R-Square is also

much better than before.

Finally the regression equation will be:

Logsale = - 0.064 (price in thousands) + 0.041(Wheelbase)

Enter & Remove by Backward Method

The same test and approximately the same result can calculate by using the backward method.

We validate the result of before and after transformation by backward method.


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How to perform Backward Method? (Without transformation)


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Variables Entered/Removedb

Model

Variables

Entered

Variables

Removed Method

1 Fuel efficiency,

4-year resale

value, Length,

Width, Engine

size, Fuel

capacity,

Wheelbase, Curb

weight,

Horsepower,

Price in

thousandsa

. Enter

2 . 4-year resale

value

Backward

(criterion:

Probability of F-

to-remove >=

.100).

3 . Price in

thousands

Backward

(criterion:

Probability of F-

to-remove >=

.100).

4 . Fuel capacity Backward

(criterion:

Probability of F-

to-remove >=

.100).

5 . Width Backward

(criterion:

Probability of F-

to-remove >=

.100).


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6 . Length Backward

(criterion:

Probability of F-

to-remove >=

.100).

7 . Fuel efficiency Backward

(criterion:

Probability of F-

to-remove >=

.100).

a. All requested variables entered.


Model Summary

Model R R Square

Adjusted R

Square

Std. Error of the

Estimate

1 .565a

.319 .255 64.798014

2 .565b

.319 .262 64.494517

3 .565c

.319 .269 64.195246

4 .565d

.319 .275 63.900650

5 .565e

.319 .282 63.614616

6 .563f

.317 .287 63.398826

7 .556g

.309 .285 63.486236




b. Predictors: (Constant), Fuel efficiency, Length, Width, Engine size,

Fuel capacity, Wheelbase, Curb weight, Horsepower, Price in

thousands

c. Predictors: (Constant), Fuel efficiency, Length, Width, Engine size,

Fuel capacity, Wheelbase, Curb weight, Horsepower

d. Predictors: (Constant), Fuel efficiency, Length, Width, Engine size,

Wheelbase, Curb weight, Horsepower

e. Predictors: (Constant), Fuel efficiency, Length, Engine size,

Wheelbase, Curb weight, Horsepower


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f. Predictors: (Constant), Fuel efficiency, Engine size, Wheelbase, Curb

weight, Horsepower

g. Predictors: (Constant), Engine size, Wheelbase, Curb weight,

Horsepower

Coefficientsa

Model

Unstandardized Coefficients

Standardized

Coefficients

t Sig.B Std. Error Beta

1 (Constant) -225.116 191.901 -1.173 .243

4-year resale value .011 2.253 .002 .005 .996

Price in thousands -.015 2.146 -.003 -.007 .994

Engine size 37.640 15.588 .529 2.415 .017

Horsepower -.612 .342 -.477 -1.788 .077

Wheelbase 6.391 1.787 .685 3.576 .001

Width -.375 3.120 -.018 -.120 .905

Length -.437 1.077 -.081 -.406 .686

Curb weight -69.476 29.602 -.553 -2.347 .021

Fuel capacity -.157 3.697 -.008 -.043 .966

Fuel efficiency -2.608 2.931 -.153 -.890 .376

2 (Constant) -224.919 186.969 -1.203 .232

Price in thousands -.006 .991 -.001 -.006 .996

Engine size 37.627 15.300 .529 2.459 .016

Horsepower -.611 .339 -.477 -1.801 .074

Wheelbase 6.392 1.769 .686 3.613 .000

Width -.374 3.099 -.018 -.121 .904

Length -.438 1.046 -.081 -.419 .676

Curb weight -69.529 27.528 -.553 -2.526 .013

Fuel capacity -.154 3.619 -.008 -.043 .966

Fuel efficiency -2.610 2.878 -.153 -.907 .367

3 (Constant) -225.048 184.706 -1.218 .226


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Engine size 37.654 14.496 .529 2.597 .011

Horsepower -.613 .213 -.478 -2.881 .005

Wheelbase 6.392 1.758 .686 3.636 .000

Width -.371 3.057 -.017 -.121 .904

Length -.437 1.019 -.081 -.429 .669

Curb weight -69.586 25.500 -.554 -2.729 .007

Fuel capacity -.156 3.590 -.008 -.043 .966

Fuel efficiency -2.613 2.827 -.153 -.924 .357

4 (Constant) -225.414 183.665 -1.227 .222

Engine size 37.728 14.328 .530 2.633 .010

Horsepower -.614 .211 -.479 -2.906 .004

Wheelbase 6.364 1.619 .683 3.929 .000

Width -.395 2.993 -.019 -.132 .895

Length -.424 .967 -.078 -.438 .662

Curb weight -70.015 23.394 -.557 -2.993 .003

Fuel efficiency -2.561 2.557 -.150 -1.002 .319

5 (Constant) -242.535 129.494 -1.873 .064

Engine size 37.238 13.775 .523 2.703 .008

Horsepower -.611 .209 -.477 -2.919 .004

Wheelbase 6.346 1.607 .681 3.949 .000

Length -.460 .924 -.085 -.498 .619

Curb weight -70.466 23.039 -.561 -3.059 .003

Fuel efficiency -2.554 2.545 -.150 -1.004 .318

6 (Constant) -247.393 128.688 -1.922 .057

Engine size 36.371 13.619 .511 2.671 .009

Horsepower -.626 .207 -.489 -3.031 .003

Wheelbase 5.745 1.057 .616 5.436 .000

Curb weight -71.956 22.767 -.572 -3.161 .002

Fuel efficiency -2.831 2.475 -.166 -1.144 .255

7 (Constant) -353.714 89.146 -3.968 .000

Engine size 39.752 13.313 .559 2.986 .003

Horsepower -.638 .207 -.498 -3.087 .003


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Wheelbase 5.556 1.045 .596 5.315 .000

Curb weight -56.888 18.597 -.453 -3.059 .003


At the end we find the same significant variables which we find in enter and remove method

with approximately the same level of significant values.

How to perform Backward Method? (After transformation)


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Variables Entered/Removedb

Model Variables Entered

Variables

Removed Method

1 Fuel efficiency, 4-year resale value, Length,

Width, Engine size, Fuel capacity, Wheelbase,

Curb weight, Horsepower, Price in thousandsa

. Enter

2 . 4-year resale value Backward (criterion: Probability of F-to-

remove >= .100).

3 . Horsepower Backward (criterion: Probability of F-to-

remove >= .100).

4 . Width Backward (criterion: Probability of F-to-

remove >= .100).

5 . Fuel efficiency Backward (criterion: Probability of F-to-

remove >= .100).

6 . Curb weight Backward (criterion: Probability of F-to-

remove >= .100).

7 . Length Backward (criterion: Probability of F-to-

remove >= .100).

8 . Engine size Backward (criterion: Probability of F-to-

remove >= .100).

9 . Fuel capacity Backward (criterion: Probability of F-to-

remove >= .100).

a. All requested variables entered.



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Model Summary

Model R R Square

Adjusted R

Square

Std. Error of the

Estimate

1 .634a

.402 .345 1.08253

2 .633b

.401 .351 1.07796

3 .632c

.400 .355 1.07419

4 .631d

.398 .360 1.07068

5 .629e

.396 .363 1.06766

6 .628f

.394 .367 1.06480

7 .621g

.386 .364 1.06719

8 .618h

.382 .365 1.06590

9 .613i

.376 .365 1.06600




b. Predictors: (Constant), Fuel efficiency, Length, Width, Engine size,

Fuel capacity, Wheelbase, Curb weight, Horsepower, Price in

thousands

c. Predictors: (Constant), Fuel efficiency, Length, Width, Engine size,

Fuel capacity, Wheelbase, Curb weight, Price in thousands

d. Predictors: (Constant), Fuel efficiency, Length, Engine size, Fuel

capacity, Wheelbase, Curb weight, Price in thousands

e. Predictors: (Constant), Length, Engine size, Fuel capacity,

Wheelbase, Curb weight, Price in thousands

f. Predictors: (Constant), Length, Engine size, Fuel capacity,

Wheelbase, Price in thousands

g. Predictors: (Constant), Engine size, Fuel capacity, Wheelbase, Price

in thousands

h. Predictors: (Constant), Fuel capacity, Wheelbase, Price in thousands

i. Predictors: (Constant), Wheelbase, Price in thousands


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Coefficientsa

Model

Unstandardized Coefficients

Standardized

Coefficients

t Sig.B Std. Error Beta

1 (Constant) -1.097 3.206 -.342 .733

4-year resale value -.012 .038 -.101 -.311 .757

Price in thousands -.036 .036 -.385 -1.015 .312

Engine size .310 .260 .244 1.190 .237

Horsepower -.003 .006 -.118 -.470 .639

Wheelbase .093 .030 .559 3.111 .002

Width -.026 .052 -.068 -.492 .624

Length -.018 .018 -.188 -1.008 .316

Curb weight .262 .495 .117 .530 .597

Fuel capacity -.059 .062 -.166 -.949 .345

Fuel efficiency .026 .049 .087 .538 .592

2 (Constant) -1.301 3.125 -.416 .678


Engine size .323 .256 .255 1.264 .209

Horsepower -.003 .006 -.124 -.497 .620

Wheelbase .092 .030 .553 3.108 .002

Width -.027 .052 -.071 -.516 .607

Length -.017 .017 -.175 -.968 .335

Curb weight .317 .460 .141 .689 .493

Fuel capacity -.062 .060 -.176 -1.027 .307

Fuel efficiency .029 .048 .095 .599 .551

3 (Constant) -1.344 3.113 -.432 .667


Engine size .238 .188 .187 1.262 .210

Wheelbase .094 .029 .564 3.210 .002

Width -.028 .052 -.073 -.537 .592

Length -.019 .017 -.199 -1.147 .254


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Curb weight .377 .442 .168 .853 .395

Fuel capacity -.062 .060 -.175 -1.024 .308

Fuel efficiency .031 .048 .103 .653 .515

4 (Constant) -2.502 2.239 -1.117 .266


Engine size .204 .177 .161 1.153 .251

Wheelbase .094 .029 .565 3.224 .002

Length -.022 .016 -.228 -1.380 .170

Curb weight .354 .439 .158 .806 .422

Fuel capacity -.068 .059 -.192 -1.150 .253

Fuel efficiency .029 .047 .096 .617 .539

5 (Constant) -1.553 1.622 -.958 .340


Engine size .167 .166 .132 1.006 .316

Wheelbase .096 .029 .579 3.340 .001

Length -.021 .016 -.218 -1.331 .186

Curb weight .262 .411 .117 .637 .526

Fuel capacity -.083 .053 -.236 -1.556 .123

6 (Constant) -1.771 1.581 -1.120 .265


Engine size .199 .158 .157 1.256 .212

Wheelbase .098 .029 .589 3.425 .001

Length -.019 .015 -.196 -1.226 .223

Fuel capacity -.063 .042 -.177 -1.473 .143

7 (Constant) -2.338 1.515 -1.542 .126


Engine size .125 .147 .099 .852 .396

Wheelbase .070 .017 .422 4.011 .000

Fuel capacity -.050 .041 -.141 -1.205 .231

8 (Constant) -2.593 1.484 -1.747 .083


Wheelbase .073 .017 .441 4.292 .000


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Fuel capacity -.040 .040 -.113 -1.011 .314

9 (Constant) -1.920 1.326 -1.448 .150


Wheelbase .061 .012 .369 4.980 .000


As here, only the two variables that is price in thousand and wheelbase are the two significant variables while

the constant term is insignificant for the model.

Finally, the regression equation will be:

logSale = - 0.049(price in thousands) + 0.061(Wheelbase)

The result which we generate by enter method is mentioned below.

Logsale = - 0.064 (price in thousands) + 0.041(Wheelbase)

After the study of the two regression model we find that both the method gives us the same variable with

slightly different in the coefficients. So it is possible that may find a minor difference in the beta coefficients.

Tolerance and Variance Inflation factor (VIF)The tolerance and VIF are reciprocal to each other. It use for the measurement of independency

of the independent variables or one can say that to measure the multicollinearity among the

independent variables. multicollinearity measure expressing the degree of explanation or

variation of one independent variable in the model due to other independent variable in the

model. This can be calculated as to take one of the independent variable as a dependent

variable and rest of the independent variables as predictors and then to regress them. The

regression will tell us the value of R2, the percentage which is explained by the other variable.

This R2

is called as R2*

, the amount of independent variable which is explained by the other

independent variable.Tolerance is calculated as 1-R

2*, It means that as far as the value of tolerance becomes high, the

multicollinearity will be low.

Variance inflation factor (VIF) is calculated as 1/Tolerance. VIF is the degree to which thestandard error has been increased due to multicollinearity.

R2*

increases then tolerance will decrease this implies VIF will increase this means

multicollinearity is increases.

Regression Equation

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