AMDA Regression Solution

8/13/2019 AMDA Regression Solution

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Question 3:

1. Checking research design of a multiple regression: The data included 80 responses from the customer base and

13 independent variables. The data can be used to detect relationships with R-square of approximately 29% at a

power of 0.80 with significance level set at 0.01.

2. Checking Assumptions of Model:

a. Linearity: Scatter plots did not indicate any non linear relationship between dependent and independentvariables.

b. Homoscedasticity: Tests for heteroscedasticity was done using comparison of variances of metricvariables (x6 to x18) across levels of non metric variables (x1 to x5). This test is called Levene Test. At

significance level of 0.05, only x1 showed patterns of heteroscedasticity.

c. Normality: Both empirical measures and normal probability plots were used to evaluate the normality ofthe metric variables. KS and Shapiro-Wilk tests indicated that six variables (x6, X7, X12, X13,X16, X17)

violate the normality assumption. Initial regressions were conducted without any transformation. To

check if normality violation was affecting the regression model, transformation was applied and another

regression was carried out.

3. Estimating the Regression Model and Assessing the Overall Model Fit

Stepwise regression procedure was employed to select variables for inclusion in the regression model. The step

wise model selects variable based on bivariate correlations. Table below provides the summary of results obtained

from stepwise regression.

Model Summary

Model R R Square Adjusted

R Square

Std

Error

Change Statistics Durbin-

Watson

R Square

Change

F

Change

df1 df2 Sig. F

Change

2.40

1 0.54 0.29 0.28 1.01 0.29 32.04 1.00 78.00 0.00

2 0.71 0.50 0.49 0.85 0.21 32.22 1.00 77.00 0.00

3 0.88 0.77 0.76 0.59 0.27 86.86 1.00 76.00 0.00

4 0.89 0.78 0.77 0.57 0.02 5.89 1.00 75.00 0.02

5 0.89 0.80 0.78 0.56 0.01 4.21 1.00 74.00 0.04

a. Predictors: (Constant), X9 - Complaint Resolution

b. Predictors: (Constant), X9 - Complaint Resolution, X12 - Salesforce Image

c. Predictors: (Constant), X9 - Complaint Resolution, X12 - Salesforce Image, X6 - Product Quality

d. Predictors: (Constant), X9 - Complaint Resolution, X12 - Salesforce Image, X6 - Product Quality, X7 - E-Commerce

Activities

e. Predictors: (Constant), X9 - Complaint Resolution, X12 - Salesforce Image, X6 - Product Quality, X7 - E-Commerce

Activities, X18 - Delivery Speed

f. Dependent Variable: X19Satisfaction

Table 1: Overall Model Fit Results of Stepwise Regression

As can be seen from the table above the first 3 variables are explaining 77% of the variance in the dependent

variable. The addition of 2 more variables increased the R-square by only 3%.

Overall Model Fit: The first variable to be entered was X9 (Complaint Resolution) as it had the highest bivariate

correlation with the dependent variable (0.54). The other variables to be added subsequently were X12, X6, X7 and

X18.The final regression model with 5 independent variables explains almost 80 percent of the variance of X19


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(Customer Satisfaction). The adjusted R-square of 78% indicates no overfitting of the models has taken place and

the results are generalizable. The table below provides the regression coefficients and their significance.

Coefficients

Model Unstandardized Coefficients Standardized

Coefficients

t Sig. Correlations Collinearity

Statistics

B Std.

Error

Beta Zero-

order

Partial Part Tolerance V

1 (Constant) 3.80 0.57 6.72 0.00

1.00 1.X9 - Complaint Resolution 0.56 0.10 0.54 5.66 0.00 0.54 0.54 0.54

2 (Constant) 1.56 0.62 2.51 0.01

X9 - Complaint Resolution 0.49 0.09 0.47 5.73 0.00 0.54 0.55 0.46 0.98 1

X12 - Salesforce Image 0.52 0.09 0.46 5.68 0.00 0.54 0.54 0.46 0.98 1

3 (Constant)

-

1.860.56

-

3.310.00



X6 - Product Quality 0.44 0.05 0.53 9.32 0.00 0.51 0.73 0.52 0.97 1

4 (Constant)

-

1.480.57

-

2.610.01




X7 - E-Commerce

Activities

-

0.370.15 -0.21

-

2.430.02 0.35 -0.27

-

0.130.37 2

5 (Constant)

-

1.88 0.59

-

3.20 0.00




X7 - E-Commerce

Activities

-

0.360.15 -0.21

-

2.420.02 0.35 -0.27

-

0.130.37 2

X18 - Delivery Speed 0.36 0.18 0.21 2.05 0.04 0.52 0.23 0.11 0.26 3

a. Dependent Variable: X19Satisfaction

Table 2: Regression Coefficients with their significance and correlation statistics

Estimated Coefficients: All the five regression coefficients are significant at .05 level but X9, X17 and X18 are not

significant at .01 level.

Multicollinearity: Multicollinearity is significant for the model. Of the five variables used by the regression model,

four have tolerance values less than 0.5 indicating that over half of their variance is accounted for by the other

variables in the equation. For example Complaint resolution which has the highest bivariate correlation of 0.540

among all the variables has a very low part correlation of 0.110 may due to the effect of multicollinearity. It has the

highest VIF among all the other variables used in the regression model. It may due to the fact that complaint

resolution is related to several other variables like Technical Support (X8), Product Quality (X6) and Sales Force


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Image (X12). Only product quality has no or insignificant multicollinearity.Examining the partial correlations of

variables not included in the model, it can be seen that none of the remaining variables have a significant partial

correlation at 0.05 level needed for entry.

4. Evaluating the Variate for the Assumptions of Regression Analysis

The measure used to evaluate the regression variate is the residual. The assumptions that have been checked are:

Linearity:The scatter plot of studentized residual and standardized predicted value (provided in the chart below)

indicates no non linear pattern to the residuals, thus ensuring that the overall equation is linear.Partial regression

plots were obtained for each independent variable in the equation. This was done to check if each independent

variables relationship is also linear. The scatter plots show that the relationships for X6, X7 and X12 are reasonably

well defined where as X9 and X18 are relatively less well defined.

Homoscedasticity: Analysis of the residual plot does not indicate any heteroscedasticity patterns.

Independence of Residuals: Assuming that the identification number represents the order in which the responses

were collected, residual were plotted with respect to the IDs to determine if any relationship is present. No

consistent pattern can be observed in the plot indicating that the residuals are independent.

Figure 1: Studentized residual plot and p-p plot for the dependent variable

Normality:The normality of error term of the variate was checked using visual inspection of the normal probability

plots of the residual (given in the chart above). As seen in the probability plot, most of the values fall along the

diagonal with no systematic departures indicating that the residuals represent a normal distribution.

Identification of Outliers: One of the ways to identify the outliers is to examine the studentized residuals. The

observations which are greater than the upper limit of 95% confidence interval (i.e. 1.96) are considered to be

outliers. As can be seen from the studentized residual plot below, ID 10, 20, 25 and 71 have significant residuals

and can be classified as outliers.


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Figure 2: Studentised residual plot used to identify outliers

5. Interpreting Regression Variate

The model obtained from step wise regression can be written as

X19 (Customer Satisfaction) = -1.88 + 0.23 * X9 (Complaint Resolution) + 0.77 * X12 (Sales force Image) + 0.45* X6

(Product Quality)0.36 * X7 (E-commerce Activities) + 0.36 * X18 (Delivery Speed)

In this model, all the coefficients including constant are significant.Constant can be interpreted as the average

customer satisfaction when all the current variables have values zero and other variables are considered which

have not been included in the current model.

Variable Importance: All variables except X7 (E-commerce activities) have positive relationship with customer

satisfaction indicating that perception that increase e-commerce activities will negatively affect customer

satisfaction. This is puzzling because bivariate correlation between X17 and X19 is positive (0.35). In order to make

direct comparison of variables, their standardized Beta coefficients were compared. It was found that Salesforce

Image (X12) was the most important variable followed by Product Quality(X6), Complaint Resolution (X9), DeliverySpeed (X9) and E-commerce activities (X7).

Multicollinearity: Tolerance values range from 0.94 to 0.26, indicating wide range of multicollinearity effects. VIF

also ranges from 1.06 to 3.91 but below the VIF cutoff of 4which would have indicated serious multicollinearity

problem. As the presence of multicollinearity would impact the interpretation of coefficients, it can be a cause of

negative sign of E-commerce activities.

6. Validating the Results

The validity of the results was determined by the two methods described below:

Examining adjusted R-Square: There is little difference between adjusted R-square (0.78) and R-square (0.80)

which indicates that overfitting hasnt taken place by addition of variables.

Split Sample Validation The sample set of 80 variables was split into two sub samples and step wise regression was

applied on both samples separately. The comparison of overall fit (given in the table below) demonstrates high

level of similarity in terms of R-square and adjusted R-square. But differences appear when individual coefficients

are compared. X9 and X12 fail to enter the model for sub sample 2 where as X18 has been dropped in the

regression model for sub sample 1. Also both the models are different to the model that was obtained for the

entire set. (Models provided in Appendix).


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Overall Model Fit

Parameters Sample 1 Sample 2

Multiple R 0.905 0.871

Coefficient of

Determination(R- Square) 0.82 0.759

Adjusted R-Square 0.799 0.739

Std Error 0.5518 0.5715

Table 3: Overall model fit for split sample validation

7. Assessing the Impact of Non-Normality of Independent Variables

As noted in section 2, some of the independent variables did not follow the normality condition. They were

transformed according to the shape characteristics they displayed. The table below provides details about the

transformations used.

Variable Transformation Required

X6 Squared Term

X7 Logarithm

X12 Not Required

X13 Cubed Term

X16 Squared Term

X17 Inverse

Table 4: Transformations used to get over non-normality

Transformation did not improve overall model fit, the adjusted R-square obtained was around 0.78 which is similar

to what was obtained in the earlier model. Constant and X9 coefficient were significant in the new model. The

coefficient for X7 was still negative in the new model. Overall transformation did not have any significant effect on

the model, so it was decided to continue with the model obtained without any transformation.

8. Evaluating Alternative Regression Models

Including all the variables: Confirmatory regression was carried out by including all the 13 perceptual measures asindependent variables. Comparison of step wise regression model and full regression model was carried out,

results of which is given below:

1. Overall Model Fit: The overall model fit decreases when all the variables are included. Even though the

multiple R and R-square increased slightly, the adjusted R-square decreased for the full model. Also,

standard error increased in the full model.

Overall Model Fit

Parameters Full Stepwise

Multiple R 0.901 0.892

Coefficient of

Determination(R- Square) 0.811 0.795

Adjusted R-Square 0.774 0.782

Std Error 0.5675 0.558


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2. Variate Interpretation: Only 3 variables (X6, X7 and X12) were found to be significant. In step wise

model, X11 was the least significant variable which was rendered insignificant in the full model due to

multicollinearity. Multicollinearity was found to be very significant for atleast 4 variables (VIF>4) which

would render the model useless. X11, X17 and X18 have VIFs greater than 50 with tolerance values less

than 0.05.

Including Dummy Variables: The model obtained through stepwise regression can be further improved by

including non metric variables. Five non metric variables are available for inclusion in the model. Stepwise

regression was carried out using the dummy variable for Firm Size. The overall model fit results are provided in the

table below and the full model has been given in the Appendix.

1. Overall model fit was better than the previous model obtained. The adjusted R-square was 0.811 which

is higher than then what was obtained in step wise regression, also the standard error was found to be

lower in the new model.

Overall Model Fit

Parameters Values

Multiple R 0.908

Coefficient of

Determination(R- Square) 0.825

Adjusted R-Square 0.811

Std Error 0.5191

Table 5: Overall Model Fit when dummy variable was included

2. All the coefficients were found to be significant. The coefficient for firm size was found to be positive

which indicates that the large firms have higher customer satisfaction. The other variables included in the

model are Complaint Resolution (X9), Salesforce Image (X12), Product Quality (X76), Delivery Speed (X18)

and Competitive Pricing (X13).

9. Managerial Overview

The regression models achieve very high levels of predictive accuracy as the amount of variance explained is about

80%.

The variables which have significant impact on customer satisfaction were Salesforce Image (X9) and Product

Quality (X6). Increases in any of these variables will increase customer satisfaction. The other three variables

(Delivery SpeedX18, Ecommerce ActivitiesX7 and Complaint Resolution-X9) also have an impact on customer

satisfaction but it is much lesser. X7 has a reversed sign which represents a result which is corollary to what is

expected.


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APPENDIX

1.Levenes TestResults

Metric

Variable

X1:Customer

Type

X2: Industry

Type X3: Firm Size X4: Region X4: Region

Levene

Statis

tic

Significa

nce

Levene

Statis

tic

Significa

nce

Levene

Statis

tic

Significa

nce

Levene

Statis

tic

Significa

nce

Levene

Statis

tic

Significa

nce

X6 - Product

Quality16.29 0.00 0.00 0.95 0.05 0.82 12.94 0.00 0.01 0.94

X7 - E-

Commerce

Activities

3.43 0.04 0.52 0.47 0.25 0.62 0.00 0.98 0.42 0.52

X8 -

Technical

Support

0.01 0.99 0.02 0.90 0.50 0.48 0.23 0.63 0.03 0.86

X9 -

Complaint

Resolution

0.12 0.88 0.53 0.47 1.57 0.21 0.06 0.81 1.27 0.26

X10 -

Advertising2.72 0.07 0.23 0.63 3.18 0.08 0.49 0.49 1.06 0.31

X11 -

Product

Line

0.04 0.96 0.70 0.41 0.01 0.94 1.46 0.23 0.99 0.32

X12 -

Salesforce

Image

6.32 0.00 0.54 0.46 0.01 0.92 0.33 0.57 0.05 0.82

X13 -

Competitive

Pricing

0.38 0.68 8.46 0.00 3.21 0.08 0.42 0.52 1.24 0.27

X14 -

Warranty &

Claims

1.11 0.34 3.61 0.06 1.80 0.18 0.12 0.73 0.00 0.96

X15 - New

Products0.71 0.50 1.40 0.24 1.33 0.25 0.10 0.76 1.26 0.26

X16 - Order

& Billing1.39 0.26 1.37 0.25 2.06 0.16 0.98 0.32 1.81 0.18

X17 - Price

Flexibility2.73 0.07 2.33 0.13 3.36 0.07 13.90 0.00 0.82 0.37

X18 -

Delivery

Speed

1.09 0.34 0.00 0.95 2.06 0.16 1.06 0.31 3.88 0.05


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2.K-S and Shapiro-Wilk Test Results

Tests of Normality

Kolmogorov-

Smirnova

Shapiro-Wilk

Statistic Sig. Statistic Sig.

X6 - Product Quality 0.1 0.02 0.9 0.00X7 - E-Commerce Activities 0.1 0.01 1.0 0.02

X8 - Technical Support 0.1 0.20 1.0 0.23

X9 - Complaint Resolution 0.1 0.20 1.0 0.30

X10 - Advertising 0.1 0.20 1.0 0.12

X11 - Product Line 0.1 0.20 1.0 0.23

X12 - Salesforce Image 0.1 0.02 1.0 0.32

X13 - Competitive Pricing 0.1 0.00 0.9 0.00

X14 - Warranty & Claims 0.1 0.20 1.0 0.65

X15 - New Products 0.1 0.20 1.0 0.48

X16 - Order & Billing 0.1 0.02 1.0 0.26

X17 - Price Flexibility 0.1 0.05 1.0 0.04

X18 - Delivery Speed 0.1 0.07 1.0 0.35

X19 - Satisfaction 0.1 0.20 1.0 0.11

a. Lilliefors Significance Correction

*. This is a lower bound of the true significance.

3. Collinearity

VariablesX19 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X1

X19 - Satisfaction 1.00

X6 - Product Quality 0.51 1.00

X7 - E-Commerce

Activities0.35

-

0.101.00

X8 - Technical Support

-

0.020.07 0.00 1.00

X9 - Complaint Resolution0.54 0.12 0.15

-

0.041.00

X10 - Advertising0.30

-

0.070.46

-

0.080.12 1.00

X11 - Product Line0.47 0.46

-

0.040.14 0.55

-

0.061.00

X12 - Salesforce Image0.54

-

0.120.79

-

0.070.16 0.54

-

0.081.00

X13 - Competitive Pricing-

0.12-

0.410.24 -

0.24-

0.020.16 -

0.460.33 1.00

X14 - Warranty & Claims0.08 0.09 0.04 0.77 0.04

-

0.010.26 0.03

-

0.261.00

X15 - New Products0.07 0.07 0.02

-

0.100.06 0.05 0.05 0.03 0.04 0.04 1.00

X16 - Order & Billing0.44 0.10 0.11

-

0.050.71 0.10 0.34 0.11

-

0.020.13 0.10 1.00

X17 - Price Flexibility 0.08 - 0.25 - 0.39 0.29 - 0.31 0.54 - 0.10 0.43 1.0


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0.47 0.23 0.39 0.19

X18 - Delivery Speed0.52 0.01 0.18

-

0.080.86 0.20 0.58 0.21 0.03 0.07 0.12 0.70 0.5

4. Stepwise Regression Results

Coefficients


Coefficients

t Sig. Correlations Collinearity

Statistics

B Std.

Error

Beta Zero-

order

Partial Part Tolerance V

1 (Constant) 3.80 0.57 6.72 0.00

1.00 1.X9 - Complaint Resolution 0.56 0.10 0.54 5.66 0.00 0.54 0.54 0.54

2 (Constant) 1.56 0.62 2.51 0.01



3 (Constant)

-

1.860.56

-

3.310.00




4 (Constant)

-

1.480.57

-

2.610.01



X6 - Product Quality 0.44 0.05 0.52 9.56 0.00 0.51 0.74 0.51 0.97 1X7 - E-Commerce

Activities

-

0.370.15 -0.21

-

2.430.02 0.35 -0.27

-

0.130.37 2

5 (Constant)

-

1.880.59

-

3.200.00




X7 - E-Commerce

Activities

-

0.360.15 -0.21

-

2.420.02 0.35 -0.27

-

0.130.37 2

X18 - Delivery Speed 0.36 0.18 0.21 2.05 0.04 0.52 0.23 0.11 0.26 3

a. Dependent Variable: X19 - Satisfaction


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5. Partial Regression Plots


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6. Checking Heteroscedasticity

7. Validation: Split Sample Check

Variables Entered

Sample 1 Sample 2

Regression Coefficients

Statistical

Significance Regression Coefficients

Statisti

Significa

BStd.

ErrorBeta t Sig B

Std.

ErrorBeta t

(Constant)-1.91 0.75

-

2.560.02 -1.90 0.88

-

2.170



X9 - Complaint Resolution 0.39 0.08 0.37 4.95 0.00

X7 - E-Commerce Activities-0.54 0.22

-

0.32

-

2.480.02

X18 - Delivery Speed 0.67 0.13 0.43 5.03 0

8. Stepwise Regression Results for Full Model

Coefficients

Model

Unstandardized Coefficients Std.

t Sig.

CorrelationsCollinearity

StatisticsStd.

ErrorBeta

Zero-

orderPartial Part Tolerance V

1 (Constant)-

1.581.27 -1.24 0.22

X6 - Product Quality 0.41 0.06 0.49 7.10 0.00 0.51 0.66 0.38 0.61 1.

X7 - E-Commerce

Activities

-

0.400.15 -0.23 -2.60 0.01 0.35 -0.31 -0.14 0.36 2.


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X8 - Technical Support 0.06 0.07 0.07 0.82 0.42 -0.02 0.10 0.04 0.35 2.

X9 - Complaint Resolution 0.16 0.12 0.15 1.31 0.19 0.54 0.16 0.07 0.22 4.

X10 - Advertising-

0.010.07 -0.01 -0.11 0.91 0.30 -0.01 -0.01 0.67 1.

X11 - Product Line 0.33 0.36 0.35 0.91 0.37 0.47 0.11 0.05 0.02 52

X12 - Salesforce Image 0.85 0.11 0.76 7.66 0.00 0.54 0.69 0.41 0.29 3.

X13 - Competitive Pricing-

0.080.06 -0.11 -1.53 0.13 -0.12 -0.18 -0.08 0.53 1.

X14 - Warranty & Claims-

0.160.14 -0.11 -1.13 0.26 0.08 -0.14 -0.06 0.33 3.

X15 - New Products-

0.010.04 -0.01 -0.17 0.86 0.07 -0.02 -0.01 0.91 1.

X16 - Order & Billing 0.13 0.12 0.09 1.09 0.28 0.44 0.13 0.06 0.40 2.

X17 - Price Flexibility 0.29 0.37 0.30 0.79 0.43 0.08 0.10 0.04 0.02 49

X18 - Delivery Speed-

0.240.71 -0.14 -0.35 0.73 0.52 -0.04 -0.02 0.02 60

a. Dependent Variable: X19 - Satisfaction

9. Stepwise Regression Results when Dummy Variable Included

Coefficients


Coefficients

t Sig. Correlations Collinearit

Statistics

Variables B Std.

Error

Beta Zero-

order

Partial Part Tolerance

6 (Constant)

-

1.320.64 -2.06 0.04

X9 - Complaint

Resolution0.22 0.10 0.21 2.21 0.03 0.54 0.25 0.11 0.26

X12 - Salesforce Image 0.55 0.06 0.49 9.03 0.00 0.54 0.73 0.44 0.80

X6 - Product Quality 0.43 0.05 0.51 9.35 0.00 0.51 0.74 0.46 0.80

X3 - Firm Size 0.54 0.13 0.23 4.03 0.00 0.24 0.43 0.20 0.76

X13 - Competitive

Pricing

-

0.130.04 -0.17 -2.91 0.00 -0.12 -0.32

-

0.140.68

X18 - Delivery Speed 0.37 0.16 0.22 2.23 0.03 0.52 0.25 0.11 0.26

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