Regression Analysis

Regression analysis

Definition: • Regression analysis is a statistical method

for fitting an equation to a data set. • It is used to estimate demand, production

and cost equations. • A regression equation can be used for

forecasting purposes, i.e., – it predicts the value of the dependent variable

for given values of the independent variable.

Regression analysis contd.

• Estimation of the parameters of an equation with more than one independent variable is called multiple regressions.

• Estimated coefficients in a regression equation measure the change in the value of the dependent variable for each one-unit change in the independent variable, holding the other independent variables constant.

Regression analysis contd.

• The standard error of the estimate is the measure of the error in prediction.

• The least-squares regression technique can be used to quantify the relationship between a dependent variable and one or more independent variables.

Coefficient of Determination

R-square

• The coefficient of determination (R-square) is used to test the explanatory power of the entire regression equation.

• This statistic measures the proportion of the total variation in the dependent variable that is explained by variations in the independent variables.

t-test

• t-statistic

• Hypotheses regarding the coefficients of individual independent variables are tested using t-statistic;

• t-statistic is computed by dividing the estimated coefficient by its standard error;

t-test contd.

• If the absolute value of this ratio is greater than the value taken from a table of the student’s distribution, the coefficient is said to be statistically significant.

t-test contd.

• If relevant variables are excluded from a regression equation, the equation is said to be mis-specified.

• The use of mis-specified equation may lead to incorrect conclusions about the relationships between the dependent and independent variables.

Multiple Regression Analysis: Example

• The first step in using multiple regression analysis is to develop a conceptual model based on economic theory.

• This model can specify the data that should be collected and can also be used to interpret results.

MRA Example• Example: the following conceptual model is

based on cross-sectional data. The specification of the model is given as:

Y = a + b1X1 + b2X2 + b3X3 + b4X4 + b5X5 + b6X6 (4.1)

Where,– Y and X1-6 represent dependent and independent

variables respectively; – b1-6 are the unknown parameters that are to be

estimated.

MRA Example contd.

• Variables are defined below:• Y = Consumer Loyalty (CL)• X1 = Self Image of the consumer (SI)• X2 = Trust towards main service provider (Tr)• X3 = Switching costs (SC)• X4 = Customer service quality provided by

service provider (CSQ)• X5 = Overall satisfaction from main cell service

provider (OS)• X6 = Know how about mobile phones (KH)

MRA Example contd.

• The empirical model will take the following form:

Y (CL) = a + b1SI + b2Tr + b3SC + b4CSQ + b5OS + b6KH (4.2)

Hypotheses Testing

• The following six alternative hypotheses (H1-6) are to be tested against six null (Ho) hypotheses:

• H1: Self image of the consumer has a positive effect on customer’s loyalty.

H1: b1 > 0

Ho: b1 = 0

Hypotheses Development contd.

• H2: Trust towards main service provider has a positive effect on customer’s loyalty.

H2: b2 > 0

Ho: b2 = 0

• H3: Switching costs has a positive effect on customer’s loyalty.

H1: b3 > 0

Ho: b3 = 0

Hypotheses Development contd.

• H4: Customer service quality provided by service provider has a positive effect on customer’s loyalty.

H1: b4 > 0

Ho: b4 = 0

Hypotheses Development contd.

• H5: Overall satisfaction from main cell service provider has a positive effect on customer’s loyalty.

H1: b5 > 0

Ho: b5 = 0

Hypotheses Development contd.

• H6: Knowledge about mobile phones has a positive effect on customer’s loyalty.

H1: b6 > 0

Ho: b6 = 0

Regression Results Discussion

• In this analysis, a long run relationship has been explored between – customer loyalty of cell phone service provider

(dependent variable) – with its construct, i.e.

• customer satisfaction, • switching cost, • self image of the mobile phone user, • customer trust on services provider, • over all service quality offered by cell services provider and • know how of customers (independent variables)

Regression Results Discussion contd.

• Since there are many constraints prior to run a multiple regression, the first thing that can be examined prior to run a regression is to check the fitness of the model , i.e.,

• how much of the variations in the dependent variable is explained by the group of independent variables.

Regression Results Discussion contd.

• In Ordinary Least Square (OLS) method, R-square is considered as a model fit.

• In the present analysis it is 65.5% (Table 1) which means, 65.5% variations in the dependent variable are explained by the set of independent variables, which is good enough for a cross sectional data.

Table 1: Model Summary

R R-square

Adjusted

R-square

Std. Error of the

Estimate

.810 .655 .645 .7894

Results discussions contd.

• In Bivariate regression, t and F tests produce the same results since t-square is equal to F.

• In multiple regressions, the F test has an overall role for the model, and each of the independent variables is evaluated with a separate t-test (Kothari 1990; Cooper and Schindler 2003).

Table 2: ANOVA

Sum of Squares

df Mean Square

F Sig.

Regression

Residual

Total

225.86

118.309

343.583

6

190

196

37.53

.62

60.228 .000

Results discussions contd.

• The analysis of variance and the ‘F’ statistic in Table 2 suggest that the model is fit and it is valid with the existing set of independent variables.

Results discussions contd.

• The coefficient estimates of the independent variables are reported in Table 3.

• The coefficient estimates, calculated through OLS method, show that all of the independent variables, except Know how, have positive relationship with the dependent variable, loyalty.

Table 3: Coefficients Unstandard

Coefficients

B Std. Error

(1) (2)

Standard

Coefficients

Beta t Sig.

(3) (4) (5)

Constant .395 .321 1.230 .220

Self Image

.127 .044 .152 2.861** .005

Trust .095 .064 .098 1.476 .141

Swit. Cost .099 .064 .076 1.558 .121

Service Quality

.129 .079 .123 1.625* .106

Overall Satisfact.

.520 .075 .522 6.940*** .000

Knowhow -.047 .040 -.054 -1.175 .241

Results discussions contd.

• Table 3 reports the coefficient estimates (b1—b6) of the construct in column 2, and standard error of the estimates in column 3. Column 4 reports the t- values for each coefficient estimate and column 5 reports the significance of the estimates at the specified levels of significance.

Results discussions contd.

• Self image of the cell phone user and over all satisfaction from current service provider are the statistically significant variables.

• Customer service quality provided by service provider is significant but at a little higher than 10 percent level of significance.

Results discussions contd.

• The t-values of switching cost to new service provider, know how of cell services, and trust towards main cell services provider are not significant contributors towards customer loyalty.

Results discussions contd.

• It is only the self image of cell phone user and the overall satisfaction from current cell service provider which satisfy the customers; and – it is obvious that only the satisfied customers

remain loyal to their service providers.

Results discussions contd.

• The six alternative hypotheses, developed earlier, have been tested against six null hypotheses, using the t-values.

• The results of the test show that the coefficient estimates of all the independent variables, except that of the Know how about cell services, are positive conveying the message that these five independent variables have positive effect on customer loyalty.

Results discussions contd.

• The coefficient estimate of know how is negative (-.047).

• The important point to note here is that mere having positive or negative signs for coefficient estimates are neither necessary nor sufficient condition for making a coefficient significant.

Results discussions contd.

• It is the t-values of the estimates, at certain critical level of significance, that make the corresponding variable an important contributor. Looking at table 3 makes this point clear.

• The only two important contributors are self image of cell users with t-value (2.86), and overall satisfaction from current cell service provider with t-value (6.94), both at a very high level of significance.

Results discussions contd.

• The case of customer service quality comes next in significance with t-value) 1.625) at low level of significance (a little more than 10%).

Results discussions contd.

• The coefficient estimates of trust towards main service provider, switching costs, and know how (.095, .099, and -.047 respectively) are not statistically different from zero as shown by their t-values ( .1.478, 1.558, and -1.175 respectively), hence the null hypotheses (Ho: b2 = 0; Ho:b3 = 0, and Ho:b3) can not be rejected, implying that they have no effect on customer loyalty.

Conclusions

• Self image of the cell phone users and over all satisfaction from current service provider are the two significant contributors to customer loyalty;

Conclusions contd.

• The insignificance of the three independent variables (trust, switching costs, and know how) may have been the result of the exclusion of some important variables - purchase intent, repeat purchase behaviour, perceived value – from the conceptual model.