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 oneunit 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 (Rsquare) 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
.810
Adjusted
R-square R-square
.655
.645
Std. Error of the
Estimate
.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
Square
s
Regression 225.86
Residual
118.309
Total
343.583
df
6
190
196
Mean F
Sig.
Squa
re
37.53 60.228 .000
.62
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
(1)
Std. Error
(2)
Constant
.395
.321
Self
Image
.127
.044
Trust
.095
Standard
Coefficients
Beta
t
(3)
(4)
Sig.
(5)
1.230
.220
.152
2.861**
.005
.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.