**
INTRODUCTION**

**Regression analysis** allows the researcher to estimate the relative importance
of independent variables in influencing a dependent variable. It also identifies
a mathematical equation that describes the relationship between the independent
and dependent variables. According to Sandy^{1},
simple linear regression is a technique that is used to describe the effects
of changes in an independent variable on a dependent variable. However, the
Multiple Regression is a technique that predicts the effect on the average level
of the dependent variable of a unit change in any independent variable while
the other independent variables are held constant. According to Gujarati^{2},
a model with two or more independent variables is known as a multiple regression
model. The term “multiple” indicates that there are multiple independent
variables involved in the analysis and multiple influences will affect the dependent
variable. According to Pasha^{3} using too
few independent variables may give a biased prediction, while using too many
independent variables, can cause a varied fluctuation prediction value.

Most cost and production functions are curves with changing slopes and not
a straight line with constant slopes^{4}. The
polynomial model enables the estimation of these curves^{5,6}.
In a polynomial model, independent variables can be squared, cubed or raised
to any power or exponential power. The studies of^{7}
is an example of a squared polynomial or quadratic model, while^{8}
fitted a third order polynomial model of the best R^{2} correlation.
The coefficients still appear in the regression in a linear fashion, allowing
the regression to be estimated as usual. As in^{2},
for example, the polynomial model is given as:

Since the same independent variables appear more than once in different forms,
it will appear to be highly correlated. The effect of multicollinearity may
not exist because the independent variables are not linearly correlated.

The main objective of this study is to obtain the best model with a polynomial
order which would represent the whole structure of the collected data so that
further analysis can be carried out. There is no unique statistical procedure
for doing this and personal judgment will be a necessary part of any of the
statistical methods discussed.

**
METHODOLOGY
**

A Hierarchical Multiple Regression Model relates a dependent variable Y,
to several independent variables W_{1}, W_{2}, …, W_{k}.
The model is in the following form:

Basic assumptions of Multiple Regression Models are made about the error terms,
u_{i} (for i=1, 2, …, n) and the values of independent variables W_{1},
W_{2}, …,
W_{k} are shown in Table 1 with Ω_{0}
denotes the constant term of the model, Ω_{j}
denotes the j-th coefficient of independent variable W_{j} (for j =
1, 2, …,
k) and u denotes the random residual of the model. The k denotes the number
of independent variables (k+1) denotes the total number of parameters.

The general Multiple Regression Model with k independent variables is defined
in Eq. 1 where Y denotes the dependent variable, W_{j }denotes
the j-th independent variable (which can be single independent quantitative
variable, or interaction variable (first-order interaction, second-order interaction,
third-order interaction, …), or generated variable (dummy or/and polynomial
or/and categorical variables) or transformed variable (Ladder transformation
and Box-Cox transformation). The Ω_{0}
denotes the constant term of the model, Ω_{j}
denotes the j-th coefficient of independent variable W_{j} (for j =
1, 2, …,
k) and u denotes the random residual of the model. The k denotes the number
of independent variables, (k+1) denotes the total number of parameters.

The Multiple Regression Model as defined in Eq. 1 is a hierarchically
well-formulated models^{9} and can be written
as a system of n equations, as shown by the following:

This system of n equations can be expressed in matrix terms as the following:

Where:

The Y, Ω
and u are in a form of vectors. The W is in a form of matrix. For example, the
regression model for two single independent variables (X_{1} and X_{2})
with possible interaction variable (X_{12 }is the product of independent
variables X_{1} and X_{2}) can be expressed as follows:

This Eq. 4 can be written in general form as in Eq.
1:

Where:

Ω_{0} |
= |
β_{0} constant |

Ω_{1} |
= |
β_{1} and W_{1}= X_{1} single independent |

Ω_{2} |
= |
β_{2} and W_{2}= X_{2} single independent |

Ω_{3} |
= |
β_{12} and W_{3}= X_{12}= X_{1}X_{2}
(First-order interaction variable) |

Ω_{4} |
= |
β_{11} and W_{4}= X_{1}^{2}= X_{1}X_{1}
(Quadratic variable) |

Ω_{5} |
= |
β_{22} and W_{5}= X_{2}^{2}= X_{2}X_{2}
(Quadratic variable) |

Ω_{6} |
= |
β_{111} and W_{6}= X_{1}^{3}= X_{1}X_{1}X_{1}
(Cubic variable) |

Ω_{7} |
= |
β_{222} and W_{7}= X_{2}^{3}= X_{2}X_{2}X_{2}
(Cubic variable) |

Ω_{8} |
= |
β_{122} and W_{8}= X_{1}X_{2}^{2}=
X_{1}X_{2}X_{2} (interaction variable) |

where, k = 8 and (k+1) = 9.

The variable X_{1}X_{2} denotes the product of variables X_{1}
and X_{2}. According to Zainodin and Khuneswari^{10},
the variable X_{1}X_{2} can also be written as X_{12}
and defined it as the first-order interaction variable. In general, the cross
product between q single independent quantitative variables, the interaction
variable is (q-1)-order interaction variable.

In the development of the **mathematical model**, there are four phases involved.
These phases are possible models, selected models, best model and goodness-of-fit
test^{11}. In the beginning, all the possible
models are listed. Once this has been done, the next step is to estimate the
coefficients for the entire possible model and then carry out the tests to get
selected models. All the possible models must be run one by one to obtain selected
models. In the process of getting the selected models from possible models,
multicollinearity test and elimination procedure (elimination of insignificant
variables) should be carried out in order to obtain a selected model. These
selected models should be free from the multicollinearity sources and insignificant
variables.

Multicollinearity arises when two independent variables are closely linearly
related. This is equivalent to saying that a coefficient of determination greater
than 0.95 represents a strong linear relation. An absolute correlation coefficient
greater than 0.95 (i.e., |r|>0.95) defines a strong multicollinearity. The
algorithm for the multicollinearity test procedures have also been described
by Zainodin *et al.*^{11}. The Global
test is then carried out as shown by Zainodin and Khuneswari^{10}.
If there are no more multicollinearity source variables, then the next step
is to carry out the elimination procedures on the insignificant variables. This
is done by performing the coefficient test for all the coefficients in the model.
According to Abdullah *et al.*^{12} the
coefficient test is carried out for each coefficient by testing the coefficient
of the corresponding variable with the value of zero. The insignificant variables
will be eliminated one at a time. Subsequent removals are carried out on all
the models until all the insignificant variables are eliminated. The Wald test
is then carried out on all the resultant selected models so as to justify the
removal of the insignificant variables^{10}.

A best model will be selected from a set of selected models that have been
obtained from previous phases in the model-building procedures. The best model
will be selected with the help of the eight selection criteria (8SC)^{13,11}.
The best model is chosen when a particular model has the most of the least criteria
value (preferably all the 8 criteria chose the same model).

Finally, the residual analysis should be carried out on the best model to verify
whether the residuals are randomly and normally distributed. Bin Mohd *et
al.*^{14 }stated that randomness test should
be carried out to investigate the randomness of the residuals produced. One
of the MR assumptions is that the residuals should follow a normal distribution.
Besides that, Shapiro-Wilk test is used (for n<50) to check the normality
assumption of the residuals. Scatter plot, histogram and box-plot of the residuals
are to a get a clear picture of distribution of the residual. These plots are
used as supporting evidence in addition to the two quantitative tests.

**
RESULTS
**

This section will describe the process in getting the best polynomial regression
model by including a numerical illustration. In this study, the data was collected
from 30 sale regions of detergent ‘FRESH’ during 1989, as published
in Bowerman and O’Connell^{15}.

In this study, the variables used are the demand (Y) for the detergent bottles
to the factors affecting, such as the "price difference" (X_{1}) between
the price offered by the enterprise and the average industry prices of competitors’
similar detergents and advertising expenditure (X_{2}). The aim is to
analyse what is the contribution of a specific attribute in determining the
demand. Table 2 shows the descriptive statistics for each
variable used. As stated by Crawley^{16}, the
distribution of a variable is said to be normal if the value of skewness fall
within [-0.4472, 0.4472] and kurtosis fall within [-0.8944, 0.8944].

There are two single quantitative independent variables involved in this data
in order to determine the demand of detergent bottles. Figure
1 shows the scatter plots of all the variables involved and their possible
trends.

Based on the plots in Fig. 1, it could be seen significantly
that Y and X_{2} have a polynomial shaped relationship. The fitted curves
show a quadratic line relationship. The linear line between Y and X_{1}
shows a linear relationship and it is supported with the Table
3 that shows Y and X_{1} is highly correlated. This shows that there
exists possible non linear variable that contributes to Y.

All the quadratic and cubic terms introduced in the models are thus shown
in Table 3.

Table 3 shows all the possible models with interaction and
polynomial variables. There are four all possible models (M1-M4) for two single
independent variable (X_{1} and X_{2}). The polynomial and interaction
with polynomial variables are added in model M1 to model M4. Therefore, there
are 12 possible models which include quadratic and cubic variables.

Table 4 shows the Pearson correlations matrix for example,
the possible model M8.0. It can be seen that there exists multicollinearity
among 9 pairs of independent variables (shaded values). The multicollinearity
source variables should firstly be removed according to the Zainodin-Noraini
Multicollinearity Remedial Techniques^{11}.

Since there is a tie, the variable X_{1}^{3} (correlation with
Y is 0.732) is removed because it has the smallest correlation coefficient with
dependent variable (this is a case B of the Zainodin-Noraini multicollinearity
remedial technique). After the removal of variable X_{1}^{3},
model then becomes model M8.1. The correlation coefficient is recalculated as
shown in Table 5. Here, it could be seen that there exists
multicollinearity among the 7 pairs of independent variables (shaded values).
There exist a tie and the variable X_{2} (correlation with Y is 0.876)
is then removed because it has the smallest correlation coefficient with the
dependent variable (this is again a case B).

After removing variable X_{2}, the new model is model M8.2. The correlation
coefficient is recalculated as shown in Table 6.

Here, it could be seen that there exists multicollinearity among the 5 pairs
of independent variables. The most commonly appearing variables are X_{1},
X_{12} and X_{1}X_{2}^{2}. Since there is a
tie, the variable X_{1}X_{2}^{2} (correlation with Y
is 0.890) is removed because it has the smallest correlation coefficient with
dependent variable (this is again a case B). The new model after removing variable
X_{1}X_{2}^{2} is model M8.3. Then the correlation coefficient
was recalculated as shown in Table 7. From Table
7, it can be seen that there exists multicollinearity among the 3 pairs
of independent variables.

There are no common variables among the multicollinearity source variables.
Therefore, the variable with the smallest correlation coefficient with dependent
variable in each pair should be removed.

The three variables X_{1}, X_{1}^{2} and X_{2}^{2}
are removed from the model M8.3 (this is a case C where frequency of all the
variables is one).

The new model after removing the variables X_{1}, X_{1}^{2}
and X_{2}^{2 }is then model M8.6. The correlation coefficient
is thus recalculated as shown in Table 8. Based on the highlighted
triangle in Table 8, it could be seen that there is no more
multicollinearity left in the model.

The next step is to carry out the coefficient test of the elimination procedures
on the insignificant variables and the Wald test on model M8.6.0. The Wald test
is carried out to justify the elimination process. There is only one variable
that is omitted from the model M8.6 as shown in Table 9.

Table 10 shows the Wald test for model M8.6.1. From Table
10, F_{cal } is 0.7103 and F_{table }= F(1, 26, 0.05) =
4.23 from the F-distribution table.

Since F_{cal }is less than F_{table}, H_{0 }is accepted.
The removal of insignificant variables in the coefficient test is therefore
justified. The selected model M8.6.1 is:

Similar procedures and tests (Global test, Coefficient test and Wald test)
are carried out on the remaining selected models. There are 8 selected models
obtained in this phase. For each selected model, the eight selection criterion
(8SC) values are obtained and the corresponding values are shown in Table
11.

It can be seen from Table 11 that all the criteria (8SC)
indicate that model M11.3.1 as the best model. Thus, the best model is M11.3.1
is given by:

Referring to Table 4, the Pearson correlation shows the correlation
where the "price difference" (X_{1}) (i.e., r =0.890) between the price
offered by the enterprise and the average industry price of competitors’ similar
detergents, are highly correlated with the demand for detergent bottles (Y).
The components of model M11.3.1 further indicate the presence of polynomial-order
(cubic) of the significant independent variables. The variable X_{2}^{3}
(i.e., r = 0.8949) are highly correlated with the demand for detergent bottles
(Y). The interpretations of the coefficients for best model M11.3.1 are depicted
in Table 12.

Based on the best model, the residuals analyses are obtained. Several tests
on the model’s goodness-of-fit are carried out. It is found that all the
basic assumptions are satisfied and the residuals plots are shown in Fig.
2-4. Using the residuals obtained, randomness test and
the normality test are carried out. Both randomness test and residuals scatter
plot, as shown in Fig. 2 and 3, indicate
that the residuals are random, independent and normal.

The total sum of residual of the best model, M11.3.1 is 0.0370 while the sum
of square error is 1.4110. The randomness test carried out on residuals shows
that resulting error term of best model M11.3.1 is random and independent. This
strengthens the belief which is reflected in the residuals plot of Fig.
3 which confirms that no obvious pattern exists. This shows that the best
model M11.3.1 is an appropriate model in determining the demand of detergent.

Besides that by taking ±3
standard deviation (i.e., 99.73%) for Upper Control Limit (UCL) and Lower Control
Limit (LCL) as in Fig. 2, the residuals are distributed between
the ±3
standard deviation lines which indicate that there are no outliers. The Shapiro-Wilk
statistics of the normality plot in Fig. 3 shows that the
residuals of model M11.3.1 are distributed normally (i.e., statistics =0.9865,
df =30 and p-value =0.9603). Figure 4 with its median positioned
at the centre of the box further shows that model M11.3.1 is a well represented
model to describe the demand of detergents. Thus, the model is ready to be used
for further analysis. Now the demand model is ready for use in forecasting or
estimating to make a logical decision in determining the appropriate demand
of detergent.

**
DISCUSSIONS
**

Other goodness-of-fit tests can be used such as the root-mean square error
(RMSE), Modelling Efficiency (EF) and many others^{17,18}.
However, in this research, the residuals analysis is based on the randomness
and normality tests. The numerical illustration of this study shows that polynomial
regression model M11.3.1 is the best model to describe the demand of detergent.
The "price difference" (=X_{1}) between the price offered by the enterprise
and the average industry price of competitors’ with similar detergents
is the main factor in determining the demand of detergent and the “advertising
expenditure” (=X_{2}) as the cubic factor. The best model M11.3.1
(i.e., )
in Table 12 shows that constant demand without any contributions
of other factors is approximately 7 bottles (since the constant value is 6.6398).
Every one unit increase in “price difference” will directly increase
the number of detergent by 1.4673 bottles and every one unit increase in advertising
expenditure will positively increase the number of detergent by 0.0052 bottles,
respectively. This means that if “price difference” and expenditure
are increased by one unit, then the number of detergent demanded will be 8 bottles
(approximate value).

Similar to Chu^{19}, the cubic polynomial approach
is used to forecast the demand of detergent. ^{20}had
also attempted the use of price in the consumer purchase decision using price
awareness with three distinct factors, namely, price knowledge, price search
in store and between stores and the Dickson-Sawyer method^{21}.
The effect of advertising tools on the behaviour of consumers of detergents
had also been surveyed and compared by Salavati *et al*.^{22}
in **developing countries**. Moreover, advance researches on promoting hygiene,
such as hand hygiene, with criteria like fast and timely, easy and skin protection,
are part of the campaigns involved in advertising expenditure^{23}.

**
CONCLUSIONS
**

The consumer buying behaviour will affect the demand for detergents. It
is imperative in today’s diverse global market that a firm can identify
consumer behavioural attributes and needs, lifestyles and purchase processes
and the influencing factors which are responsible for the consumer decisions.
Hence, devising good marketing plans is necessary for a firm. While serving
its targeted markets, it minimises dissatisfaction and still stay ahead of other
competitors. It was seen from the best model that there is conclusive increase
in production due to price difference and expenditure. In other words, mathematically,
a detergent manufacturing company has to project its marketing and production
plans to incorporate the impacts of pricing and expenditure on the consumers.
Indirectly, the firm succeeded in meeting the demands of the consumers in a
win-win relationship.