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April 2008 - Volume 2, Issue
2
MARRIAGE MIGRATION ASSOCIATED
WITH DISTANCE IN BANGLADESH: AN APPLICATION OF POLYNOMIAL
MODEL
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Md.
Rafiqul Islam
Correspondence:
Dr. Rafiqul Islam, Associate Professor, Dept. of Population
Science and Human Resource Development, Rajshahi University,
Bangladesh.
E-mail: rafique_pops@yahoo.com
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| ABSTRACT
In this study an effort has
been made to fit a mathematical model to marriage migration
associated with distance in Comilla district in Bangladesh.
For this, data have been taken from Yadava, Soni and
Sabina (2002) but the data is also available in Hosain
(2000). It is to be noted that Hossain (2000) applied
a Pareto-Exponential model (Morril and Pitts, 1967).
Yadava, Soni and Sabina (2002) also applied exponential
distribution to the same data and they showed that exponential
distribution provided good approximation. In this study
an attempt has been made to show that the polynomial
model is also applicable to the same data set. It is
found that marriage migration associated with distance
follows a polynomial model. To verify the stability
of the model, cross validity prediction power is employed
in the model.
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Key words and phrases: Marriage
migration Mathematical modeling Polynomial Variance explained
(R2) Cross validity prediction power (CVPP)
F-test.
INTRODUCTION
It is to be mentioned here that mathematical
modeling in Population Studies especially in Demography (Fertility,
Mortality, Migration) in Bangladesh has been worked on a very
limited scale. In the era of globalization, mathematical models
are very realistic and sophisticated mechanisms to express
data in mathematical terms. Mathematical models are of great
use to demographers in realizing the process in differentiating
among various variables to find out the functional relationships
and their dynamic behaviors among various demographic phenomena.
Finally, a model is important for prediction purposes. Mathematical
models in demography are mainly of two groups: stochastic
and deterministic.
A deterministic model has only been
discussed in the present study. Deterministic models are used
to describe the functional relationship between variables
that take definite values. Traditionally, one can draw graphs
of the demographic parameters but very few of us know in the
context of Bangladesh, which models are more appropriate for
the parameters.
Islam and Ali (2004) found that age
specific fertility rates (ASFRs) follow a slightly modified
biquadratic polynomial model whereas forward and backward
cumulative ASFRs follow quadratic and cubic polynomial models,
respectively in the rural community of Bangladesh. To observe
the distribution or pattern of marriage migration associated
with distance in Bangladesh, India and other countries of
the world a number of models have been fitted to the data
set (Libbee and Sopher, 1975; Morril and Pitts, 1967; Perry,
1969a and 1969b; Samuel, 1994; Sharma, 1984; Yadava et. al.
1988). Hossain gave attention to building up the model of
Sharma (1984) and Yadava et. al (1988). But these models did
not provide a good fit and then Hossain used the Pareto-Exponential
model proposed by Morril and Pitts (1967) to present the marriage
migration related to distance for his data of Bangladesh.
Although the Pareto-Exponential model supplied better approximation
than the models of Sharma (1984) and Yadava et. al. (1988)
it did not significantly fit to the utilised data set. It
is to be noted that the proposed models of Sharma (1984) and
Yadava et. al (1988) are suitable for the Hindu community
in India.
For this, Yadava et. al. (2002) tried
to show that exponential distribution provides a better fit
to the distribution of marriage migration associated with
distance than the Pareto-Exponential function as applied by
Hossain (2000). Also Yadava et. al (2002) compared their model
with the Pareto-exponential function applied by Yadava et.
al (1998).
In this study an effort has been
made to build a mathematical model to total marriage migration
associated with distance, that is, the same data aggregate
which was already used by Yadava et. al. (2002). For this
purpose, a polynomial model was chosen to be applied here.
A brief discussion about the polynomial model is given below:
A general expression of the form:
 (an  0) (Waerden, 1948)
where ao is
the constant term; ai is the coefficient
of xi (i =1, 2, 3, ..., n) but a1,
a2,..., an are also constants but these
belong to a field (a field means a non-empty set in which
group for addition, group for multiplication and left as well
as right distributive law hold) and n is the positive integer,
is
called a polynomial of degree n and the symbol x is called
an indeterminate.
An
effort has been made here to find out what types of models
are more appropriate to total marriage migration by distance
in Comilla of Bangladesh. Thus, the fundamental objectives
of this study are briefly mentioned below:
i) to
build up mathematical models to total marriage migration by
distance and
ii) to apply cross-validity
prediction power (CVPP),  , to the model to verify
how much the model is valid or not.
METHODOLOGY
Sources of Data
The data on total marriage
migration associated with distance in Comilla district in Bangladesh
have been taken from Yadava et. al. (2002). This data was also
available in Hossain (2000) and shown in Table 1.
Mathematical
Model Fitting
Using the scattered plot of marriage
migration associated with distance in Bangladesh (Figure. 1),
it is observed that marriage migration can be fitted by polynomial
model with respect to distance. Therefore, an nth degree polynomial
model is considered and the form of the model is 
(Gupta and Kapoor, 1997)
where, x is distance; y is marriage
migration; a0 is the constant; ai is
the coefficient of xi
(i = 1, 2, 3, ..., n) and u is the stochastic error term of
the model. Here a suitable n has been selected for which the
error sum of square is minimum.
The software STATISTICA was used
to fit the mathematical model.
Checking Model Validation
To check how much the model is
stable, the cross validity prediction power (CVPP), , is applied. Here,
 ; where, n
is the number of cases, k = the number of regressors in the
model and the cross-validated R is the correlation between observed
and predicted values of the dependent variable (Stevens, 1996).
The shrinkage of the model is the absolute value of the
difference of and R2. The
stability of R2 of this model is equal to 1- shrinkage.
F-test
To verify the measure of the overall
significance of the fitted model as well as the significance
of R2, the F-test is employed to this model. The
formula for F-test in mathematics is as follows:
where k is the number of parameters
to be estimated, n is the number of cases and R2 is
the coefficient of determination in the model (Gujarati, 1998).
Application of the Model and Results
The polynomial model is assumed
for marriage migration due to distance in Comilla in Bangladesh
and the fitted equation is
y = 1025.557-169.5126x+9.613215x2-0.182286x3
t-stats- (105.562) (-47.561) (27.80423) (-19.2117)
p-value- (0.000) (0.000) (0.00001) (0.0000)
providing R2=0.999714324 and =0.998875.
This is the polynomial of degree three i.e. cubic polynomial.
From these statistics we see that
the fitted model is highly cross-validated and its shrinkage
is 0.000839. These imply that the fitted model is 99.8875%
stable. Moreover, all the parameters of the fitted model are
also highly statistically significant with 99.9714324% of
variance explained. Moreover, the stability of R2
of this model is also more than 99%.
In this study the calculated value
of F-test is 4665.96, that is, a large quantity which means
that the fitted model is overall highly significant at 1%
level of significance. Therefore, from these statistics we
see that the fitted model and corresponding R2
are highly statistically significant. As a result, the model
is a good fit. Thereafter, the prediction is done and the
predicted values of the model are also demonstrated in the
last column of Table1.
| Table 1: Distribution
of Marriage Migration Associated with Distance in Comilla
in Bangladesh |
| Distance (in miles) |
Number of Migrants |
Predicted Values |
| 0-3 |
792 |
792.3030 |
| 3-6 |
442 |
440.8074 |
| 6-9 |
219 |
218.0541 |
| 9-12 |
87 |
94.5130 |
| 12-15 |
48 |
40.6537 |
| 15-18 |
29 |
26.9459 |
| 18-21 |
18 |
23.8593 |
| 21-24 |
4 |
1.86360 |
Fig. 1 Observed and
Fitted Marriage Migration Associated with Distance in Comilla
in Bangladesh
CONCLUSION
In this paper it is found that a
third degree polynomial model is fitted to the distribution
of marriage migration associated with distance in a Muslim
community in Bangladesh. The results show that this model
is also applicable or suitable even if Hossain fitted the
Pareto-Exponential model and Yadava et. al. showed that the
exponential distribution provided better approximation than
Hossain. Hence it is concluded that the pattern of marriage
migration due to distance follows a 3rd degree polynomial
model.
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