Research Journal of Applied Sciences, Engineering and Technology 4(17): 3125-3129, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: March 14, 2012
Accepted: March 26, 2012
Published: September 01, 2012
Mathematical Modeling of Age and of Income Distribution Associated with Female
Marriage Migration in Rajshahi, Bangladesh
Rafiqul Islam
Department of Population Science and Human Resource Development, Rajshahi University,
Bangladesh
Abstract: An effort has been made, in this study, to fit mathematical models to age and income distribution
associated with female marriage migration in Rajshahi district, Bangladesh. For this, the data is taken under
the project entitled “Strengthening the Department of Population Science and Human Resource Development”
in collaboration with UNFPA, Bangladesh. It is found that marriage migration associated with age follows
polynomial model and income distribution associated with female marriage migration follows two parameters
positive exponential model. To verify the adequacy and steadiness situation of the model, Cross Validity
Prediction Power (CVPP) and F-test are employed to these models. The contribution of this paper to knowledge
is the fitted cubic polynomial model and positive exponential model to the migration data aggregate.
Keywords: Cross Validity Prediction Power (CVPP), exponential model, F-test, female marriage migration
associated with age and income, polynomial model
INTRODUCTION
Models are very realistic and sophisticated tools to
express data in mathematical structure in the era of
globalization process. Model is of great help to
demographers as well as social researchers for realizing
the process to distinguish among various important and
unnecessary variables to find out the functional
relationships among various demographic phenomena and
social indicators. In the current study, deterministic model
is used to explain the functional association between the
dependent and independent variables that can take exact
values. A number of models were fitted to the data
aggregate (Morril and Pitts, 1967; Perry, 1969a, 1969b;
Samuel, 1994; Sharma, 1984; Yadava et al., 1988) related
to migration in Bangladesh, India and other countries of
this globe. Hossain attempted to develop the model
proposed by Sharma (1984) and Yadava et al. (1988). But
these models did not afford good fit. Hossain (2000) used
the Pareto-Exponential model proposed by Morril and
Pitts (1967). Although Pareto-Exponential model
provided better approximation than the models of Sharma
(1984) and Yadava et al. (1988), it was not significantly
well to the utilized data. In India, Sharma (1984) and
Yadava et al. (1988) fitted well for Hindu community.
Yadavs et al. (2002) showed that exponential distribution
provides a better fit to the distribution of marriage
migration associated with distance than ParetoExponential as applied by Hossain (2000). Moreover,
Yadavs et al. (2002) compared their findings with those
of the Pareto-exponential function applied by Yadavs
et al. (1998). Ali et al. (2004) showed that age associated
marriage migration followed exponential distribution.
In this study, an n degree polynomial model
(Waerden and Van Der, 1948; Spiegel, 1992; Gupta and
Kapoor, 1997) for age related marriage migration and
exponential model for income distribution associated with
marriage migration is chosen to be applied here. It is to be
mentioned that polynomial model has also been applied
by Islam et al. (2003), Islam and Ali (2004) and Islam
(2005). Moreover, exponential model was fitted for age
structure of male population of Bangladesh in 1991
census (Islam et al., 2003)
Therefore, the fundamental objectives of this study
are to build up mathematical models for female marriage
migration associated with age and income.
METHODOLOGY
Data sources of the study: To accomplish the objectives
mentioned above the data on female marriage migration
associated with age and income distribution in Rajshahi
district, Bangladesh is taken under the project entitled
“Strengthening the Department of Population Science and
Human Resource Development” sponsored by UNFPA,
Bangladesh. These data have been used to fit the proposed
model and shown in Table 1 and Table 2, respectively.
Model fitting: Using the scattered plot of marriage
migration associated with age (Fig. 1, 2 and 3), it is seen
that marriage migration with respect to age can be
fitted by polynomial model. So, an nth degree
polynomial model is considered and the model is
3125
Res. J. App. Sci. Eng. Technol., 4(17): 3125-3129, 2012
Table 1: Observed and predicted of age associated with female migrants due to marriage
Rural
Suburban
------------------------------------------------------------------------------------Age
Observed
Predicted
Observed
Predicted
15-19
151
158
56
65
20-24
285
270
266
239
25-29
303
298
281
303
30-34
237
264
286
288
35-39
202
194
232
227
40-44
126
113
159
154
45-49
39
46
96
99
Total (rural and suburban jointly)
---------------------------------------------Observed
Predicted
207
223
551
510
584
601
523
552
434
422
285
267
135
145
Table 2: Observed and predicted of income distribution associated with female migrants due to marriage
Rural
Suburban
----------------------------------------------------------------------------------------Income (in taka)
Observed
Predicted
Observed
Predicted
0-500
48
65.14
3
35.51
500-1000
75
98.01
20
63.96
1000-1500
152
147.46
101
115.23
1500-2000
228
221.85
219
207.57
2000-2500
321
333.79
343
373.93
2500-3000
519
502.20
690
673.61
Total (rural and suburban jointly)
---------------------------------------------Observed
Predicted
51
90.86
95
151.61
253
252.96
447
422.06
664
704.22
1209
1175.00
Observed
Predicted
350
Predicted
300
600
250
500
200
400
150
300
100
200
50
100
0
15-19 20-24 25-29
0
30-34 35-39 40-44 45-49
Fig. 1: Observed and predicted values of age associated with
female migrants due to marriage of rural area in
Rajshahi, Bangladesh. X axis represents age and Y axis
represents female migrants
Observed
Predicted
350
Observed
700
15-19
30-34 35-39 40-44 45-49
Fig. 3: Observed and predicted values of age associated with
female migrants due to marriage of rural and suburban
area jointly in Rajshahi, Bangladesh. X axis represents
age and Y axis represents female migrants
300
600
250
500
200
400
150
300
Observed
Predicted
100
200
50
100
n
y = a0 +
∑a
i =1
i
x i + u in which x be the mid value of
age, y indicates marriage migration, a0 is the constant, ai
is the coefficient of xi (i = 1, 2, 3, ..., n) and u is the
00
25
00
-30
250
0
20
00-
00
00
15
00
-20
10
00
-15
Fig. 2: Observed and predicted values of age associated with
female migrants due to marriage of suburban area in
Rajshahi, Bangladesh. X axis represents age and Y axis
represents female migrants
500
-10
00
30-34 35-39 40-44 45-49
00
0
15-19 20-24 25-29
0-5
0
20-24 25-29
Fig. 4: Observed and predicted values of income distribution
associated with female migrants due to marriage of rural
area in Rajshahi, Bangladesh. X axis represents income
(in Taka) and Y axis represents female migrants
stochastic error term of the model. Here a suitable n has
been selected for which the error sum of squares is
minimum.
3126
Res. J. App. Sci. Eng. Technol., 4(17): 3125-3129, 2012
Observed
800
700
Observed
1400
Predicted
Predicted
1200
600
1000
500
800
400
600
300
400
200
200
100
25
00-
30
00
0-2
5
00
2 00
00
- 20
00
15
10
00
- 15
00
50 0
- 10
00
0-5
0
30
0
0
25
0 0-
0- 2
50
2 00
-20
15
00
-15
10
00
00
00
0
10
0
50
0-
00
0 -5
Fig. 5: Observed and predicted values of income distribution
associated with female migrants due to marriage of
suburban area in Rajshahi, Bangladesh. X axis
represents income (in taka) and Y axis represents female
migrants
00
0
0
Fig. 6: Observed and predicted values of income distribution
associated with female migrants due to marriage of rural
and suburban area jointly in Rajshahi, Bangladesh. X
axis represents income (in taka) and Y axis represents
female migrants
Table 3: The results of CVPP and information on model fittings
Model
(i)
(ii)
(iii)
(iv)
(v)
(vi)
n
7
7
7
6
6
6
k
3
3
3
1
1
1
D2cv
0.8608
R2
0.97564
0.97463
Shrinkage
0.11484
0.855029
0.98115
0.892286
0.99143
0.983336
0.9865
0.97375
0.99134
Significant
probability (p)
a0
a1
a2
a3
0.03
0. 0264
0.03755
0.061251
a0
a1
a2
a3
0.01
0.0123
0.0193
0.0320
a0
a1
a2
a3
0.01
0.0106
0.01622
0.02735
a0
a1
0.00007
0.00000
a0
a1
0.00023
0.00016
a0
a1
0.00008
0.00001
0.11960
0.08886
0.0081
0.01275
0.983161
If female marriage migration associated with income
is plotted in the graph paper (Fig. 4, 5 and 6), then it
appears that income distribution due to marriage
migration is positively exponentially distributed.
Therefore, a two-parameter positive exponential model is
considered and the form of the model is:
Parameters
0.0082
Model validation: To check the adequacy and soundness
of the model, the CVPP is applied here. The mathematical
formula for CVPP is given by:
2
ρcv
= 1−
(n − 1)(n − 2)(n + 1) 1 − R2
n(n − k − 1)(n − k − 2)
(
)
y = e( a0 x + a1 + u )
In which x represents the central value of income (in
taka); y represents marriage migration; a0, a1 are unknown
parameters and u is the stochastic disturbance term of the
model. These mathematical models are fitted employing
the software STATISTICA.
where, n is the number of cases, k is the number of
predictors in the model and the cross-validated R is the
correlation between observed and predicted values of the
dependent variable (Stevens, 1996). The positive value of
the difference of D2cv and R2 is the shrinkage of the model.
1-shrinkage is the stability of R2 of the fitted model. It
3127
Res. J. App. Sci. Eng. Technol., 4(17): 3125-3129, 2012
should be noted here that to clarify the validation of the
model the CVPP was also employed by Islam and Ali
(2004), Islam et al. (2003) and Islam (2005). The
estimated CVPP analogous to their R2 and information on
model fittings are presented in Table 3.
where as with (1, 4) d. f. at 1% level of significance the
tabulated value of F is 21.2. That is, the fitted models are
highly significant. Thus these models provide good fit to
the female marriage migration with age and income. The
observed and fitted values of these models are shown in
Fig. (1) to (6) correspondingly.
F-test: To verify the overall measure of the significance
of the fitted model as well as the significance of R2, the Ftest is employed here (Gujarati, 1998).
APPLICATION OF MODELS AND RESULTS
The polynomial model is assumed to fit the model for
female marriage migration due to ages in Rajshahi,
Bangladesh and the fitted models are as follows:
CONCLUSION
In this study it is found, on the one hand, that four
parameters 3rd degree polynomial model is fitted for the
distribution of female marriage migration associated with
age in Rajshahi district, Bangladesh. On the other hand,
distribution of female marriage migration associated with
income follow two parameters positive exponential
model.
y = -1263.79+139.8992x-3.92714x2+0.032889x3 for Rural
area ...
(1)
y = -1893.34+188.8944x-5.16238x2+0.043556x3 for Suburban area ...
(2)
y = -3157.13+328.7937x-9.08952x2+0.076444x3 for both
Rural and Sub-urban area ...
(3)
The positive exponential model is considered to be fitted
the model for female marriage migration associated with
income in Rajshahi, Bangladesh and the fitted models are
presented below:
y = exp(0.00082x+3.97230)
for Rural area…
(4)
y = exp(0.00118x+3.27543)
for Sub-urban area … (5)
y = exp(0.00102x+4.25339)
for both Rural and
Sub-urban area…
(6)
It is seen from the statistics of Table 3 that the fitted
models are highly cross-validated and their shrinkages are
0.11484, 0.1196; 0.08886, 0.0081, 0.01275 and 0.0082
for the models (1)-(6) respectively. And the fitted models
are more than 86, 85, 89, 98, 97 and 98% stable
respectively for the models (1)-(6). Moreover, all the
parameters of the fitted models are also highly statistically
significant with more than 97% proportion of variation
explained. Furthermore, the stability of R2 of these models
(1) to (3) is also more than 88%. And the stability of R2 of
the models (4) to (5) is also more than 99%.
In this study the calculated value of F-statistic are
40.05, 38.42 and 52.05 for the models (1) to (3),
respectively while with 3 and 3 degrees of freedom (d.f.)
at 1% level of significance the tabulated value of F is
29.5. The enumerated value of F-statistic for the models
(4) to (5) are 462.75, 292.30 and 455.77, respectively
ACKNOWLEDGMENT
I would like to give special thanks to UNFPA,
Country Representative to Bangladesh for providing
financial assistance for the project entitled “Strengthening
the Department of Population Science and Human
Resource Development” to strengthen the research
activities of research students as well as teachers in the
Department of Population Science and Human Resource
Development.
REFERENCES
Ali, K.M. Earfan, M.S. Hossain and M.A. Mannan, 2004.
Distribution of on. J. Appl. Sci., 4(2): 205-207.
Gujarati, D.N., 1998. Age Associated with Marriage
Migrati Basic Econometrics. 3rd Edn., McGraw Hill,
Inc., New York.
Gupta, S.C. and V.K. Kapoor, 1997. Fundamentals of
Mathematical Statistics. 9th Extensively Revised
(Ed.), Sultan Chand and Sons, Educational
Publishers, New Delhi.
Hossain, M.Z., 2000. Some demographic models and their
applications with special reference to Bangladesh. An
Unpublished Ph.D. Thesis, in Statistics, Banaras
Hindu University, Varanasi, India.
Islam, R., N. Islam, A. Ali and G. Mostofa, 2003.
Construction of male life table from female widowed
information of Bangladesh. Int. J. Stat. Sci., Dept. of
Statistics, University of Rajshahi, Bangladesh,
2: 69-82.
Islam, R. and M.K. Ali, 2004. Mathematical modeling of
agespecific fertility rates and study the reproductivity
in the rural area of bangladesh during 1980-1998.
Pak. J. Stat., 20(3): 381-394.
Islam, R., 2005. Mathematical modeling of age specific
marital fertility rates in the urban area of Bangladesh.
Pak. J. Stat., 21(3): 289-295.
3128
Res. J. App. Sci. Eng. Technol., 4(17): 3125-3129, 2012
Morril, R.L. and F.R. Pitts, 1967. Marriage, migration and
the mean information field. Ann. Assoc. Am. Geogr.,
57: 401-422.
Perry, P., 1969a. Marriage distance relationsship in north
otago 1975-1914. New Zealand’s Geogr., 25(1): 3643.
Perry, P., 1969b. Working class isolation and mobility in
rural dorset 1837-1936: A study of marriage distance.
Trans. Inst. Brit. Geogr., 47: 121-140.
Samuel, M.J., 1994. Patterns of Female Migration. In:
Maithili, V., (Ed.), Women and Society. IV Printwell,
Jaipur, India.
Sharma, L., 1984. A study of the pattern of out-migration
from rural areas. Unpublished., Ph.D. Thesis, in
Statistics, Banaras Hindu University. India.
Spiegel, M.R., 1992. Theory and Problems of Statistics.
2nd Edn., SI Units, Schaum's Outline Series,
McGraw-Hill Book Co., London.
Stevens, J., 1996. Applied Multivariate Statistics for the
Social Sciences. 3rd Edn., Lawrence Erlbaum
Associates, Inc., Publishers, New Jersey.
Waerden and B.L. Van Der, 1948. Modern Algebra. ICK
Ungar Publishing Co., New York, Vol. 1.
Yadavs, K.N.S., S. Srivastava and S. Islam, 2002.
Distribution of distance associated with marriage
migration. Int. J. Stat. Sci., Dept. of Statistics,
University of Rajshahi, Bangladesh, 1: 49-54.
Yadavs, K.N.S., M.Z. Hossain and S. Islam, 1998.
Distribution of distance associated with marriage
migration in rural areas of Bangladesh. Bangladesh
J. Sci. Res., 16(2): 201-207.
Yadava, K.N.S., K.N.M. Raju and G.S. Yadava, 1988. On
the distribution of distance associated with marriage
migration in rural areas of uttar Pradesh India. Rural
Demogr., 15(1): 7-18.
3129