Introduction
Recent (2005-2007) estimates of Bangladesh's population range from 142 to 159 million, making it the 7th most populous nation in the world. With a land area of 144,000 square kilometers, ranked 94th), the population density is remarkable. A striking comparison is offered by the fact that Russia's population is slightly smaller. Indeed Bangladesh has the highest population density in the world, excluding a handful of city-state (COTW, 2006). Despite sustained domestic and international efforts to improve economic and demographic prospects, Bangladesh remains a developing nation, in part due to its large population (UNFPA, 2007). Its per capita income in 2006 was US$2300 (on purchasing power parity basis) compared to the world average of $10,200 (CIAWF, 2007).[2] Yet, as the World Bank notes in its July 2005 Country Brief, the country has made significant progress in human development in the areas of literacy, gender parity in schooling, and reduction of population growth (WB, 2005).

Like many other developing countries it needs poverty alleviation and empowering women as well. Since the empowerment of women is an essential precondition for the elimination of world poverty and the upholding of human rights (DFID, 2000:8). Keller and Mbwewe (1991) describe Women's empowerment as "a process whereby women become able to organize themselves to increase their own self-reliance, to assert their independent right to make choices and to control resources which will assist in challenging and eliminating their own subordination". According to ICPD there are five components of women's empowerment: 1) women's sense of self worth 2) their right to determine choices 3) their right to have access to opportunities and resources 4) their right to have the power to control their own lives, both within and outside the home and 5) their ability to influence the direction of social change to create a more just social and economic order, nationally and internationally. Hashemi and Schuler (1993) defined the empowerment of women through the use of six spheres: 1) sense of self and a vision of the future, including resisting negative behaviors of the husband; 2) mobility and visibility, including how women are treated when they are traveling; 3) economic security, including cash income, new skills and knowledge; 4) status and decision-making power within the household, including making purchases on their own; 5) ability to interact effectively in the public sphere, such as joining credit programs, and 6) participation in non-family groups, such as credit programs and solidarity movements.

Women's empowerment is a matter of basic human rights. Interest in women's empowerment among demographers and population policy makers was heightened during the 1994 International Conference on Population and Development (ICPD) held in Cairo, at which the empowerment of women was legitimated as a social goal and enshrined as a necessary condition for population stabilization (Hodgson and Watkins 1997). Since then, critiques of demographers' views of gender and women (Presser 1997; Watkins 1993) have grown apace with the wealth of empirical studies investigating women's empowerment and its demographic consequences (Amin et al. 1994; Balk 1994, 1997; Chowdhury and Trovato 1994; Dharmalingam and Morgan 1996; Greenhalgh and Li 1995; Jejeebhoy 1995; Malhotra, Vanneman, and Kishor 1995; Morgan and Niraula 1995; Schuler and Hashemi 1994). Thus here efforts have been made to measure the women empowerment index (WEI) score for Bangladeshi women and to find out which types of models is more appilicable to WEI score. Thus the specific objectives of this study are:
i) to measure the WEI score by age
ii) to build up mathematical models to WEI score and
iii) to apply CVPP to the model to verify the validity of the model

Materials and Methods
Women empowerment is multidimensional and is very difficult to measure. For constructing WEI and other analysis, a nationally representative survey, Bangladesh Demographic and Health Survey (BDHS - 2004) (Mitra and Associates, 2005), of 11,440 women of age 10-49 is used. The survey was conducted under the authority of the National Institute for Population Research and Training (NIPORT) over a five month period from 1 January to 25 May 2004 using multistage cluster sampling.

In this study we have tried to measure women empowerment in the domestic sphere by making women empowerment index using the dimensions in accordance with Mason and Smith (2003). The particular aspects or dimensions of domestic empowerment we take are:

1. Women's economic decision making power.
2. Their household decision making power.
3. Their physical freedom of movement.

The detailed description of these three dimensions with their relevant indicators is given in table 1. Then the index of each dimension was constructed where minimum and maximum values were chosen for each underlying indicator. Performance in each indicator is expressed as the minimum and maximum value between 0 and 1 in accordance with the construction method of the Human Development Index (UNDP, 2005) as follows:

IVij =

Where, (X_ij), Min (X_ij), Max (X_ij) and IV_ij are, respectively, the actual, minimum, maximum and dimension index.

The Women Empowerment Index (WEI) is then computed by averaging these three indices.

Table 1: Description of dimensions and indicators with their measurement

Model Fitting

We plotted the women empowerment index value by the respondent's current age in figure 1 which depicts that the empowerment of women increases as age of women increases. But to fit a specific mathematical model to WEI score, we plotted the mean score of WEI by age groups which showed that mean score of WEI can be distributed by polynomial model for different ages. Therefore a polynomial is briefly discussed as an expression of the form:

Y = f(X) = a₀ + a₁X + a₂X² + ..............+ a_nXⁿ ; (a_n ? 0)

(Waerden, 1948), where, x is age group; y is mean score of WEI; a₀ is the constant; a_i is the coefficient of xⁱ (i =1, 2, 3,.........., n). If n = 0 then it becomes constant function. If n=1 then it is polynomial of degree 1 i.e. simple linear function. If n = 2 then it is polynomial of degree 2 i.e. quadratic polynomial, etc. (Spiegel, 1992; Gupta and Kapoor, 1997).

Figure 1: Graphical Representation of WEI by age

Model Validation Technique
To test the stability of the model, the cross validity prediction power (CVPP), ²cv, is applied here. The method for CVPP is given by
;
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. The shrinkage of the model is the absolute value of the difference between r2cv and R². Moreover, the stability of R² of the model is equal to (1- shrinkage) (Stevens, 1996).

Results and Discussion

Though BDHS- 2004 collected data from 11440 women of age 10 - 49, we get only one women's relevant data for constructing WEI in age group 10 - 14. That is why, mean scores of WEI are presented in table 2 excepting score of age group 10 - 14.

We observe from table 2 that there is an upward trend in the mean score of WEI from age 15 - 44 years. That is in average the women who are under 20 years of age are much lower empowered than the women who are in 40 - 44 age group.

Table 2: Mean score of WEI by selected demographic variables

Now for fitting more appropriate model to the WEI score, we utilize usual models such as Makehams, Logistic, Gompertz, log-linear and semi-log linear. Among them, polynomial model of order two best fits according to its shrinkages. So, only the outputs of the poynomial model are exhibited in table 3 and figure 2.

The polynomial model for WEI in Bangladesh is
y = - 0.12013 + 0.04774x - 0.64e-3x² ........................................(A)
which is the polynomial of degree two i. e. quadratic polynomial.

From table 3, it is shown that two parameters of the fitted model are statistically significant with 93% of variance explained and the model's shrinkage is 0.34824. This model will be stable more than 57%.

Table 3: Information of Model Fitting

Figure 2: Graphical Representation Observed and Fitted mean score of WEI
Note: C presents observed values

Conclusion
The level of women's empowerment in Bangladesh is not satisfactory for any of the age group. Older women have more independence and empowerment than younger women because they have more experience with life, a better understanding of how to get what they want or need, a closer relationship with the husband, or because they have fulfilled certain social obligations to the husband and his family (for example, bearing children or sons) and thus are more trusted than are young wives, over whom tighter controls are maintained (Tareque et. al., 2007). And the mathematical model of WEI score in Bangladesh follows a quadratic polynomial model.

Further Scope
Table 2 and figure 2 show that not all women in Bangladesh are equally empowered i. e. young aged women are less empowered than their older counterparts. Now some questions will be arisen - why it is happened and what the consequences are.

On average, girls who marry as adolescents attain lower schooling levels, which cause lower self confidence, bargaining power, freedom of choices etc. as a result they have less reproductive control and suffer higher rates of maternal mortality and domestic violence and thereafter less empowerment. We guess some other causes like early female marriage make women low empowered. Thus further research is needed to find out the rationale behind the above questions.

References

Amin, Ruhul, A. U. Ahmed, J. Chowdhury, and M. Ahmed. "Poor Women's Participation in Income-Generating Projects and Their Fertility Regulation in Rural Bangladesh: Evidence from a Recent Survey." World Development, 1994; 44:555-65.

Balk, Deborah. "Individual and Community Aspects of Women's Status and Fertility in Rural Bangladesh." Population Studies, 1994; 48(March):21-45.
"Defying Gender Norms in Rural Bangladesh: A Social Demographic Analysis." Population Studies, 1997; 51(July):153-72.

Chowdhury, Fakhrul I. and Frank Trovato. "The Role and Status of Women and the Timing of Marriage in Five Asian countries." Journal of Comparative Family Studies, 1994; 25:143-57.

CIAWF, "CIA World Factbook 2007", accessed through http://en.wikipedia.org/wiki/Bangladesh 2007

COTW, "Countries of the World (by highest population density)", WorldAtlus.com, accessed through http://en.wikipedia.org/wiki/Bangladesh2006.

DFID "Report on Poverty Alleviation and the Empowerment of women, UK: Stairway Communications", 2000.

Dharmalingam, A. and S. Philip Morgan. "Women's Work, Empowerment, and Birth Control: Evidence from Two South Indian Villages." Population Studies, 1996; 50:187-201.

Greenhalgh, Susan and Jiali Li. "Engendering reproductive policy and practice in peasant China: For a feminist demography of reproduction." Signs: Journal of Women in Culture and Society, 1995; 20:601-41.

Gupta, S. C. & Kapoor, V. K. "Fundamentals of Mathematical Statistics" RevisedEdition, Sultan Chand & Sons, Educational Publishers, New Delhi, 1997.

Hashemi, Syed M. and Sidney Ruth Schuler "Defining and Studying Empowerment of Women: A Research Note from Bangladesh." JSI Working Paper No. 3. Washington DC: John Snow, Inc. 1993.

Hodgson, Dennis and Susan Cotts Watkins. "Feminists and Neo-Malthusians: Past and Present Alliances." Population and Development Review, 1997; 23(3):469-523.

Jejeebhoy, Shireen J. "Women's Education, Empowerment, and Reproductive Behaviour: Experience from Developing Countries". Oxford: Clarendon Press, 1995.

Keller, B. and D.C. Mbwewe "Policy and planning for the empowerment of Zambia's women farmers." Canadian Journal of Development Studies, 1991; 12(1):75-88.

Malhotra, Anju, Reeve Vanneman, and Sunita Kishor. "Fertility, Dimensions of Patriarchy, and Development in India." Population and Development Review, 1995; 21:281-305

Mason, Karen Oppenheim and Smith "Women's Empowerment and Social Context: Result from Five Asian Countries", 2003.

Mitra and Associates. "Bangladesh Demographic and Health Survey (BDHS-2004)" Dhaka, Bangladesh and Calverton, Maryland [USA]: National Institute of Population Research and Training, Mitra and Associates, and ORC Macro, 2005.

Morgan, S. Philip and Banu Niraula. "Gender Inequality and Fertility in Two Nepali Villages." Population and Development Review, 1995: 21:541-61.

Presser, Harriet B. "Demography, Feminism, and the Science-Policy Nexus." Population and Development Review, 1997: 23(2):295-331.

Schuler, Sidney Ruth and Syed M. Hashemi. "Credit Programs, Women's Empowerment, and Contraceptive Use in Rural Bangladesh." Studies in Family Planning, 1994; 25:65-76.

Spiegel, M. R. "Theory and Problems of Statistics", Second Edition in SI Units., Schaum's Outline Series, McGraw -Hill Book Company, London, 1992.

Stevens, J. "Applied Multivariate Statistics for the Social Sciences." Third Edition, Lawrence Erlbaum Associates, Inc., Publishers, New Jersey, 1996.

Tareque M. I., Haque, M.M., Mostofa, M. G. and Islam, T. M, "Age, Age at Marriage, Age Difference between Spouses and Women Empowerment: Bangladesh Context" Middle East Journal of Age and Ageing, 2007; 4 (6): 8 - 14.

UNFPA, "Reproductive Health and Rights is Fundamental for Sound Economic Development and Poverty Alleviation," United Nations Population Fund. accessed through http://en.wikipedia.org/wiki/Bangladesh, 2007.

United Nations Development Programme (UNDP) "Human Development Report 2005, United Nations Development Programme, New York, 2005.

Waerden, B.L.Van Der. "Modern Algebra", Vol. 1, ICK Ungar Publishing Co. New York, 1948.

WB, "Bangladesh - Country Brief", World Bank, accessed through http://en.wikipedia.org/wiki/Bangladesh 2005.

Wikipedia, The free encyclopedia http://en.wikipedia.org/wiki/Bangladesh 17.12.07