fertilmodel (version 1.4)

Hadwiger: Fertility models

Description

Fertility models.

Usage

Hadwiger(rate, age)
Gama(rate, age)
Model1(rate, age)
Model2(rate, age)

Value

A list including:

param

The vector of the estimated parameters.

sse

The sum of squars of the errors \(\sum_{i=1}^n(f_x-\hat{f}(x))^2\), where \(f_x\) denotes the observed age-specific fertility rates and \(\hat{f}(x)\) denote the fitted age-specific fertility rates.

fx

The fitted values, the fitted age-specific fertility rates \(\hat{f}(x)\).

res

The residuals, \(f_x-\hat{f}_x\).

Arguments

rate

A vector with the age-specific fertility rates.

age

A vector with the age of the women.

Author

Michail Tsagris.

R implementation and documentation: Michail Tsagris .

Details

The following fertility models are fitted: Hadwiger: $$ f(x)=\frac{ab}{c}(\frac{c}{x})^{3/2}\exp[-b^2(\frac{c}{x}+\frac{x}{c}-2)], $$ where \(x\) is the age of the mother at birth, \(a\) is associated with total fertility, the parameter \(b\) determines the height of the curve and the parameter \(c\) is related to the mean age of motherhood.

Gama: $$ f(x)=R\frac{1}{\Gamma(b)c^b}(x-d)^{b-1}\exp(-\frac{x-d}{c}), $$ where \(d\) represents the lower age at childbearing, while the parameter \(R\) determines the level of fertility.

Model1: $$ f(x)=c_1\exp[-\frac{(x-\mu)^2}{\sigma^2(x)}], $$ where \(\sigma(x)=\sigma_{11}\) if \(x \leq \mu\) and \(\sigma(x)=\sigma_{12}\) if \(x>\mu\). The parameter \(c_1\) describes the base level of the fertility curve and is associated with the total fertility rate, \(\mu\) reflects the location of the distribution, i.e. the modal age and \(\sigma_{11}\) and \(\sigma_{12}\) reflect the spread of the distribution before and after its peak, respectively.

Model2: $$ f(x)=c_1\exp[-\frac{(x-\mu_1)^2}{\sigma_1^2}] + c_2\exp[-\frac{(x-\mu_2)^2}{\sigma_2^2}], $$ where the parameters \(c_1\) and \(c_2\) express the severity i.e. the total fertility rates of the first and the second hump respectively, \(\mu_1\) and \(\mu_2\) are related to the mean ages of the two subpopulations the one with earlier fertility and the other with fertility at later ages, while \(\sigma_1\) and \(\sigma_2\) reflect the variances of the two humps.

References

Peristera P. and Kostaki A. (2007). Modeling fertility in modern populations. Demographic Research, 16(6): 141--194.

See Also

fertil.plot, comb

Examples

rate <- c(0.0001, 0.0006, 0.0033, 0.0111, 0.0263, 0.0412, 0.0544, 0.0622,
0.0660, 0.0704,0.0723, 0.0753, 0.0814, 0.0873, 0.0924, 0.0962, 0.0989,
0.1006, 0.0990, 0.0933,0.0831, 0.0747, 0.0634, 0.0529, 0.0424, 0.0326,
0.0242, 0.0172, 0.0115, 0.0073, 0.0040, 0.0022, 0.0012, 0.0006, 0.0003,
0.0002, 0.0001)
age <- 13:49
mod1 <- Hadwiger(rate, age)
mod2 <- Gama(rate, age)
mod3 <- Model1(rate, age)
mod4 <- Model2(rate, age)