Hadwiger {fertilmodel}

Fertility models

Description

Fertility models.

Usage

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

Arguments

rate

A vector with the age-specific fertility rates.

age

A vector with the age of the women.

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.

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.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris [e-mail].

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)