Fertility models for ASFR approximation

Usage

fert.approx(fx, age, model, start = NULL, se = FALSE, alpha = 0.05, bn = 1000)

Arguments

fx: Numeric vector of age specific fertility rates.
age: Numeric vector of ages.
model: Character. Model name to be estimated. Now "Hadwiger", "Gamma", "Brass" and "Beta" are supported.
start: Numeric vector with user-specific values of parameters for optimization. Default is NULL (choose automatically)
se: Logical. Should bootstrapped variance for ASFR approximation be calculated. Default is FALSE for no bootstrap.
alpha: Numeric. Used if se = TRUE, the level of uncertainty. By default, alpha = 0.05 for 95% CI.
bn: Numeric. Used if se = TRUE, number of bootstrap samples. By default, bn = 1000.

Value

A list with two components: model, a list describing the fitted fertility model (type, fitted params, rmse, and, if se = TRUE, bootstrap covmat and parameter percentile intervals prc); and predicted, a data frame with observed and fitted age-specific fertility rates. When se = TRUE, predicted also includes bootstrap standard errors and percentile intervals.

Details

This function runs least squares optimization (using default optim) of the selected fertility function with 1e-06 as tolerance parameter.

$f_x$ is age-specific fertility rate for age $x$.

Hadwiger model

The model is as follows: $$f_x = \frac{ab}{c} \frac{c}{x}^{3/2} exp[-b^2(\frac{c}{x}+\frac{x}{c}-2)]$$ where $a,b,c$ are estimated parameters that do not have demographic interpretation. Sometimes $c$ is interpreted as mean age at childbearing.

Gamma model

The model is as follows: $$f_x = \frac{R}{\Gamma(b)c^b}(x-d)^{b-1} exp[-(\frac{x-d}{c})]$$ where $R,b,c,d$ are estimated parameters. $\Gamma$ is gamma function. $R$ can be interpreted as fertility level (TFR) and $d$ as mean age at childbearing.

Brass model

The model is as follows: $$f_x = c(x-d)(d+w-x)$$ where $c,d,w$ are estimated parameters.

Beta model

The model is as follows: $$f_x = \frac{R}{B(A,C)}(\beta - \alpha)^{-(A+C-1)}(x-\alpha)^{(A-1)}(\beta-x)^{(B-1)}$$ where $B(A, C)$ is beta function, $R, \beta, \alpha$ are estimated parameters, which can be interpreted as fertility level (TFR) and max and min age of childbearing respectively. $A,C$ are $$C = (\frac{(v - \alpha)(\beta - v)}{\tau^2} - 1)\frac{\beta - v}{\beta - \alpha}$$ $$A = C\frac{v-\alpha}{v - \beta}$$ where $v, \tau^2$ are estimated parameters, where $v$ can be interpreted as mean age at childbearing. Thus, Beta model uses 5 parameters $R, \beta, \alpha, v, \tau^2$, where only $\tau^2$ has no demographic interpretation.

References

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

Examples

age <- seq(15, 45, 5)
fx <- c(0.03, 0.10, 0.14, 0.12, 0.07, 0.03, 0.01)
fert.approx(fx = fx, age = age, model = "Hadwiger", se = FALSE)
#> $model
#> $model$type
#> [1] "Hadwiger"
#> 
#> $model$params
#>         a         b         c 
#>  1.442418  2.544989 28.091324 
#> 
#> $model$rmse
#> [1] 0.005297732
#> 
#> 
#> $predicted
#>   age   fx.model   fx
#> 1  15 0.02403464 0.03
#> 2  20 0.10226576 0.10
#> 3  25 0.14252027 0.14
#> 4  30 0.11513875 0.12
#> 5  35 0.06861307 0.07
#> 6  40 0.03395900 0.03
#> 7  45 0.01489530 0.01
#>