Fertility models for ASFR approximation

Usage

fert.approx(
  fx,
  age,
  model,
  start = NULL,
  se = FALSE,
  se.method = c("asymptotic", "bootstrap"),
  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 variance for ASFR approximation be calculated. Default is FALSE for no uncertainty estimates.
se.method: Character. Method for uncertainty estimation. Can be "asymptotic" (by default) or "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 and se.method = "bootstrap", number of bootstrap samples. By default, bn = 1000.

Value

A list of class fert.approx with two components: model, a list describing the fitted fertility model (type, fitted params, rmse, and, if se = TRUE, variance-covariance matrix vcov and parameter percentile intervals prc); and predicted, a data frame with observed and fitted age-specific fertility rates. When se = TRUE, predicted also includes standard errors and 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: $$f_x = c(x-d)(d+w-x)^2$$ 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)
fit <- fert.approx(fx = fx, age = age, model = "Hadwiger", se = FALSE)
predict(fit, age = 15:49)
#>    age    fx.model
#> 1   15 0.024034642
#> 2   16 0.036974591
#> 3   17 0.052334040
#> 4   18 0.069126477
#> 5   19 0.086166055
#> 6   20 0.102265760
#> 7   21 0.116399423
#> 8   22 0.127804670
#> 9   23 0.136024556
#> 10  24 0.140898693
#> 11  25 0.142520267
#> 12  26 0.141175349
#> 13  27 0.137277876
#> 14  28 0.131309554
#> 15  29 0.123769935
#> 16  30 0.115138749
#> 17  31 0.105850364
#> 18  32 0.096278886
#> 19  33 0.086731854
#> 20  34 0.077450287
#> 21  35 0.068613069
#> 22  36 0.060343940
#> 23  37 0.052719761
#> 24  38 0.045779070
#> 25  39 0.039530262
#> 26  40 0.033959003
#> 27  41 0.029034648
#> 28  42 0.024715618
#> 29  43 0.020953737
#> 30  44 0.017697630
#> 31  45 0.014895296
#> 32  46 0.012495965
#> 33  47 0.010451388
#> 34  48 0.008716664
#> 35  49 0.007250697