Fertility models for ASFR approximation
fert.approx.RdFertility models for ASFR approximation
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
FALSEfor no bootstrap.- alpha
Numeric. Used if
se = TRUE, the level of uncertainty. By default,alpha = 0.05for 95% CI.- bn
Numeric. Used if
se = TRUE, number of bootstrap samples. By default,bn = 1000.
Value
list with estimated model (parameters, variance-covariance matrix, percentiles of parameters) and dataframe with predicted and observed ASFR as well as SE and percentile of predictions
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: $$f_x = \frac{R}{\Beta(A,C)}(\beta - \alpha)^{-(A+C-1)}(x-\alpha)^{(A-1)}(\beta-x)^{(B-1)}$$ where \(\Beta\) 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.