CalcFert {paramDemo}

Calculating Parametric Age-Specific Fertility.

Description

CalcFert is used to calculate age-specific fertility from different parametric models.

Usage

CalcFert(beta, x, modelFert = "quadratic", checkBeta = TRUE)

Arguments

beta

Numerical vector of age-specific fertility parameters (see details).

x

Numerical vector of ages at which to calculate fertility.

modelFert

Age-specific fertility model. Options are “quadratic” (default), “PeristeraKostaki”, “ColcheroMuller”, “Hadwiger”, “gamma”, “beta”, “skewNormal”, “gammaMixture”, “HadwigerMixture”, “skewSymmetric”, “skewLogistic” (see details)

checkBeta

Logical to verify that the beta parameters conform with the model's specification (see details).

Details

1) Age-specific fertility models:

The age-specific fertility models correspond to the expected number of offspring produced by adults of a given age. Therefore, for a random variable Y_{x} with realizations y_x for the number of offspring produced by adults of age x, we have that E(Y_x) = g(x | \beta), where g: \mathbb{R}_{\ge 0} \rightarrow \mathbb{R}_{\ge 0} is a non-negative smooth fertility function and \beta is a vector of parameters to be estimated. The functional forms of function g fall in two main categories, namely polynomial and distributional models.

2.a) Polynomial:

Of the models available in paramDemo, the “quadratic”, “PeristeraKostaki” (Peristera and Kostaki 2007), and “ColcheroMuller” (Colchero et al. 2021, Muller et al. 2020) fall within the polynomial category. Both, the “PeristeraKostaki” and “ColcheroMuller” are non-symmetric around the age at maximum fertility.

2.b) Distributional:

The distributional models are of the form

g(x | \beta) = R f(x | \beta_1),

where f: \mathbb{R}_{\ge 0} \rightarrow \mathbb{R}_{\ge 0} is a probability density function, R > 0 is a parameter for the total fertility rate, and \beta_1 \in \beta is a vector of parameters. The “Hadwiger” (Hadwiger 1940), “gamma” (Hoem et al. 1981), “beta” (Hoem et al. 1981), “skewNormal” (Mazzuco and Scarpa 2011, 2015), “gammaMixture” (Hoem et al. 1981), “HadwigerMixture” (Chandola et al. 1991), “skewSymmetric” (Mazzuco and Scarpa 2011, 2015), and “skewLogistic” (Asili et al. 2014) all fall in this category. Notably, the “gammaMixture”, “HadwigerMixture”, “skewSymmetric”, “skewLogistic” are appropriate when fertility might be bimodal (Hoem et al. 1981, Chandola et al. 1999, Mazzuco and Scarpa 2011, 2015, Asili et al. 2014).

2) Specifying beta parameters:

Argument beta requires a vector of parameters. For instance, for a quadratic model with parameter vector \beta^{\top} = [0.5, 0.01, 10], the argument should be specified as theta = c(b0 = 0.5, b1 = 0.01, b2 = 10). Note that in this example the parameter names are specified directly, this is required when checkBeta = FALSE. Although assigning the names to each parameter is not necessary when checkBeta = TRUE, it is advisable to ensure that the right values are assigned to the right parameter.

If argument checkBeta is set to TRUE, then the vector of beta parameters is verified for consistency with the requirements of the model and shape selected.

Value

CalcFert returns a vector of class “numeric” with the fertility values at the specified ages.

Author(s)

Fernando Colchero [e-mail]

References

Asili, S., Rezaei, S. & Najjar, L. (2014) Using Skew-Logistic Probability Density Function as a Model for Age-Specific Fertility Rate Pattern. BioMed Research International, 2014, 790294.

Azzalini, A. (1985) A Class of Distributions Which Includes the Normal Ones. Scandinavian Journal of Statistics, 2, 171-178.

Chandola, T., Coleman, D.A. & Hiorns, R.W. (1999) Recent European fertility patterns: Fitting curves to ‘distorted’ distributions. Population Studies, 53, 317-329.

Colchero, F. (In prep.) Inference on age-specific fertility in ecology and evolution. Learning from other disciplines and improving the state of the art.

Colchero, F., Eckardt, W. & Stoinski, T. (2021) Evidence of demographic buffering in an endangered great ape: Social buffering on immature survival and the role of refined sex-age classes on population growth rate. Journal of Animal Ecology, 90, 1701-1713.

Hadwiger, H. (1940) Eine analytische Reproduktionssunktion fur biologische Gesamtheiten. Scandinavian Actuarial Journal, 1940, 101-113.

Hoem, J.M., Madien, D., Nielsen, J.L., Ohlsen, E.M., Hansen, H.O. & Rennermalm, B. (1981) Experiments in modelling recent Danish fertility curves. Demography, 18, 231-244.

Mazzuco, S. & Scarpa, B. (2011) Fitting age-specific fertility rates by a skew- symmetric probability density function. (Working paper 10), University of Padua.

Mazzuco, S. & Scarpa, B. (2015) Fitting age-specific fertility rates by a flexible generalized skew normal probability density function. Journal of the Royal Statistical Society: Series A, 178, 187-203.

Muller, M. N., Blurton Jones, N. G, Colchero, F., Thompson, M. E., Enigk, D. K. (2020) Sexual dimorphism in chimpanzee (Pan troglodytes schweinfurthii) and human age-specific fertility. Journal of human evolution, 144, 102795.

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

See Also

CalcMort to calculate age-specific mortality, CalcSurv to calculate age-specific survival

Examples

# Age specific fertility based on quadratic model (default):
fert <- CalcFert(beta = c(b0 = 0.5, b1 = 0.01, b2 = 10), x = 10)

# Age specific fertility based on gamma model:
fert <- CalcFert(beta = c(b0 = 13, b1 = 2, b2 = 0.15), x = 10,
                  modelFert = "gamma")