cNORM (version 3.6.2)

predictMoments: Model-Implied Distributional Moments at Specific Ages

Description

Computes the mean, standard deviation, variance, skewness and (excess) kurtosis of the raw score distribution implied by a fitted cNORM model at one or more ages (or, more generally, values of the explanatory variable). The moments are model-implied population moments of the conditional raw score distribution, censored at the bounds of the raw score range [minRaw, maxRaw] for consistency across model families.

Usage

predictMoments(model, age, ...)
# S3 method for default
predictMoments(model, age, ...)
# S3 method for cnorm
predictMoments(model, age, nNodes = 100, ...)
# S3 method for cnormBetaBinomial
predictMoments(model, age, ...)
# S3 method for cnormBetaBinomial2
predictMoments(model, age, ...)
# S3 method for cnormShash
predictMoments(model, age, nNodes = 100, ...)

Value

A data.frame with one row per age and the columns

age, mean, sd, variance, skewness

and kurtosis (excess). The computation method is stored in the attribute "method".

Arguments

model

A model object of class cnorm, cnormBetaBinomial, cnormBetaBinomial2 or cnormShash.

age

A numeric vector of ages (values of the explanatory variable) at which to compute the moments.

...

Additional parameters passed to the methods, e.g. nNodes.

nNodes

Number of Gauss-Hermite quadrature nodes (default 100). Only relevant for the Taylor and SHASH methods; ignored for beta-binomial models, which are computed exactly by summation.

Details

The computation strategy depends on the model family:

Taylor polynomial (cnorm)

The bivariate regression function is collapsed at the specified age into a univariate polynomial in the norm score (location) variable. Moments are then obtained by Gauss-Hermite quadrature, which is mathematically exact for polynomial quantile functions (up to the censoring at minRaw/maxRaw).

Beta-binomial (cnormBetaBinomial, cnormBetaBinomial2)

Moments are computed exactly by summation over the discrete probability mass function on the support 0:n, using the age-specific predicted \(\alpha\) and \(\beta\) parameters. This respects the discreteness of the distribution; no continuity approximation is involved.

SHASH (cnormShash)

Moments are obtained by Gauss-Hermite quadrature of the quantile function qshash evaluated at the age-specific distribution parameters, censored at minRaw/maxRaw.

Kurtosis is reported as excess kurtosis (0 for the normal distribution).

Note that the skewness and kurtosis of the censored distribution are reported. For well-fitting models whose raw score range covers the probability mass of the conditional distribution, censoring effects are negligible; for distributions with substantial floor or ceiling effects, the censored moments are the substantively meaningful ones.

References

Isserlis, L. (1918). On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables. Biometrika, 12(1/2), 134-139.

Jones, M. C. & Pewsey, A. (2009). Sinh-arcsinh distributions. Biometrika, 96(4), 761-780.

See Also

normTable, predictNorm, predictRaw

Examples

if (FALSE) {
# Taylor polynomial model
model <- cnorm(raw = elfe$raw, group = elfe$group)
predictMoments(model, age = c(2.25, 2.75, 3.25, 3.75, 4.25))

# Beta-binomial model
bb <- cnorm.betabinomial(age = ppvt$age, score = ppvt$raw, n = 228)
predictMoments(bb, age = seq(4, 16, by = 2))
}