cNORM (version 3.3.1)

cnorm.betabinomial: Fit a beta-binomial regression model for continuous norming

Description

This function fits a beta-binomial regression model where both the \(\alpha\) and \(\beta\) parameters of the beta-binomial distribution are modeled as polynomial functions of the predictor variable (typically age). Setting mode to 1 fits a beta-binomial model on the basis of \(\mu\) and \(\sigma\), setting it to 2 (default) fits a beta-binomial model directly on the basis of \(\alpha\) and \(\beta\).

Usage

cnorm.betabinomial(
  age,
  score,
  n = NULL,
  weights = NULL,
  mode = 2,
  alpha = 3,
  beta = 3,
  control = NULL,
  scale = "T",
  plot = T
)

Value

A list of class "cnormBetaBinomial" or "cnormBetaBinomial2". In case of mode 2 containing:

alpha_est

Estimated coefficients for the alpha model

beta_est

Estimated coefficients for the beta model

se

Standard errors of the estimated coefficients

alpha_degree

Degree of the polynomial for the alpha model

beta_degree

Degree of the polynomial for the beta model

result

Full result from the optimization procedure

Arguments

age

A numeric vector of predictor values (e.g., age).

score

A numeric vector of response values.

n

The maximum score (number of trials in the beta-binomial distribution). If NULL, max(score) is used.

weights

A numeric vector of weights for each observation. Default is NULL (equal weights).

mode

Integer specifying the mode of the model. Default is 2 (direct modelling of \(\gamma\) and \(\beta\)). If set to 1, the model is fitted on the basis of \(\mu\) and \(\sigma\), the predicted mean and standard deviation over age.

alpha

Integer specifying the degree of the polynomial for the alpha model. Default is 3. If mode is set to 1, this parameter is used to specify the degree of the polynomial for the \(\mu\) model.

beta

Integer specifying the degree of the polynomial for the beta model. Default is 3. If mode is set to 1, this parameter is used to specify the degree of the polynomial for the \(\sigma\) model.

control

A list of control parameters to be passed to the `optim` function. If NULL, default values are used, namely control = list(reltol = 1e-8, maxit = 1000) for mode 1 and control = list(factr = 1e-8, maxit = 1000) for mode 2.

scale

Type of norm scale, either "T" (default), "IQ", "z" or a double vector with the mean and standard deviation.

plot

Logical indicating whether to plot the model. Default is TRUE.

Details

The function standardizes the input variables, fits polynomial models for both the alpha and beta parameters, and uses maximum likelihood estimation to find the optimal parameters. The optimization is performed using the L-BFGS-B method.

Examples

if (FALSE) {
# Fit a beta-binomial regression model to the PPVT data
model <- cnorm.betabinomial(ppvt$age, ppvt$raw, n = 228)
summary(model)

# Use weights for post-stratification
marginals <- data.frame(var = c("sex", "sex", "migration", "migration"),
                        level = c(1,2,0,1),
                        prop = c(0.51, 0.49, 0.65, 0.35))
weights <- computeWeights(ppvt, marginals)
model <- cnorm.betabinomial(ppvt$age, ppvt$raw, n = 228, weights = weights)
}