Item and Test Information from Zero-Inflated Negative Binomial Model.

Estimate item and test information from Bayesian zero-inflated negative binomial model that was fit using the brms package.

Usage

deriv_d_negBinom(n, alpha, beta, theta, phi)

d_negBinom(n, alpha, beta, theta, phi)

log_gen_binom(n, phi)

deriv_logd_negBinom(n, alpha, beta, theta, phi)

info_neg_binom_analytical(
  theta = seq(-2.5, 2.5, length.out = 101),
  alpha,
  beta,
  phi,
  varpi
)

item_info_NB_zero_analytical(theta, alpha, beta, phi, varpi)

item_info_NB_analytical(theta, alpha, beta, phi, varpi = NULL, zero = FALSE)

test_info_NB(theta, alpha, beta, phi, varpi = NULL, zero = FALSE)

error_variance_NB(
  lower = -Inf,
  upper = Inf,
  alpha,
  beta,
  phi,
  varpi = NULL,
  zero = FALSE
)

reliability_NB(alpha, beta, phi, varpi = NULL, zero = FALSE)

Arguments

n

Integer. The observed count, representing the the event frequency.

alpha

Numeric. The slope/discrimination parameter of the item, indicating how steeply the item response changes with the person's (theta).

beta

Numeric. The intercept/easiness parameter of the item, indicating the expected count at a given level on the construct (theta).

theta

Numeric. The respondent's level on the latent factor/construct.

phi

Numeric. The shape/overdispersion parameter of the negative binomial distribution, indicating the variance beyond what is expected from a negative binomial distribution.

varpi

Numeric. The probability of observing a zero count due to a separate zero-inflation process.

zero

TRUE/FALSE. Whether the item is a from a zero-inflated model.

lower

Numeric. The lower range of theta, for estimating error variance or reliability.

upper

Numeric. The upper range of theta, for estimating error variance or reliability.

Value

The amount of information for a given item (or the test as a whole) at each of the values of theta specified. Based on test information, one can estimate error variance and marginal reliability using error_variance_NB() and reliability_NB(), respectively.

Details

Created by Philipp Doebler (doebler@statistik.tu-dortmund.de) and Loreen Sabel (loreen.sabel@tu-dortmund.de).

See also

Other bayesian: pA()

Other IRT: discriminationToFactorLoading(), fourPL(), itemInformation(), reliabilityIRT(), standardErrorIRT(), test_info_4PL()

Examples

if (FALSE) { # \dontrun{
library("brms")
library("rstan")

coef_bayesianMixedEffectsGRM_gam <- coef(bayesianMixedEffectsGRM_gam)
str(coef_bayesianMixedEffectsGRM_gam)
itempars <- coef_bayesianMixedEffectsGRM_gam$item[,1,1:4]

# define a grid of thetas for the computations:
theta_seq <- seq(-4, 4, length.out = 201)

# item information for all items
# The resulting matrix has length(theta_seq) columns and a row per item.
# We use a loop for the calculations
item_info <- matrix(NA, nrow = nrow(itempars), ncol = length(theta_seq))
for(i in 1:nrow(itempars)){
  item_info[i, ] <- item_info_NB_zero_analytical(
    theta = theta_seq,
    alpha = itempars[i, "alpha_Intercept"],
    beta = itempars[i, "beta_Intercept"],
    phi = exp(itempars[i, "shape_Intercept"]),
    varpi = plogis(itempars[i, "zi_Intercept"]))
}

test_info <- data.frame(
  theta = theta_seq,
  testInformation = colSums(item_info)
)

# Or, alternatively:
test_info_NB(
  theta = compareTestInfo$theta,
  alpha = itempars[,"alpha_Intercept"],
  beta = itempars[,"beta_Intercept"],
  phi = exp(itempars[,"shape_Intercept"]),
  varpi = plogis(itempars[,"zi_Intercept"]),
  zero = TRUE)

# Test Standard Error of Measurement in Different Theta Ranges
error_variance_NB(
  lower = -4,
  upper = 4,
  alpha = itempars[,"alpha_Intercept"],
  beta = itempars[,"beta_Intercept"],
  phi = exp(itempars[,"shape_Intercept"]),
  varpi = plogis(itempars[,"zi_Intercept"]),
  zero = TRUE
)

error_variance_NB(
  lower = -4,
  upper = 0,
  alpha = itempars[,"alpha_Intercept"],
  beta = itempars[,"beta_Intercept"],
  phi = exp(itempars[,"shape_Intercept"]),
  varpi = plogis(itempars[,"zi_Intercept"]),
  zero = TRUE
)

error_variance_NB(
  lower = 0,
  upper = 1.5,
  alpha = itempars[,"alpha_Intercept"],
  beta = itempars[,"beta_Intercept"],
  phi = exp(itempars[,"shape_Intercept"]),
  varpi = plogis(itempars[,"zi_Intercept"]),
  zero = TRUE
)

error_variance_NB(
  lower = 1.5,
  upper = 4,
  alpha = itempars[,"alpha_Intercept"],
  beta = itempars[,"beta_Intercept"],
  phi = exp(itempars[,"shape_Intercept"]),
  varpi = plogis(itempars[,"zi_Intercept"]),
  zero = TRUE
)

# One-Number Summary of Test Reliability
reliability_NB(
  alpha = itempars[,"alpha_Intercept"],
  beta = itempars[,"beta_Intercept"],
  phi = exp(itempars[,"shape_Intercept"]),
  varpi = plogis(itempars[,"zi_Intercept"]),
  zero = TRUE)
} # }