Package {irtbem2pl}

Compute Likelihood Information for the 2PL IRT Model

Description

' Calculates the log-likelihood, deviance, and expected frequencies (f, fz, rz) required for the E-step of the Expectation-Maximization (EM) algorithm under a two-parameter logistic (2PL) model.

Usage

LikelihoodInfo2pl(
  data.simple,
  CountNum,
  Model,
  Par.est0,
  n.Quadpts,
  node.Quadpts,
  weight.Quadpts,
  D
)

Arguments

Value

A list containing the following statistical components:

• LH The log-likelihood value of the model given the data.

• Deviance The model deviance statistic calculated as -2 * LH.

• f A numeric matrix of expected frequencies per quadrature point.

• fz A numeric matrix of expected correct response frequencies based on P*.

• rz A numeric matrix of expected correct joint frequencies.

Compute Item Response Probabilities for Various IRT Models

Description

Calculates the probability of a correct response for a given latent trait level (Theta) under specified Item Response Theory (IRT) models.

Usage

Prob.model(X, Model, Par.est0, D = 1)

Arguments

Value

A numeric vector or matrix containing the calculated correct response probabilities, bounded between 1e-12 and (1 - 1e-12) for numerical stability.

Dichotomous Response Dataset for IRT Calibration

Description

A sample dataset containing binary examination responses used to demonstrate 2PL model parameter estimation.

Usage

data(dat01)

Format

A data frame with rows representing examinees and 20 columns representing items (ITEM001 to ITEM023). Responses are dichotomous where 1 indicates a correct response and 0 indicates an incorrect response.

Estimation of the 2PL Model via Marginal Bayesian Modal Estimation

Description

Estimates item discrimination and difficulty parameters for the two-parameter logistic (2PL) model using an Expectation-Maximization (EM) algorithm with Bayesian priors.

Usage

irt2pl(
  Model = Model,
  data = data,
  data.simple = data.simple,
  CountNum = CountNum,
  n.class = n.class,
  Prior = Prior,
  Par.est0 = Par.est0,
  Par.SE0 = Par.SE0,
  D = D,
  np,
  Tol = Tol,
  max.ECycle = max.ECycle,
  max.MCycle = max.MCycle,
  n.Quadpts = n.Quadpts,
  n.decimal = n.decimal,
  Theta.lim = Theta.lim,
  Missing = Missing,
  ParConstraint = ParConstraint,
  BiasSE = BiasSE,
  I = I,
  J = J,
  Time.Begin = Time.Begin
)

Arguments

Value

A list containing the calibrated item parameters, information matrices, and convergence metadata.

Calibrate 2PL Model via Marginal Bayesian Modal Estimation

Description

' Estimates item parameters of the two-parameter logistic (2PL) model using marginal Bayesian modal estimation via the Expectation-Maximization (EM) algorithm. The results are similar to those of "BILOG-MG", from the book Item Response Theory Parameter Estimation Techniques.

Usage

irtbem2pl(
  data,
  PriorA = c(0, 0.25),
  PriorB = c(0, 4),
  InitialA = 1,
  InitialB = 0,
  Tol = 0.01,
  max.ECycle = 50L,
  max.MCycle = 10L,
  n.decimal = 5L,
  n.Quadpts = 50L,
  Theta.lim = c(-4, 4),
  Missing = -9,
  ParConstraint = FALSE,
  BiasSE = FALSE,
  D = 1
)

Arguments

Details

The two-parameter logistic (2PL) model proposed by Birnbaum (1968) is defined as:

P(x=1|\Theta,a,b) = \frac{1}{1 + exp(-D \cdot a \cdot (\Theta - b))}

where x=1 is the correct response, Theta is the examinee's ability, a is the item discrimination, and b is the difficulty parameter.

Value

A list containing the following components:

• Est.ItemPars A data frame with estimates of parameters a and b, along with their standard errors.

• Est.Theta A data frame with EAP ability estimates and their standard errors.

• Loglikelihood The final log-likelihood value.

• Iteration The number of iterations until convergence.

• EM.Map The parameter estimation history across iterations.

• fits.test Model fit statistics including G2, AIC, BIC, and RMSEA.

• Elapsed.time Total execution time of the program.

• InitialValues The initial values used for item calibration.

Author(s)

Juan Luis Legorreta Torres jlegorreta2002@yahoo.com.mx

References

Baker, F. B., & Kim, S.-H. (2004). Item Response Theory: Parameter Estimation Techniques (2nd ed.). Marcel Dekker.

Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In F. M. Lord & M. R. Novick (Eds.), Statistical theories of mental test scores (pp. 395-479). Addison-Wesley.

Examples

data(dat01)

# It is recommended to use:
# InitialA = point-biserial correlation (Classical Test Theory)
# InitialB = item difficulty index (Classical Test Theory)

mod_2PL <- irtbem2pl(data = dat01, PriorA = c(0, 0.25), PriorB = c(0, 4),
                     InitialA = 1, InitialB = 0, Tol = 0.01, max.ECycle = 50L,
                     max.MCycle = 10L, n.decimal = 5L, n.Quadpts = 50L,
                     Theta.lim = c(-4, 4), Missing = -9, ParConstraint = FALSE,
                     BiasSE = FALSE, D = 1)

mod_2PL$Est.ItemPars       # show item estimates
mod_2PL$Est.Theta          # show ability estimates

Checking user speciflied input variables

Description

Verifies whether the input variables provided by the user are correct based on the 2PL model specifications. If acceptable, the function formats and returns them as a combined list; otherwise, it throws an error

Usage

irtchek2pl(
  Model = "2PL",
  data,
  PriorA = c(0, 0.25),
  PriorB = c(0, 4),
  InitialA = 1,
  InitialB = 0,
  Tol = 0.01,
  max.ECycle = 50L,
  max.MCycle = 10L,
  n.decimal = 5L,
  n.Quadpts = 50L,
  Theta.lim = c(-4, 4),
  Missing = -9,
  ParConstraint = FALSE,
  BiasSE = FALSE,
  D = 1
)

Arguments

Value

A combined list containing formatted data arrays, sample sizes, prior specifications, initial parameter values, and convergence controls.

Author(s)

Juan Luis Legorreta Torres jlegorreta2002@yahoo.com.mx

References

Item Response Theory Parameter Estimation Techniques Second Edition, Revised and Expanded Frank B. Baker University ofWisconsin Madison, Wisconsin, U.S.A. Seock-Ho Kim The University ofGeorgia Athens, Georgia, u.S.A.

Examples

data(dat01)
library(irtbem2pl)
 Checking_2PL<-irtchek2pl(Model = "2PL", dat01, D = 1)