BEMM.4PL: Calibrating 4PL model via Bayesian Expectation-Maximization-Maximization (BEMM) algorithm.

Description

This function can estimate the item parameters of the 4PL model via Bayesian Expectation-Maximization-Maximization (BEMM) algorithm proposed by Zhang, Guo, & Zheng (2018, April). Both Bayesan modal estimates and maximum likelihood estimates are available. In addition, the examinees' ability and a few model fits information can be also obtained through this function.

Usage

BEMM.4PL(data, PriorA = c(0, 0.25), PriorB = c(0, 4), PriorC = c(4, 16), 
	PriorS = c(4, 16), InitialA = NA, InitialB = NA, InitialC = NA, 
	InitialS = NA, Tol = 0.0001, max.ECycle = 2000L, max.MCycle = 100L, 
	n.decimal = 3L, n.Quadpts = 31L, Theta.lim = c(-6, 6), 
	Missing = -9, ParConstraint = FALSE, BiasSE=FALSE)

Value

This function will return a list includes following:

Est.ItemPars: A dataframe consists of the estimates of a, b, c and s parameters and corresponding estimated standard errors.
Est.Theta: A dataframe consists of the estimates of theta and corresponding estimated standard errors (EAP method).
Loglikelihood: The loglikelihood.
Iteration: The number of iterations.
EM.Map: The parameter estimation history of iterations.
fits.test: The model fits information includes G2 test, AIC, BIC and RMSEA.
Elapsed.time: The running time of the program.
InitialValues: The initial values of item parameters.

Arguments

data

A matrix or data.frame consists of dichotomous data (1 for correct and 0 for wrong response), with missing data coded as in Missing (by default, Missing=-9). Each row of data represents a examinne' responses, and each column represents an item.

PriorA

The user specified logarithmic normal distribution prior for item discrimation (a) parameters in the 3PL and 4PL models. Can be:

A numeric with two hyperparameters mean and variance of logarithmic normal distribution for all a parameters. By default, PriorA=c(0,0.25), which means a log normal prior of mean=0 and variance=0.25 will be used for all item discrimation parameters.
A NA, refers to no priors will be used, so maximum likelihood estimates for item discrimation parameter will be obtained.
A matrix with two columns, and each row of matrix consists of two hyperparameters of log normal prior (mean and variance) for single item a parameter.

PriorB

The user specified normal distribution prior for item difficulty (b) parameters in the 3PL and 4PL models. Can be:

A numeric with two hyperparameters mean and variance of normal distribution for all b parameters. By default, PriorB=c(0,4), which means a normal prior of mean=0 and variance=4 will be used for all item difficulty parameters.
A NA, refers to no priors will be used, so maximum likelihood estimates for item difficulty parameter will be obtained.
A matrix with two columns, and each row of matrix consists of two hyperparameters of normal prior (mean and variance) for single item b parameter.

PriorC

The user specified Beta(x,y) distribution prior for item guessing (c) parameters in the 3PL and 4PL models. Can be:

A numeric with two hyperparameters x and y of Beta distribution for all c parameters. By default, PriorC=c(4,16), which means a Beta prior of mean=4/(4+16)=0.2 and variance=0.008 will be used for all item guessing parameters.
A NA, refers to no priors will be used, so maximum likelihood estimates for item guessing parameter will be obtained.
A matrix with two columns, and each row of matrix consists of two hyperparameters of Beta prior (x and y) for single item c parameter.

PriorS

The user specified Beta(x,y) distribution prior for item slipping (s) parameters in the 4PL model. Can be:

A numeric with two hyperparameters x and y of Beta distribution for all s parameters. By default, PriorS=c(4,16), which means a Beta prior of mean=4/(4+16)=0.2 and variance=0.008 will be used for all item slipping parameters.
A NA, refers to no priors will be used, so maximum likelihood estimates for item slipping parameter will be obtained.
A matrix with two columns, and each row of matrix consists of two hyperparameters of Beta prior (x and y) for single item s parameter.

InitialA

The user specified starting values for item discrimation (a) parameters in the 3PL and 4PL models. Can be:

A NA (default), refers to no specified starting values for a parameter.
A single number (numeric), refers to set this number to be the starting values of a for all items.
A numeric consists of starting values for each a parameter.

InitialB

The user specified starting values for item difficulty (b) parameters in the 3PL and 4PL models. Can be:

A NA (default), refers to no specified starting values for b parameter.
A single number (numeric), refers to set this number to be the starting values of b for all items.
A numeric consists of starting values for each b parameter.

InitialC

The user specified starting values for item guessing (c) parameters in the 3PL and 4PL models. Can be:

A NA (default), refers to no specified starting values for c parameter.
A single number (numeric), refers to set this number to be the starting values of c for all items.
A numeric consists of starting values for each c parameter.

InitialS

The user specified starting values for item slipping (s) parameters in the 4PL model. Can be:

A NA (default), refers to no specified starting values for s parameter.
A single number (numeric), refers to set this number to be the starting values of s for all items.
A numeric consists of starting values for each s parameter.

Tol

A single number (numeric), refers to convergence threshold for E-step cycles; defaults are 0.0001.

max.ECycle

A single integer, refers to maximum number of E-step cycles; defaults are 2000L.

max.MCycle

A single integer, refers to maximum number of M-step cycles; defaults are 100L.

n.Quadpts

A single integer, refers to number of quadrature points per dimension (must be larger than 5); defaults are 31L.

n.decimal

A single integer, refers to number of decimal places when outputs results.

Theta.lim

A numeric with two number, refers to the range of integration grid for each dimension; default is c(-6, 6).

Missing

A single number (numeric) to indicate which elements are missing; default is -9. The Missing cannot be 0 or 1.

ParConstraint

A logical value to indicate whether estimates parametes in a reasonable range; default is FALSE. If ParConstraint=TRUE: a in [0.001, 6], b in [-6, 6], c in [0.0001, 0.5], s in [0.0001, 0.5].

BiasSE

A logical value to determine whether directly estimating SEs from inversed Hession matrix rather than USEM method, default is FALSE.

Details

Four parameter logistic (4PL) model proposed by Barton & Lord's (1981). Transfer the unslipping (upper asymptote) parameter d to slipping parameter s by set s=1-d: $$P(x = 1|\theta, a, b, c, s) = c + (1 - s - c) / (1 + exp(-D * a * (\theta - b))),$$ where x=1 is the correct response; theta is examinne's ability. a, b, c and s are the item discrimination, difficulty guessing and slipping parameter, respectively; D is the scaling constant 1.702. These parameter labels are capitalized in program for emphasis.

References

Barton, M. A., & Lord, F. M. (1981). An upper asymptote for the three-parameter logistic item response model. ETS Research Report Series, 1981(1), 1-8. tools:::Rd_expr_doi("10.1002/j.2333-8504.1981.tb01255.x")

Zhang, C., Guo, S., & Zheng, C. (2018, April). Bayesian Expectation-Maximization-Maximization Algorithm for the 4PLM. Paper presented at the 80th NCME Annual Meeting, New York, NY.

Examples

Run this code

###Example: A brief simulation study###

#generate true values and response matrix
set.seed(10)
library(IRTBEMM)
I=500    #set the number of examinees is 500
J=10      #set the number of items is 10
true.a=runif(J,0.4,2)     #simulate true discrimination parameters
true.b=rnorm(J,0,1)       #simulate true difficulty parameters
true.c=rbeta(J,2,8)       #simulate true guessing parameters
true.s=rbeta(J,2,8)       #simulate true slipping parameters
true.th=rnorm(I,0,1)      #simulate true theta parameters
true.par=list(A=true.a, B=true.b, C=true.c, S=true.s)   #make a list
response=matrix(NA,I,J)       #Create a array to save response data
for (i in 1:I){
  #calucate the probability of 4PL
  P=Prob.model(X=true.th[i], Model='4PL', Par.est0=true.par, D=1.702)  
  response[i,]=rbinom(J,1,P)   #simulate the response
}

#To save example running time, we set the Tol to 0.1
#Obtain the Bayesian modal estimation (BME) using default priors

#Estimate model via BEMM algorithm
bme.res=BEMM.4PL(response, Tol=0.1) 

bme.res$Est.ItemPars       #show item estimates
bme.res$Est.Theta          #show ability estimates
bme.res$Loglikelihood      #show log-likelihood
bme.res$EM.Map             #show EM iteration history
bme.res$fits.test 		   #show model fits information


#Obtain the maximum likelihood estimation (MLE) by setting Prior=NA

#Estimate model via EMM algorithm
mle.res=BEMM.4PL(response, Tol=0.1,
		PriorA=NA, PriorB=NA, PriorC=NA, PriorS=NA)

mle.res$Est.ItemPars       #show item estimates
mle.res$Est.Theta          #show ability estimates
mle.res$Loglikelihood      #show log-likelihood
mle.res$EM.Map             #show EM iteration history
mle.res$fits.test 		   #show model fits information

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