BEMM.1PLG: Calibrating 1PLG model via Bayesian Expectation-Maximization-Maximization (BEMM) algorithm.

Description

This function can estimate the item parameters of the 1PLG model via Bayesian Expectation-Maximization-Maximization (BEMM) algorithm proposed by Guo, Wu, Zheng, & Wang (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.1PLG(data, PriorBeta = c(0, 4), PriorGamma = c(-1.39, 0.25), 
	InitialBeta = NA, InitialGamma = 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 beta and gamma 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.

PriorBeta

The user specified normal distribution prior for item difficulty (beta) parameters in the 1PLAG and 1PLG model. Can be:

A numeric with two hyperparameters mean and variance of normal distribution for all beta parameters. By default, PriorBeta=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 beta parameter.

PriorGamma

The user specified normal distribution prior for item guessing (gamma) parameters in the 1PLAG and 1PLG model. Can be:

A numeric with two hyperparameters mean and variance of normal distribution for all gamma parameters. By default, PriorGamma=c(-1.39,0.25), which means a normal prior of mean=-1.39 and variance=0.25 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 normal prior (mean and variance) for single item gamma parameter.

InitialBeta

The user specified starting values for item difficulty (beta) parameters in the 1PLAG and 1PLG models. Can be:

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

InitialGamma

The user specified starting values for item guessing (gamma) parameters in the 1PLAG and 1PLG models. Can be:

A NA (default), refers to no specified starting values for gamma parameter.
A single number (numeric), refers to set this number to be the starting values of gamma for all items.
A numeric consists of starting values for each gamma 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: beta in [-6, 6], gamma in [-7, 0].

BiasSE

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

Details

One parameter logsitc guessing (1PLG) model proposed by San Martín et al.(2006). Let invlogit(x)=1 / (1 + exp(-x)): $$P(x = 1|\theta, \beta, \gamma) = invlogit(\theta - \beta) + (1 - invlogit(\theta - \beta)) * invlogit(\gamma)$$ where x=1 is the correct response, theta is examinne's ability; beta and gamma are the item difficulty and guessing parameter, respectively. These parameter labels are capitalized in program for emphasis.

References

Guo, S., Wu, T., Zheng, C., & Wang, W.-C. (2018, April). Bayesian Expectation-Maximization-Maximization for 1PL-AG Model. Paper presented at the 80th NCME Annual Meeting, New York, NY.

San Martín, E., Del Pino, G., & De Boeck, P. (2006). IRT models for ability-based guessing. Applied Psychological Measurement, 30(3), 183-203. tools:::Rd_expr_doi("10.1177/0146621605282773")

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.beta=rnorm(J,0,1)     #simulate true difficulty parameters
true.gamma=rnorm(J,-1.39,0.5)    #simulate true guessing parameters
true.th=rnorm(I,0,1)       #simulate true theta parameters
true.par=list(Beta=true.beta, Gamma=true.gamma)   #make a list
response=matrix(NA,I,J)       #Create a array to save response data
for (i in 1:I){
   #calucate the probability of 1PLG
   P=Prob.model(X=true.th[i], Model='1PLG', Par.est0=true.par)  
   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.1PLG(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.1PLG(response, PriorBeta=NA, PriorGamma=NA, Tol=0.1) 

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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