R: Calibrating 3PL model via Bayesian...

BEMM.3PL {IRTBEMM}

R Documentation

Calibrating 3PL model via Bayesian Expectation-Maximization-Maximization (BEMM) algorithm.

Description

This function can estimate the item parameters of the 3PL model via Bayesian Expectation-Maximization-Maximization (BEMM) algorithm proposed by Guo & Zheng(2019) and Zheng, Meng, Guo, & Liu (2018). 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.3PL(data, PriorA = c(0, 0.25), PriorB = c(0, 4), PriorC = c(4, 16), 
	InitialA = NA, InitialB = NA, InitialC = 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)

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

Details

Three parameter logistic (3PL) model proposed by Birnbaum(1968):

P(x = 1|\theta, a, b, c) = c + (1 - c) / (1 + exp(-D * a * (\theta - b))),

where x=1 is the correct response, theta is examinne's ability; a, b and c are the item discrimination, difficulty and guessing parameter, respectively; D is the scaling constant 1.702. These parameter labels are capitalized in program for emphasis.

Value

This function will return a list includes following:

Est.ItemPars: A dataframe consists of the estimates of a, b and c 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.

References

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). MA: Adison-Wesley.

Guo, S., & Zheng, C. (2019). The Bayesian Expectation-Maximization-Maximization for the 3PLM. Frontiers in Psychology, 10(1175), 1-11. doi:10.3389/fpsyg.2019.01175

Zheng, C., Meng, X., Guo, S., & Liu, Z. (2018). Expectation-Maximization-Maximization: A feasible MLE algorithm for the three-parameter logistic model based on a mixture modeling reformulation. Frontiers in Psychology, 8(2302), 1-10. doi:10.3389/fpsyg.2017.02302

Examples

###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.th=rnorm(I,0,1)      #simulate true theta parameters
true.par=list(A=true.a, B=true.b, C=true.c)   #make a list
response=matrix(NA,I,J)       #Create a array to save response data
for (i in 1:I){
  #calucate the probability of 3PL
  P=Prob.model(X=true.th[i], Model='3PL', 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.3PL(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.3PL(response, PriorA=NA, PriorB=NA, PriorC=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

[Package IRTBEMM version 1.0.8 Index]