BEMM.4PL: Calibrating 4PL model via Bayesian...
In IRTBEMM: Family of Bayesian EMM Algorithm for Item Response Models
BEMM.4PL R Documentation
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)
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.
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.
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. \Sexpr[results=rd]{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
###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
IRTBEMM documentation built on June 7, 2023, 6:08 p.m.
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| BEMM.4PL | R Documentation |
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)
Arguments
data |
A |
PriorA |
The user specified logarithmic normal distribution prior for item discrimation (a) parameters in the 3PL and 4PL models. Can be:
|
PriorB |
The user specified normal distribution prior for item difficulty (b) parameters in the 3PL and 4PL models. Can be:
|
PriorC |
The user specified Beta(x,y) distribution prior for item guessing (c) parameters in the 3PL and 4PL models. Can be:
|
PriorS |
The user specified Beta(x,y) distribution prior for item slipping (s) parameters in the 4PL model. Can be:
|
InitialA |
The user specified starting values for item discrimation (a) parameters in the 3PL and 4PL models. Can be:
|
InitialB |
The user specified starting values for item difficulty (b) parameters in the 3PL and 4PL models. Can be:
|
InitialC |
The user specified starting values for item guessing (c) parameters in the 3PL and 4PL models. Can be:
|
InitialS |
The user specified starting values for item slipping (s) parameters in the 4PL model. Can be:
|
Tol |
A single number ( |
max.ECycle |
A single |
max.MCycle |
A single |
n.Quadpts |
A single |
n.decimal |
A single |
Theta.lim |
A |
Missing |
A single number ( |
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.
Value
This function will return a list includes following:
- Est.ItemPars
A
dataframeconsists of the estimates of a, b, c and s parameters and corresponding estimated standard errors.- Est.Theta
A
dataframeconsists 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
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. \Sexpr[results=rd]{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
###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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