simCRM: Generating Data under the Continuous Response Model
In EstCRM: Calibrating Parameters for the Samejima's Continuous IRT Model
simCRM R Documentation
Generating Data under the Continuous Response Model
Description
Generating data under the Continuous Response Model
Usage
simCRM(thetas, true.param, max.item)
Arguments
thetas
a vector of length N with N denoting the number of examinees. Each element of the vector is the true ability level for an examinee
true.param
a matrix of true item parameters with m rows and three columns, with m denoting the number of items. The first column is the a parameters, the second column is the b parameters, and the third column is the alpha parameters
max.item
a vector of length m indicating the maximum possible score for each hypothetical item.
Details
The simCRM generates data under Continuous Response Model as described in Shojima(2005). Given the true ability level for person i and the true item parameters for item j, the transformed response of person i for item j follows a normal distribution with a mean of α((θ-β)) and a standard deviation of α^2/a^-2.
Value
a data frame with N rows and m columns with N denoting the number of observations and m denoting the number of items.
Author(s)
Cengiz Zopluoglu
References
Shojima, K.(2005). A Noniterative Item Parameter Solution in Each EM Cycyle of the Continuous Response Model. Educational Technology Research, 28, 11-22.
See Also
EstCRMitem for estimating item parameters,
EstCRMperson for estimating person parameters,
fitCRM for computing item-fit statistics and drawing empirical 3D item response curves,
plotCRM for drawing theoretical 3D item category response curves,
Examples
## Not run:
#####################################################
# Example 1: #
# Basic data generation and parameter recovery #
#####################################################
#Generate true person ability parameters for 1000 examinees from
#a standard normal distribution
true.thetas <- rnorm(1000,0,1)
#Generate the true item parameter matrix for the hypothetical items
true.par <- matrix(c(.5,1,1.5,2,2.5,
-1,-.5,0,.5,1,1,.8,1.5,.9,1.2),
nrow=5,ncol=3)
true.par
#Generate the vector maximum possible scores that students can
#get for the items
max.item <- c(30,30,30,30,30)
#Generate the response matrix
simulated.data <- simCRM(true.thetas,true.par,max.item)
#Let's examine the simulated data
head(simulated.data)
summary(simulated.data)
#Let's try to recover the item parameters
min.item <- c(0,0,0,0,0)
CRM <- EstCRMitem(simulated.data,max.item, min.item,
max.EMCycle=500,converge=0.01)
#Compare the true item parameters with the estimated item parameters.
#The first three column is the true item parameters, and the second
#three column is the estimated item parameters
cbind(true.par,CRM$param)
#Let's recover the person parameters
par <- CRM$param
CRMthetas <- EstCRMperson(simulated.data,par,min.item,max.item)
theta.par <- CRMthetas$thetas
#Compare the true person ability parameters to the estimated person
##ability parameters.The first column is the true parameters and the
##second column is the estimated parameters
thetas <- cbind(true.thetas,theta.par[,2])
head(thetas)
cor(thetas)
plot(thetas[,1],thetas[,2])
#RMSE for the estimated person parameters
sqrt(sum((thetas[,1]-thetas[,2])^2)/nrow(thetas))
#RMSE is comparable and similar to the standard error of the
#theta estimates. Standard error of the theta estimate is the square
#root of the reciprocal of the total test information which is the sum
#of square of the "a" parameters
sqrt(1/sum(CRM$param[,1]^2))
#####################################################
# Example 2: #
# Item fit Residuals, Empirical and Theoretical #
# Item Category Response Curves for the Simulated #
# Data Above #
#####################################################
#Because of the run time issues during the package development,
#I run the fit analysis for a subset of simulated data above.
#The simulated data has 1000 examinees, but I run the fit analysis
#for the first 100 subjects of the simulated data. Please ignore the
#following line and run the analysis for whole data
simulated.data <- simulated.data[1:100,] #Ignore this line
par <- CRM$param
max.item <- c(30,30,30,30,30)
min.item <- c(0,0,0,0,0)
CRMthetas <- EstCRMperson(simulated.data,par,min.item,max.item)
theta.par <- CRMthetas$thetas
mean(theta.par[,2])
sd(theta.par[,2])
hist(theta.par[,2])
fit <- fitCRM(simulated.data,par, CRMthetas, max.item, group=10)
#Item-Fit Residuals
fit$fit.stat
#Empirical Item Category Response Curves
fit$emp.irf[[1]] #Item 1
fit$emp.irf[[5]] #Item 5
#Theoretical Item Category Response Curves
plotCRM(par,1,min.item, max.item) #Item 1
#####################################################
# Example 3: #
# The replication of Shojima's simulation study #
# 2005 #
#####################################################
#In Shojima's simulation study published in 2005
#true person parameters were generated from a standard normal distribution.
#The natural logarithm of the true "a" parameters were generated from a N(0,0.09)
#The true "b" parameters were generated from a N(0,1)
#The natural logarithm of the true "alpha" parameters were generated from a N(0,0.09)
#The independent variables were the number of items and sample size
#in the simulation study
#There were 9 different conditions and 100 replications for each condition.
#In Table 1 (Shojima,2005), the RMSD statistics were reported for each condition.
#The code below replicates the same study. The results are comparable to the Table 1.
#The user
#should only specify the sample size and the number of items. Then, the user should
#run the rest of the code.
#At the end, RMSEa, RMSEb, RMSEalp are the item parameter recovery statistics which is
#comparable to Table 1
#Set the conditions for the simulation study.
#It takes longer to run for big number of replications
N=500 #sample size
n=10 #number of items
replication=1 #number if replications for each condition
############################################################
# Run the rest of the code from START to END #
############################################################
#START
true.person <- vector("list",replication)
true.item <- vector("list",replication)
est.person <- vector("list",replication)
est.item <- vector("list",replication)
simulated.datas <- vector("list",replication)
for(i in 1:replication) {
true.person[[i]] <- rnorm(N,0,1)
true.item[[i]] <- cbind(exp(rnorm(n,0,.09)),rnorm(n,0,1),1/exp(rnorm(n,0,.09)))
}
max.item <- rep(50,n)
min.item <- rep(0,n)
for(i in 1:replication) {
simulated.datas[[i]] <- simCRM(true.person[[i]],true.item[[i]],max.item)
}
for(i in 1:replication) {
CRM<-EstCRMitem(simulated.datas[[i]],max.item,min.item,max.EMCycle= 500,converge=0.01)
est.item[[i]]=CRM$par
}
for(i in 1:replication) {
persontheta <- EstCRMperson(simulated.datas[[i]],est.item[[i]],min.item,max.item)
est.person[[i]]<- persontheta$thetas[,2]
}
#END
############################################################
#RMSE for parameter "a"
RMSEa <- c()
for(i in 1:replication) {
RMSEa[i]=sqrt(sum((true.item[[i]][,1]-est.item[[i]][,1])^2)/n)
}
mean(RMSEa)
#RMSE for parameter "b"
RMSEb <- c()
for(i in 1:replication) {
RMSEb[i]=sqrt(sum((true.item[[i]][,2]-est.item[[i]][,2])^2)/n)
}
mean(RMSEb)
#RMSE for parameter "alpha"
RMSEalp <- c()
for(i in 1:replication) {
RMSEalp[i]=sqrt(sum((true.item[[i]][,3]-est.item[[i]][,3])^2)/n)
}
mean(RMSEalp)
#RMSE for person parameter
RMSEtheta <- c()
for(i in 1:replication) {
RMSEtheta[i]=sqrt(sum((true.person[[i]]-est.person[[i]])^2)/N)
}
mean(RMSEtheta)
## End(Not run)
EstCRM documentation built on Sept. 17, 2022, 1:06 a.m.
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| simCRM | R Documentation |
Generating Data under the Continuous Response Model
Description
Generating data under the Continuous Response Model
Usage
simCRM(thetas, true.param, max.item)
Arguments
thetas |
a vector of length N with N denoting the number of examinees. Each element of the vector is the true ability level for an examinee |
true.param |
a matrix of true item parameters with m rows and three columns, with m denoting the number of items. The first column is the a parameters, the second column is the b parameters, and the third column is the alpha parameters |
max.item |
a vector of length m indicating the maximum possible score for each hypothetical item. |
Details
The simCRM generates data under Continuous Response Model as described in Shojima(2005). Given the true ability level for person i and the true item parameters for item j, the transformed response of person i for item j follows a normal distribution with a mean of α((θ-β)) and a standard deviation of α^2/a^-2.
Value
a data frame with N rows and m columns with N denoting the number of observations and m denoting the number of items.
Author(s)
Cengiz Zopluoglu
References
Shojima, K.(2005). A Noniterative Item Parameter Solution in Each EM Cycyle of the Continuous Response Model. Educational Technology Research, 28, 11-22.
See Also
EstCRMitem for estimating item parameters,
EstCRMperson for estimating person parameters,
fitCRM for computing item-fit statistics and drawing empirical 3D item response curves,
plotCRM for drawing theoretical 3D item category response curves,
Examples
## Not run:
#####################################################
# Example 1: #
# Basic data generation and parameter recovery #
#####################################################
#Generate true person ability parameters for 1000 examinees from
#a standard normal distribution
true.thetas <- rnorm(1000,0,1)
#Generate the true item parameter matrix for the hypothetical items
true.par <- matrix(c(.5,1,1.5,2,2.5,
-1,-.5,0,.5,1,1,.8,1.5,.9,1.2),
nrow=5,ncol=3)
true.par
#Generate the vector maximum possible scores that students can
#get for the items
max.item <- c(30,30,30,30,30)
#Generate the response matrix
simulated.data <- simCRM(true.thetas,true.par,max.item)
#Let's examine the simulated data
head(simulated.data)
summary(simulated.data)
#Let's try to recover the item parameters
min.item <- c(0,0,0,0,0)
CRM <- EstCRMitem(simulated.data,max.item, min.item,
max.EMCycle=500,converge=0.01)
#Compare the true item parameters with the estimated item parameters.
#The first three column is the true item parameters, and the second
#three column is the estimated item parameters
cbind(true.par,CRM$param)
#Let's recover the person parameters
par <- CRM$param
CRMthetas <- EstCRMperson(simulated.data,par,min.item,max.item)
theta.par <- CRMthetas$thetas
#Compare the true person ability parameters to the estimated person
##ability parameters.The first column is the true parameters and the
##second column is the estimated parameters
thetas <- cbind(true.thetas,theta.par[,2])
head(thetas)
cor(thetas)
plot(thetas[,1],thetas[,2])
#RMSE for the estimated person parameters
sqrt(sum((thetas[,1]-thetas[,2])^2)/nrow(thetas))
#RMSE is comparable and similar to the standard error of the
#theta estimates. Standard error of the theta estimate is the square
#root of the reciprocal of the total test information which is the sum
#of square of the "a" parameters
sqrt(1/sum(CRM$param[,1]^2))
#####################################################
# Example 2: #
# Item fit Residuals, Empirical and Theoretical #
# Item Category Response Curves for the Simulated #
# Data Above #
#####################################################
#Because of the run time issues during the package development,
#I run the fit analysis for a subset of simulated data above.
#The simulated data has 1000 examinees, but I run the fit analysis
#for the first 100 subjects of the simulated data. Please ignore the
#following line and run the analysis for whole data
simulated.data <- simulated.data[1:100,] #Ignore this line
par <- CRM$param
max.item <- c(30,30,30,30,30)
min.item <- c(0,0,0,0,0)
CRMthetas <- EstCRMperson(simulated.data,par,min.item,max.item)
theta.par <- CRMthetas$thetas
mean(theta.par[,2])
sd(theta.par[,2])
hist(theta.par[,2])
fit <- fitCRM(simulated.data,par, CRMthetas, max.item, group=10)
#Item-Fit Residuals
fit$fit.stat
#Empirical Item Category Response Curves
fit$emp.irf[[1]] #Item 1
fit$emp.irf[[5]] #Item 5
#Theoretical Item Category Response Curves
plotCRM(par,1,min.item, max.item) #Item 1
#####################################################
# Example 3: #
# The replication of Shojima's simulation study #
# 2005 #
#####################################################
#In Shojima's simulation study published in 2005
#true person parameters were generated from a standard normal distribution.
#The natural logarithm of the true "a" parameters were generated from a N(0,0.09)
#The true "b" parameters were generated from a N(0,1)
#The natural logarithm of the true "alpha" parameters were generated from a N(0,0.09)
#The independent variables were the number of items and sample size
#in the simulation study
#There were 9 different conditions and 100 replications for each condition.
#In Table 1 (Shojima,2005), the RMSD statistics were reported for each condition.
#The code below replicates the same study. The results are comparable to the Table 1.
#The user
#should only specify the sample size and the number of items. Then, the user should
#run the rest of the code.
#At the end, RMSEa, RMSEb, RMSEalp are the item parameter recovery statistics which is
#comparable to Table 1
#Set the conditions for the simulation study.
#It takes longer to run for big number of replications
N=500 #sample size
n=10 #number of items
replication=1 #number if replications for each condition
############################################################
# Run the rest of the code from START to END #
############################################################
#START
true.person <- vector("list",replication)
true.item <- vector("list",replication)
est.person <- vector("list",replication)
est.item <- vector("list",replication)
simulated.datas <- vector("list",replication)
for(i in 1:replication) {
true.person[[i]] <- rnorm(N,0,1)
true.item[[i]] <- cbind(exp(rnorm(n,0,.09)),rnorm(n,0,1),1/exp(rnorm(n,0,.09)))
}
max.item <- rep(50,n)
min.item <- rep(0,n)
for(i in 1:replication) {
simulated.datas[[i]] <- simCRM(true.person[[i]],true.item[[i]],max.item)
}
for(i in 1:replication) {
CRM<-EstCRMitem(simulated.datas[[i]],max.item,min.item,max.EMCycle= 500,converge=0.01)
est.item[[i]]=CRM$par
}
for(i in 1:replication) {
persontheta <- EstCRMperson(simulated.datas[[i]],est.item[[i]],min.item,max.item)
est.person[[i]]<- persontheta$thetas[,2]
}
#END
############################################################
#RMSE for parameter "a"
RMSEa <- c()
for(i in 1:replication) {
RMSEa[i]=sqrt(sum((true.item[[i]][,1]-est.item[[i]][,1])^2)/n)
}
mean(RMSEa)
#RMSE for parameter "b"
RMSEb <- c()
for(i in 1:replication) {
RMSEb[i]=sqrt(sum((true.item[[i]][,2]-est.item[[i]][,2])^2)/n)
}
mean(RMSEb)
#RMSE for parameter "alpha"
RMSEalp <- c()
for(i in 1:replication) {
RMSEalp[i]=sqrt(sum((true.item[[i]][,3]-est.item[[i]][,3])^2)/n)
}
mean(RMSEalp)
#RMSE for person parameter
RMSEtheta <- c()
for(i in 1:replication) {
RMSEtheta[i]=sqrt(sum((true.person[[i]]-est.person[[i]])^2)/N)
}
mean(RMSEtheta)
## End(Not run)
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