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One booklet PC items
tmatta edited this page Oct 17, 2017
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We examined item parameter recovery under the following conditions: 1 (IRT model) x 3 (IRT R packages) x 3 (sample sizes) x 4 (test lengths) x 1 (test booklet)
- One IRT model was included: partial credit (PC) model
- Item parameters were randomly generated
- The bounds of the item difficulty parameter, b, are constrained to
b_bounds = (-2, 2)where -2 is the lowest generating value and 2 is the highest generating value
- Three IRT R packages were evaluated:
TAM(version 2.4-9),mirt(version 1.25), andltm(version, 1.0-0) - Three sample sizes were used: 500, 1000, and 5000
- Simulated samples were based on one ability level from distribution N(0, 1)
- Four test lengths were used: 40, 60, 80, and 100
- A single booklet was used.
- One hundred replications were used for each condition for the calibration
- Summary of item parameter recovery:
-
TAM,mirt, andltmdemonstrated a similar level of accuracy - b1-parameter recovered well, with correlation ranging from 0.983 to 0.998, with bias ranging from -0.066 to 0.010, and with RMSE ranging from 0.041 to 0.197
- b2-parameter recovered well, with correlation ranging from 0.985 to 0.999, with bias ranging from -0.049 to 0.021, and with RMSE ranging from 0.041 to 0.194
- For b1- and b2-parameters, sample sizes of 5000 consistently produced the most accurate results
- For b1- and b2-parameters, four levels of test lengths performed very similarly
-
# Load libraries
if(!require(lsasim)){
install.packages("lsasim")
library(lsasim) #version 1.0.1
}
if(!require(mirt)){
install.packages("mirt")
library(mirt) #version 1.25
}
if(!require(TAM)){
install.packages("TAM")
library(TAM) #version 2.4-9
}
if(!require(ltm)){
install.packages("ltm")
library(ltm) #version 1.0-0
}# Set up conditions
N.cond <- c(500, 1000, 5000) #number of sample sizes
I.cond <- c(40, 60, 80, 100) #number of items
K.cond <- 1 #number of booklets
# Set up number of replications
reps <- 100
# Create space for outputs
results <- NULL#==============================================================================#
# START SIMULATION
#==============================================================================#
for (N in N.cond) { #sample size
for (I in I.cond) { #number of items
# generate item parameters for a PC model
set.seed(4366) # fix item parameters across replications
item_pool <- lsasim::item_gen(n_1pl = I,
thresholds = 2,
b_bounds = c(-2, 2))
for (K in K.cond) { #number of booklets
for (r in 1:reps) { #replication
#------------------------------------------------------------------------------#
# Data simulation
#------------------------------------------------------------------------------#
set.seed(8088*(r+4))
# generate thetas
theta <- rnorm(N, mean=0, sd=1)
# assign items to block
block_bk1 <- lsasim::block_design(n_blocks = K,
item_parameters = item_pool)
#assign block to booklet
book_bk1 <- lsasim::booklet_design(item_block_assignment =
block_bk1$block_assignment,
book_design = matrix(K))
#assign booklet to subjects
book_samp <- lsasim::booklet_sample(n_subj = N,
book_item_design = book_bk1,
book_prob = NULL)
# generate item responses
cog <- lsasim::response_gen(subject = book_samp$subject,
item = book_samp$item,
theta = theta,
b_par = item_pool$b,
d_par = list(item_pool$d1,
item_pool$d2))
# extract item responses (excluding "subject" column)
resp <- cog[, c(1:I)]
#------------------------------------------------------------------------------#
# Item calibration
#------------------------------------------------------------------------------#
# fit PC model using mirt package
mirt.mod <- NULL
mirt.mod <- mirt::mirt(resp, 1, itemtype = 'Rasch', verbose = F,
technical = list( NCYCLES = 500))
# fit PC model using TAM package
tam.mod <- NULL
tam.mod <- TAM::tam.mml(resp, irtmodel = "PCM2", control = list(maxiter = 200))
# fit PC model using ltm package
ltm.mod <- NULL
ltm.mod <- ltm::gpcm(resp, constraint = "rasch", IRT.param=T,
control = list(iter.qN = 1000))
#------------------------------------------------------------------------------#
# Item parameter extraction
#------------------------------------------------------------------------------#
# extract b1, b2 in mirt package
mirt_b1 <- coef(mirt.mod, IRTpars = TRUE, simplify=TRUE)$items[,"b1"]
mirt_b2 <- coef(mirt.mod, IRTpars = TRUE, simplify=TRUE)$items[,"b2"]
# convert TAM output into PCM parametrization
tam_b1 <- tam.mod$item$AXsi_.Cat1
tam_b2 <- (tam.mod$item$AXsi_.Cat2) - (tam.mod$item$AXsi_.Cat1)
# extract Catgr.1 and Catgr.2 in ltm package
ltm_b1 <- (data.frame(coef(ltm.mod)))$Catgr.1
ltm_b2 <- (data.frame(coef(ltm.mod)))$Catgr.2
#------------------------------------------------------------------------------#
# Item parameter recovery
#------------------------------------------------------------------------------#
# summarize results
itempars <- data.frame(matrix(c(N, I, K, r), nrow=1))
colnames(itempars) <- c("N", "I", "K", "rep")
# retrieve generated item parameters
genPC.b1 <- item_pool$b + item_pool$d1
genPC.b2 <- item_pool$b + item_pool$d2
# calculate corr, bias, RMSE for item parameters in mirt pacakge
itempars$corr_mirt_b1 <- cor( genPC.b1, mirt_b1)
itempars$bias_mirt_b1 <- mean( mirt_b1 - genPC.b1 )
itempars$RMSE_mirt_b1 <- sqrt(mean( ( mirt_b1 - genPC.b1 )^2 ))
itempars$corr_mirt_b2 <- cor( genPC.b2, mirt_b2)
itempars$bias_mirt_b2 <- mean( mirt_b2 - genPC.b2 )
itempars$RMSE_mirt_b2 <- sqrt(mean( ( mirt_b2 - genPC.b2 )^2 ))
# calculate corr, bias, RMSE for item parameters in TAM pacakge
itempars$corr_tam_b1 <- cor( genPC.b1, tam_b1)
itempars$bias_tam_b1 <- mean( tam_b1 - genPC.b1 )
itempars$RMSE_tam_b1 <- sqrt(mean( ( tam_b1 - genPC.b1 )^2 ))
itempars$corr_tam_b2 <- cor( genPC.b2, tam_b2)
itempars$bias_tam_b2 <- mean( tam_b2 - genPC.b2 )
itempars$RMSE_tam_b2 <- sqrt(mean( ( tam_b2 - genPC.b2 )^2 ))
# calculate corr, bias, RMSE for item parameters in ltm pacakge
itempars$corr_ltm_b1 <- cor( genPC.b1, ltm_b1)
itempars$bias_ltm_b1 <- mean( ltm_b1 - genPC.b1 )
itempars$RMSE_ltm_b1 <- sqrt(mean( ( ltm_b1 - genPC.b1 )^2 ))
itempars$corr_ltm_b2 <- cor( genPC.b2, ltm_b2)
itempars$bias_ltm_b2 <- mean( ltm_b2 - genPC.b2 )
itempars$RMSE_ltm_b2 <- sqrt(mean( ( ltm_b2 - genPC.b2 )^2 ))
# combine results
results <- rbind(results, itempars)
}
}
}
}
- Correlation, bias, and RMSE for item parameter recovery in
mirtpackage
mirt_recovery <- aggregate(cbind(corr_mirt_b1, bias_mirt_b1, RMSE_mirt_b1,
corr_mirt_b2, bias_mirt_b2, RMSE_mirt_b2) ~ N + I,
data=results, mean, na.rm=TRUE)
names(mirt_recovery) <- c("Sample Size", "Test Length",
"corr_b1", "bias_b1", "RMSE_b1",
"corr_b2", "bias_b2", "RMSE_b2")
round(mirt_recovery, 3)## Sample Size Test Length corr_b1 bias_b1 RMSE_b1 corr_b2 bias_b2 RMSE_b2
## 1 500 40 0.984 -0.009 0.130 0.987 0.003 0.129
## 2 1000 40 0.992 -0.007 0.092 0.994 -0.008 0.089
## 3 5000 40 0.998 -0.002 0.041 0.999 -0.003 0.041
## 4 500 60 0.984 -0.007 0.131 0.985 -0.002 0.133
## 5 1000 60 0.992 -0.007 0.093 0.992 -0.007 0.093
## 6 5000 60 0.998 -0.002 0.041 0.998 -0.002 0.042
## 7 500 80 0.984 -0.006 0.129 0.985 -0.002 0.134
## 8 1000 80 0.992 -0.006 0.092 0.993 -0.006 0.092
## 9 5000 80 0.998 -0.003 0.041 0.998 -0.002 0.042
## 10 500 100 0.983 -0.004 0.132 0.986 -0.004 0.132
## 11 1000 100 0.991 -0.007 0.093 0.993 -0.005 0.095
## 12 5000 100 0.998 -0.003 0.042 0.999 -0.003 0.042
- Correlation, bias, and RMSE for item parameter recovery in
TAMpackage
tam_recovery <- aggregate(cbind(corr_tam_b1, bias_tam_b1, RMSE_tam_b1,
corr_tam_b2, bias_tam_b2, RMSE_tam_b2) ~ N + I,
data=results, mean, na.rm=TRUE)
names(tam_recovery) <- c("Sample Size", "Test Length",
"corr_b1", "bias_b1", "RMSE_b1",
"corr_b2", "bias_b2", "RMSE_b2")
round(tam_recovery, 3)## Sample Size Test Length corr_b1 bias_b1 RMSE_b1 corr_b2 bias_b2 RMSE_b2
## 1 500 40 0.984 -0.003 0.134 0.987 0.010 0.132
## 2 1000 40 0.992 -0.009 0.095 0.994 -0.009 0.092
## 3 5000 40 0.998 -0.003 0.043 0.999 -0.002 0.042
## 4 500 60 0.984 -0.015 0.149 0.985 -0.005 0.148
## 5 1000 60 0.992 -0.003 0.110 0.992 0.002 0.108
## 6 5000 60 0.998 -0.002 0.046 0.998 0.003 0.047
## 7 500 80 0.984 -0.019 0.160 0.985 -0.006 0.165
## 8 1000 80 0.992 -0.016 0.127 0.993 -0.008 0.128
## 9 5000 80 0.998 -0.004 0.059 0.998 0.006 0.058
## 10 500 100 0.983 0.010 0.171 0.986 0.021 0.172
## 11 1000 100 0.991 -0.020 0.130 0.993 -0.008 0.127
## 12 5000 100 0.998 -0.009 0.068 0.999 0.003 0.067
- Correlation, bias, and RMSE for item parameter recovery in
ltmpackage
ltm_recovery <- aggregate(cbind(corr_ltm_b1, bias_ltm_b1, RMSE_ltm_b1,
corr_ltm_b2, bias_ltm_b2, RMSE_ltm_b2) ~ N + I,
data=results, mean, na.rm=TRUE)
names(ltm_recovery) <- c("Sample Size", "Test Length",
"corr_b1", "bias_b1", "RMSE_b1",
"corr_b2", "bias_b2", "RMSE_b2")
round(ltm_recovery, 3)## Sample Size Test Length corr_b1 bias_b1 RMSE_b1 corr_b2 bias_b2 RMSE_b2
## 1 500 40 0.984 -0.006 0.144 0.987 0.009 0.141
## 2 1000 40 0.992 -0.011 0.099 0.994 -0.008 0.096
## 3 5000 40 0.998 -0.004 0.045 0.999 -0.001 0.045
## 4 500 60 0.984 -0.040 0.162 0.985 -0.027 0.160
## 5 1000 60 0.992 -0.001 0.118 0.992 0.008 0.119
## 6 5000 60 0.998 -0.004 0.052 0.998 0.006 0.053
## 7 500 80 0.984 -0.053 0.173 0.985 -0.037 0.175
## 8 1000 80 0.992 -0.034 0.141 0.993 -0.021 0.139
## 9 5000 80 0.998 -0.006 0.070 0.998 0.010 0.070
## 10 500 100 0.983 -0.066 0.197 0.986 -0.049 0.194
## 11 1000 100 0.991 -0.047 0.146 0.993 -0.028 0.142
## 12 5000 100 0.998 -0.009 0.074 0.999 0.009 0.075