One booklet 2PL items - tmatta/lsasim GitHub Wiki
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: 2PL 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 - The bounds of the item discrimination parameter, a, are constrained to
a_bounds = (0.75, 1.25)where 0.75 is the lowest generating value and 1.25 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- b-parameter recovered well, with correlation ranging from 0.989 to 0.999, with bias ranging from -0.013 to -0.001, and with RMSE ranging from 0.057 to 0.206
- a-parameter recovered moderately, with correlation ranging from 0.699 to 0.962, with bias ranging from -0.035 to 0.015, and with RMSE ranging from 0.044 to 0.152
- For b-parameter, sample sizes of 5000 consistently produced the most accurate results
- For a-parameter, when sample size increased, recovery accuracy improved further
- For b- and a-parameters, the 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 2PL model
set.seed(4364) # fix item parameters across replications
item_pool <- lsasim::item_gen(n_2pl = I,
thresholds = 1,
b_bounds = c(-2, 2),
a_bounds = c(0.75, 1.25))
for (K in K.cond) { #number of booklets
for (r in 1:reps) { #replication
#------------------------------------------------------------------------------#
# Data simulation
#------------------------------------------------------------------------------#
set.seed(8088*(r+2))
# 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,
a_par = item_pool$a)
# extract item responses (excluding "subject" column)
resp <- cog[, c(1:I)]
#------------------------------------------------------------------------------#
# Item calibration
#------------------------------------------------------------------------------#
# fit 2PL model using mirt package
mirt.mod <- NULL
mirt.mod <- mirt::mirt(resp, 1, itemtype = '2PL', verbose = F)
# fit 2PL model using TAM package
tam.mod <- NULL
tam.mod <- TAM::tam.mml.2pl(resp)
# fit 2PL model using ltm package
ltm.mod <- NULL
ltm.mod <- ltm::ltm(resp ~ z1, IRT.param=T)
#------------------------------------------------------------------------------#
# Item parameter extraction
#------------------------------------------------------------------------------#
# extract b and a in mirt package
mirt_b <- coef(mirt.mod, IRTpars = TRUE, simplify=TRUE)$items[,"b"]
mirt_a <- coef(mirt.mod, IRTpars = TRUE, simplify=TRUE)$items[,"a"]
# convert TAM output into 2PL parametrization
tam_b <- (tam.mod$item$AXsi_.Cat1/tam.mod$item$B.Cat1.Dim1)
tam_a <- (tam.mod$item$B.Cat1.Dim1)
# extract Dffclt and Dscrmn in ltm package
ltm_b <- data.frame(coef(ltm.mod))$Dffclt
ltm_a <- data.frame(coef(ltm.mod))$Dscrmn
#------------------------------------------------------------------------------#
# Item parameter recovery
#------------------------------------------------------------------------------#
# summarize results
itempars <- data.frame(matrix(c(N, I, K, r), nrow=1))
colnames(itempars) <- c("N", "I", "K", "rep")
# calculate corr, bias, RMSE for item parameters in mirt pacakge
itempars$corr_mirt_b <- cor( item_pool$b, mirt_b)
itempars$bias_mirt_b <- mean( mirt_b - item_pool$b )
itempars$RMSE_mirt_b <- sqrt(mean( ( mirt_b - item_pool$b)^2 ))
itempars$corr_mirt_a <- cor( item_pool$a, mirt_a)
itempars$bias_mirt_a <- mean( mirt_a - item_pool$a )
itempars$RMSE_mirt_a <- sqrt(mean( ( mirt_a - item_pool$a)^2 ))
# calculate corr, bias, RMSE for item parameters in TAM pacakge
itempars$corr_tam_b <- cor( item_pool$b, tam_b)
itempars$bias_tam_b <- mean( tam_b - item_pool$b )
itempars$RMSE_tam_b <- sqrt(mean( ( tam_b - item_pool$b)^2 ))
itempars$corr_tam_a <- cor( item_pool$a, tam_a)
itempars$bias_tam_a <- mean( tam_a - item_pool$a )
itempars$RMSE_tam_a <- sqrt(mean( ( tam_a - item_pool$a)^2 ))
# calculate corr, bias, RMSE for item parameters in ltm pacakge
itempars$corr_ltm_b <- cor( item_pool$b, ltm_b)
itempars$bias_ltm_b <- mean( ltm_b - item_pool$b )
itempars$RMSE_ltm_b <- sqrt(mean( ( ltm_b - item_pool$b)^2 ))
itempars$corr_ltm_a <- cor( item_pool$a, ltm_a)
itempars$bias_ltm_a <- mean( ltm_a - item_pool$a )
itempars$RMSE_ltm_a <- sqrt(mean( ( ltm_a - item_pool$a)^2 ))
# combine results
results <- rbind(results, itempars)
}
}
}
}
- Correlation, bias, and RMSE for item parameter recovery in
mirtpackage
mirt_recovery <- aggregate(cbind(corr_mirt_b, bias_mirt_b, RMSE_mirt_b,
corr_mirt_a, bias_mirt_a, RMSE_mirt_a) ~ N + I,
data=results, mean, na.rm=TRUE)
names(mirt_recovery) <- c("Sample Size", "Test Length",
"corr_b", "bias_b", "RMSE_b",
"corr_a", "bias_a", "RMSE_a")
round(mirt_recovery, 3)
## Sample Size Test Length corr_b bias_b RMSE_b corr_a bias_a RMSE_a
## 1 500 40 0.990 -0.005 0.199 0.699 0.013 0.152
## 2 1000 40 0.995 -0.009 0.133 0.814 0.005 0.106
## 3 5000 40 0.999 -0.002 0.058 0.952 0.000 0.047
## 4 500 60 0.990 -0.004 0.203 0.703 0.014 0.152
## 5 1000 60 0.995 -0.008 0.139 0.814 0.007 0.107
## 6 5000 60 0.999 -0.002 0.062 0.952 0.001 0.047
## 7 500 80 0.990 -0.003 0.192 0.728 0.012 0.146
## 8 1000 80 0.995 -0.004 0.131 0.838 0.006 0.101
## 9 5000 80 0.999 -0.002 0.057 0.960 0.000 0.045
## 10 500 100 0.989 -0.004 0.191 0.739 0.014 0.143
## 11 1000 100 0.995 -0.010 0.132 0.848 0.006 0.099
## 12 5000 100 0.999 -0.002 0.058 0.962 0.000 0.044
- Correlation, bias, and RMSE for item parameter recovery in
TAMpackage
tam_recovery <- aggregate(cbind(corr_tam_b, bias_tam_b, RMSE_tam_b,
corr_tam_a, bias_tam_a, RMSE_tam_a) ~ N + I,
data=results, mean, na.rm=TRUE)
names(tam_recovery) <- c("Sample Size", "Test Length",
"corr_b", "bias_b", "RMSE_b",
"corr_a", "bias_a", "RMSE_a")
round(tam_recovery, 3)
## Sample Size Test Length corr_b bias_b RMSE_b corr_a bias_a RMSE_a
## 1 500 40 0.990 -0.004 0.199 0.699 0.013 0.152
## 2 1000 40 0.995 -0.008 0.133 0.814 0.005 0.106
## 3 5000 40 0.999 -0.001 0.058 0.952 0.000 0.047
## 4 500 60 0.990 -0.004 0.203 0.703 0.015 0.152
## 5 1000 60 0.995 -0.008 0.139 0.814 0.007 0.107
## 6 5000 60 0.999 -0.001 0.062 0.952 0.000 0.047
## 7 500 80 0.990 -0.005 0.194 0.728 0.010 0.145
## 8 1000 80 0.995 -0.003 0.132 0.839 0.002 0.100
## 9 5000 80 0.999 -0.003 0.058 0.960 -0.004 0.045
## 10 500 100 0.989 -0.005 0.196 0.738 0.002 0.141
## 11 1000 100 0.995 -0.012 0.135 0.848 -0.006 0.097
## 12 5000 100 0.999 -0.003 0.062 0.962 -0.013 0.045
- Correlation, bias, and RMSE for item parameter recovery in
ltmpackage
ltm_recovery <- aggregate(cbind(corr_ltm_b, bias_ltm_b, RMSE_ltm_b,
corr_ltm_a, bias_ltm_a, RMSE_ltm_a) ~ N + I,
data=results, mean, na.rm=TRUE)
names(ltm_recovery) <- c("Sample Size", "Test Length",
"corr_b", "bias_b", "RMSE_b",
"corr_a", "bias_a", "RMSE_a")
round(ltm_recovery, 3)
## Sample Size Test Length corr_b bias_b RMSE_b corr_a bias_a RMSE_a
## 1 500 40 0.990 -0.005 0.199 0.699 0.013 0.152
## 2 1000 40 0.995 -0.009 0.133 0.814 0.005 0.106
## 3 5000 40 0.999 -0.002 0.058 0.952 0.000 0.047
## 4 500 60 0.990 -0.004 0.205 0.703 0.012 0.151
## 5 1000 60 0.995 -0.009 0.140 0.814 0.005 0.106
## 6 5000 60 0.999 -0.002 0.062 0.952 -0.003 0.047
## 7 500 80 0.990 -0.008 0.199 0.727 -0.001 0.143
## 8 1000 80 0.995 -0.001 0.136 0.838 -0.010 0.099
## 9 5000 80 0.999 -0.002 0.063 0.960 -0.016 0.047
## 10 500 100 0.989 -0.012 0.206 0.737 -0.018 0.140
## 11 1000 100 0.995 -0.013 0.145 0.847 -0.027 0.098
## 12 5000 100 0.999 -0.002 0.075 0.962 -0.035 0.055