1. Dichotomous items
The function DataGeneration can be used for the
pre-analysis step. This function returns artificial data and some useful
objects for analysis (i.e., theta, data_D,
item_D, and initialitem_d).
In the parameter estimation process, the initialitem can be
used for an input of the function IRTest_Dich (i.e.,
initialitem = initialitem). The data is an
artificial item response data that could be unnecessary if user-imported
item response data is used. The theta and item
are not used for the estimation process. They can be considered as the
true parameters only if the artificial data (data) is used
for analysis.
Alldata <- DataGeneration(seed = 123456789,
model_D = rep(1:2, each=5),
N=500,
nitem_D = 10,
nitem_P = 0,
d = 1.664,
sd_ratio = 2,
prob = 0.3)
data <- Alldata$data_D
item <- Alldata$item_D
initialitem <- Alldata$initialitem_D
theta <- Alldata$theta
If the artificial data (data) is used, the true latent
distribution looks like,
- Analysis (parameter estimation)
###### ######
###### Empirical histogram method ######
###### ######
Mod1 <- IRTest_Dich(initialitem = initialitem,
data = data,
model = rep(1:2, each=5),
latent_dist = "EHM",
max_iter = 200,
threshold = .0001)
###### ######
###### Kernel density estimation method ######
###### ######
# Mod1 <- IRTest_Dich(initialitem = initialitem,
# data = data,
# model = rep(1:2, each=5),
# latent_dist = "KDE",
# bandwidth = "SJ-ste",
# max_iter = 200,
# threshold = .001)
###### ######
###### Normality assumption ######
###### ######
# Mod1 <- IRTest_Dich(initialitem = initialitem,
# data = data,
# model = rep(1:2, each=5),
# latent_dist = "Normal",
# max_iter = 200,
# threshold = .0001)
###### ######
###### Two-component Gaussian mixture distribution ######
###### ######
# Mod1 <- IRTest_Dich(initialitem = initialitem,
# data = data,
# model = rep(1:2, each=5),
# latent_dist = "Mixture",
# max_iter = 200,
# threshold = .0001)
###### ######
###### Davidian curve (for an arbitrarily chosen case of h=4)######
###### ######
# Mod1 <- IRTest_Dich(initialitem = initialitem,
# data = data,
# model = rep(1:2, each=5),
# latent_dist = "DC",
# max_iter = 200,
# threshold = .0001,
# h=4)
### The estimated item parameters
Mod1$par_est
#> a b c
#> [1,] 1.000000 -0.74508109 0
#> [2,] 1.000000 0.51116617 0
#> [3,] 1.000000 0.80336947 0
#> [4,] 1.000000 0.53158883 0
#> [5,] 1.000000 -0.39455364 0
#> [6,] 1.700258 0.01012807 0
#> [7,] 1.526307 0.89420050 0
#> [8,] 2.203176 -1.12635131 0
#> [9,] 1.566285 0.39084407 0
#> [10,] 1.385190 0.76792824 0
### The asymptotic standard errors of item parameters
Mod1$se
#> a b c
#> [1,] NA 0.10304406 NA
#> [2,] NA 0.10098172 NA
#> [3,] NA 0.10355444 NA
#> [4,] NA 0.10112153 NA
#> [5,] NA 0.10034057 NA
#> [6,] 0.1446766 0.06711641 NA
#> [7,] 0.1537574 0.08561540 NA
#> [8,] 0.2434543 0.07258288 NA
#> [9,] 0.1404522 0.07275011 NA
#> [10,] 0.1388418 0.08841969 NA
### The estimated ability parameters
head(Mod1$theta)
#> [1] -0.8551470 -0.6567867 -0.7198031 -1.0039922 -1.2714637 -0.7823374
### The estimated latent distribution
plot_LD(Mod1, xlim = c(-6,6))
2. Polytomous items
As in the case of dichotomous items, the function
DataGeneration can be used for the pre-analysis step. This
function returns artificial data and some useful objects for analysis
(i.e., theta, data_P, item_P, and
initialitem_P).
In the parameter estimation process, the initialitem can be
used for an input of the function IRTest_Poly (i.e.,
initialitem = initialitem). The data is an
artificial item response data that could be unnecessary if user-imported
item response data is used. The theta and item
are not used for the estimation process. They can be considered as the
true parameters only if the artificial data (data) is used
for analysis.
Alldata <- DataGeneration(seed = 123456789,
model_P = "GPCM",
categ = rep(c(3,7), each = 5),
N=1000,
nitem_D = 0,
nitem_P = 10,
d = 1.414,
sd_ratio = 2,
prob = 0.5)
data <- Alldata$data_P
item <- Alldata$item_P
initialitem <- Alldata$initialitem_P
theta <- Alldata$theta
If the artificial data (data) is used, the true latent
distribution looks like,
- Analysis (parameter estimation)
###### ######
###### Kernel density estimation method ######
###### ######
Mod1 <- IRTest_Poly(initialitem = initialitem,
data = data,
model = "GPCM",
latent_dist = "KDE",
bandwidth = "SJ-ste",
max_iter = 200,
threshold = .001)
###### ######
###### Normality assumption ######
###### ######
# Mod1 <- IRTest_Poly(initialitem = initialitem,
# data = data,
# model = "GPCM",
# latent_dist = "Normal",
# max_iter = 200,
# threshold = .001)
###### ######
###### Empirical histogram method ######
###### ######
# Mod1 <- IRTest_Poly(initialitem = initialitem,
# data = data,
# model = "GPCM",
# latent_dist = "EHM",
# max_iter = 200,
# threshold = .001)
###### ######
###### Two-component Gaussian mixture distribution ######
###### ######
# Mod1 <- IRTest_Poly(initialitem = initialitem,
# data = data,
# model = "GPCM",
# latent_dist = "Mixture",
# max_iter = 200,
# threshold = .001)
###### ######
###### Davidian curve (for an arbitrarily chosen case of h=4) ######
###### ######
# Mod1 <- IRTest_Poly(initialitem = initialitem,
# data = data,
# model = "GPCM",
# latent_dist = "DC",
# max_iter = 200,
# threshold = .001,
# h=4)
### The estimated item parameters
Mod1$par_est
#> a b_1 b_2 b_3 b_4 b_5
#> [1,] 1.9320419 0.396698909 0.46603651 NA NA NA
#> [2,] 1.6880632 -0.303763927 0.05188864 NA NA NA
#> [3,] 2.0504530 -0.376056390 -0.24941074 NA NA NA
#> [4,] 1.0605873 -0.159282350 0.16815184 NA NA NA
#> [5,] 0.7835049 0.003729855 0.22658723 NA NA NA
#> [6,] 1.8907281 0.253908544 0.29706486 0.5563451 0.56108999 0.7435219
#> [7,] 1.5439636 -1.603051232 -1.12357477 -1.0013097 -1.06426560 -0.7066684
#> [8,] 0.9479375 -0.332417438 -0.17196848 0.3086190 0.06595696 -0.2510836
#> [9,] 1.4230350 -0.784687384 -0.69487372 -0.6851528 -0.46813907 -0.5758896
#> [10,] 2.5827215 0.546921194 0.94939512 0.7099745 1.04467562 1.0309654
#> b_6
#> [1,] NA
#> [2,] NA
#> [3,] NA
#> [4,] NA
#> [5,] NA
#> [6,] 0.8505778
#> [7,] -0.6593098
#> [8,] 0.1465100
#> [9,] -0.4351178
#> [10,] 1.2435975
### The asymptotic standard errors of item parameters
Mod1$se
#> a b_1 b_2 b_3 b_4 b_5
#> [1,] 0.11560425 0.06407208 0.06475432 NA NA NA
#> [2,] 0.09873818 0.05692682 0.05873271 NA NA NA
#> [3,] 0.12254146 0.05222493 0.05290338 NA NA NA
#> [4,] 0.06700339 0.08275252 0.08438107 NA NA NA
#> [5,] 0.05524448 0.10839246 0.10990234 NA NA NA
#> [6,] 0.11338544 0.06983861 0.08428678 0.1076463 0.11530990 0.10459915
#> [7,] 0.09071435 0.14185389 0.12458750 0.1208955 0.10620807 0.08435877
#> [8,] 0.05380619 0.11427616 0.13330931 0.1692535 0.18844552 0.17129060
#> [9,] 0.08124190 0.10135122 0.11385157 0.1183858 0.11737822 0.10510023
#> [10,] 0.16806178 0.05840518 0.08680555 0.1003863 0.09807777 0.09198491
#> b_6
#> [1,] NA
#> [2,] NA
#> [3,] NA
#> [4,] NA
#> [5,] NA
#> [6,] 0.08789139
#> [7,] 0.07567987
#> [8,] 0.13258224
#> [9,] 0.08666445
#> [10,] 0.07509506
### The estimated ability parameters
head(Mod1$theta)
#> [1] -0.5375323 -0.5787214 -0.2605974 -1.0428218 -0.9306040 -1.2750381
### The estimated latent distribution
plot_LD(Mod1, xlim = c(-6,6))
3. Mixed-format test
As in the case of dichotomous and polytomous items, the function
DataGeneration can be used for the pre-analysis step. This
function returns artificial data and some useful objects for analysis
(i.e., theta, data_D, item_D,
initialitem_D, data_P, item_P,
and initialitem_P).
In the parameter estimation process, the initialitem can be
used for an input of the function IRTest_Mix (i.e.,
initialitem = initialitem). The data is an
artificial item response data that could be unnecessary if user-imported
item response data is used. The theta and item
are not used for the estimation process. They can be considered as the
true parameters only if the artificial data (data) is used
for analysis.
Alldata <- DataGeneration(seed = 123456789,
model_D = rep(2,5),
model_P = "GPCM",
categ = rep(3,5),
N=1000,
nitem_D = 5,
nitem_P = 5,
d = 1.664,
sd_ratio = 1,
prob = 0.5)
DataD <- Alldata$data_D
DataP <- Alldata$data_P
itemD <- Alldata$item_D
itemP <- Alldata$item_P
initialitemD <- Alldata$initialitem_D
initialitemP <- Alldata$initialitem_P
theta <- Alldata$theta
If the artificial data (data) is used, the true latent
distribution looks like,
- Analysis (parameter estimation)
###### ######
###### Kernel density estimation method ######
###### ######
Mod1 <- IRTest_Mix(initialitem_D = initialitemD,
initialitem_P = initialitemP,
data_D = DataD,
data_P = DataP,
model_D = rep(2,5),
model_P = "GPCM",
latent_dist = "KDE",
bandwidth = "SJ-ste",
max_iter = 200,
threshold = .001)
###### ######
###### Normality assumption ######
###### ######
# Mod1 <- IRTest_Mix(initialitem_D = initialitemD,
# initialitem_P = initialitemP,
# data_D = DataD,
# data_P = DataP,
# model_D = rep(2,5),
# model_P = "GPCM",
# latent_dist = "Normal",
# max_iter = 200,
# threshold = .001)
###### ######
###### Empirical histogram method ######
###### ######
# Mod1 <- IRTest_Mix(initialitem_D = initialitemD,
# initialitem_P = initialitemP,
# data_D = DataD,
# data_P = DataP,
# model_D = rep(2,5),
# model_P = "GPCM",
# latent_dist = "EHM",
# max_iter = 200,
# threshold = .001)
###### ######
###### Two-component Gaussian mixture distribution ######
###### ######
# Mod1 <- IRTest_Mix(initialitem_D = initialitemD,
# initialitem_P = initialitemP,
# data_D = DataD,
# data_P = DataP,
# model_D = rep(2,5),
# model_P = "GPCM",
# latent_dist = "Mixture",
# max_iter = 200,
# threshold = .001)
###### ######
###### Davidian curve (for an arbitrarily chosen case of h=4) ######
###### ######
# Mod1 <- IRTest_Mix(initialitem_D = initialitemD,
# initialitem_P = initialitemP,
# data_D = DataD,
# data_P = DataP,
# model_D = rep(2,5),
# model_P = "GPCM",
# latent_dist = "DC",
# max_iter = 200,
# threshold = .001,
# h = 4)
### The estimated item parameters
Mod1$par_est
#> $Dichotomous
#> a b c
#> [1,] 2.200954 0.9115074 0
#> [2,] 1.984903 -1.0332999 0
#> [3,] 1.110728 0.4980855 0
#> [4,] 1.258792 0.5392781 0
#> [5,] 2.287480 1.4436252 0
#>
#> $Polytomous
#> a b_1 b_2 b_3 b_4 b_5 b_6
#> [1,] 1.966701 0.3937469 0.42467910 NA NA NA NA
#> [2,] 2.007852 -0.3261131 0.06844343 NA NA NA NA
#> [3,] 2.056885 -0.3897486 -0.22236003 NA NA NA NA
#> [4,] 1.030497 -0.1731280 0.19751324 NA NA NA NA
#> [5,] 0.785141 0.2433940 0.08376259 NA NA NA NA
### The asymptotic standard errors of item parameters
Mod1$se
#> $Dichotomous
#> a b c
#> [1,] 0.15475720 0.04668629 NA
#> [2,] 0.14171212 0.05355501 NA
#> [3,] 0.08520384 0.06947386 NA
#> [4,] 0.09141200 0.06332382 NA
#> [5,] 0.19665345 0.06471224 NA
#>
#> $Polytomous
#> a b_1 b_2 b_3 b_4 b_5 b_6
#> [1,] 0.11688799 0.05981212 0.05958615 NA NA NA NA
#> [2,] 0.11456065 0.05304505 0.05249768 NA NA NA NA
#> [3,] 0.11996224 0.05626596 0.05611498 NA NA NA NA
#> [4,] 0.06432937 0.08543217 0.08539457 NA NA NA NA
#> [5,] 0.05401889 0.11481480 0.11424988 NA NA NA NA
### The estimated ability parameters
head(Mod1$theta)
#> [1] -0.5775496 -0.7558405 0.6185433 -0.8691925 -1.3624809 -1.5623822
### The estimated latent distribution
plot_LD(Mod1, xlim = c(-6,6))
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