Itempool Objects"

In irt: Item Response Theory and Computerized Adaptive Testing Functions

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

library(irt)

An Itempool object is a collection of Item and Testlet objects. It contains information of all items in a test or all items that will be used in an item pool of an adaptive test.

Creating an Itempool Object

Dichotomous Items (Rasch, 1PL, 2PL, 3PL, 4PL)

There are multiple ways to create an Itempool object. If all of the items are from the same psychometric model, then using a data.frame object to create an Itempool object is the most straightforward way.

# Create an item pool of 2PL items:
ip_dtf <- data.frame(
  a = c(1.1821, 0.6645, 0.8994, 1.0731, 1.0252, 1.2325, 0.9278, 1.0967), 
  b = c(0.4185, -0.5992, 0.2193, 0.8823, 0.4652, 1.4006, -1.1193, -0.3747))

ip <- itempool(ip_dtf, model = "2PL")
ip

If desired, item_id's and content areas of each item can be specified:

ip_dtf <- data.frame( 
  a = c(1.1821, 0.6645, 0.8994, 1.0731, 1.0252, 1.2325, 0.9278, 1.0967), 
  b = c(0.4185, -0.5992, 0.2193, 0.8823, 0.4652, 1.4006, -1.1193, -0.3747),
  item_id = c("i1", "i2", "i3", "i4", "i5", "i6", "i7", "i8"), 
  content = c("Geometry", "Geometry", "Algebra", "Algebra", "Algebra", 
              "Geometry", "Algebra", "Algebra")
  )

ip <- itempool(ip_dtf, model = "2PL")
ip

By default, the scaling parameter D for most of the models are set to 1. This can be overwritten:

ip <- itempool(ip_dtf, model = "2PL", D = 1.7)
ip

Please review psychometric models for other models and required parameters. For example, for 3PL model:

ip_dtf <- data.frame(a = c(0.9303, 1.9423, 0.8417, 1.2622), 
                     b = c(1.3515, -0.5039, -1.7263, 1.3125), 
                     c = c(0.2301, 0.2224, 0.0967, 0.0112))
ip <- itempool(ip_dtf, model = "3PL", D = 1.7)
ip

Polytomous Items (GRM, GPCM, PCM)

For polytomous items, usually a vector of item thresholds should be specified. So, threshold (or step) parameters can given as the following example:

ip_dtf <- data.frame(a = c(1.8619, 1.2458, 1.3213, 0.6174, 1.3625), 
                     b1 = c(-0.3666, -0.9717, -1.1588, 0.1093, 0.0858), 
                     b2 = c(0.3178, 0.2458, -0.4978, 0.6437, 0.5161), 
                     b3 = c(1.0384, 1.2382, 1.2787, 1.3609, 1.2145))

ip <- itempool(ip_dtf, model = "GPCM")
ip

The number of threshold parameters can be differ among items. In those cases, simply fill the rest of the values using NA:

ip_dtf <- data.frame(a = c(1.175, 0.981, 1.0625, 0.9545, 0.7763), 
                     b1 = c(-0.9633, -0.4098, -0.298, 0.0576, -0.5342), 
                     b2 = c(-0.6213, NA, 0.4792, 0.538, 0.0363), 
                     b3 = c(0.5938, NA, NA, 0.9815, NA), 
                     b4 = c(NA, NA, NA, 1.3351, NA))
ip <- itempool(ip_dtf, model = "GRM", D = 1.702)
ip

An example reparameterized "GPCM" item pool (see psychometric models vignette for details of this model):

ip_dtf <- data.frame(a = c(1.1152, 0.8231, 0.9527, 0.6423), 
                     b = c(0.234, 0.0219,  0.7424, -0.3426), 
                     d1 = c(0.0081, 0.8569, -1.5181, -0.8458), 
                     d2 = c(0.3392, NA, -0.1978, 0.3756), 
                     d3 = c(NA, NA, 0.1677, NA))
ip <- itempool(ip_dtf, model = "GPCM2", D = 1.702)
ip

Mixture of Models

If an Itempool object should consist of mixture of items, then the item parameter of each item should be specified as the examples above. But, model should be specified for each item in the data.frame.

For example, the following item pool consists of three 3PL items and two GPCM items:

ip_dtf <- data.frame(
  model = c("3PL", "3PL", "3PL", "GPCM", "GPCM"), 
  a = c(1.6242, 0.9471, 1.4643, 0.6582, 1.0234), 
  b = c(0.4563, -0.2994, -0.3027, NA, NA), 
  c = c(0.0156, 0.0339, 0.1243, NA, NA), 
  b1 = c(NA, NA, NA, -1.1532, -1.2171), 
  b2 = c(NA, NA, NA, -0.5384, -0.3992), 
  b3 = c(NA, NA, NA, 0.0591, 0.1431), 
  b4 = c(NA, NA, NA, NA, 1.52), 
  D = c(1.7, 1.7, 1.7, 1, 1))
ip <- itempool(ip_dtf)
ip

Other Ways to Create Itempool Objects

An Itempool object can be created using individual Item and Testlet objects. For example, in the the following code, individual Item objects are combined to create an Itempool object.

ip <- c(item(model = "3PL", a = 2.09, b = 1.17, c = 0.25, item_id = "i1"), 
        item(model = "3PL", a = 0.59, b = 0.77, c = 0.13, item_id = "i2"), 
        item(model = "3PL", a = 1.67, b = 1.05, c = 0.04, item_id = "i3"), 
        item(model = "3PL", a = 0.84, b = -1.8, c = 0.24, item_id = "i4"), 
        item(model = "GPCM", a = 1.96, b = c(-0.94, -0.09, 0.25), item_id = "i5"), 
        item(model = "GPCM", a = 0.59, b = c(0.07, 1.46), item_id = "i6"), 
        item(model = "GPCM", a = 0.73, b = c(-1.2, -0.78, 0.2, 1.8), item_id = "i7"))
ip

Using this method, Testlet objects can be added to the item pool.

# Create a testlet object with three items. 
t1 <- testlet(c(item(model = "3PL", a = 2.09, b = 1.17, c = 0.25, item_id = "i1"), 
                item(model = "3PL", a = 0.59, b = 0.77, c = 0.13, item_id = "i2"), 
                item(model = "3PL", a = 1.67, b = 1.05, c = 0.04, item_id = "i3")), 
              testlet_id = "Testlet-932")
# Create another testlet object with two items. 
t2 <- testlet(c(item(model = "3PL", a = 0.84, b = -1.8, c = 0.24, item_id = "i4"), 
                item(model = "GPCM", a = 1.96, b = c(-0.94, -0.09, 0.25), 
                     item_id = "i5")), 
              testlet_id = "Testlet-77")
# Standalone items to be added:
i6 <- item(model = "GPCM", a = 0.59, b = c(0.07, 1.46), item_id = "i6")
i7 <- item(model = "GPCM", a = 0.73, b = c(-1.2, -0.78, 0.2, 1.8), item_id = "i7")

# Combine all items and testlets:
ip_testlet <- c(t1, t2, i6, i7)
ip_testlet

Operations with Itempool Objects

We will show the operations that can be performed on an Itempool object using the following three example item pools:

ip1 <- itempool(data.frame( 
  a = c(1.1821, 0.6645, 0.8994, 1.0731, 1.0252, 1.2325, 0.9278, 1.0967), 
  b = c(0.4185, -0.5992, 0.2193, 0.8823, 0.4652, 1.4006, -1.1193, -0.3747),
  item_id = c("i1", "i2", "i3", "i4", "i5", "i6", "i7", "i8"), 
  content = c("Geometry", "Geometry", "Algebra", "Algebra", "Algebra", 
              "Geometry", "Algebra", "Algebra")
  ))

ip_mixed <- itempool(data.frame(
  model = c("3PL", "3PL", "3PL", "GPCM", "GPCM"), 
  a = c(1.6242, 0.9471, 1.4643, 0.6582, 1.0234), 
  b = c(0.4563, -0.2994, -0.3027, NA, NA), 
  c = c(0.0156, 0.0339, 0.1243, NA, NA), 
  b1 = c(NA, NA, NA, -1.1532, -1.2171), 
  b2 = c(NA, NA, NA, -0.5384, -0.3992), 
  b3 = c(NA, NA, NA, 0.0591, 0.1431), 
  b4 = c(NA, NA, NA, NA, 1.52), 
  D = c(1.7, 1.7, 1.7, 1.7, 1.7)))

ip_testlet <- c(
  testlet(c(item(model = "3PL", a = 2.09, b = 1.17, c = 0.25, item_id = "i1"), 
            item(model = "3PL", a = 0.59, b = 0.77, c = 0.13, item_id = "i2"), 
            item(model = "3PL", a = 1.67, b = 1.05, c = 0.04, item_id = "i3")), 
          testlet_id = "Testlet-932"), 
  item(model = "GPCM", a = 0.59, b = c(0.07, 1.46), item_id = "i6"),
  testlet(c(item(model = "3PL", a = 0.84, b = -1.8, c = 0.24, item_id = "i4"), 
            item(model = "GPCM", a = 1.96, b = c(-0.94, -0.09, 0.25), 
                 item_id = "i5")), 
          testlet_id = "Testlet-77"), 
  item(model = "GPCM", a = 0.73, b = c(-1.2, -0.78, 0.2, 1.8), item_id = "i7"))

Combining Item Pools

Two Itempool objects can be combined using c() function.

ip <- c(ip1, ip_mixed)
ip

Subsetting

Itempool objects can be subsetted using brackets [ ]. This operation will always return an Itempool object.

# Subset only the first element of the item pool
ip1[1]

# Create an Itempool using the first and third element:
ip1[c(1, 3)] # Order is important
ip1[c(3, 1)]

# Create an Itempool using all but the second element: 
ip1[-2]

# Subsetting using item ID's:
ip1[c("i2", "i1")]

# Subsetting using logical operators:
ip1[ip1$b < 0]

# Select items with information values larger than 0.2 at theta = 1:
ip1[info(ip1, theta = 1) > 0.2]

Extracting Items

An Item object can be extracted from an Itempool using double bracket operator [[ ]]. This operation will return an Item or Testlet object.

# Extract the second element
ip1[[2]]

# Extract a testlet
ip_testlet[[3]]

Replacing Items of an Itempool

Using the double bracket with assignment operator, individual items of an Itempool object can be replaced.

ip_new <- ip1
# Replace the second item with a new item
ip_new[[2]] <- item(a = 1, b = c(-1, 0, 1), model = "GRM", item_id = "NewItm4",
                    D = 1.7, content = "Quadratic Functions")
ip_new

Accessing Parts of an Itempool object

$ operator can be used to access parts of an Itempool object. Here, we will discuss the most common uses of this operator. The full list of available options can be found in the help page:

?`$,Itempool-method`

# Extract the ID's of the items within an item pool
ip1$item_id

# Extract the contents of the items within an item pool
ip1$content

# Extract the models of the items within an item pool
ip1$model
ip_mixed$model

# Maximum possible score of items
ip1$max_score
ip_mixed$max_score
ip_testlet$max_score

# Maximum scores of each standalone item
ip1$item_max_score
ip_mixed$item_max_score
ip_testlet$item_max_score

Item parameters can also be accessed:

ip1$a
ip1$b
ip1$c
ip1$D

ip_mixed$a
ip_mixed$b
ip_mixed$b1
ip_mixed$b4
ip_mixed$D

Using the operator $n, we can extract a detailed summary of the number of Items and Testlets in the item pool. This is especially useful if the item pool has both standalone and testlet items.

• $n$elements gives the total number of testlets and standalone items.
• $n$testlets gives the total number of testlets.
• $n$items gives the total number of standalone items and items within testlets.

# Extract the number of items within an item pool
ip1$n
# In ip_testlet, there are two testlets and two standalone items. Within those
# two testlets, there are a total of 5 items. At total there are 7 items.
ip_testlet$n

Updating Parts of an Itempool object

Using $<- operator, the parts of an item pool can be updated. For example, using the following code, we can change the item_ids and contents of the items of an item pool.

ip_new <- ip1
ip_new$item_id <- paste0("Question-", 1:length(ip_new))
ip_new$content <- c("M", "M", "R", "M", "E", "R", "E", "M")

New item parameters can be set this way as well:

ip_new$a <- 1
ip_new
ip_new$b <- rnorm(length(ip_new))
ip_new

ip <- generate_ip(model = c("3PL", "3PL", "3PL", "3PL", "GPCM", "GPCM", "GPCM"))
dput(ip)
dput(as.data.frame(generate_ip(model = c("3PL", "3PL", "3PL", "3PL", "3PL"))))
dput(as.data.frame(generate_ip(model = "GPCM", n = 5)))
dput(as.data.frame(generate_ip(model = "GPCM2", n = 4, n_categories = c(3, 2, 4, 3))))
dput(as.data.frame(generate_ip(model = "GPCM", n = 5, n_categories = c(5, 2, 3, 4, 3))))

ip <- generate_ip(model = c("3PL", "3PL", "3PL", "GPCM", "GPCM"))
dput(data.frame(ip))
# list(key = c("A", "C", "D", "B", "B", "A", "D", "C"), 
#                            grade = c(11, 9, 9, 10, 10, 10, 11, 9))

irt documentation built on May 29, 2024, 12:02 p.m.