Quick Start Guide"

In irt: Item Response Theory and Computerized Adaptive Testing Functions

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

library(irt)

Basic Objects

irt package contains many useful functions commonly used in psychometrics.

Item parameters are defined within three main objects types:

• Item object contains all of the information about a test/survey item. It is the building block of Testlet and Itempool objects. It mainly contains item parameters and item's psychometric model. But it can also contain many other item attributes such as item content, item_id. User can also define additional attributes, for example, key, enemies, word_count, seed_group, etc.

• Testlet object is a set of Item objects. It can also have it's own psychometric model. User can also define additional attributes.

• Itempool object is a collection of Item and Testlet objects.

Item

In order to create an Item object, the psychometric model and item parameter values is sufficient. Specifying an item_id field is required if Item will be used within an Itempool or Testlet.

3PL Model

A three parameter logistic model item (3PL) requires a, b and c parameters to be specified:

item1 <- item(a = 1.2, b = -.8, c = .33, model = "3PL")
item1

a is the item discrimination, b is the item difficulty and c is the pseudo-guessing parameter.

By default, the value of scaling constant D is specified as 1. But it can be overridden:

item1 <- item(a = 1.2, b = -.8, c = .33, D = 1.7, model = "3PL")
item1

item_id and content field can be specified as well:

item1 <- item(a = 1.2, b = -.8, c = .33, D = 1.7, model = "3PL", 
              item_id = "ITM384", content = "Quadratic Equations")
item1

Additional fields can be added through misc field:

item1 <- item(a = 1.2, b = -.8, c = .33, D = 1.7, model = "3PL", 
              item_id = "ITM384", content = "Quadratic Equations", 
              misc = list(key = "A", 
                          enemies = c("ITM664", "ITM964"), 
                          seed_year = 2020, 
                          target_grade = "11")
              )
item1

An item characteristic curve can be plotted using plot function:

plot(item1)

Rasch Model

Rasch model item requires b parameter to be specified:

item2 <- item(b = -.8, model = "Rasch")
item2

For Rasch model, D parameter cannot be specified.

1PL Model

A one-parameter model item requires b parameter to be specified:

item3 <- item(b = -.8, D = 1.7, model = "1PL")
item3

2PL Model

A two-parameter model item requires a and b parameters to be specified:

item4 <- item(a = 1.2, b = -.8, D = 1.702, model = "2PL")
item4

4PL Model

A four-parameter model item requires a, b, c and d parameters to be specified:

item5 <- item(a = 1.06, b = 1.76, c = .13, d = .98, model = "4PL", 
              item_id = "itm-5")
item5

d is the upper-asymptote parameter.

Graded Response Model (GRM)

A Graded Response model item requires a and b parameters to be specified. b parameters is ascending vector of threshold parameters:

item6 <- item(a = 1.22, b = c(-1.9, -0.37, 0.82, 1.68), model = "GRM", 
              item_id = "itm-6")
item6
plot(item6)

D parameter can also be specified.

Generalized Partial Credit Model (GPCM)

A Generalized Partial Credit model item requires a and b parameters to be specified. b parameters is ascending vector of threshold parameters:

item7 <- item(a = 1.22, b = c(-1.9, -0.37, 0.82, 1.68), D = 1.7, model = "GPCM", 
              item_id = "itm-7")
item7

Partial Credit Model (PCM)

A Partial Credit model item requires b parameters to be specified. b parameters is ascending vector of threshold parameters:

item8 <- item(b = c(-1.9, -0.37, 0.82, 1.68), model = "PCM")
item8

Generating Random Item Parameters

An item with random item parameters can be generated using generate_item function:

generate_item("3PL")
generate_item("2PL")
generate_item("Rasch")
generate_item("GRM")
# The number of categories of polytomous items can be specified:
generate_item("GPCM", n_categories = 5)

Testlet

A testlet is simply a collection of Item objects:

item1 <- item(a = 1.2, b = -.8, c = .33, D = 1.7, model = "3PL", 
              item_id = "ITM384", content = "Quadratic Equations")
item2 <- item(a = 0.75, b = 1.8, c = .21, D = 1.7, model = "3PL", 
              item_id = "ITM722", content = "Quadratic Equations")
item3 <- item(a = 1.06, b = 1.76, c = .13, d = .98, model = "4PL", 
              item_id = "itm-5")
t1 <- testlet(c(item1, item2, item3))
t1

An testlet_id field is required if testlet will be used in an item pool.

t1 <- testlet(item1, item2, item3, testlet_id = "T1")
t1

Itempool

An Itempool object is the most frequently used object type in irt package. It is a collection of Item and Testlet objects.

item1 <- generate_item("3PL", item_id = "I1") 
item2 <- generate_item("3PL", item_id = "I2") 
item3 <- generate_item("3PL", item_id = "I3") 
ip1 <- itempool(item1, item2, item3)

Item pools can be composed of items from different psychometric models and testlets:

item4 <- generate_item("GRM", item_id = "I4") 
item5 <- generate_item("3PL", item_id = "T1-I1") 
item6 <- generate_item("3PL", item_id = "T1-I2") 
t1 <- testlet(item5, item6, item_id = "T1")
ip2 <- itempool(item1, item2, item3, item4, t1)

Most of the time item pools are generated using data frames:

n_item <- 6 # Number of items
ipdf <- data.frame(a = rlnorm(n_item), b = rnorm(n_item), 
                   c = runif(n_item, 0, .3))
ip3 <- itempool(ipdf)
ip3

# Scaling constant can be specified
ip4 <- itempool(ipdf, D = 1.7)
ip4

ipdf <- data.frame(
  item_id = c("Item_1", "Item_2", "Item_3", "Item_4", "Item_5", "Item_6"), 
  model = c("3PL", "3PL", "3PL", "GPCM", "GPCM", "GPCM"), 
  a = c(1.0253, 1.3609, 1.6617, 1.096, 0.9654, 1.3995), 
  b1 = c(NA, NA, NA, -1.112, -0.1709, -1.1324), 
  b2 = c(NA, NA, NA, -0.4972, 0.2778, -0.5242), 
  b3 = c(NA, NA, NA, -0.0077, 0.9684, NA), 
  D = c(1.7, 1.7, 1.7, 1.7, 1.7, 1.7), 
  b = c(0.7183, -0.4107, -1.5452, NA, NA, NA), 
  c = c(0.0871, 0.0751, 0.0589, NA, NA, NA), 
  content = c("Geometry", "Algebra", "Algebra", "Geometry", "Algebra", 
              "Algebra") 
)

ip5 <- itempool(ipdf)

Itempool objects can also be converted to a data frame:

as.data.frame(ip2)

Basic IRT Functions

Probability

Probability of correct response (for dichotomous items) and probability of each category (for polytomous items) can be calculated using prob function:

item1 <- generate_item("3PL")
theta <- 0.84
# The probability of correct and incorrect response for `item1` at theta = 0.84
prob(item1, theta)

# Multiple theta values
prob(item1, theta = c(-1, 1))

# Polytomous items:
item2 <- generate_item(model = "GPCM")
prob(item2, theta = 1)
prob(item2, theta = c(-1, 0, 1))

Probability of correct response (or category) for each item in an item pool can be calculated as:

ip <- generate_ip(model = "3PL", n = 7)
ip
prob(ip, theta = 0)
# When there are multiple theta values, a list where each element corresponds
# to a theta value returned. 
prob(ip, theta = c(-2, 0, 1))

Item characteristic curves (ICC) can be plotted:

# Plot ICC of each item in the item pool
plot(ip)

# Plot test characteristic curve
plot(ip, type = "tcc")

Information

Information value of an item at a given $\theta$ value can also be calculated:

item1 <- generate_item("3PL")
info(item1, theta = -2)

# Multiple theta values
info(item1, theta = c(-1, 1))

# Polytomous items:
item2 <- generate_item(model = "GPCM")
info(item2, theta = 1)
info(item2, theta = c(-1, 0, 1))

Information values for each item in an item pool can be calculated as:

ip <- generate_ip(model = "3PL", n = 7)
ip
info(ip, theta = 0)
info(ip, theta = c(-2, 0, 1))

Information functions can be plotted:

# Plot information function of each item
plot_info(ip)
# Plot test information function
plot_info(ip, tif = TRUE)

Ability Estimation

For a given set of item parameters and item responses, the ability ($\theta$) estimates can be calculated using est_ability function.

# Generate an item pool 
ip <- generate_ip(model = "2PL", n = 10)
true_theta <- rnorm(5)
resp <- sim_resp(ip = ip, theta = true_theta, output = "matrix")

# Calculate raw scores
est_ability(resp = resp, ip = ip, method = "sum_score")
# Estimate ability using maximum likelihood estimation:
est_ability(resp = resp, ip = ip, method = "ml")
# Estimate ability using EAP estimation:
est_ability(resp = resp, ip = ip, method = "eap")
# Estimate ability using EAP estimation with a different prior 
# (prior mean = 0, prior standard deviation = 2):
est_ability(resp = resp, ip = ip, method = "eap", prior_pars = c(0, 2))

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