area_between_icc {irt}

Calculate the area between two ICC curves

Description

This function calculates the area between two item characteristic curves (ICC) for unidimensional dichotomous IRT models.

There are two types of area calculation methods. The first one is type = "exact" where the exact area from negative infinity to positive infinity between the two ICC curves will be calculated. This method implements the approach in Raju's 1988 paper. This method works for 'Rasch', '1PL', '2PL', '3PL' models but when the pseudo-guessing parameters of the items differ for '3PL' model, the area will be infinity. In such cases it is advisable to use type = "closed".

The area can only be calculated for 'Rasch', '1PL', '2PL', '3PL' or '4PL' models.

Usage

area_between_icc(
  ...,
  type = c("closed", "exact"),
  theta_range = c(-5, 5),
  signed_area = FALSE
)

Arguments

...

An Itempool-class object; or a combination of Item-class and Testlet-class objects.

type

A string representing the method that will be used to calculate the area between two ICC curves. Available values are:

"exact"

The exact area between the whole theta scale -Inf and Inf. This method implements Raju's (1988) approach. When the pseudo-guessing parameters of the items differ for '3PL' model, the area will be infinity. See Raju (1988) for details.

"closed"

The area within a closed interval defined by theta_range argument will be calculated. This method always returns a finite value. See Kim and Cohen (1991) for details.

The default method is "closed".

theta_range

A numeric vector of length two with the first element smaller than the second element. The values define the boundaries in which the area between two ICC's will be calculated. The default value is c(-5, 5).

signed_area

A logical value for whether the signed or unsigned area between two curves will be calculated. When signed = TRUE, the area under the second item is subtracted from the area under the first item. The result can be negative if the first item is mostly under the second item. When signed = FALSE, the distance between two ICC curves will be calculated. The default value is signed = TRUE.

Value

A matrix where the values in cells are the areas between items. The rows represent the first item and the columns represents the second item and the area of second item is subtracted from the first item when "signed" area is desired. For example, the value corresponding to the cell where row is for "Item_4" and column is for "Item_2", the value in the cell is the area of "Item_4 - Item_2".

Author(s)

Emre Gonulates

References

Kim, S.-H., & Cohen, A. S. (1991). A comparison of two area measures for detecting differential item functioning. Applied Psychological Measurement, 15(3), 269–278.

Raju, N. S. (1988). The area between two item characteristic curves. Psychometrika, 53(4), 495–502.

Examples

# Closed area example:
ip <- generate_ip(model = c("3PL", "3PL", "3PL"))
# plot(ip) # See the ICCs
area_between_icc(ip, type = "closed")
area_between_icc(ip, type = "closed", signed_area = TRUE)

# The result is infinite because 'c' parameters are not equal
area_between_icc(ip, type = "exact")

# Exact area example:
ip <- generate_ip(model = c("2PL", "2PL", "2PL"))
area_between_icc(ip, type = "exact")
area_between_icc(ip, type = "exact", signed_area = TRUE)
# The 'closed' area is very close to the 'exact' area with a wide theta range
area_between_icc(ip, type = "closed", theta_range = c(-10, 10))