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Four-Parameter Logistic Model

Source: R/ICCtoTIC.R
LogisticModel.Rd

Computes the probability of a correct response using the four-parameter logistic model (4PLM) in Item Response Theory.

Usage

LogisticModel(x, a = 1, b, c = 0, d = 1)

Arguments

x

Numeric. The ability parameter (theta).

a

Numeric. The slope (discrimination) parameter. Default is 1.

b

Numeric. The location (difficulty) parameter.

c

Numeric. The lower asymptote (guessing) parameter. Default is 0.

d

Numeric. The upper asymptote (carelessness) parameter. Default is 1.

Value

A numeric value representing the probability of a correct response.

Details

The four-parameter logistic model extends the 3PLM by adding an upper asymptote parameter d, which accounts for careless errors by high-ability examinees.

The model formula is: $$P(\theta) = c + \frac{d - c}{1 + \exp(-a(\theta - b))}$$

Special cases:

  • 1PLM: a = 1, c = 0, d = 1

  • 2PLM: c = 0, d = 1

  • 3PLM: d = 1

See also

ItemInformationFunc

Examples

# Compute probability for ability = 0, difficulty = 0
LogisticModel(x = 0, a = 1, b = 0, c = 0, d = 1) # Returns 0.5
#> [1] 0.5

# 3PLM with guessing parameter
LogisticModel(x = -3, a = 1.5, b = 0, c = 0.2, d = 1)
#> [1] 0.2087896