Prob.model {IRTBEMM}

Calculate the probabilites based on a given model and parameters.

Description

Based on the given model, return the correct probabilities of a single examinne with ability X answering each item.

Usage

Prob.model(X, Model, Par.est0, D=1.702)

Arguments

X

A numeric with length=1 consists of an examinee's ability theta.

Model

A character to declare the type of items to be modeled. The parameter labels follow conventional uses, can be:

• '3PL' - Three parameter logistic (3PL) model proposed by Birnbaum(1968):

  P(x = 1|\theta, a, b, c) = c + (1 - c) / (1 + exp(-D * a * (\theta - b)))

  where x=1 is the correct response, theta is examinne's ability; a, b and c are the item discrimination, difficulty and guessing parameter, respectively; D is the scaling constant 1.702.

• '4PL' - Four parameter logistic (4PL) model proposed by Barton & Lord's(1981). Transfer the unslipping (upper asymptote) parameter d to slipping parameter s by set s=1-d:

  P(x = 1|\theta, a, b, c, s) = c + (1 - s - c) / (1 + exp(-D * a * (\theta - b)))

  where x=1 is the correct response; theta is examinne's ability. a, b, c and s are the item discrimination, difficulty guessing and slipping parameter, respectively; D is the scaling constant 1.702.

• '1PLG' - One parameter logsitc guessing (1PLG) model proposed by San Martín et al.(2006). Let invlogit(x)=1 / (1 + exp(-x)):

  P(x = 1|\theta, \beta, \gamma) = invlogit(\theta - \beta) + (1 - invlogit(\theta - \beta)) * invlogit(\gamma)

  where x=1 is the correct response, theta is examinne's ability; beta and gamma are the item difficulty and guessing parameter, respectively.

• '1PLAG' - One parameter logsitc ability-based guessing (1PLAG) model proposed by San Martín et al.(2006). Let invlogit(x)=1 / (1 + exp(-x)):

  P(x = 1|\theta, \alpha, \beta, \gamma) = invlogit(\theta - \beta) + (1 - invlogit(\theta - \beta)) * invlogit(\alpha * \theta + \gamma)

  where x=1 is the correct response, theta is examinne's ability; alpha is the weight of the ability in the guessing component; beta and gamma are the item difficulty and guessing parameter, respectively.

These parameter labels are capitalized in program for emphasis.

Par.est0

A list that consists of item parameters for each item based on the given model. Can be:

• For 3PL model, list(A, B, C) - A, B, C are numeric refer to item discrimination, difficulty and pseudo guessing parameters for each item, respectively.

• For 4PL model, list(A, B, C, S) - A, B, C, S are numeric refer to item discrimination, difficulty, pseudo guessing and slipping parameters for each item, respectively.

• For 1PLG model, list(Beta, Gamma) - Beta, Gamma are numeric refer to item difficulty and guessing (on the logistic scales) parameters for each item, respectively.

• For 1PLAG model, list(Alpha, Beta, Gamma) - Alpha refers to the weight of the ability in the guessing component, and Beta and Gamma are numeric refer to item difficulty and guessing (on the logistic scales) parameters for each item, respectively.

Please note these capitalized parameter lables are transformed from the Model section.

D

A single numeric refers to the scaling constant only used in the 3PL and 4PL model. By default, D=1.702.

Value

A numeric consists of the correct probabilities of a single examinne with ability X answering each item.

References

Barton, M. A., & Lord, F. M. (1981). An upper asymptote for the three-parameter logistic item response model. ETS Research Report Series, 1981(1), 1-8. doi:10.1002/j.2333-8504.1981.tb01255.x

Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In F. M. Lord & M. R. Novick (Eds.), Statistical theories of mental test scores (pp. 395-479). MA: Adison-Wesley.

San Martín, E., Del Pino, G., & De Boeck, P. (2006). IRT models for ability-based guessing. Applied Psychological Measurement, 30(3), 183-203. doi:10.1177/0146621605282773

Examples

#Obtain the correct probabilities of five 3PL model items when theta=1.2 and D=1.702. 
library(IRTBEMM)
th=1.2                           #Examinee's ability parameter theta
A=c(1.5, 2, 0.5, 1.2, 0.4)       #item discrimination parameters
B=c(-0.5, 0, 1.5, 0.3, 2.8)      #item difficulty parameters
C=c(0.1, 0.2, 0.3, 0.15, 0.25)   #item pseudo guessing parameters
Par3PL=list(A=A, B=B, C=C)       #Create a list for 3PL
P.3pl=Prob.model(X=th, Model='3PL', Par.est0=Par3PL)   #Obtain the 3PL probabilities

#Obtain the correct probabilities of five 4PL model items when theta=1.2 and D=1. 
S=c(0.3, 0.1, 0.13, 0.09, 0.05)  #item pseudo slipping parameters
Par4PL=list(A=A, B=B, C=C, S=S)  #Create a list for 4PL
P.4pl=Prob.model(X=th, Model='4PL', Par.est0=Par4PL, D=1)   #Obtain the 4PL probabilities

#Obtain the correct probabilities of three 1PLG model items when theta=0.3.
th=0.3
Beta=c(0.8, -1.9, 2.4)
Gamma=c(-1.31, -0.89, -0.18)
Par1PLG=list(Beta=Beta, Gamma=Gamma)                 #Create a list for 1PLG
P.1plg=Prob.model(X=th, Model='1PLG', Par.est0=Par1PLG)   #Obtain the 1PLG probabilities

#Obtain the correct probabilities of three 1PLAG model items when theta=0.3.
Alpha=0.2
Par1PLAG=list(Alpha=Alpha, Beta=Beta, Gamma=Gamma)     #Create a list for 1PLAG
P.1plag=Prob.model(X=th, Model='1PLAG', Par.est0=Par1PLAG)   #Obtain the 1PLAG probabilities