sirt (version 0.31-20)

sim.qm.ramsay: Simulate from Ramsay's Quotient Model

Description

This function simulates dichotomous item response data according to Ramsay's quotient model (Ramsay, 1989).

Usage

sim.qm.ramsay(theta, b, K)

Arguments

theta

Vector of of length $N$ person parameters (must be positive!)

b

Vector of length $I$ of item difficulties (must be positive)

K

Vector of length $I$ of guessing parameters (must be positive)

Value

• An $N$ times $I$ data frame with dichotomous item responses.

Details

Ramsay's quotient model (Ramsay, 1989) is defined by the equation $P(X_{pi} = 1 | \theta_p ) = \frac{ exp{ ( \theta_p / b_i ) } } { K_i + exp{ ( \theta_p / b_i ) } }$

References

Ramsay, J. O. (1989). A comparison of three simple test theory models. Psychometrika, 54, 487-499. van der Maas, H. J. L., Molenaar, D., Maris, G., Kievit, R. A. & Borsboom, D. (2011). Cognitive psychology meets psychometric theory: On the relation between process models for decision making and latent variable models for individual differences. Psychological Review, 318, 339-356.

See Also

See rasch.mml2 for estimating Ramsay's quotient model. See sim.raschtype for simulating response data from the generalized logistic item response model.

Examples

#***
set.seed(657)
# simulate data according to the Ramsay model
N <- 1000       # persons
I <- 11         # items
theta <- exp( rnorm( N ) )  # person ability
b <- exp( seq(-2,2,len=I))  # item difficulty
K <- rep( 3 , I )           # K parameter (=> guessing)

# apply simulation function
dat <- sim.qm.ramsay( theta , b , K )

#***
# analysis
mmliter <- 50       # maximum number of iterations
I <- ncol(dat)
fixed.K <- rep( 3 , I ) 

# Ramsay QM with fixed K parameter (K=3 in fixed.K specification)
mod1 <- rasch.mml2( dat , mmliter = mmliter , irtmodel = "ramsay.qm", 
	numdiff.parm = .0001 ,  fixed.K = fixed.K )                  
summary(mod1)

# Ramsay QM with joint estimated K parameter
mod2 <- rasch.mml2( dat , mmliter = mmliter , irtmodel = "ramsay.qm" , 
	numdiff.parm = .0001 ,  est.K = rep(1,I)  )                  
summary(mod2)

# Ramsay QM with itemwise estimated K parameter
mod3 <- rasch.mml2( dat , mmliter = mmliter , irtmodel = "ramsay.qm" , 
	numdiff.parm = .0001 ,   est.K = 1:I  )                  
summary(mod3)

# Rasch model
mod4 <- rasch.mml2( dat )
summary(mod4)

# generalized logistic model
mod5 <- rasch.mml2( dat , est.alpha = TRUE )
summary(mod5) 

# 2PL model
mod6 <- rasch.mml2( dat , est.a = rep(1,I) )
summary(mod6) 

# Difficulty + Guessing (b+c) Model
mod7 <- rasch.mml2( dat , est.c = rep(1,I) )
summary(mod7) 

# estimate separate guessing (c) parameters
mod8 <- rasch.mml2( dat , est.c = 1:I  )
summary(mod8)