sirt (version 4.1-15)

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)

Value

An \(N \times I\) data frame with dichotomous item responses.

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)

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

#############################################################################
# EXAMPLE 1: Estimate Ramsay Quotient Model with rasch.mml2
#############################################################################

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

# apply simulation function
dat <- sirt::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 <- sirt::rasch.mml2( dat, mmliter=mmliter, irtmodel="ramsay.qm",
              fixed.K=fixed.K )
summary(mod1)

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

if (FALSE) {
# Ramsay QM with itemwise estimated K parameters
mod3 <- sirt::rasch.mml2( dat, mmliter=mmliter, irtmodel="ramsay.qm",
              est.K=1:I  )
summary(mod3)

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

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

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

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

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

#*** estimate Model 1 with user defined function in mirt package

# create user defined function for Ramsay's quotient model
name <- 'ramsayqm'
par <- c("K"=3, "b"=1 )
est <- c(TRUE, TRUE)
P.ramsay <- function(par,Theta){
     eps <- .01
     K <- par[1]
     b <- par[2]
     num <- exp( exp( Theta[,1] ) / b )
     denom <- K + num
     P1 <- num / denom
     P1 <- eps + ( 1 - 2*eps ) * P1
     cbind(1-P1, P1)
}

# create item response function
ramsayqm <- mirt::createItem(name, par=par, est=est, P=P.ramsay)
# define parameters to be estimated
mod1m.pars <- mirt::mirt(dat, 1, rep( "ramsayqm",I),
                   customItems=list("ramsayqm"=ramsayqm), pars="values")
mod1m.pars[ mod1m.pars$name=="K", "est" ] <- FALSE
# define Theta design matrix
Theta <- matrix( seq(-3,3,len=10), ncol=1)
# estimate model
mod1m <- mirt::mirt(dat, 1, rep( "ramsayqm",I), customItems=list("ramsayqm"=ramsayqm),
               pars=mod1m.pars, verbose=TRUE,
               technical=list( customTheta=Theta, NCYCLES=50)
                )
print(mod1m)
summary(mod1m)
cmod1m <- sirt::mirt.wrapper.coef( mod1m )$coef
# compare simulated and estimated values
dfr <- cbind( b, cmod1m$b, exp(mod1$item$b ) )
colnames(dfr) <- c("simulated", "mirt", "sirt_rasch.mml2")
round( dfr, 2 )
  ##      simulated mirt sirt_rasch.mml2
  ## [1,]      0.14 0.11            0.11
  ## [2,]      0.20 0.17            0.18
  ## [3,]      0.30 0.27            0.29
  ## [4,]      0.45 0.42            0.43
  ## [5,]      0.67 0.65            0.67
  ## [6,]      1.00 1.00            1.01
  ## [7,]      1.49 1.53            1.54
  ## [8,]      2.23 2.21            2.21
  ## [9,]      3.32 3.00            2.98
  ##[10,]      4.95 5.22            5.09
  ##[11,]      7.39 5.62            5.51
}