sirt (version 1.9-0)

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

#############################################################################
# SIMULATED 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( 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", 
              fixed.K = fixed.K )                  
summary(mod1)

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

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

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

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

#*** 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$name == "K" , "est" ] <- FALSE
# define Theta design matrix
Theta <- matrix( seq(-3,3,len=10) , ncol=1)
# estimate model
mod1m <- mirt::mirt(dat, 1, rep( "ramsayqm",I) , customItems=list("ramsayqm"=ramsayqm), 
               pars = mod1m.pars , verbose=TRUE , 
               technical = list( customTheta=Theta , NCYCLES=50) 
                )
print(mod1m)                
summary(mod1m)
cmod1m <- mirt.wrapper.coef( mod1m )$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