plausible.value.imputation.raschtype: Plausible Value Imputation in Generalized Logistic Item...

In sirt: Supplementary Item Response Theory Models

Plausible Value Imputation in Generalized Logistic Item Response Model

Description

This function performs unidimensional plausible value imputation (Adams & Wu, 2007; Mislevy, 1991).

Usage

plausible.value.imputation.raschtype(data=NULL, f.yi.qk=NULL, X,
   Z=NULL, beta0=rep(0, ncol(X)), sig0=1, b=rep(1, ncol(X)),
   a=rep(1, length(b)), c=rep(0, length(b)), d=1+0*b,
   alpha1=0, alpha2=0, theta.list=seq(-5, 5, len=50),
   cluster=NULL, iter, burnin, nplausible=1, printprogress=TRUE)

Arguments

data	An N \times I data frame of dichotomous responses
f.yi.qk	An optional matrix which contains the individual likelihood. This matrix is produced by rasch.mml2 or rasch.copula2. The use of this argument allows the estimation of the latent regression model independent of the parameters of the used item response model.
X	A matrix of individual covariates for the latent regression of \theta on X
Z	A matrix of individual covariates for the regression of individual residual variances on Z
beta0	Initial vector of regression coefficients
sig0	Initial vector of coefficients for the variance heterogeneity model
b	Vector of item difficulties. It must not be provided if the individual likelihood f.yi.qk is specified.
a	Optional vector of item slopes
c	Optional vector of lower item asymptotes
d	Optional vector of upper item asymptotes
alpha1	Parameter \alpha_1 in generalized item response model
alpha2	Parameter \alpha_2 in generalized item response model
theta.list	Vector of theta values at which the ability distribution should be evaluated
cluster	Cluster identifier (e.g. schools or classes) for including theta means in the plausible imputation.
iter	Number of iterations
burnin	Number of burn-in iterations for plausible value imputation
nplausible	Number of plausible values
printprogress	A logical indicated whether iteration progress should be displayed at the console.

Details

Plausible values are drawn from the latent regression model with heterogeneous variances:

\theta_p=X_p \beta + \epsilon_p \quad, \quad \epsilon_p \sim N( 0, \sigma_p^2 ) \quad, \quad \log( \sigma_p )=Z_p \gamma + \nu_p

Value

A list with following entries:

coefs.X	Sampled regression coefficients for covariates X
coefs.Z	Sampled coefficients for modeling variance heterogeneity for covariates Z
pvdraws	Matrix with drawn plausible values
posterior	Posterior distribution from last iteration
EAP	Individual EAP estimate
SE.EAP	Standard error of the EAP estimate
pv.indexes	Index of iterations for which plausible values were drawn

References

Adams, R., & Wu. M. (2007). The mixed-coefficients multinomial logit model: A generalized form of the Rasch model. In M. von Davier & C. H. Carstensen: Multivariate and Mixture Distribution Rasch Models: Extensions and Applications (pp. 57-76). New York: Springer.

Mislevy, R. J. (1991). Randomization-based inference about latent variables from complex samples. Psychometrika, 56, 177-196.

See Also

For estimating the latent regression model see latent.regression.em.raschtype.

Examples

#############################################################################
# EXAMPLE 1: Rasch model with covariates
#############################################################################

set.seed(899)
I <- 21     # number of items
b <- seq(-2,2, len=I)   # item difficulties
n <- 2000       # number of students

# simulate theta and covariates
theta <- stats::rnorm( n )
x <- .7 * theta + stats::rnorm( n, .5 )
y <- .2 * x+ .3*theta + stats::rnorm( n, .4 )
dfr <- data.frame( theta, 1, x, y )

# simulate Rasch model
dat1 <- sirt::sim.raschtype( theta=theta, b=b )

# Plausible value draws
pv1 <- sirt::plausible.value.imputation.raschtype(data=dat1, X=dfr[,-1], b=b,
            nplausible=3, iter=10, burnin=5)
# estimate linear regression based on first plausible value
mod1 <- stats::lm( pv1$pvdraws[,1] ~ x+y )
summary(mod1)
  ##               Estimate Std. Error t value Pr(>|t|)
  ##   (Intercept) -0.27755    0.02121  -13.09   <2e-16 ***
  ##   x            0.40483    0.01640   24.69   <2e-16 ***
  ##   y            0.20307    0.01822   11.15   <2e-16 ***

# true regression estimate
summary( stats::lm( theta ~ x + y ) )
  ## Coefficients:
  ##             Estimate Std. Error t value Pr(>|t|)
  ## (Intercept) -0.27821    0.01984  -14.02   <2e-16 ***
  ## x            0.40747    0.01534   26.56   <2e-16 ***
  ## y            0.18189    0.01704   10.67   <2e-16 ***

## Not run: 
#############################################################################
# EXAMPLE 2: Classical test theory, homogeneous regression variance
#############################################################################

set.seed(899)
n <- 3000       # number of students
x <- round( stats::runif( n, 0,1 ) )
y <- stats::rnorm(n)
# simulate true score theta
theta <- .4*x + .5 * y + stats::rnorm(n)
# simulate observed score by adding measurement error
sig.e <- rep( sqrt(.40), n )
theta_obs <- theta + stats::rnorm( n, sd=sig.e)

# define theta grid for evaluation of density
theta.list <- mean(theta_obs) + stats::sd(theta_obs) * seq( - 5, 5, length=21)
# compute individual likelihood
f.yi.qk <- stats::dnorm( outer( theta_obs, theta.list, "-" ) / sig.e )
f.yi.qk <- f.yi.qk / rowSums(f.yi.qk)
# define covariates
X <- cbind( 1, x, y )
# draw plausible values
mod2 <- sirt::plausible.value.imputation.raschtype( f.yi.qk=f.yi.qk,
                  theta.list=theta.list, X=X, iter=10, burnin=5)

# linear regression
mod1 <- stats::lm( mod2$pvdraws[,1] ~ x+y )
summary(mod1)
  ##             Estimate Std. Error t value Pr(>|t|)
  ## (Intercept) -0.01393    0.02655  -0.525      0.6
  ## x            0.35686    0.03739   9.544   <2e-16 ***
  ## y            0.53759    0.01872  28.718   <2e-16 ***

# true regression model
summary( stats::lm( theta ~ x + y ) )
  ##             Estimate Std. Error t value Pr(>|t|)
  ## (Intercept) 0.002931   0.026171   0.112    0.911
  ## x           0.359954   0.036864   9.764   <2e-16 ***
  ## y           0.509073   0.018456  27.584   <2e-16 ***

#############################################################################
# EXAMPLE 3: Classical test theory, heterogeneous regression variance
#############################################################################

set.seed(899)
n <- 5000       # number of students
x <- round( stats::runif( n, 0,1 ) )
y <- stats::rnorm(n)
# simulate true score theta
theta <- .4*x + .5 * y + stats::rnorm(n) * ( 1 - .4 * x )
# simulate observed score by adding measurement error
sig.e <- rep( sqrt(.40), n )
theta_obs <- theta + stats::rnorm( n, sd=sig.e)

# define theta grid for evaluation of density
theta.list <- mean(theta_obs) + stats::sd(theta_obs) * seq( - 5, 5, length=21)
# compute individual likelihood
f.yi.qk <- stats::dnorm( outer( theta_obs, theta.list, "-" ) / sig.e )
f.yi.qk <- f.yi.qk / rowSums(f.yi.qk)
# define covariates
X <- cbind( 1, x, y )
# draw plausible values (assuming variance homogeneity)
mod3a <- sirt::plausible.value.imputation.raschtype( f.yi.qk=f.yi.qk,
                  theta.list=theta.list, X=X, iter=10, burnin=5)
# draw plausible values (assuming variance heterogeneity)
#  -> include predictor Z
mod3b <- sirt::plausible.value.imputation.raschtype( f.yi.qk=f.yi.qk,
                  theta.list=theta.list, X=X, Z=X, iter=10, burnin=5)

# investigate variance of theta conditional on x
res3 <- sapply( 0:1, FUN=function(vv){
        c( stats::var(theta[x==vv]), stats::var(mod3b$pvdraw[x==vv,1]),
              stats::var(mod3a$pvdraw[x==vv,1]))})
rownames(res3) <- c("true", "pv(hetero)", "pv(homog)" )
colnames(res3) <- c("x=0","x=1")
  ## > round( res3, 2 )
  ##             x=0  x=1
  ## true       1.30 0.58
  ## pv(hetero) 1.29 0.55
  ## pv(homog)  1.06 0.77
## -> assuming heteroscedastic variances recovers true conditional variance

## End(Not run)

sirt documentation built on May 29, 2024, 8:43 a.m.