Plausible Value Imputation in Generalized Logistic Item Response Model

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 ***

if (FALSE) {
#############################################################################
# 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
}