Plausible Value Imputation in Generalized Logistic Item Response Model
This function performs unidimensional plausible value imputation (Adams & Wu, 2007; Mislevy, 1991).
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)
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 |
X |
A matrix of individual covariates for the latent regression of θ 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 |
a |
Optional vector of item slopes |
c |
Optional vector of lower item asymptotes |
d |
Optional vector of upper item asymptotes |
alpha1 |
Parameter α_1 in generalized item response model |
alpha2 |
Parameter α_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. |
Plausible values are drawn from the latent regression model with heterogeneous variances:
θ_p=X_p β + ε_p \quad, \quad ε_p \sim N( 0, σ_p^2 ) \quad, \quad \log( σ_p )=Z_p γ + ν_p
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 |
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.
For estimating the latent regression model see
latent.regression.em.raschtype.
#############################################################################
# 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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.