Plausible Value Imputation

Plausible value imputation for objects of the classes tam and tam.mml (Adams & Wu, 2007). For converting generated plausible values into a list of multiply imputed datasets see tampv2datalist and the Examples 2 and 3 of this function.

The function tam.pv.mcmc employs fully Bayesian estimation for drawing plausible values and is recommended in cases when the latent regression model is unreliably estimated (multidimensional model with stochastic nodes). The parameters of the latent regression (regression coefficients and residual covariance matrices) are drawn by Bayesian bootstrap (Rubin, 1981). Either case probabilities (i.e., non-integer weights for cases in resampling; argument sample_integers=FALSE) or ordinary bootstrap (i.e., sampling cases with replacement; argument sample_integers=TRUE) can be used for the Bootstrap step by obtaining posterior draws of regression parameters.

tam.pv(tamobj, nplausible=10, ntheta=2000, normal.approx=FALSE,
    samp.regr=FALSE, theta.model=FALSE, np.adj=8, na.grid=5, verbose=TRUE)

tam.pv.mcmc( tamobj, Y=NULL, group=NULL, beta_groups=TRUE, nplausible=10, level=.95,
    n.iter=1000, n.burnin=500, adj_MH=.5, adj_change_MH=.05, refresh_MH=50,
    accrate_bound_MH=c(.45, .55), sample_integers=FALSE, theta_init=NULL,
    print_iter=20, verbose=TRUE, calc_ic=TRUE)

# S3 method for tam.pv.mcmc
summary(object, file=NULL, ...)

# S3 method for tam.pv.mcmc
plot(x, ...)

Arguments

tamobj

Object of class tam or tam.mml. For tam.pv.mcmc, it must not be an object of this class but rather a list with (at least the) entries AXsi, B, resp.

nplausible

Number of plausible values to be drawn

ntheta

Number of ability nodes for plausible value imputation. Note that in this function ability nodes are simulated for the whole sample, not for every person (contrary to the software ConQuest).

normal.approx

An optional logical indicating whether the individual posterior distributions should be approximated by a normal distribution? The default is FALSE. In the case normal.approx=TRUE (normal distribution approximation), the number of ability nodes ntheta can be substantially smaller than 2000, say 200 or 500. The normal approximation is implemented for unidimensional and multidimensional models.

samp.regr

An optional logical indicating whether regression coefficients should be fixed in the plausible value imputation or also sampled from their posterior distribution? The default is FALSE. Sampled regression coefficients are obtained by nonparametric bootstrap.

theta.model

Logical indicating whether the theta grid from the tamobj object should be used for plausible value imputation. In case of normal.approx=TRUE, this should be sufficient in many applications.

np.adj

This parameter defines the "spread" of the random theta values for drawing plausible values when normal.approx=FALSE. If \(s_{EAP}\) denotes the standard deviation of the posterior distribution of theta (in the one-dimensional case), then theta is simulated from a normal distribution with standard deviation np.adj times \(s_{EAP}\).

na.grid

Range of the grid in normal approximation. Default is from -5 to 5.

...

Further arguments to be passed

Y

Optional matrix of regressors

group

Optional vector of group identifiers

beta_groups

Logical indicating whether group specific beta coefficients shall be estimated.

level

Confidence level

n.iter

Number of iterations

n.burnin

Number of burnin-iterations

adj_MH

Adjustment factor for Metropolis-Hastings (MH) steps which controls the variance of the proposal distribution for \(\theta\). Can be also a vector of length equal to the number of persons.

adj_change_MH

Allowed change for MH adjustment factor after refreshing

refresh_MH

Number of iterations after which the variance of the proposal distribution should be updated

accrate_bound_MH

Bounds for target acceptance rates of sampled \(\theta\) values.

sample_integers

Logical indicating whether weights for complete cases should be sampled in bootstrap

theta_init

Optional matrix with initial \(\bold{\theta}\) values

print_iter

Print iteration progress every print_iterth iteration

verbose

Logical indicating whether iteration progress should be displayed.

calc_ic

Logical indicating whether information criteria should be computed.

object

Object of class tam.pv.mcmc

x

Object of class tam.pv.mcmc

file

A file name in which the summary output will be written

Value

The value of tam.pv is a list with following entries:

pv

A data frame containing a person identifier (pid) and plausible values denoted by PVxx.Dimyy which is the xxth plausible value of dimension yy.

hwt

Individual posterior distribution evaluated at the ability grid theta

hwt1

Cumulated individual posterior distribution

theta

Simulated ability nodes

The value of tam.pv.mcmc is a list containing entries

pv

Data frame containing plausible values

parameter_samples

Sampled regression parameters

ic

Information criteria

beta

Estimate of regression parameters \(\bold{\beta}\)

variance

Estimate of residual variance matrix \(\bold{\Sigma}\)

correlation

Estimate of residual correlation matrix corresponding to variance

theta_acceptance_MH

Acceptance rates and acceptance MH factors for each individual

theta_last

Last sampled \(\bold{\theta}\) value

...

Further values

References

Adams, R. J., & Wu, M. L. (2007). The mixed-coefficients multinomial logit model. A generalized form of the Rasch model. In M. von Davier & C. H. Carstensen (Eds.): Multivariate and mixture distribution Rasch models: Extensions and applications (pp. 55-76). New York: Springer. doi:10.1007/978-0-387-49839-3_4

Rubin, D. B. (1981). The Bayesian bootstrap. The Annals of Statistics, 9(1), 130-134.

See also

See tam.latreg for further examples of fitting latent regression models and drawing plausible values from models which provides an individual likelihood as an input.

Examples

#############################################################################
# EXAMPLE 1: Dichotomous unidimensional data sim.rasch
#############################################################################

data(data.sim.rasch)
resp <- data.sim.rasch[ 1:500, 1:15 ]  # select subsample of students and items

# estimate Rasch model
mod <- TAM::tam.mml(resp)

# draw 5 plausible values without a normality
# assumption of the posterior and 2000 ability nodes
pv1a <- TAM::tam.pv( mod, nplausible=5, ntheta=2000 )

# draw 5 plausible values with a normality
# assumption of the posterior and 500 ability nodes
pv1b <- TAM::tam.pv( mod, nplausible=5, ntheta=500, normal.approx=TRUE )

# distribution of first plausible value from imputation pv1
hist(pv1a$pv$PV1.Dim1 )
# boxplot of all plausible values from imputation pv2
boxplot(pv1b$pv[, 2:6 ] )

if (FALSE) {
# draw plausible values with tam.pv.mcmc function
Y <- matrix(1, nrow=500, ncol=1)
pv2 <- TAM::tam.pv.mcmc( tamobj=mod, Y=Y, nplausible=5 )
str(pv2)

# summary output
summary(pv2)
# assessing convergence with traceplots
plot(pv2, ask=TRUE)

# use starting values for theta and MH factors which fulfill acceptance rates
# from previously fitted model
pv3 <- TAM::tam.pv.mcmc( tamobj=mod, Y=Y, nplausible=5, theta_init=pv2$theta_last,
            adj_MH=pv2$theta_acceptance_MH$adj_MH )

#############################################################################
# EXAMPLE 2: Unidimensional plausible value imputation with
#            background variables; dataset data.pisaRead from sirt package
#############################################################################

data(data.pisaRead, package="sirt")
dat <- data.pisaRead$data
  ##   > colnames(dat)
  ##    [1] "idstud"   "idschool" "female"   "hisei"    "migra"    "R432Q01"
  ##    [7] "R432Q05"  "R432Q06"  "R456Q01"  "R456Q02"  "R456Q06"  "R460Q01"
  ##   [13] "R460Q05"  "R460Q06"  "R466Q02"  "R466Q03"  "R466Q06"

## Note that reading items have variable names starting with R4

# estimate 2PL model without covariates
items <- grep("R4", colnames(dat) )    # select test items from data
mod2a <- TAM::tam.mml.2pl( resp=dat[,items] )
summary(mod2a)

# fix item parameters for plausible value imputation
   # fix item intercepts by defining xsi.fixed
xsi0 <- mod2a$xsi$xsi
xsi.fixed <- cbind( seq(1,length(xsi0)), xsi0 )
   # fix item slopes using mod2$B
# matrix of latent regressors female, hisei and migra
Y <- dat[, c("female", "hisei", "migra") ]
mod2b <- TAM::tam.mml( resp=dat[,items], B=mod2a$B, xsi.fixed=xsi.fixed, Y=Y,
            pid=dat$idstud)

# plausible value imputation with normality assumption
# and ignoring uncertainty about regression coefficients
#    -> the default is samp.regr=FALSE
pv2c <- TAM::tam.pv( mod2b, nplausible=10, ntheta=500, normal.approx=TRUE )
# sampling of regression coefficients
pv2d <- TAM::tam.pv( mod2b, nplausible=10, ntheta=500, samp.regr=TRUE)
# sampling of regression coefficients, normal approximation using the
# theta grid from the model
pv2e <- TAM::tam.pv( mod2b, samp.regr=TRUE, theta.model=TRUE, normal.approx=TRUE)

#--- create list of multiply imputed datasets with plausible values
# define dataset with covariates to be matched
Y <- dat[, c("idstud", "idschool", "female", "hisei", "migra") ]

# define plausible value names
pvnames <- c("PVREAD")
# create list of imputed datasets
datlist1 <- TAM::tampv2datalist( pv2e, pvnames=pvnames, Y=Y, Y.pid="idstud")
str(datlist1)

# create a matrix of covariates with different set of students than in pv2e
Y1 <- Y[ seq( 1, 600, 2 ), ]
# create list of multiply imputed datasets
datlist2 <- TAM::tampv2datalist( pv2e, pvnames=c("PVREAD"), Y=Y1, Y.pid="idstud")

#--- fit some models in lavaan and semTools
library(lavaan)
library(semTools)

#*** Model 1: Linear regression
lavmodel <- "
   PVREAD ~ migra + hisei
   PVREAD ~~ PVREAD
        "
mod1 <- semTools::lavaan.mi( lavmodel, data=datlist1, m=0)
summary(mod1, standardized=TRUE, rsquare=TRUE)

# apply lavaan for third imputed dataset
mod1a <- lavaan::lavaan( lavmodel, data=datlist1[[3]] )
summary(mod1a, standardized=TRUE, rsquare=TRUE)

# compare with mod1 by looping over all datasets
mod1b <- lapply( datlist1, FUN=function(dat0){
    mod1a <- lavaan( lavmodel, data=dat0 )
    coef( mod1a)
        } )
mod1b
mod1b <- matrix( unlist( mod1b ), ncol=length( coef(mod1)), byrow=TRUE )
mod1b
round( colMeans(mod1b), 3 )
coef(mod1)   # -> results coincide

#*** Model 2: Path model
lavmodel <- "
   PVREAD ~ migra + hisei
   hisei ~ migra
   PVREAD ~~ PVREAD
   hisei ~~ hisei
        "
mod2 <- semTools::lavaan.mi( lavmodel, data=datlist1 )
summary(mod2, standardized=TRUE, rsquare=TRUE)
# fit statistics
inspect( mod2, what="fit")

#--- using mitools package
library(mitools)
# convert datalist into an object of class imputationList
datlist1a <- mitools::imputationList( datlist1 )
# fit linear regression
mod1c <- with( datlist1a, stats::lm( PVREAD ~ migra + hisei ) )
summary( mitools::MIcombine(mod1c) )

#--- using mice package
library(mice)
library(miceadds)
# convert datalist into a mids object
mids1 <- miceadds::datalist2mids( datlist1 )
# fit linear regression
mod1c <- with( mids1, stats::lm( PVREAD ~ migra + hisei ) )
summary( mice::pool(mod1c) )

#############################################################################
# EXAMPLE 3: Multidimensional plausible value imputation
#############################################################################

# (1) simulate some data
set.seed(6778)
library(mvtnorm)
N <- 1000
Y <- cbind( stats::rnorm(N), stats::rnorm(N) )
theta <- mvtnorm::rmvnorm( N, mean=c(0,0), sigma=matrix( c(1,.5,.5,1), 2, 2 ))
theta[,1] <- theta[,1] + .4 * Y[,1] + .2 * Y[,2]  # latent regression model
theta[,2] <- theta[,2] + .8 * Y[,1] + .5 * Y[,2]  # latent regression model
I <- 20
p1 <- stats::plogis( outer( theta[,1], seq( -2, 2, len=I ), "-" ) )
resp1 <- 1 * ( p1 > matrix( stats::runif( N*I ), nrow=N, ncol=I ) )
p1 <- stats::plogis( outer( theta[,2], seq( -2, 2, len=I ), "-" ) )
resp2 <- 1 * ( p1 > matrix( stats::runif( N*I ), nrow=N, ncol=I ) )
resp <- cbind(resp1,resp2)
colnames(resp) <- paste("I", 1:(2*I), sep="")

# (2) define loading Matrix
Q <- array( 0, dim=c( 2*I, 2 ))
Q[cbind(1:(2*I), c( rep(1,I), rep(2,I) ))] <- 1

# (3) fit latent regression model
mod <- TAM::tam.mml( resp=resp, Y=Y, Q=Q )

# (4) draw plausible values
pv1 <- TAM::tam.pv( mod, theta.model=TRUE )

# (5) convert plausible values to list of imputed datasets
Y1 <- data.frame(Y)
colnames(Y1) <- paste0("Y",1:2)
pvnames <- c("PVFA","PVFB")
# create list of imputed datasets
datlist1 <- TAM::tampv2datalist( pv1, pvnames=pvnames, Y=Y1 )
str(datlist1)

# (6) apply statistical models
library(semTools)
# define linear regression
lavmodel <- "
   PVFA ~ Y1 + Y2
   PVFA ~~ PVFA
        "
mod1 <- semTools::lavaan.mi( lavmodel, data=datlist1 )
summary(mod1, standardized=TRUE, rsquare=TRUE)

# (7) draw plausible values with tam.pv.mcmc function
Y1 <- cbind( 1, Y )
pv2 <- TAM::tam.pv.mcmc( tamobj=mod, Y=Y1, n.iter=1000, n.burnin=200 )

# (8) group-specific plausible values
set.seed(908)
# create artificial grouping variable
group <- sample( 1:3, N, replace=TRUE )
pv3 <- TAM::tam.pv.mcmc( tamobj, Y=Y1, n.iter=1000, n.burnin=200, group=group )

# (9) plausible values with no fitted object in TAM

# fit IRT model without covariates
mod4a <- TAM::tam.mml( resp=resp, Q=Q )
# define input for tam.pv.mcmc
tamobj1 <- list( AXsi=mod4a$AXsi, B=mod4a$B, resp=mod4a$resp )
pmod4 <- TAM::tam.pv.mcmc( tamobj1, Y=Y1 )

#############################################################################
# EXAMPLE 4: Plausible value imputation with measurement errors in covariates
#############################################################################

library(sirt)
set.seed(7756)
N <- 2000    # number of persons
I <- 10     # number of items

# simulate covariates
X <- mvrnorm( N, mu=c(0,0), Sigma=matrix( c(1,.5,.5,1),2,2 ) )
colnames(X) <- paste0("X",1:2)
# second covariate with measurement error with variance var.err
var.err <- .3
X.err <- X
X.err[,2] <-X[,2] + rnorm(N, sd=sqrt(var.err) )
# simulate theta
theta <- .5*X[,1] + .4*X[,2] + rnorm( N, sd=.5 )
# simulate item responses
itemdiff <- seq( -2, 2, length=I)  # item difficulties
dat <- sirt::sim.raschtype( theta, b=itemdiff )

#***********************
#*** Model 0: Regression model with true variables
mod0 <- stats::lm( theta ~ X )
summary(mod0)

#***********************
#*** Model 1: latent regression model with true covariates X
xsi.fixed <- cbind( 1:I, itemdiff )
mod1 <- TAM::tam.mml( dat, xsi.fixed=xsi.fixed, Y=X)
summary(mod1)

# draw plausible values
res1a <- TAM::tam.pv( mod1, normal.approx=TRUE, ntheta=200, samp.regr=TRUE)
# create list of multiply imputed datasets
library(miceadds)
datlist1a <- TAM::tampv2datalist( res1a, Y=X )
imp1a <- miceadds::datalist2mids( datlist1a )

# fit linear model
# linear regression with measurement errors in X
lavmodel <- "
   PV.Dim1 ~ X1 + X2true
   X2true=~ 1*X2
   X2 ~~ 0.3*X2  #=var.err
   PV.Dim1 ~~ PV.Dim1
   X2true ~~ X2true
        "
mod1a <- semTools::lavaan.mi( lavmodel, datlist1a)
summary(mod1a, standardized=TRUE, rsquare=TRUE)

#***********************
#*** Model 2: latent regression model with error-prone covariates X.err
mod2 <- TAM::tam.mml( dat, xsi.fixed=xsi.fixed, Y=X.err)
summary(mod2)

#***********************
#*** Model 3: Adjustment of covariates

cov.X.err <- cov( X.err )
# matrix of variance of measurement errors
measerr <- diag( c(0,var.err) )
# true covariance matrix
cov.X <- cov.X.err - measerr
# mean of X.err
mu <- colMeans(X.err)
muM <- matrix( mu, nrow=nrow(X.err), ncol=ncol(X.err), byrow=TRUE)
# reliability matrix
W <- solve( cov.X.err ) %*% cov.X
ident <- diag(2)
# adjusted scores of X
X.adj <- ( X.err - muM ) %*% W   + muM %*% ( ident - W )

# fit latent regression model
mod3 <- TAM::tam.mml( dat, xsi.fixed=xsi.fixed, Y=X.adj)
summary(mod3)

# draw plausible values
res3a <- TAM::tam.pv( mod3, normal.approx=TRUE, ntheta=200, samp.regr=TRUE)

# create list of multiply imputed datasets
library(semTools)

#*** PV dataset 1
# datalist with error-prone covariates
datlist3a <- TAM::tampv2datalist( res3a, Y=X.err )
# datalist with adjusted covariates
datlist3b <- TAM::tampv2datalist( res3a, Y=X.adj )

# linear regression with measurement errors in X
lavmodel <- "
   PV.Dim1 ~ X1 + X2true
   X2true=~ 1*X2
   X2 ~~ 0.3*X2  #=var.err
   PV.Dim1 ~~ PV.Dim1
   X2true ~~ X2true
        "
mod3a <- semTools::lavaan.mi( lavmodel, datlist3a)
summary(mod3a, standardized=TRUE, rsquare=TRUE)

lavmodel <- "
   PV.Dim1 ~ X1 + X2
   PV.Dim1 ~~ PV.Dim1
        "
mod3b <- semTools::lavaan.mi( lavmodel, datlist3b)
summary(mod3b, standardized=TRUE, rsquare=TRUE)
#=> mod3b leads to the correct estimate.

#*********************************************
# plausible value imputation for abilities and error-prone
# covariates using the mice package

library(mice)
library(miceadds)

# creating the likelihood for plausible value for abilities
mod11 <- TAM::tam.mml( dat, xsi.fixed=xsi.fixed )
likePV <- IRT.likelihood(mod11)
# creating the likelihood for error-prone covariate X2
lavmodel <- "
  X2true=~ 1*X2
  X2 ~~ 0.3*X2
    "
mod12 <- lavaan::cfa( lavmodel, data=as.data.frame(X.err) )
summary(mod12)
likeX2 <- TAM::IRTLikelihood.cfa( data=X.err, cfaobj=mod12)
str(likeX2)

#-- create data input for mice package
data <- data.frame( "PVA"=NA, "X1"=X[,1], "X2"=NA  )
vars <- colnames(data)
V <- length(vars)
predictorMatrix <- 1 - diag(V)
rownames(predictorMatrix) <- colnames(predictorMatrix) <- vars
imputationMethod <- rep("norm", V )
names(imputationMethod) <- vars
imputationMethod[c("PVA","X2")] <- "plausible.values"

#-- create argument lists for plausible value imputation
# likelihood and theta grid of plausible value derived from IRT model
like <- list( "PVA"=likePV, "X2"=likeX2 )
theta <- list( "PVA"=attr(likePV,"theta"), "X2"=attr(likeX2, "theta") )
#-- initial imputations
data.init <- data
data.init$PVA <- mod11$person$EAP
data.init$X2 <- X.err[,"X2"]

#-- imputation using the mice and miceadds package
imp1 <- mice::mice( as.matrix(data), predictorMatrix=predictorMatrix, m=4, maxit=6,
             method=imputationMethod,  allow.na=TRUE,
             theta=theta, like=like, data.init=data.init )
summary(imp1)

# compute linear regression
mod4a <- with( imp1, stats::lm( PVA ~ X1 + X2 ) )
summary( mice::pool(mod4a) )
}