est_theta {lazy.irt}

R Documentation

Estimation of Theta

Description

Estimation of Theta

Usage

est_theta(
  Uc = NULL,
  U = NULL,
  param = NULL,
  method = "EAP",
  min_theta = -7,
  max_theta = 7,
  theta = NULL,
  thd = NULL,
  thdist = "NORMAL",
  thmin = -4,
  thmax = 4,
  npoints = 21,
  thmean = 0,
  thstd = 1,
  print = 0,
  plot = 0
)

Arguments

Uc	The compressed item response data.
U	The uncompressed item response data.
param	The parameter data frame.
method	= "EAP", "MAP" or "ML"
min_theta	Minimum value of theta for method="MLE" or "MAP".
max_theta	Maximum value of theta for method="MLE" or "MAP".
theta	Discrete theta values.
thd	pdf of theta normalized to sum to unity. When NULL, N(0,1) will be used. When thd == 1, locally uniform prior will be used.
thdist	= "NORMAL" or "UNIFORM"
thmin	Minimum value of discrete thata value.
thmax	Maximum value of discrete thata value.
npoints	# of discrete points for theta.
thmean	The prior mean of normal theta distribution.
thstd	The prior standard deviation of normal theta distribution.
print	= 2 to print the result.
plot	= 1 to show the histogram of theta.

Details

The core part of EAP is as follows:

 ULP=U%*%t(logP)
 eULP=exp(ULP)*matrix(1,nrow(ULP))%*%t(thd)
 rseULP=rowSums(eULP)
 H=eULP/rseULP
 thetahat=H%*%theta

where logP is the log of irf matrix (ntheta x sum(ncat)),
U is the un-compressed item response matrix( N x sum(ncat)),
and thd is the normalized prior theta pdf vector (ntheta).
Therefore, H is N x ntheta matrix of normalized posterior pdf
First two lines calculate the unnormalized joint pdf of U and theta.
The next line line normalizes the posterior distribution of theta given U stored in H matrix.
The last line calculates the posterior mean.

When thd is a scalar, the locally uniform theta distribution is used.

method = "ML" or method ="MAP" use native optimize function and may be very slow.

When method is EAP or ML, the posterior standard deviation of theta or the standard error of theta will be calculated.

Value

A list of
thetahat Estimated theta values
thetastd Standard Deviation of the Estimated theta values
theta Theta values for EAP
thd Theta disrtibution for EAP
method
thdist
thmin and thmax
npoints
thmean
thstd
min_theta and max_theta

Examples

# compressed item response at equally spaced 21 theta points in [-3,3]
set.seed(1701)
resg <- gendataIRT( 1, paramS3, npoints=21, compress=1 )
Uc <- resg$U
theta <- resg$theta
thetaEAP <- est_theta( Uc, param=paramS3, print=1 )$thetahat
thetaMAP <- est_theta( Uc, param=paramS3, print=1, method="MAP" )$thetahat
thetaML <- est_theta( Uc, param=paramS3, print=1, method="ML" )$thetahat
cor( cbind(theta,thetaEAP,thetaMAP,thetaML) )
plot(theta, thetaEAP)

# uncompressed item response at equally spaced 101 theta points in [-3,3]
set.seed(1701)
resg <- gendataIRT( 1, paramS3, npoints=101, compress=0 )
U <- resg$U
theta <- resg$theta
res <- est_theta( U=U, param=paramS3, print=1 )
plot(theta,res$thetahat)

# 100 normally distriuted theta
set.seed(1701)
resg <- gendataIRT( 1, paramS3, npoints=100, compress=1, thdist="rnorm" )
Uc <- resg$U
theta <- resg$theta
res <- est_theta( Uc=Uc, param=paramS3, print=1 )
plot(theta,res$thetahat)

---