Standard Error Estimation of WLE by Jackknifing
wle.rasch.jackknife.RdThis function calculates standard errors of WLEs (Warm, 1989) for stratified item designs and item designs with testlets for the Rasch model.
Usage
wle.rasch.jackknife(dat, b, itemweights=1 + 0 * b, pid=NULL,
testlet=NULL, stratum=NULL, size.itempop=NULL)Arguments
- dat
An \(N \times I\) data frame of item responses
- b
Vector of item difficulties
- itemweights
Weights for items, i.e. fixed item discriminations
- pid
Person identifier
- testlet
A vector of length \(I\) which defines which item belongs to which testlet. If some items does not belong to any testlet, then define separate testlet labels for these single items.
- stratum
Item stratum
- size.itempop
Number of items in an item stratum of the finite item population.
Value
A list with following entries:
- wle
Data frame with some estimated statistics. The column
wleis the WLE andwle.jackseits corresponding standard error estimated by jackknife.- wle.rel
WLE reliability (Adams, 2005)
References
Adams, R. J. (2005). Reliability as a measurement design effect. Studies in Educational Evaluation, 31(2-3), 162-172. doi:10.1016/j.stueduc.2005.05.008
Gershunskaya, J., Jiang, J., & Lahiri, P. (2009). Resampling methods in surveys. In D. Pfeffermann and C.R. Rao (Eds.). Handbook of Statistics 29B; Sample Surveys: Inference and Analysis (pp. 121-151). Amsterdam: North Holland. doi:10.1016/S0169-7161(09)00228-4
Wainer, H., & Wright, B. D. (1980). Robust estimation of ability in the Rasch model. Psychometrika, 45(3), 373-391. doi:10.1007/BF02293910
Warm, T. A. (1989). Weighted likelihood estimation of ability in item response theory. Psychometrika, 54(3), 427-450. doi:10.1007/BF02294627
Examples
#############################################################################
# EXAMPLE 1: Dataset Reading
#############################################################################
data(data.read)
dat <- data.read
# estimation of the Rasch model
res <- sirt::rasch.mml2( dat, parm.conv=.001)
# WLE estimation
wle1 <- sirt::wle.rasch(dat, b=res$item$thresh )
# simple jackknife WLE estimation
wle2 <- sirt::wle.rasch.jackknife(dat, b=res$item$thresh )
## WLE Reliability=0.651
# SE(WLE) for testlets A, B and C
wle3 <- sirt::wle.rasch.jackknife(dat, b=res$item$thresh,
testlet=substring( colnames(dat),1,1) )
## WLE Reliability=0.572
# SE(WLE) for item strata A,B, C
wle4 <- sirt::wle.rasch.jackknife(dat, b=res$item$thresh,
stratum=substring( colnames(dat),1,1) )
## WLE Reliability=0.683
# SE (WLE) for finite item strata
# A (10 items), B (7 items), C (4 items -> no sampling error)
# in every stratum 4 items were sampled
size.itempop <- c(10,7,4)
names(size.itempop) <- c("A","B","C")
wle5 <- sirt::wle.rasch.jackknife(dat, b=res$item$thresh,
stratum=substring( colnames(dat),1,1),
size.itempop=size.itempop )
## Stratum A (Mean) Correction Factor 0.6
## Stratum B (Mean) Correction Factor 0.42857
## Stratum C (Mean) Correction Factor 0
## WLE Reliability=0.876
# compare different estimated standard errors
a2 <- stats::aggregate( wle2$wle$wle.jackse, list( wle2$wle$wle), mean )
colnames(a2) <- c("wle", "se.simple")
a2$se.testlet <- stats::aggregate( wle3$wle$wle.jackse, list( wle3$wle$wle), mean )[,2]
a2$se.strata <- stats::aggregate( wle4$wle$wle.jackse, list( wle4$wle$wle), mean )[,2]
a2$se.finitepop.strata <- stats::aggregate( wle5$wle$wle.jackse,
list( wle5$wle$wle), mean )[,2]
round( a2, 3 )
## > round( a2, 3 )
## wle se.simple se.testlet se.strata se.finitepop.strata
## 1 -5.085 0.440 0.649 0.331 0.138
## 2 -3.114 0.865 1.519 0.632 0.379
## 3 -2.585 0.790 0.849 0.751 0.495
## 4 -2.133 0.715 1.177 0.546 0.319
## 5 -1.721 0.597 0.767 0.527 0.317
## 6 -1.330 0.633 0.623 0.617 0.377
## 7 -0.942 0.631 0.643 0.604 0.365
## 8 -0.541 0.655 0.678 0.617 0.384
## 9 -0.104 0.671 0.646 0.659 0.434
## 10 0.406 0.771 0.706 0.751 0.461
## 11 1.080 1.118 0.893 1.076 0.630
## 12 2.332 0.400 0.631 0.272 0.195