sirt (version 1.9-0)

wle.rasch.jackknife: Standard Error Estimation of WLE by Jackknifing

Description

This 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 wle is the WLE and wle.jackse its corresponding standard error estimated by jackknife.
• wle.rel
  WLE reliability (Adams, 2005)

Details

The idea of Jackknife in item response models can be found in Wainer and Wright (1980).

References

Adams, R. J. (2005). Reliability as a measurement design effect. Studies in Educational Evaluation, 31, 162-172. 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. Wainer, H., & Wright, B. D. (1980). Robust estimation of ability in the Rasch model. Psychometrika, 45, 373-391. Warm, T. A. (1989). Weighted likelihood estimation of ability in item response theory. Psychometrika, 54, 427-450.

See Also

wle.rasch

Examples

#############################################################################
# EXAMPLE 1: Dataset Reading
#############################################################################
data(data.read)
dat <- data.read

# estimation of the Rasch model
res <- rasch.mml2( dat , parm.conv = .001)

# WLE estimation
wle1 <- wle.rasch(dat, b = res$item$thresh )

# simple jackknife WLE estimation
wle2 <- wle.rasch.jackknife(dat, b =res$item$thresh )
  ## WLE Reliability = 0.651 

# SE(WLE) for testlets A, B and C
wle3 <- 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 <- 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 <- 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 <- aggregate( wle2$wle$wle.jackse , list( wle2$wle$wle) , mean )
colnames(a2) <- c("wle" , "se.simple")
a2$se.testlet <- aggregate( wle3$wle$wle.jackse , list( wle3$wle$wle) , mean )[,2]
a2$se.strata <- aggregate( wle4$wle$wle.jackse , list( wle4$wle$wle) , mean )[,2]
a2$se.finitepop.strata <- aggregate( wle5$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