HybridEquate: Hybrid Levine-PSE equating.

In CambridgeAssessmentResearch/KernEqWPS: Kernel Equating Without Pre-Smoothing

Hybrid Levine-PSE equating.

Description

This function applies equating using the hybrid method described by von Davier and Chen (2013). Like Levine equating, it attempts to account for measurement error in the anchor test whilst equating. However, unlike Leveine equating, it is also designed to retain the nonlinear nature of equipercentile equating. The last of the examples given below provides a situation where we might expect this method to be superior to both chained and post-stratification equating.

Usage

HybridEquate(
  dx,
  dy,
  target = NA,
  order = 5,
  internal = FALSE,
  hX = NA,
  hY = NA
)

Arguments

dx	Data frame with variables "x" and "a" representing scores for individual candidates on form X and on the anchor test.
dy	Data frame with variables "y" and "a" representing scores for individual candidates on form Y and on the anchor test.
target	A character denoting the synthetic population. Can be "x" to denote weighting to the form X population, "y" to denote weighting to the form Y population or left blank (default) to indicate the data should be weighted to the average across all data.
order	Numeric (integer) input denoting the order up to which the anchor distributions should be matched across populations during the PSE step. For example, 2 would indicate that the means and the means of the squared values (related to the standard deviation) should match. An order of 4 essentially indicates that the means, SDs, skewness and kurtosis of anchor test scores should be matched. The default is 5. Setting this to a lower value (2, 3 or 4) may be useful in the event of error messages (often related to non-convergence). Higher values may be valuable if the distributions of anchor test scores follow an extremely different shape on the two populations.
internal	Logical value denoting whether an internal or an external anchor test is being used. The default (FALSE) assumes that it is an external anchor test.
hX	Bandwidth for form X used during the PSE step. By default a plug-in estimator from Andersson and von Davier (2014) is used.
hY	Bandwidth for form Y used during the PSE step. By default a plug-in estimator from Andersson and von Davier (2014) is used.

Value

The function returns a list with the following elements:

hybridFunc

A function that translates any vector of scores on form X into equivalent scores on form Y.

EqTable

A data frame combining the sorted unique scores on form X in the data and their equated values on form Y.

References

Andersson, B., & von Davier, A. A. (2014). Improving the bandwidth selection in kernel equating. Journal of Educational Measurement, 51(3), 223-238.

von Davier, A. A., & Chen, H. (2013). The Kernel Levine Equipercentile Observed-Score Equating Function. ETS Research Report Series.

Examples

#example using some real data (but no criterion equate to compare to)
dx2=data.frame(x=rowSums(mathsdata[1:250,1:35]),a=rowSums(mathsdata[1:250,41:50]))
summary(dx2)
dy2=data.frame(y=rowSums(mathsdata[251:500,51:90]),a=rowSums(mathsdata[251:500,41:50]))
summary(dy2)
hybrideq2=HybridEquate(dx2, dy2,target="y")$EqTable
chainedeq2=KernelChainedEquate(dx2, dy2)$EqTable
plot(chainedeq2$x,chainedeq2$equiyx,type='l')#chained equating function
lines(hybrideq2$x,hybrideq2$hybridyx,lty=2)#Hybrid equating estimated equating function

#demonstrate point of hybrid equating on a 30 item test with an internal 5 item anchor
#define 30 rasch item difficulties as equally spread
itedifs=rep(seq(-2,2,length=5),6)
#set up ensures that last 5 items (anchor) are minitest of first 25

#simulate population one item scores (and then form scores)
n1=3000
t1=rnorm(n1,0.5,1)
ites1=0+(plogis(t1%*%t(rep(1,30))-rep(1,length(t1))%*%t(itedifs))>matrix(runif(n1*30),nrow=n1))
scoresX1=rowSums(ites1[,1:30])
scoresA1=rowSums(ites1[,26:30])
#simulate parallel tests in population two
n2=3000
t2=rnorm(n2,0,1)
ites2=0+(plogis(t2%*%t(rep(1,30))-rep(1,length(t2))%*%t(itedifs))>matrix(runif(n2*30),nrow=n2))
scoresY2=rowSums(ites2[,1:30])
scoresA2=rowSums(ites2[,26:30])

chainedeq3=KernelChainedEquate(data.frame(x=scoresX1,a=scoresA1),data.frame(y=scoresY2,a=scoresA2))$EqTable
pseeq3=PSEObservedEquate(data.frame(x=scoresX1,a=scoresA1),data.frame(y=scoresY2,a=scoresA2))$EqTable
hybrideq3=HybridEquate(data.frame(x=scoresX1,a=scoresA1),data.frame(y=scoresY2,a=scoresA2),internal=TRUE)$EqTable
plot(chainedeq3$x,chainedeq3$equiyx,type='l')#chained equating function
lines(pseeq3$x,pseeq3$equiyx,lty=3,col="red")#PSE estimated equating function
lines(hybrideq3$x,hybrideq3$hybridyx,lty=2)#Hybrid equating estimated equating function
lines(0:30,0:30,col="blue",lty=3,lwd=4)#true equating function (identity as set up to be parallel tests)

CambridgeAssessmentResearch/KernEqWPS documentation built on Feb. 23, 2024, 9:34 p.m.