index_distn: Compute score density

In TestGardener: Information Analysis for Test and Rating Scale Data

Compute score density

Description

Computes the cumulated density for distribution function, the probability density function, and the log probability density function as fd objects by spline smoothing of the score values indexdens using the basis object logdensbasis. The norming constant C is also output.

The score values may score index values index, expected test score values mu, or arc length locations on the test information or scale curve. The argument functional data object logdensfd should have a range that is appropriate for the score values being represented: For score indices, [0,100], for expected test scores, the range of observed or expected scores; and for test information curve locations in the interval [0,infoSurp].

Usage

  index_distn(indexdens, logdensbasis,  
                pvec=c(0.05, 0.25, 0.50, 0.75, 0.95), nfine = 101)

Arguments

indexdens	A vector of score index, test score, or arc length values. In the score index case, these are usually only the values in the interior of the interval [0,100].
logdensbasis	A functional basis object for representing the log density function. The argument may also be a functional parameter object (fdPar) or a functional basis object (Sbasis).
pvec	A vector length NL containing the marker percentages.
nfine	The number of values in a fine grid, default as 101.

Value

A named list containing:

pdf_fd:	An fd object for the probability density function values over the fine mesh.
cdffine:	A vector of cumulative probability values beginning with zero and ending with 1. It must not have ties.
pdffine:	A vector of probability values.
logdensfd:	A functional data object (fd) representing the log of the probability function for input index.
C:	The normalization constant for computing the probability density function with the command densityfd = exp(logdensfd)/C.
denscdf:	A set of unique values of the cumulative probability function defined over an equally spaced mesh of score index values of the same length as denscdf.
indcdf:	A vector of values within [0,100] corresponding to the values in denscdf.

Author(s)

Juan Li and James Ramsay

References

Ramsay, J. O., Li J. and Wiberg, M. (2020) Full information optimal scoring. Journal of Educational and Behavioral Statistics, 45, 297-315.

Ramsay, J. O., Li J. and Wiberg, M. (2020) Better rating scale scores with information-based psychometrics. Psych, 2, 347-360.

See Also

index_fun, index2info, mu, scoreDensity

Examples

#  Example 1.  Display the item power curves for the 
#  short SweSAT multiple choice test with 24 items and 1000 examinees
#  Assemble information for estimating index density
indfine   <- seq(0,100,len=101)
SfdList   <- Quant_13B_problem_parmList$SfdList
index     <- Quant_13B_problem_parmList$index
N         <- length(index)
#  Define the density for only interior index values
inside    <- index > 0 & index < 100
logdensbasis    <- Quant_13B_problem_parmList$logdensfd$basis
index_distnList <- index_distn(index[inside], logdensbasis)
index.densfine  <- eval.fd(indfine, index_distnList$pdf_fd)
#  Add number of values at each boundary
indexzero       <- length(index[index == 0])
indexcent       <- length(index[index == 100])
#  Plot both the interior density and the boundary proportions
oldpar <- par(no.readonly=TRUE)
on.exit(oldpar)
plot(indfine, index.densfine, type="l", lwd=2, ylim=c(0,0.025), 
     xlab="Percentile Index", ylab="Density")
lines(c(  0,100), c(1/100,1/100), lty=2)
lines(c(  0,  0), c(0,indexzero/N), lty=1, lwd=4)
lines(c(100,100), c(0,indexcent/N), lty=1, lwd=4)

TestGardener documentation built on Nov. 24, 2023, 5:08 p.m.