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isop.scoring

Scoring Persons and Items in the ISOP Model


Description

This function does the scoring in the isotonic probabilistic model (Scheiblechner, 1995, 2003, 2007). Person parameters are ordinally scaled but the ISOP model also allows specific objective (ordinal) comparisons for persons (Scheiblechner, 1995).

Usage

isop.scoring(dat,score.itemcat=NULL)

Arguments

dat

Data frame with dichotomous or polytomous item responses

score.itemcat

Optional data frame with scoring points for every item and every category (see Example 2).

Details

This function extracts the scoring rule of the ISOP model (if score.itemcat !=NULL) and calculates the modified percentile score for every person. The score s_{ik} for item i and category k is calculated as

s_{ik}=∑_{j=0}^{k-1} f_{ij} - ∑_{j=k+1}^K f_{ij}=P( X_i < k ) - P( X_i > k )

where f_{ik} is the relative frequency of item i in category k and K is the maximum category. The modified percentile score ρ_p for subject p (mpsc in person) is defined by

ρ_p=\frac{1}{I} ∑_{i=1}^I ∑_{j=0}^K s_{ik} \mathbf{1}( X_{pi}=k )

Note that for dichotomous items, the sum score is a sufficient statistic for ρ_p but this is not the case for polytomous items. The modified percentile score ρ_p ranges between -1 and 1.

The modified item P-score ρ_i (Scheiblechner, 2007, p. 52) is defined by

ρ_i=\frac{1}{I-1} \cdot ∑_j ≤ft[ P( X_j < X_i ) - P( X_j > X_i ) \right ]

Value

A list with following entries:

person

A data frame with person parameters. The modified percentile score ρ_p is denoted by mpsc.

item

Item statistics and scoring parameters. The item P-scores ρ_i are labeled as pscore.

p.itemcat

Frequencies for every item category

score.itemcat

Scoring points for every item category

distr.fct

Empirical distribution function

References

Scheiblechner, H. (1995). Isotonic ordinal probabilistic models (ISOP). Psychometrika, 60, 281-304.

Scheiblechner, H. (2003). Nonparametric IRT: Scoring functions and ordinal parameter estimation of isotonic probabilistic models (ISOP). Technical Report, Philipps-Universitaet Marburg.

Scheiblechner, H. (2007). A unified nonparametric IRT model for d-dimensional psychological test data (d-ISOP). Psychometrika, 72, 43-67.

See Also

For fitting the ISOP and ADISOP model see isop.dich or fit.isop.

Examples

#############################################################################
# EXAMPLE 1: Dataset Reading
#############################################################################

data( data.read )
dat <- data.read

# Scoring according to the ISOP model
msc <- sirt::isop.scoring( dat )
# plot student scores
boxplot( msc$person$mpsc ~ msc$person$score )

#############################################################################
# EXAMPLE 2: Dataset students from CDM package | polytomous items
#############################################################################

library("CDM")
data( data.Students, package="CDM")
dat <- stats::na.omit(data.Students[, -c(1:2) ])

# Scoring according to the ISOP model
msc <- sirt::isop.scoring( dat )
# plot student scores
boxplot( msc$person$mpsc ~ msc$person$score )

# scoring with known scoring rule for activity items
items <- paste0( "act", 1:5 )
score.itemcat <- msc$score.itemcat
score.itemcat <- score.itemcat[ items, ]
msc2 <- sirt::isop.scoring( dat[,items], score.itemcat=score.itemcat )

sirt

Supplementary Item Response Theory Models

v3.10-118
GPL (>= 2)
Authors
Alexander Robitzsch [aut,cre] (<https://orcid.org/0000-0002-8226-3132>)
Initial release
2021-09-22 17:45:34

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