isop.scoring.Rd
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).
isop.scoring(dat,score.itemcat=NULL)
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). |
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
$$ s_{ik}=\sum_{j=0}^{k-1} f_{ij} -
\sum_{j=k+1}^K f_{ij}=P( X_i < k )
- P( X_i > k ) $$
where \(f_{ik}\) is the relative frequency of item \(i\)
in category \(k\) and \(K\) is the maximum category.
The modified percentile score \(\rho_p\)
for subject \(p\) (mpsc
in person
) is
defined by
$$ \rho_p=\frac{1}{I} \sum_{i=1}^I
\sum_{j=0}^K s_{ik} \mathbf{1}( X_{pi}=k ) $$
Note that for dichotomous items, the sum score is a
sufficient statistic for \(\rho_p\) but this is
not the case for polytomous items.
The modified percentile score \(\rho_p\)
ranges between -1 and 1.
The modified item P-score \(\rho_i\) (Scheiblechner, 2007, p. 52) is defined by $$ \rho_i=\frac{1}{I-1} \cdot \sum_j \left[ P( X_j < X_i ) - P( X_j > X_i ) \right ] $$
A list with following entries:
A data frame with person parameters. The modified
percentile score \(\rho_p\) is denoted by mpsc
.
Item statistics and scoring parameters.
The item P-scores \(\rho_i\)
are labeled as pscore
.
Frequencies for every item category
Scoring points for every item category
Empirical distribution function
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.
############################################################################# # 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 )