mudfold (version 1.1.21)

ISO: Iso statistic for a given unfolding scale.

Description

This function calculates the iso statistic based on the conditional adjacency matrix (CAM) of a given scale. In order to quantify if the rows of the CAM show a weakly unimodal pattern, the iso statistic was introduced (Post, 1992). Iso statistic (ISO), is a measure for the degree of unimodality violation in the rows of CAM. ISO can be obtained for each item (\({ISO}_j\)) and their summation results in the total ISO for the scale (\({ISO}_{tot}\)).

To come up with an ISO value for an item j, one should first locate the maximum in each row of the CAM. If we index \(m^*\) the maximum in row j of CAM, the ISO measures deviations from unimodality to the left and right of \(m^*\). The function takes as input objects of class "cam.mdf" obtained from the function CAM or objects of class "mdf" obtained from the function mudfold

Usage

ISO(x, type)

Value

A vector with the ISO statistic for each item. The sum of the individual ISO statistics for each of the items yield the ISO statistic for the whole scale.

Arguments

x

: A matrix of class 'cam.mdf' obtained from the function CAM(). Alternatively, x can be a fitted object of class "mdf" resulted from the mudfold() function.

type

: This argument controls the type of the statistic that is returned. If type="item" (default) then the ISO statisic for each item in the scale. When type="scale" the ISO statistic for the whole scale will be returned.

Author

Spyros E. Balafas ()

Details

\( ISO_j = \sum_{h\leq k \leq m^*} max(0, {CAM}_{jh} - {CAM}_{jk}) + \sum_{m^* \leq h \leq k} max(0,{CAM}_{jk} - {CAM}_{jh} ) \)

References

W.J. Post. (1992). Nonparametric Unfolding Models: A Latent Structure Approach. M & T series. DSWO Press.

See Also

CAM

Examples

## load the ANDRICH data
data("ANDRICH")

## Calculate the CAM for the ANDRICH scale
CAM_andrch <- CAM(ANDRICH)

## Use the CAM to calculate the ISO statistic
## for the ANDRICH scale
ISO(CAM_andrch)