check.iio {mokken}

Check of Invariant Item Ordering

Description

Returns a list (of class iio.class) with results from the investigation of invariant item ordering. Three methods may be used for the investigation of invariant item ordering. (1) Method MIIO (manifest invariant item ordering: investigates the manifest item response functions for all pairs of items). For polytomous items, t-tests are used to test violations, for dichotomous items z-tests are used to test are used to test violations. (2) Method MS-CPM (manifest scale - cumulative probability model: investigates the manifest item step response functions for all pairs of items). Z-tests are used to test violations. (3) Method IT (increasingness in transposition: investigates all bivariate joint probabilities for all pairs of items). Chi-square tests are used to test violations.

For a complete description of Method MIIO, see Ligtvoet, Van der Ark, Te Marvelde, and Sijtsma (2010); for a complete description of the Method MS-CPM and Method IT with reference to Method MIIO, see Ligtvoet, Van der Ark, Bergsma, and Sijtsma (2011). For a discription of investigating the ordering structure of clustered items, see Koopman & Braeken (2024).

For two-level test data (clustered respondents) argument level.two.var exist, for clustered item data, argument fixed.item.order exist. For both arguments, two lists are returned, containing the results for level 1 (person or item level) and level 2 (group or cluster level), respectively. Only method MIIO is implemented for two-level and clustered-item test data.

Usage

check.iio(X, method="MIIO", minvi = default.minvi, minsize = default.minsize, 
alpha = .05, item.selection=TRUE, verbose=FALSE, fixed.item.order = NULL,
level.two.var = NULL)

Arguments

X

matrix or data frame of numeric data containing the responses of nrow(X) respondents to ncol(X) items. Missing values are not allowed

method

Either "MIIO" (default), "MSCPM", or "IT". Partial matching is allowed (e.g. method="ms" is equivalent to method="MSCPM")

minvi

minimum size of a violation that is reported. By default minvi = .03 times the number of item step response functions (m) for Method MIIO; minvi = .03 for Method IT and Method MSCPM

minsize

minimum size of a rest score group. By default minsize = N/10 if N \ge 500; minsize = N/5 if 250 \le N < 500; and minsize = max(N/3,50) if N < 250

alpha

Nominal Type I error for t test (Method MIIO), z test (Method MSCPM), or McNemar test (Method IT). Default alpha = .05

item.selection

Conduct backward item selection procedure (see Ligtvoet et al., 2010). Default item.selection=TRUE

verbose

Show the results of the backward item selection algorithm on screen. Default verbose=FALSE

fixed.item.order

Matrix or vector containing J numeric values to indicate the item ordering from easy to difficult, to perform a confirmatory analysis of manifest invariant item/cluster ordering. For clustered items, the cluster numbers are given, which are repeated for each item of that cluster, such that the length is still J, for example c(1, 1, 2, 2, 3, 3) for three clusters of two items, of which the first cluster is easiest and the last cluster is most difficult (see Koopman & Braeken, 2024).

level.two.var

Add respondent-clustering variable to get results for Level 1 (person level) and Level 2 (cluster level; see Koopman et al., 2023a,b)

.

Details

The output is of class iio.class, and is often numerous. Functions plot and summary can be used to summarize the output. See Van der Ark (2014) for an example. For an example of clustered items, see Koopman & Braeken (2024).

Value

results

A list with as many components as there are item pairs. Each component itself is also a list containing the results of the investigation of IIO.

violations

A matrix: Summary of the backward item selection (Corresponds to Table 4 in Ligtvoet et al., 2010, and Table 1 in Ligtvoet et al., 2011). The first column gives, for each item, the number of violations of IIO. If the number of violations is nonzero, then the item with the largest number of violations is removed. If two or more items have the maximum number of violations, then from those items the item producing the lowest value of Loevinger's H is removed. The second column shows the number of violations with one item removed, the third column shows the number of violations with two items removed, etc.

items.removed

List of the items removed in chronological order

HT

Coefficient HT for the remaining items. For the use of coefficient HT see Ligtvoet et al. (2010). If the sample size is extremely large coefficient HT is estimated using a random subsample. For clustered items, coefficient HBT and ratio HBT/HT is also estimated, see Koopman & Braeken (2024).

method

The argument method

item.mean

The mean item scores

Author(s)

L. A. van der Ark [e-mail]

References

Koopman, L. & Braeken, J. (2024). Investigating the Ordering Structure of Clustered Items Using Nonparametric Item Response Theory. Manuscript submitted for publication.

Koopman, L., Zijlstra, B. J. H., & Van der Ark, L. A. (2023a). Assumptions and Properties of Two-Level Nonparametric Item Response Theory Models. Manuscript submitted for publication.

Koopman, L., Zijlstra, B. J. H., & Van der Ark, L. A. (2023b). Evaluating Model Fit in Two-Level Mokken Scale Analysis. Manuscript submitted for publication.

Ligtvoet, R., L. A. van der Ark, J. M. te Marvelde, & K. Sijtsma (2010). Investigating an invariant item ordering for polytomously scored items. Educational and Psychological Measurement, 70, 578-595. doi:10.1177/0013164409355697

Ligtvoet, R., L. A. van der Ark, W. P. Bergsma, & K. Sijtsma (2011). Polytomous latent scales for the investigation of the ordering of items. Psychometrika, 76, 200-216. doi:10.1007/s11336-010-9199-8

Sijtsma, K., R. R. Meijer, & Van der Ark, L. A. (2011). Mokken scale analysis as time goes by: An update for scaling practitioners. Personality and Individual Differences, 50, 31-37. doi:10.1016/j.paid.2010.08.016

Sijtsma, K., & Molenaar, I. W. (2002) Introduction to nonparametric item response theory. Sage.

Van der Ark, L. A. (2007). Mokken scale analysis in R. Journal of Statistical Software, 20 (11), 1-19. doi:10.18637/jss.v020.i11

Van der Ark, L. A. (2012). New developments in Mokken scale analysis in R. Journal of Statistical Software, 48(5), 1-27. doi:10.18637/jss.v048.i05

See Also

check.errors, check.monotonicity, check.pmatrix, check.reliability check.restscore, coefH, plot.iio.class, summary.iio.class,

Examples

# Examples from Ligtvoet et al. (2010).

data(acl)            

Order <- acl[,41:50]               
summary(check.iio(Order))
plot(check.iio(Order))

Autonomy <- acl[,91:100]
summary(check.iio(Autonomy))
plot(check.iio(Autonomy))

# Examples from Ligtvoet et al. (2011).
 
data(cavalini)
X1 <- cavalini[,c(3,5,6,7,9,11,13,14)]

# Use Method MIIO and remove items violating MIIO
iio.list1 <- check.iio(X1)
summary(iio.list1)
X2 <- X1[,is.na(charmatch(dimnames(X1)[[2]],names(iio.list1$items.removed)))]

# Use Method MSCPM and remove items violating MSCPM
iio.list2 <- check.iio(X2,method="MSCPM")
summary(iio.list2)
X3 <- X2[,is.na(charmatch(dimnames(X2)[[2]],names(iio.list2$items.removed)))]

# Use Method IT
iio.list3 <- check.iio(X3,method="IT")
summary(iio.list3)

# Examples for investigating the ordering structure of a clustered item set 
# (Koopman & Braeken, 2024)
      
data("trog")
clusters <- rep(1:20, each = 4)
ico <- check.iio(trog, item.selection = FALSE, fixed.item.order = clusters)
summary(ico)


# Compute two-level fit statistics (Koopman et al., 2023a, 2023b)
      
data("autonomySupport")
dat <- autonomySupport[, -1]
groups <- autonomySupport[, 1]
autonomyMIIO <- check.iio(dat, item.selection = FALSE, level.two.var = groups)
summary(autonomyMIIO)
plot(autonomyMIIO)