+ Statement of need
+ Common to most item response theory (IRT) models is the assumption
+ of unidimensionality, i.e., that a test or item
+ measures simple structures
+ (Hambleton
+ & Rogers, 1991). There are, however, many occasions where
+ this may be improper. Consider a mathematical word problem
+ (Reckase,
+ 1985,
+ 2009;
+ Reckase
+ & McKinley, 1991). To solve a mathematical word problem,
+ one must often have verbal and mathematical skills, referred to as
+ abilities (denoted
+
+ θ)
+ in the literature on IRT. In other words, one’s resulting answer would
+ be a function based on one’s ability to read, on the one hand, and
+ one’s ability to perform numerical manipulations, on the other.
+ Accordingly, instead of a person’s location on a unidimensional latent
+ variable, the mathematical word problem illustrates a situation where
+ it seems more reasonable to assume that a correct response is due to
+ the respondent’s location in a multidimensional latent variable
+ space.
+ Descriptive multidimensional item response theory (DMIRT)
+ (Reckase,
+ 1985,
+ 2009;
+ Reckase
+ & McKinley, 1991) was developed to handle the just
+ mentioned situation. The method is based on using a
+ compensatory model, i.e., a type of measurement model
+ in multidimensional IRT that uses linear combinations of
+
+
+ θ-values
+ for ability assessment. This model assumes that the same probability
+ score for a correct response can be reached by using different
+ combinations of
+
+ θ-values,
+ as opposed to assuming that the relation is one-to-one. In turn, this
+ implies that the compensatory model allows items to be unidimensional,
+ i.e., that they measure a single ability, or
+ within-multidimensional, i.e., that the items can
+ measure more than one ability in the model space.
+ Note that the D3mirt approach is limited to
+ two types of compensatory models, depending on item type. If
+ dichotomous items are used, the analysis is based on the
+ multidimensional extension of the two-parameter logistic model
+ (McKinley
+ & Reckase, 1983) as the compensatory model. If polytomous
+ items are used, the analysis is based on the two-parameter
+ multidimensional graded response model
+ (Muraki
+ & Carlson, 1995) as the compensatory model.
+ Compared to other software, D3mirt is unique
+ in implementing DMIRT methodology explicitly and comprehensively in a
+ three-dimensional interactive environment. For instance, a
+ compensatory model can be fitted with software dedicated to
+ multidimensional IRT that supports dichotomous or polytomous data and
+ allows the user to specify the necessary factor structure. This
+ includes software such as Mplus
+ (Muthén
+ & Muthén, 1998-2017) or IRTPRO 2.1
+ (Thissen
+ & du Toit, 2011) and flexMIRT
+ (Houts
+ & Cai, 2020) for the Windows environment. Using these
+ software programs, the DMIRT estimates can then be derived manually
+ with the help of general mathematical software, such as
+ MATLAB
+ (MATLAB,
+ 2022) or Mathematica
+ (Mathematica,
+ 2023), and plotted with built-in options for creating vector
+ plots in two or three dimensions. This is, however, often
+ time-consuming, and the plotting methods are not optimized for DMIRT
+ analysis and test development. Regarding R, the
+ mirt package
+ (Chalmers,
+ 2012) can be used to fit the compensatory multidimensional
+ model and derive the basic DMIRT item and person parameters while the
+ vector plot options are limited. There is also the
+ R package plink
+ (Weeks,
+ 2010) that offers two-dimensional vector plots suitable for
+ DMIRT analysis but only for dichotomous items. Another more general
+ limitation is that none of the formerly mentioned software
+ applications provides a function to help identify the DMIRT model.
+ The D3mirt package was designed to counter
+ many of the just mentioned shortages by implementing specialized
+ functions for identifying the DMIRT model, calculation of the
+ necessary DMIRT estimates, and plotting the results in an interactive
+ three-dimensional latent environment (see
+ [fig:anes]). An
+ example of the utility of using the package in an empirical context
+ for item and scale analysis has been presented in Forsberg &
+ Sjöberg
+ (2024).
+
+ A still shot of the graphical output from
+ D3mirt. The Figure illustrates a
+ three-dimensional vector plot for items in the
+ anes0809offwaves data set included in the
+ package. The output also shows three construct vector arrows:
+ Compassion, Fairness, and Conformity (solid black
+ arrows).
+ ![]()
+
+
+ A still shot of the graphical output from
+ D3mirt illustrating respondent scores in the
+ latent space separated on sex (male in blue and female in red) from
+ the anes0809offwaves data set included in the
+ package.
+ ![]()
+
+
+
+ Multidimensional item parameters
+ The theoretical framework for DMIRT rests foremost on three
+ assumptions
+ (Reckase,
+ 1985). Firstly, ability maps the probability monotonically,
+ such that a higher level of ability implies a higher probability of
+ answering an item correctly. Second, we wish to locate an item at a
+ singular point at which it is possible to derive item characteristics
+ for the multidimensional case conceptually similar to the
+ unidimensional case. Thirdly, an item’s maximum level of
+ discrimination, i.e., its highest possible capacity to separate
+ respondents on level of ability, is the best option for the singular
+ point estimation. The most important parameter equations regarding the
+ assumptions just mentioned will be briefly presented below.
+ Firstly, by using the discrimination score
+
+
+ ai
+ on item
+
+ i
+ from the compensatory model, we can define the multidimensional
+ discrimination index (MDISC) as follows.
+
+
+ MDISC:=∑k=1maik2,
+ on
+
+ m
+ dimensions with the slope constant
+
+ 14
+ omitted
+ (Reckase,
+ 2009;
+ Reckase
+ & McKinley, 1991). The MDISC is sometimes denoted
+
+
+ Ai
+ to highlight the connection to the unidimensional
+
+
+ ai
+ parameter
+ (Reckase,
+ 2009;
+ Reckase
+ & McKinley, 1991). Importantly, the MDISC sets the
+ orientation of the item vectors in the multidimensional space
+ (Reckase,
+ 2009;
+ Reckase
+ & McKinley, 1991).
+
+
+ ωil=cos−1(ail∑k=1maik2),
+ on latent axis
+
+ l
+ in the model. Note, the
+
+ ωil
+ is in this solution a characteristic of the item
+
+
+ i
+ that tells in what direction
+
+ i
+ has its highest level of discrimination, assuming a multidimensional
+ latent space
+ (Reckase,
+ 2009;
+ Reckase
+ & McKinley, 1991). This gives us the following criteria to
+ use as a rule of thumb. Assume a two-dimensional space, an orientation
+ of
+
+ 0∘
+ with respect to any of the model axes indicates that the item is
+ unidimensional. Such an item describes a singular trait only. In
+ contrast, an orientation of
+
+ 45∘
+ indicated that the item is within-multidimensional. Such an item
+ describes both traits in the two-dimensional model equally well. The
+ same criteria are extended to the three-dimensional case. The MDISC is
+ also used in the graphical output to scale the length of the vector
+ arrows representing the item response functions, e.g., so that longer
+ vector arrows indicate higher discrimination, shorter arrows lower
+ discrimination in the model, and so on.
+ Next, to assess multidimensional difficulty, the distance from the
+ origin is calculated using the multidimensional difficulty (MDIFF),
+ denoted
+
+ Bi
+ to highlight the connection to the unidimensional
+
+
+ bi
+ parameter
+ (Reckase,
+ 1985).
+
+
+ MDIFF:=−di∑k=1maik2,
+ in which
+
+ di
+ is the
+
+ di-parameter
+ from the compensatory model. The MDIFF is, therefore, a difficulty
+ characteristic of item
+
+ i,
+ such that higher MDIFF values indicate that higher levels of ability
+ are necessary for a correct response
+ (Reckase,
+ 2009;
+ Reckase
+ & McKinley, 1991). Observe that the denominator in
+ Equation 3 is the
+ same expression as
+ Equation 1.
+ Importantly, in DMIRT analysis, the MDISC and MDIFF only apply in
+ the direction set by
+
+ ωil
+ and Equation 2
+ (Reckase,
+ 2009;
+ Reckase
+ & McKinley, 1991). Thus, we cannot compare these estimates
+ directly across items, as would be the case in the unidimensional
+ model. This is because DMIRT seeks to maximize item discrimination as
+ a global characteristic in a multidimensional environment. To estimate
+ item discrimination as a local characteristic in the multidimensional
+ space, it is, however, possible to select a common direction for the
+ items and then recalculate the discrimination, i.e., to estimate the
+ directional discrimination (DDISC), as follows.
+
+
+ DDISC:=∑k=1maikcosωik.
+ Since the DDISC is a local characteristic in the model, it is
+ always the case that
+
+ DDISC≤MDISC.
+ In D3mirt, the DDISC is optional and
+ implemented in D3mirt as optional
+ construct vectors indicated by a subset of items or
+ using spherical coordinates.
+ The results include tables for the MDISC and MDIFF estimates as
+ well as spherical coordinates describing the location of the vector
+ arrows. If construct vectors are used, the output also includes DDISC
+ scores for all items showing the constrained discrimination. It is
+ also possible to plot individual scores (i.e., profile
+ analysis) in the three-dimensional latent space (see
+ [fig:p1]). This can be
+ useful for studying respondents’ location conditioned on some external
+ variable, e.g., sex, age, political preference, and so on.
+ Instructions on the method, such as model identification, model
+ estimation, plotting, and profile analysis, are given in the package
+ vignette.
+ To report issues, seek support, or for developers wishing to
+ contribute to the software, contact the author via the dedicated
+ GitHub page
+ (https://github.com/ForsbergPyschometrics/D3mirt)
+ or email (forsbergpsychometrics@gmail.com).
+
+