`D3MIRT`

ModelingThe `D3mirt`

analysis is based on descriptive
multidimensional item response theory (DMIRT; Reckase, 2009, 1985;
Reckase & McKinley, 1991) and can be used to analyze dichotomous and
polytomous items (Muraki & Carlson, 1995) in a three-dimensional
ability (\(\theta\)) space. The method
is foremost visual and illustrates item characteristics with the help of
vector geometry in which items are represented by vector arrows.

In DMIRT analysis, also called within multidimensional modeling, it
is assumed that items in a multidimensional ability space can measure
single or multiple latent traits (Reckase, 2009, 1985; Reckase &
McKinley, 1991). The methodology is a type of data reduction technique
based on the compensatory model (Reckase, 2009), i.e., a type of
measurement model that uses linear combinations of \(\theta\)-values for ability assessment. The
method seeks to maximize item discrimination and so is
*descriptive* because the results describe the extent to which
items in a test are unidimensional, i.e., that the items discriminate on
one dimension only, or are within-multidimensional, i.e., that the items
discriminate on more than one dimension.

Regarding vector orientation, the angle of the vector arrows
indicates what traits, located along the orthogonal axes in the model,
an item can be said to describe (Reckase, 2009, 1985, Reckase &
McKinley, 1991). For instance, in a two-dimensional space, an item is
*unidimensional* if its item vector arrow is at \(0°\) with respect to one of the axes in the
model, and \(90°\) with respect to the
other. Such an item describes a singular trait only. In contrast, an
item is *within-multidimensional* if its item vector arrow is
oriented at \(45°\) in relation to the
axes in the model. Such an item describes both traits in the model
equally well. The same criteria are extended to the three-dimensional
case.

The DMIRT approach uses two types of item models, dependent on item type. If dichotomous items are used, the analysis is based on the multidimensional two-parameter logistic model (M2PL). If polytomous items are used, the analysis is based on the multidimensional two-parameter graded response model (MGRM; Muraki & Carlson, 1995). The method is therefore limited to items that fit these item models.

The estimation process begins by first fitting and extracting the discrimination \(a\) and difficulty parameters \(d\) from a compensatory model. Next, the DMIRT estimation uses the former to compute the multidimensional discrimination (\(MDISC\)) parameter and the multidimensional difficulty (\(MDIFF\)) parameter that is used to locate the items in a vector space.

The \(MDIFF\) is interpreted similarly as the difficulty parameter in the unidimensional model, i.e., it shows the level of ability that is required for a higher or correct response. Note, if polytomous items are used, such as Likert items, the items will be represented by multiple vector arrows (one for each response function). The \(MDIFF\) will, in such a case, show the multidimensional range of difficulty for an item as located in a multidimensional latent trait space.

The \(MDISC\) shows the highest level of discrimination an item can achieve in the multidimensional model. It is, therefore, a global item characteristic assuming a multidimensional latent space. In addition, the \(MDISC\) score is visualized in the graphical output by scaling the length of the vector arrows representing the item response functions, such that longer arrows indicate higher discrimination (and vice versa).

A novel theoretical contribution to DMIRT is the use of constructs in
the `D3mirt`

functions. Constructs, in this context, refer to
the assumption that a subset of items can measure a higher-order latent
variable. In `D3mirt`

, constructs are implemented as optional
vectors whose orientation is calculated as the average direction, i.e.,
the average multidimensional discrimination of a subset of items (from
one item to all items in the set) in the model. A construct vector will,
therefore, point in the direction of the maximum slope of an imaginary
item response function indicated by the items chosen by the user.

If constructs are used, the output will include reporting of the directional discrimination (\(DDISC\)) parameter that shows how well the items discriminate under the assumption that they measure one of the constructs used in the analysis. That is, while the \(MDISC\) represents the maximum level of discrimination in the model, the \(DDISC\) represents the local discrimination that makes it possible to compare item discrimination in a specific direction set by the constructs. The constructs are, therefore, like unidimensional models nested in the multidimensional latent space and are visually represented with construct vector arrows scaled to an arbitrary length.

The package includes the following main functions.

`modid()`

: D3mirt Model Identification`D3mirt()`

: 3D DMIRT Model Estimation`plot()`

: Graphical Output for`D3mirt()`

You can install the `D3mirt`

package from CRAN, or try the
development version of the package, by using the following codes for
`R`

.

```
# Install from CRAN depository
install.packages('D3mirt')
# Install development version from Github
# install.packages("devtools")
# To include package vignette in the installation add: build_vignettes = TRUE
::install_github("ForsbergPyschometrics/D3mirt") devtools
```

In what follows, the `D3mirt`

procedure, including the
main functions and some of the more essential arguments, will be
described very briefly using the built-in data set “anes0809offwaves”.
The data set (\(N = 1046, M_{age} = 51.33, SD
= 14.56, 57\%\) Female) is a subset from the American National
Election Survey (ANES) from the 2008-2009 Panel Study Off Wave
Questionnaires, December 2009 (DeBell, et al, 2010; https://electionstudies.org/data-center/2008-2009-panel-study/).
All items measure moral preferences and are positively scored of Likert
type, ranging from 1 = *Strongly Disagree* to 6 = *Strongly
Agree*. Demographic variables include age and gender
(male/female).

The D3mirt approach largely consists of the following three steps:

- Model Identification
- D3mirt model estimation
- Plotting

For more details on the `D3mirt`

package, including
extended examples of analysis and functions, please see the vignette
included in the package.

As a first step in the analysis, the three-dimensional compensatory model must be identified (Reckase, 2009). In the three-dimensional case, this implies locating the \(x\) and \(y\)-axis by selecting two items from the item set. The first item should not load on the second and third axes (\(y\) and \(z\)), while the second item should not load on the third axis (\(z\)). Consequently, if the model is not known beforehand it is necessary to explore the data with exploratory factor analysis (EFA), preferably with the help of the EFA methods suitable for DMIRT.

The `modid()`

function was designed to help with this step
in the process. The function first performs an EFA using
multidimensional item response theory and then select items so that the
strongest loading item, from the strongest factor, always aligns
perfectly with the x-axis, and the remaining items follow thereafter.
This helps makes the result maximally interpretable while also avoiding
imposing an unempirical structure on the data.

Note, the EFA is only used to find model identification items that meet the necessary DMIRT model specification requirements. The EFA model itself is discarded after this step in the procedure. This implies that the rotation method is less crucial and the user is encouraged to try different rotation methods and compare the results.

Begin by loading the item data. Note, all outputs are available as ready-made package files that can be loaded directly into the R session.

```
# Load data
data("anes0809offwaves")
<- anes0809offwaves
x <- x[,3:22] # Remove columns for age and gender x
```

The `modid()`

can take in raw item data or a data frame
with item factor loadings. In the default mode
(`efa = TRUE`

), using raw data, the function performs an EFA
using the EFA option in the `mirt()`

(Chalmers, 2012)
function with three factors as default (`factors = 3`

). The
function then finishes with the model identification. If, however, item
factor loadings are already available, the function can jump directly to
the model identification by setting `efa = FALSE`

.

The output consists of an \(S3\)
object of class `modid`

containing data frames with model
identification items, order of factor strength (based on sum of
squares), and item factor loadings. The function has two arguments: the
*lower* and *upper* bound. In brief, the lower bound
increase the item pool used in the procedure while the upper bound acts
as a filter that removes items that do not meet the necessary
statistical requirements. This implies that the upper bound should not,
in general, be manipulated.

```
# Optional: Load the EFA data for this example directly from the package file
load(system.file("extdata/efa.Rdata", package = "D3mirt"))
# Call to modid() with x, containing factors scores from the EFA
# Observe that the efa argument is set to false
<- modid(x, efa = FALSE)
a summary(a)
#>
#> modid: 20 items and 3 factors
#>
#> Model identification items:
#> Item 1 W7Q3
#> Item 2 W7Q20
#>
#> Item.1 ABS
#> W7Q3 0.8547 0.0174
#> W7Q5 0.8199 0.0648
#> W7Q1 0.7589 0.0772
#> W7Q10 0.7239 0.0854
#>
#> Item.2 ABS
#> W7Q20 0.7723 0.0465
#> W7Q19 0.6436 0.0526
#> W7Q18 0.6777 0.0782
#>
#> SS Loadings
#> F2 5.3505
#> F1 2.1127
#> F3 1.6744
#>
#> F2 F1 F3
#> W7Q1 0.7589 0.0407 -0.0365
#> W7Q2 0.8901 -0.0263 -0.0838
#> W7Q3 0.8547 -0.0096 -0.0078
#> W7Q4 0.6628 0.0272 0.1053
#> W7Q5 0.8199 -0.0390 -0.0258
#> W7Q6 0.6654 0.0525 0.1054
#> W7Q7 0.5603 -0.0148 0.2087
#> W7Q8 0.5731 0.0390 0.1966
#> W7Q9 0.6151 0.0697 0.0918
#> W7Q10 0.7239 0.0371 -0.0483
#> W7Q11 0.2085 0.0959 0.5488
#> W7Q12 0.0755 -0.0853 0.5559
#> W7Q13 -0.0176 -0.0153 0.7654
#> W7Q14 -0.0407 0.1439 0.5629
#> W7Q15 0.1087 0.4556 -0.1111
#> W7Q16 0.1759 0.2100 0.1152
#> W7Q17 0.2160 0.5816 0.0261
#> W7Q18 -0.0560 0.6777 -0.0782
#> W7Q19 0.0589 0.6436 0.0526
#> W7Q20 -0.0735 0.7723 0.0465
```

The `summary()`

function prints the number of items and
the number of factors used in the analysis together with the suggested
model identification items. As can be seen, the items suggested by
`modid()`

are the items “W7Q3” and “W7Q20”. The output also
includes data frames that hold all the model identification items
(`Item.1...Item.n`

) selected by `modid()`

together
with the items’ absolute sum score (`ABS`

), one frame for the
sum of squares for factors sorted in descending order, and one frame for
item factor loadings.

The order of the factors follows the model identification items so that item 1 comes from the strongest factor (sorted highest up), item 2 from the second strongest (sorted second), and so on. The absolute sum scores indicate statistical fit to the structural assumptions of the DMIRT model and the items are, therefore, sorted with the lowest absolute sum score highest up. The top items are the items that best meet the necessary statistical requirements for the model identification. For a three-dimensional model this implies that the item highest up in the first data frame should be used to identify the \(x\)-axis, and the item highest up in the second data frame should be used to identify the \(y\)-axis, and so on. For more on the model identification procedure (e.g., trouble shooting, criteria, or limitations), see package vignette.

The `D3mirt()`

function takes model parameters from a
three-dimensional compensatory model (either in the form of a data frame
or an S4 object of class ‘SingleGroupClass’ exported from the
`mirt()`

(Chalmers, 2012) function) and returns an \(S3\) object of class `D3mirt`

with lists of \(a\) and \(d\), \(MDISC\), and \(MDIFF\) parameters, direction cosines, and
spherical coordinates. Regarding the latter, spherical coordinates are
represented by \(\theta\) and \(\phi\). The \(\theta\) coordinate is the positive or
negative angle in degrees, starting from the \(x\)-axis, of the vector projections from
the vector arrows in the \(xz\)-plane
up to \(\pm 180°\). Note, the \(\theta\) angle is oriented following the
positive pole of the \(x\) and \(z\) axis so that the angle increases
clockwise in the graphical output. The \(\phi\) coordinate is the positive angle in
degrees from the \(y\)-axis and the
vectors. Note, the \(\rho\) coordinate
from the spherical coordinate system is in DMIRT represented by the
MDIFF, and so is reported separately.

If constructs are used, the function also returns construct direction cosines, spherical coordinates for the construct vector arrows, and \(DDISC\) parameters (one index per construct).

The three-dimensional compensatory model is specified so that all
items load on all three factors in the model, and that the factors are
constrained to be orthogonal (see below). The fitting of the model is
preferably done with the `mirt()`

(Chalmers, 2012) function.
Please note very carefully regarding the model specification in the
example below. However, because the fitting of the compensatory model in
the `mirt()`

function takes a long time, the item parameters
for this example are contained in a data frame that is available in the
package file “mod1.Rdata”.

```
# Load data
data("anes0809offwaves")
<- anes0809offwaves
x <- x[,3:22] # Remove columns for age and gender
x
# Fit a three-dimensional graded response model with orthogonal factors
# Example below uses Likert items from the built-in data set "anes0809offwaves"
# Item W7Q3 and item W7Q20 was selected with modid()
# The model specification set all items in the data set (1-20)
# to load on all three factors (F1-F3)
# The START and FIXED commands are used on the two items to identify the DMIRT model
<- ' F1 = 1-20
spec F2 = 1-20
F3 = 1-20
START=(W7Q3,a2,0)
START=(W7Q3,a3,0)
START=(W7Q20,a3,0)
FIXED=(W7Q3,a2)
FIXED=(W7Q3,a3)
FIXED=(W7Q20,a3) '
<- mirt::mirt(x,
mod1
spec, itemtype = 'graded',
SE = TRUE,
method = 'QMCEM')
```

Constructs can be included in the analysis by creating one or more
nested lists that indicate what items belong to what construct. Such a
nested list can contain all items in the set down to a single item. From
this, the `D3mirt()`

function finds the average direction of
the subset of items contained in each nested list by adding and
normalizing the direction cosines for the items and scaling the
construct direction vector to an arbitrary length (length can be
adjusted by the user) so that the arrows can be seen when plotting.

The construct vector arrows can contribute to the analysis by (a) visualizing the average direction for a subset set of items, and (b) showing how all items discriminate locally in the direction of the construct vector with the help of the \(DDISC\) index.

The `summary()`

function is used to inspect the DMIRT
estimates. The constructs included below were grouped based on
exploratory reasons, i.e., because these items cluster in the model
(observable in the graphical output below).

```
# Optional: Load the mod1 data as data frame directly from the package file
load(system.file("extdata/mod1.Rdata", package = "D3mirt"))
# Call to D3mirt(), including optional nested lists for three constructs
# Item W7Q16 is not included in any construct because of model violations
# The model violations for the item can be seen when plotting the model
<- list(list(1,2,3,4,5,6,7,8,9,10),
c list(11,12,13,14),
list(15,17,18,19,20))
<- D3mirt(mod1, c)
g summary(g)
#>
#> D3mirt: 20 items and 5 levels of difficulty
#>
#> Constructs:
#> Vector 1: W7Q1, W7Q2, W7Q3, W7Q4, W7Q5, W7Q6, W7Q7, W7Q8, W7Q9, W7Q10
#> Vector 2: W7Q11, W7Q12, W7Q13, W7Q14
#> Vector 3: W7Q15, W7Q17, W7Q18, W7Q19, W7Q20
#>
#> a1 a2 a3 d1 d2 d3 d4 d5
#> W7Q1 2.0298 0.1643 -0.1233 8.0868 7.0642 5.9877 3.2015 -0.4836
#> W7Q2 2.6215 -0.0027 -0.2585 9.2889 6.6187 4.5102 1.6648 -2.4440
#> W7Q3 2.7917 0.0000 0.0000 10.4835 7.5865 5.6764 2.7167 -1.1788
#> W7Q4 1.9046 0.1874 0.1491 7.3754 6.0467 4.9814 2.4830 -1.1146
#> W7Q5 2.2423 -0.0287 -0.0841 8.4266 6.6706 4.9047 1.8252 -1.8316
#> W7Q6 2.0022 0.2390 0.1567 8.0687 6.3578 4.9520 2.3300 -1.0189
#> W7Q7 1.6286 0.1033 0.3593 6.0178 4.8974 3.6908 1.6326 -1.3484
#> W7Q8 1.7775 0.2252 0.3528 6.9171 5.1822 3.7661 1.4844 -1.8332
#> W7Q9 1.7199 0.2493 0.1278 7.5587 4.9755 3.3648 0.9343 -2.2094
#> W7Q10 1.7696 0.1272 -0.1407 8.3639 5.7396 4.2862 1.9646 -0.6642
#> W7Q11 1.4237 0.4673 1.0433 6.2180 4.6920 3.5430 1.1918 -1.8573
#> W7Q12 0.7605 0.0409 0.9366 4.1360 2.8770 2.3419 1.1790 -0.4239
#> W7Q13 1.1285 0.2910 1.6943 5.8922 4.4009 3.4430 1.8955 -0.6009
#> W7Q14 0.7448 0.4828 0.9785 5.3891 3.9333 3.0258 0.8144 -1.5868
#> W7Q15 0.4551 0.7870 -0.1606 4.3206 3.0544 2.3969 0.9187 -0.9705
#> W7Q16 0.6237 0.4139 0.1799 3.7249 2.0305 1.1658 -0.0612 -1.8085
#> W7Q17 1.1893 1.3412 0.0564 6.9011 5.8022 4.9344 2.7915 -0.0041
#> W7Q18 0.4107 1.3542 -0.1368 3.7837 2.0985 1.4183 0.1828 -1.9855
#> W7Q19 0.8580 1.4098 0.2279 4.4978 2.6483 1.6731 0.3740 -1.9966
#> W7Q20 0.7357 1.9068 0.0000 4.6378 2.3633 1.2791 -0.3431 -2.9190
#>
#> MDISC MDIFF1 MDIFF2 MDIFF3 MDIFF4 MDIFF5
#> W7Q1 2.0402 -3.9638 -3.4625 -2.9348 -1.5692 0.2370
#> W7Q2 2.6343 -3.5262 -2.5125 -1.7121 -0.6320 0.9278
#> W7Q3 2.7917 -3.7553 -2.7176 -2.0333 -0.9731 0.4222
#> W7Q4 1.9196 -3.8421 -3.1500 -2.5950 -1.2935 0.5806
#> W7Q5 2.2441 -3.7550 -2.9725 -2.1856 -0.8133 0.8162
#> W7Q6 2.0225 -3.9894 -3.1435 -2.4485 -1.1520 0.5038
#> W7Q7 1.6710 -3.6013 -2.9308 -2.2087 -0.9770 0.8070
#> W7Q8 1.8261 -3.7880 -2.8379 -2.0624 -0.8129 1.0039
#> W7Q9 1.7425 -4.3377 -2.8553 -1.9310 -0.5362 1.2679
#> W7Q10 1.7797 -4.6995 -3.2249 -2.4083 -1.1039 0.3732
#> W7Q11 1.8259 -3.4055 -2.5697 -1.9404 -0.6527 1.0172
#> W7Q12 1.2071 -3.4263 -2.3834 -1.9400 -0.9767 0.3512
#> W7Q13 2.0564 -2.8653 -2.1401 -1.6743 -0.9218 0.2922
#> W7Q14 1.3211 -4.0794 -2.9773 -2.2904 -0.6164 1.2011
#> W7Q15 0.9232 -4.6800 -3.3085 -2.5963 -0.9951 1.0513
#> W7Q16 0.7699 -4.8381 -2.6373 -1.5142 0.0795 2.3490
#> W7Q17 1.7934 -3.8481 -3.2353 -2.7514 -1.5566 0.0023
#> W7Q18 1.4217 -2.6613 -1.4760 -0.9976 -0.1286 1.3966
#> W7Q19 1.6661 -2.6996 -1.5895 -1.0042 -0.2245 1.1984
#> W7Q20 2.0438 -2.2693 -1.1563 -0.6259 0.1679 1.4282
#>
#> D.Cos X D.Cos Y D.Cos Z Theta Phi
#> W7Q1 0.9949 0.0805 -0.0604 -3.4748 85.3808
#> W7Q2 0.9952 -0.0010 -0.0981 -5.6305 90.0597
#> W7Q3 1.0000 0.0000 0.0000 0.0000 90.0000
#> W7Q4 0.9922 0.0976 0.0777 4.4767 84.3967
#> W7Q5 0.9992 -0.0128 -0.0375 -2.1474 90.7326
#> W7Q6 0.9900 0.1182 0.0775 4.4765 83.2140
#> W7Q7 0.9746 0.0618 0.2150 12.4409 86.4543
#> W7Q8 0.9734 0.1233 0.1932 11.2272 82.9174
#> W7Q9 0.9870 0.1431 0.0734 4.2512 81.7735
#> W7Q10 0.9943 0.0715 -0.0791 -4.5468 85.9010
#> W7Q11 0.7797 0.2560 0.5714 36.2355 75.1698
#> W7Q12 0.6300 0.0339 0.7759 50.9236 88.0565
#> W7Q13 0.5488 0.1415 0.8239 56.3330 81.8637
#> W7Q14 0.5638 0.3655 0.7407 52.7234 68.5629
#> W7Q15 0.4929 0.8525 -0.1739 -19.4324 31.5149
#> W7Q16 0.8102 0.5376 0.2336 16.0853 57.4764
#> W7Q17 0.6631 0.7478 0.0315 2.7156 41.5968
#> W7Q18 0.2888 0.9525 -0.0962 -18.4194 17.7246
#> W7Q19 0.5150 0.8462 0.1368 14.8767 32.1990
#> W7Q20 0.3600 0.9330 0.0000 0.0000 21.0997
#>
#> C.Cos X C.Cos Y C.Cos Z Theta Phi
#> C1 0.9970 0.0687 0.0364 2.0923 86.0608
#> C2 0.6412 0.2026 0.7402 49.1006 78.3129
#> C3 0.4720 0.8814 -0.0207 -2.5136 28.1932
#>
#> DDISC1 DDISC2 DDISC3
#> W7Q1 2.0305 1.2435 1.1054
#> W7Q2 2.6040 1.4890 1.2403
#> W7Q3 2.7832 1.7899 1.3177
#> W7Q4 1.9171 1.3695 1.0611
#> W7Q5 2.2305 1.3696 1.0348
#> W7Q6 2.0183 1.4482 1.1524
#> W7Q7 1.6439 1.3311 0.8523
#> W7Q8 1.8004 1.4464 1.0301
#> W7Q9 1.7364 1.2478 1.0289
#> W7Q10 1.7679 1.0562 0.9503
#> W7Q11 1.4895 1.7797 1.0622
#> W7Q12 0.7951 1.1891 0.3756
#> W7Q13 1.2068 2.0366 0.7541
#> W7Q14 0.8113 1.2996 0.7568
#> W7Q15 0.5019 0.3324 0.9118
#> W7Q16 0.6568 0.6169 0.6555
#> W7Q17 1.2799 1.0759 1.7422
#> W7Q18 0.4975 0.4364 1.3902
#> W7Q19 0.9606 1.0044 1.6428
#> W7Q20 0.8645 0.8580 2.0278
```

The `D3mirt()`

function prints a short report containing
the number of items used and the levels of difficulty of the items when
the estimation is done. As can be seen, when construct vectors are used,
the function also prints the number of construct vectors and the names
of the items included in each construct. Next, the factor loadings and
the difficulty parameters from the compensatory model are reported in
data frames followed by all necessary DMIRT estimates.

The `plot()`

method for objects of class
`D3mirt`

is built on the `rgl`

package (Adler
& Murdoch, 2023) for visualization with OpenGL. Graphing in default
mode by calling `plot()`

will return an RGL device that will
appear in an external window as a three-dimensional interactive object,
containing vector arrows with the latent dimensions running along the
orthogonal axes, that can be rotated. In this illustration, however, all
RGL devices are plotted inline as still shots displayed from two angles,
\(15^{\circ}\) (clockwise; default plot
angle) and \(90^{\circ}\). To change
the plot output to \(90^{\circ}\), use
the `view`

argument in the `plot()`

function and
change the first indicator from \(15\)
to \(90\).

```
# Plot RGL device with constructs visible and named
plot(g, constructs = TRUE,
construct.lab = c("Compassion", "Fairness", "Conformity"))
```

Figure 1: Three-dimensional vector plot for all items and the three constructs Compassion, Fairness, and Conformity (solid black arrows) plotted with the model rotated \(15^{\circ}\) clockwise.

Figure 2: Three-dimensional vector plot for all items and the three constructs Compassion, Fairness, and Conformity (solid black arrows) plotted with the model rotated \(90^{\circ}\) clockwise.

An example of how the output can be described could be as follows.

As can be seen in Figures 1 and 2, the pattern in the data indicates the presence of foremost two main nested latent constructs indicated by the items, one aligned with the \(x\)-axis and one approaching the \(y\)-axis. We might also suspect the presence of a third construct located close to the \(xy\)-plane, between the \(x\) and \(z\) axes. Studying the content of the items, the labels

Compassion,Fairness, andConformitywere introduced. The angles of the constructs inform us that Compassion (\(\theta = 2.092^{\circ}\), \(\phi = 86.061^{\circ}\)) and Conformity (\(\theta = -2.514 ^{\circ}\), \(\phi = 28.193^{\circ}\)) have some within-multidimensional tendencies. However, they are both more or less orthogonal to the \(z\)-axis. Next, we find Fairness (\(\theta = 49.101^{\circ}\), \(\phi = 78.313^{\circ}\)) with clear within-multidimensional tendencies with respect to the \(x\)-axis. Thus, the output indicates that Compassion and Conformity could be independent constructs but that Fairness seems not to be.

`items`

A subset of items can be plotted for a more thorough investigation
using the `items`

argument. In the example below, all
constructs are plotted together with the items used for the conformity
construct. In the function call, the numerical indicators in the
`items`

argument follow the item order in the original data
frame (see `?anes0809offwaves`

).

```
# The Conformity items from the model plotted with construct vector arrows
plot(g, constructs = TRUE,
items = c(15,17,18,19,20),
construct.lab = c("Compassion", "Fairness", "Conformity"),
view = c(15, 20, 0.6))
```

Figure 7: The items from the Conformity construct plotted with the model rotated \(15^{\circ}\) clockwise.

Figure 8: The items from the Conformity construct plotted the model rotated \(90^{\circ}\) clockwise.

The `plot()`

function also allows plotting a single item
by entering a single number indicating what item that should be
displayed. As was mentioned above, the W7Q16 was not included in any of
the constructs because the item showed signs of measurement problems.
For example, the short vector arrows indicate high amounts of model
violations and the location of the item in the model also indicates that
the item is within-multidimensional and that it does not seem to belong
to any construct explicitly. Typing in \(16\) in the `items`

argument
allows for a closer look.

```
# Item W7Q16 has location 16 in the data set (gender and age excluded)
# The item is plotted together with construct to aid the visual interpretation
plot(g, constructs = TRUE,
items = 16,
construct.lab = c("Compassion", "Fairness", "Conformity"))
```

Figure 3: The item W7Q16 plotted with the three constructs and with the model rotated \(15^{\circ}\) clockwise.

Figure 4: The item W7Q16 plotted with the three constructs and with the model rotated \(90^{\circ}\) clockwise.

An example of how the output for analysis of the single item could be as follows.

The Figures 3 and 4 shows that item W7Q16 is located at \(\theta = 16.085^{\circ}\), \(\phi = 57.476^{\circ}\), indicating that the item is within-multidimensional with respect to the \(x\) and \(y\)-axis; but much less so with respect to the \(z\)-axis. In addition, the directional discrimination further underscores that the item does not seem to measure any particular construct (\(DDISC_1 = .657\), \(DDISC_2 = .617\), \(DDISC_3 = .656\)). The global discrimination (\(MDISC = .770\), \(MDIFF_{range} = [-4.838, 2.349]\)) is also the lowest of all discrimination scores in the model. This, combined, implies that the item in question does not seem to fit the three-dimensional DMIRT model used in this analysis and should therefore be removed or adapted. On a side note, we can also note that item W7Q15, \(MDISC = .923\), \(MDIFF_{range} = [-4.680, 1.051]\)) has the second lowest global discrimination score. However, this item does seem to belong to the Conformity construct, observable when comparing angle orientation (\(\theta = -19.432^{\circ}, \phi = 31.515^{\circ}\)) and direction discrimination (\(DDISC_1 = .502\), \(DDISC_2 = .332\), \(DDISC_3 = .912\)).

`diff.level`

The user has the option of plotting on one level of difficulty at a
time with the `diff.level`

argument studying the entire
scale, a subset of items, or on one item at a time. Note,
*difficulty* refers to the number of item response functions in
the items, i.e., the total number of response options minus one. In this
case, \(6\) response options were used
which means that the model has \(5\)
levels of difficulty.

```
# Plot RGL device on item difficulty level 5
plot(g, diff.level = 5,
view = c(15, 20, 0.6))
```

Figure 5: All items plotted on difficulty level 5 and with the model rotated \(15^{\circ}\) clockwise.

Figure 6: All items plotted on difficulty level 5 and with the model rotated \(90^{\circ}\) clockwise.

`scale`

The `D3mirt()`

function returns item vector coordinates
estimated with and without the \(MDISC\) as a scalar for the arrow length.
When the \(MDISC\) is not used for the
arrow length, all item vector arrows are scaled to one unit length. This
allows the user to graph the item vector arrows with `plot()`

set to a uniform length. This can help reduce visual clutter in the
graphical output. To view the item vector arrows without the \(MDISC\), set `scale = TRUE`

.

```
# Plot RGL device with items in uniform length and constructs visible and named
plot(g, scale = TRUE,
constructs = TRUE,
construct.lab = c("Compassion", "Fairness", "Conformity"),
view = c(15, 20, 0.6))
```

Figure 9: All items scaled to uniform length and plotted with the model rotated \(15^{\circ}\) clockwise.

Figure 10: All items scaled to uniform length and plotted with the model rotated \(90^{\circ}\) clockwise.

`D3mirt`

Profile
AnalysisThe `plot()`

function can also display respondents in the
three-dimensional model represented as spheres whose coordinates are
derived from the respondent’s factor scores. This allows for a profile
analysis in which respondents can be separated, or subset, conditioned
on single or multiple criteria and then plotted. The resulting output
shows where the respondents are located in the model, and, accordingly,
what model profile best describes them. Similarly, respondent categories
can be simultaneously compared to see if a group-level effect can be
visually observed.

To do this, the user must first extract respondent factor scores with
`fscores()`

(Chalmers, 2012) and then separate or select a
subset of respondent rows based on one or more criteria. The resulting
data frame is imputed in the `profiles`

argument. Generally,
it can be useful to hide vector arrows with `hide = TRUE`

when plotting respondent profiles to avoid visual cluttering. The
example below separates respondents using the gender variable included
in the built-in data set.

```
# Extract respondent factor scores from mod1 with fscores()
<- mirt::fscores(mod1,
f method="EAP",
full.scores = TRUE,
full.scores.SE = FALSE, QMC = TRUE)
```

```
# Optional: Load the respondent factor scores for this example directly from the package file
load(system.file("extdata/fscores.Rdata", package = "D3mirt"))
# Attach f to the gender variable (column 2 from anes0809offwaves data set; "W3XGENDER")
# Use cbind with fscores() output attached first
data("anes0809offwaves")
<- anes0809offwaves
x <- data.frame(cbind(f, x[,2])) z
```

The `plot()`

function uses `as.factor()`

to
count the number of factor levels in the data imputed in the
`levels`

argument. This means that raw data can be used as is
but the number of colors in the color vectors argument
(`sphere.col`

) may need to be adapted. In the example below,
the criteria variable for gender only hold two factor levels and
therefore only two colors in the color vector are needed.

Call `plot()`

with the respondent data frame \(z\) in the `profiles`

argument
and the `levels`

argument with the levels column subset from
\(z\). In the function call below, the
axes in the model are named using the `x.lab`

,
`y.lab`

, and `z.lab`

arguments following the
direction of the constructs. Note, the model axes represent
unidimensional singular structures, or traits, in this case borrowing
the names of the constructs.

```
# Plot profiles with item vector arrows hidden with hide = TRUE
# Score levels: 1 = Blue ("male") and 2 = Red ("female")
plot(g, hide = TRUE,
profiles = z,
levels = z[,4],
sphere.col = c("blue", "red"),
x.lab = "Compassion",
y.lab="Conformity",
z.lab="Fairness")
```

Figure 11: Gender profile for the `anes0809offwaves`

data
set plotted with the model rotated \(15^{\circ}\) clockwise.

Figure 12: Gender profile for the `anes0809offwaves`

data
set plotted with the model rotated \(90^{\circ}\) clockwise.

An example of how the output can be described could be as follows.

In Figures 5 and 6, it can be observed a simple profile on gender in which more women tend to have higher levels of Compassion. When rotating the model \(90^{\circ}\) clockwise, there seems not to be any easily observable gender difference related to Conformity or Fairness.

It is also possible to plot a confidence interval in the shape of an
ellipse surrounding the spheres. In the example below, the younger
individuals (\(\leq30\)) are selected
and plotted together with a \(95\%\)
*CI*.

```
# Column bind fscores() with age variable ("W3Xage") from column 1
<- data.frame(cbind(f, x[,1]))
y
# Subset data frame y conditioned on age <= 30
<- subset(y, y[,4] <= 30) z1
```

When a criterion variable has a wide data range, such as an age
variable, `rep()`

can be used to set the appropriate size of
the color vector for `sphere.col`

by repeating color names
with `rep()`

. When plotting, the `plot()`

function
will pick colors from the `sphere.col`

argument following the
factor order in the levels argument. To do this, the first step is to
count the number of factors in the criterion variable. This can be done
with `nlevels()`

, as can be seen below.

```
# Check number of factor levels with nlevels() and as.factor()
nlevels(as.factor(z1[,4]))
#> [1] 14
# Use rep() to create a color vector to color groups based on the nlevels() output
# z1 has 14 factor levels
<- c(rep("red", 14)) colvec
```

To plot the *CI*, the `ci`

argument is set to
`TRUE`

. The color of the sphere was also changed from default
`grey80`

to `orange`

in the example below. Note,
the *CI* limit can be adjusted with the `ci.level`

argument.

```
# Call plot() with profile data on age with item vector arrows hidden
plot(g, hide = TRUE,
profiles = z1,
levels = z1[,4],
sphere.col = colvec,
x.lab = "Compassion",
y.lab="Conformity",
z.lab="Fairness",
ci = TRUE,
ci.level = 0.95,
ellipse.col = "orange")
```

Figure 13: Adults less than or equal to age 30 from the
`anes0809offwaves`

data set plotted surrounded by a \(95\%\,CI\) and with the model rotated \(15^{\circ}\) clockwise.

Figure 14: Adults less than or equal to age 30 from the
`anes0809offwaves`

data set plotted surrounded by a \(95\%\,CI\) and with the model rotated \(90^{\circ}\) clockwise.

An example of how the output can be described could be as follows.

In Figures 7 and 8 we can see a tendency for a profile on age in which younger individuals could be described as less oriented towards Conformity. We can also observe a tendency for what could be an interaction effect in which higher levels of Conformity seem to be associated with lower levels of Fairness.

Some options for exporting the RGL device are shown below. Over and
above these, it is also possible to export graphical devices in R
Markdown documents with `rgl::hookwebgl()`

together with
graphical options for knitr, as was done when creating the package
vignette.

```
# Export an open RGL device to the console that can be saved as an html or image file
plot(g, constructs = TRUE)
<- scene3d()
s ::rglwidget(s,
rglwidth = 1040,
height = 1040)
# Export a snap shoot of an open RGL device directly to file
plot(g, constructs = TRUE)
::rgl.snapshot('RGLdevice.png',
rglfmt = 'png')
```

If you encounter a bug, please file an issue with a minimal reproducible example on GitHub (https://github.com/ForsbergPyschometrics/D3mirt). For questions please contact me on Github or via email (forsbergpsychometrics@gmail.com).

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Michigan.

Muraki, E., & Carlson, J. E. (1995). Full-Information Factor
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Reckase, M. D.(2009).*Multidimensional Item Response Theory*.
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Reckase, M. D., & McKinley, R. L. (1991). The Discriminating
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