# D3MIRT Modeling

The 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 to measure latent traits (denoted $$\theta$$) in a three-dimensional ability space. The method is foremost visual and illustrates item characteristics with the help of vector geometry.

In DMIRT analysis, it is assumed that items in a multidimensional ability space can measure single or multiple latent traits (Reckase, 2009, 1985). This assumption is realized in the compensatory model, i.e., a type of measurement model that uses linear combinations of $$\theta$$-values for ability assessment (Reckase, 2009). DMIRT builds on the results from the former model and seeks to maximize item discrimination in the latent space by relaxing the assumption of unidimensionality. The method is descriptive because the results describe both the strength of discrimination of the items and the extent to which the items are unidimensional, i.e., that the items discriminate on one dimension only, or are within-multidimensional, i.e., that the maximum item discrimination is reached when the item measures more than one dimension.

Regarding vector orientation, the angle of the vector arrows indicates what traits, located along the axes in the model, an item measures (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 single trait only. In contrast, an item is perfectly 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 $$n$$-dimensional case.

The DMIRT approach uses two types of item models depending on item type. If dichotomous items are used, the analysis is based on the multidimensional two-parameter logistic model (M2PL) (McKinley & Reckase, 1983; Reckase, 1985). If polytomous items are used, the analysis is extended to 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 begins by first fitting and extracting the loadings $$a$$ and difficulty parameters $$d$$ from a compensatory model. Next, the DMIRT estimation takes the former estimates and computes the multidimensional discrimination ($$MDISC$$) parameter and the multidimensional difficulty ($$MDIFF$$) parameter used to locate the items in a multidimensional vector space.

The $$MDIFF$$ is interpreted similarly to the difficulty parameter in the unidimensional model. It shows the level of ability required for a higher or correct response, defined as the distance from the origin. Note that 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 then show an item’s multidimensional range of difficulty as located in the latent trait space.

The $$MDISC$$ shows the highest level of discrimination an item can achieve in the multidimensional latent model. It is, therefore, a global item characteristic referring only to the discrimination achieved in the specific direction of the item in the latent space. The $$MDISC$$ score is visualized in the graphical output by scaling the length of the vector arrows (longer arrows indicate higher discrimination and vice versa), which in turn describe the domain of item response functions.

In D3mirtit also possible to use constructs in the analysis. Constructs, in this context, refer to the assumption that a subset of items or a particular orientation in the model space can measure a higher-order latent variable. This is implemented in D3mirt as optional vectors whose orientation is either calculated as the average direction of a subset of items (from all items down to a single item) or indicated by spherical coordinates. A construct vector will, therefore, point in the direction of the maximum slope of what we might think of as an imaginary item response function indicated by the items or coordinates chosen by the user based on exploratory or theoretical reasons.

If constructs are used, the output will include reporting the directional discrimination ($$DDISC$$) parameter that shows how well the items discriminate, assuming 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.

# Overview

The package includes the following main functions.

• modid(): D3mirt Model Identification
• D3mirt(): 3D DMIRT Model Estimation
• plot(): Graphical Output for D3mirt()

## Installation

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 the development version from GitHub
# install.packages("devtools")
# To include package vignette in the installation, add: build_vignettes = TRUE
devtools::install_github("ForsbergPyschometrics/D3mirt")

In what follows, the D3mirt procedure, including the main functions and some of the more essential arguments, will be briefly described 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:

1. Model Identification
2. D3mirt model estimation
3. Plotting

Please see the vignette included in the package for more details on the D3mirt package, including extended examples of analysis and functions.

# 1. Model Identification

As a first step in the analysis, the three-dimensional compensatory model must be identified (Reckase, 2009). In the three-dimensional case, this implies identifying the $$x$$ and $$y$$-axis in the DMIRT model by selecting two items from the item set. The first item should not load on the second and third axes ($$y$$ and $$z$$-axes), 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. In brief, the function first performs an EFA based on multidimensional item response theory. It then selects items so that the strongest loading item, from the strongest factor, is always parallel with the x-axis, and the remaining items follow thereafter. This helps make the result maximally interpretable while also avoiding imposing an unempirical structure on the data.

Note that 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.

# Load Package
library(D3mirt)
data("anes0809offwaves")
x <- anes0809offwaves
x <- x[, 3:22] # Remove columns for age and gender

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 with the help of the mirt() function, from the mirt package (Chalmers, 2012), with three factors as the default (factors = 3). After sorting the results following the model identification procedure, the function finishes by presenting the model identification items. The function throws an error if no items can be located. Note that it is possible to use available factor loadings assigned to a data frame in the function call. This can be done by setting efa to FALSE, which allows the function to skip the EFA and jump directly to the model identification procedure.

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 increases 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 generally be manipulated.

# Optional: Load the EFA data for this example directly from the package file
load(system.file("extdata/id.Rdata", package = "D3mirt"))
# Call to modid() with x, containing factors scores from the EFA
# Observe that the efa argument is set to false
id <- modid(x)
summary(id)
#>
#> 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
#>
#> 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   0.5604 -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 sorted with the lowest absolute sum score highest up. Thus, 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. See the package vignette for more on the model identification procedure (e.g., troubleshooting, criteria, or limitations).

# 2. D3mirt Model Estimation

The D3mirt() function holds two built-in models. The first model is the default model, i.e., the three-dimensional compensatory model, and the latter is the orthogonal model, i.e., a restricted model in which the assumption of within-multidimensionality is removed. In this context, orthogonal refers to the strict perpendicular orientation of the items in the latent space. The choice of model depends on the vector used in the modid argument when calling D3mirt() (see examples below).

Calling D3mirt() returns an $$S3$$ object of class D3mirt with lists containing $$a$$ and $$d$$ from the compensatory model, and the $$MDISC$$, and $$MDIFF$$ parameters, direction cosines, and spherical coordinates for the item vectors from the DMIRT model. 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 that 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.

Constructs can be included in the analysis by creating one or more nested lists that indicate what items belong to what construct (see example below) and using it in the con.items argument. 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 (the length can be adjusted by the user) so that the arrows can be seen when plotting. To use spherical coordinates, the user must create a nested list containing spherical coordinate pairs in the same format as the item indicators. For example: con <- list(c(0, 45)), and so on. The con vector, in the latter case, gives one construct vector located $$45^{\circ}$$ strictly above the $$x$$-axis in the latent space.

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 summary() function presents 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 mod3 data as a data frame directly from the package file
load(system.file("extdata/mod3.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
con <- list(c(1:10),
c(11:14),
c(15:20))
mod3 <- D3mirt(x, modid = c("W7Q3", "W7Q20"), con.items = con)
summary(mod3)
#>
#> D3mirt: 20 items and 5 levels of difficulty
#>
#> Compensatory model
#> Model identification items: W7Q3, W7Q20
#>
#> Constructs
#> Item vector 1: W7Q1, W7Q2, W7Q3, W7Q4, W7Q5, W7Q6, W7Q7, W7Q8, W7Q9, W7Q10
#> Item vector 2: W7Q11, W7Q12, W7Q13, W7Q14
#> Item vector 3: W7Q15, W7Q16, W7Q17, W7Q18, W7Q19, W7Q20
#>
#>
#>           a1      a2      a3      d1     d2     d3      d4      d5
#> W7Q1  2.0297  0.1645 -0.1227  8.0868 7.0641 5.9876  3.2016 -0.4834
#> W7Q2  2.6215 -0.0025 -0.2576  9.2884 6.6186 4.5103  1.6650 -2.4437
#> W7Q3  2.7932  0.0000  0.0000 10.4884 7.5922 5.6797  2.7181 -1.1790
#> W7Q4  1.9043  0.1877  0.1502  7.3749 6.0464 4.9812  2.4830 -1.1144
#> W7Q5  2.2427 -0.0285 -0.0836  8.4293 6.6722 4.9055  1.8256 -1.8316
#> W7Q6  2.0020  0.2392  0.1578  8.0687 6.3578 4.9521  2.3301 -1.0187
#> W7Q7  1.6284  0.1036  0.3600  6.0180 4.8976 3.6909  1.6326 -1.3483
#> W7Q8  1.7773  0.2254  0.3536  6.9174 5.1824 3.7663  1.4845 -1.8331
#> W7Q9  1.7197  0.2495  0.1286  7.5588 4.9756 3.3649  0.9344 -2.2093
#> W7Q10 1.7697  0.1274 -0.1404  8.3641 5.7399 4.2865  1.9648 -0.6641
#> W7Q11 1.4237  0.4680  1.0449  6.2204 4.6938 3.5443  1.1923 -1.8579
#> W7Q12 0.7601  0.0413  0.9370  4.1361 2.8772 2.3420  1.1791 -0.4240
#> W7Q13 1.1265  0.2914  1.6908  5.8838 4.3950 3.4385  1.8931 -0.6004
#> W7Q14 0.7444  0.4832  0.9787  5.3893 3.9334 3.0259  0.8144 -1.5869
#> W7Q15 0.4551  0.7871 -0.1607  4.3208 3.0545 2.3970  0.9187 -0.9705
#> W7Q16 0.6236  0.4141  0.1798  3.7249 2.0305 1.1658 -0.0612 -1.8084
#> W7Q17 1.1891  1.3414  0.0561  6.9016 5.8026 4.9348  2.7917 -0.0040
#> W7Q18 0.4106  1.3543 -0.1372  3.7838 2.0986 1.4183  0.1829 -1.9855
#> W7Q19 0.8578  1.4100  0.2278  4.4979 2.6484 1.6731  0.3741 -1.9966
#> W7Q20 0.7355  1.9066  0.0000  4.6376 2.3632 1.2791 -0.3430 -2.9188
#>
#>        MDISC  MDIFF1  MDIFF2  MDIFF3  MDIFF4 MDIFF5
#> W7Q1  2.0401 -3.9640 -3.4627 -2.9350 -1.5693 0.2370
#> W7Q2  2.6341 -3.5262 -2.5127 -1.7123 -0.6321 0.9277
#> W7Q3  2.7932 -3.7550 -2.7181 -2.0334 -0.9731 0.4221
#> W7Q4  1.9194 -3.8423 -3.1502 -2.5952 -1.2936 0.5806
#> W7Q5  2.2444 -3.7557 -2.9728 -2.1857 -0.8134 0.8160
#> W7Q6  2.0224 -3.9897 -3.1437 -2.4486 -1.1521 0.5037
#> W7Q7  1.6710 -3.6015 -2.9310 -2.2088 -0.9771 0.8069
#> W7Q8  1.8261 -3.7881 -2.8380 -2.0625 -0.8130 1.0038
#> W7Q9  1.7425 -4.3380 -2.8555 -1.9311 -0.5362 1.2679
#> W7Q10 1.7798 -4.6995 -3.2251 -2.4084 -1.1040 0.3731
#> W7Q11 1.8269 -3.4048 -2.5692 -1.9400 -0.6526 1.0170
#> W7Q12 1.2073 -3.4260 -2.3832 -1.9399 -0.9766 0.3512
#> W7Q13 2.0525 -2.8667 -2.1413 -1.6753 -0.9223 0.2925
#> W7Q14 1.3211 -4.0792 -2.9772 -2.2903 -0.6164 1.2011
#> W7Q15 0.9232 -4.6800 -3.3085 -2.5963 -0.9951 1.0512
#> W7Q16 0.7699 -4.8384 -2.6375 -1.5143  0.0795 2.3490
#> W7Q17 1.7935 -3.8482 -3.2354 -2.7515 -1.5566 0.0022
#> W7Q18 1.4218 -2.6613 -1.4761 -0.9976 -0.1286 1.3965
#> W7Q19 1.6661 -2.6997 -1.5896 -1.0042 -0.2245 1.1984
#> W7Q20 2.0436 -2.2694 -1.1564 -0.6259  0.1678 1.4283
#>
#>       D.Cos X D.Cos Y D.Cos Z    Theta     Phi
#> W7Q1   0.9949  0.0806 -0.0601  -3.4590 85.3756
#> W7Q2   0.9952 -0.0010 -0.0978  -5.6133 90.0554
#> W7Q3   1.0000  0.0000  0.0000   0.0000 90.0000
#> W7Q4   0.9921  0.0978  0.0783   4.5098 84.3885
#> W7Q5   0.9992 -0.0127 -0.0372  -2.1341 90.7284
#> W7Q6   0.9899  0.1183  0.0780   4.5068 83.2064
#> W7Q7   0.9745  0.0620  0.2154  12.4657 86.4441
#> W7Q8   0.9733  0.1235  0.1936  11.2521 82.9082
#> W7Q9   0.9869  0.1432  0.0738   4.2769 81.7671
#> W7Q10  0.9943  0.0716 -0.0789  -4.5357 85.8965
#> W7Q11  0.7793  0.2561  0.5719  36.2765 75.1588
#> W7Q12  0.6296  0.0342  0.7761  50.9493 88.0399
#> W7Q13  0.5488  0.1420  0.8238  56.3262 81.8384
#> W7Q14  0.5634  0.3657  0.7408  52.7448 68.5472
#> W7Q15  0.4929  0.8525 -0.1740 -19.4451 31.5154
#> W7Q16  0.8101  0.5378  0.2335  16.0800 57.4627
#> W7Q17  0.6630  0.7479  0.0313   2.7031 41.5881
#> W7Q18  0.2888  0.9525 -0.0965 -18.4742 17.7264
#> W7Q19  0.5148  0.8463  0.1367  14.8712 32.1870
#> W7Q20  0.3599  0.9330  0.0000   0.0000 21.0939
#>
#>    C.Cos X C.Cos Y C.Cos Z   Theta     Phi
#> C1  0.9970  0.0688  0.0368  2.1119 86.0548
#> C2  0.6409  0.2029  0.7404 49.1207 78.2961
#> C3  0.5405  0.8411  0.0226  2.3974 32.7476
#>
#>       DDISC1 DDISC2 DDISC3
#> W7Q1  2.0304 1.2433 1.2326
#> W7Q2  2.6038 1.4887 1.4088
#> W7Q3  2.7847 1.7901 1.5096
#> W7Q4  1.9169 1.3697 1.1905
#> W7Q5  2.2308 1.3696 1.1862
#> W7Q6  2.0181 1.4484 1.2868
#> W7Q7  1.6438 1.3312 0.9754
#> W7Q8  1.8004 1.4465 1.1582
#> W7Q9  1.7364 1.2479 1.1422
#> W7Q10 1.7679 1.0560 1.0604
#> W7Q11 1.4900 1.7809 1.1867
#> W7Q12 0.7951 1.1892 0.4668
#> W7Q13 1.2053 2.0328 0.8922
#> W7Q14 0.8113 1.2997 0.8308
#> W7Q15 0.5019 0.3324 0.9043
#> W7Q16 0.6568 0.6168 0.6894
#> W7Q17 1.2799 1.0758 1.7722
#> W7Q18 0.4975 0.4363 1.3578
#> W7Q19 0.9605 1.0044 1.6547
#> W7Q20 0.8644 0.8581 2.0011

The D3mirt() function prints a short report containing the number of items used and the number of difficulty levels 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.

Regarding the orthogonal model, the function call must include nested lists indicating which items should be constrained to either of the axes. Below is an example in which items $$1$$ to $$10$$ are constrained to the $$x$$-axes, items $$15$$ to $$19$$ to the $$y$$-axis, and items $$11$$ to $$14$$ to the $$z$$-axis. Note that fitting the orthogonal model often requires removing misfitting items. Otherwise, the orthogonal model most likely will amplify the misfit tendencies. In this case, item W7Q16 was removed before fitting the model.

# Optional: Load the mod2 data as a data frame directly from the package file
load(system.file("extdata/mod2.Rdata", package = "D3mirt"))
y <- data.frame(x[, -16])# Remove misfitting item W7Q16
# Call D3mirt() and using the orthogonal model
mod2 <- D3mirt(y, modid = list(c(1:10),   # x-axis
c(15:19),  # y-axis
c(11:14))) # z-axis
summary(mod2)
#>
#> D3mirt: 19 items and 5 levels of difficulty
#>
#> Orthogonal model
#> Item vector 1: W7Q1, W7Q2, W7Q3, W7Q4, W7Q5, W7Q6, W7Q7, W7Q8, W7Q9, W7Q10
#> Item vector 2: W7Q15, W7Q17, W7Q18, W7Q19, W7Q20
#> Item vector 3: W7Q11, W7Q12, W7Q13, W7Q14
#>
#>           a1     a2     a3      d1     d2     d3      d4      d5
#> W7Q1  2.0183 0.0000 0.0000  8.0211 7.0101 5.9451  3.1762 -0.4884
#> W7Q2  2.5545 0.0000 0.0000  9.0536 6.4408 4.3879  1.6178 -2.3976
#> W7Q3  2.7483 0.0000 0.0000 10.3241 7.4782 5.6007  2.6793 -1.1828
#> W7Q4  1.9121 0.0000 0.0000  7.3485 6.0230 4.9697  2.4782 -1.1279
#> W7Q5  2.1971 0.0000 0.0000  8.2981 6.5636 4.8289  1.7951 -1.8163
#> W7Q6  2.0070 0.0000 0.0000  8.0685 6.3532 4.9445  2.3214 -1.0367
#> W7Q7  1.5979 0.0000 0.0000  5.9310 4.8301 3.6411  1.5984 -1.3322
#> W7Q8  1.7491 0.0000 0.0000  6.8381 5.1224 3.7094  1.4480 -1.8096
#> W7Q9  1.7410 0.0000 0.0000  7.5706 4.9930 3.3754  0.9282 -2.2171
#> W7Q10 1.7488 0.0000 0.0000  8.3054 5.6856 4.2405  1.9417 -0.6614
#> W7Q11 0.0000 0.0000 1.5285  5.9525 4.4684 3.3585  1.1019 -1.7346
#> W7Q12 0.0000 0.0000 1.1511  4.1002 2.8570 2.3265  1.1659 -0.4295
#> W7Q13 0.0000 0.0000 1.8866  5.7126 4.2787 3.3450  1.8306 -0.5882
#> W7Q14 0.0000 0.0000 1.3482  5.4784 4.0172 3.1011  0.8363 -1.6128
#> W7Q15 0.0000 0.8870 0.0000  4.3049 3.0375 2.3797  0.9084 -0.9651
#> W7Q17 0.0000 1.5464 0.0000  6.6917 5.6269 4.7725  2.6404 -0.0421
#> W7Q18 0.0000 1.3316 0.0000  3.6937 2.0504 1.3881  0.1841 -1.9387
#> W7Q19 0.0000 1.6471 0.0000  4.5704 2.6646 1.6572  0.3352 -2.0165
#> W7Q20 0.0000 1.9471 0.0000  4.5578 2.3173 1.2468 -0.3427 -2.8570
#>
#>        MDISC  MDIFF1  MDIFF2  MDIFF3  MDIFF4 MDIFF5
#> W7Q1  2.0183 -3.9742 -3.4733 -2.9456 -1.5737 0.2420
#> W7Q2  2.5545 -3.5442 -2.5214 -1.7178 -0.6333 0.9386
#> W7Q3  2.7483 -3.7565 -2.7210 -2.0379 -0.9749 0.4304
#> W7Q4  1.9121 -3.8432 -3.1500 -2.5992 -1.2961 0.5899
#> W7Q5  2.1971 -3.7768 -2.9873 -2.1978 -0.8170 0.8267
#> W7Q6  2.0070 -4.0202 -3.1655 -2.4636 -1.1566 0.5165
#> W7Q7  1.5979 -3.7118 -3.0228 -2.2787 -1.0003 0.8337
#> W7Q8  1.7491 -3.9096 -2.9287 -2.1208 -0.8279 1.0346
#> W7Q9  1.7410 -4.3485 -2.8680 -1.9388 -0.5332 1.2735
#> W7Q10 1.7488 -4.7492 -3.2511 -2.4248 -1.1103 0.3782
#> W7Q11 1.5285 -3.8944 -2.9234 -2.1973 -0.7209 1.1348
#> W7Q12 1.1511 -3.5620 -2.4820 -2.0211 -1.0129 0.3731
#> W7Q13 1.8866 -3.0279 -2.2679 -1.7730 -0.9703 0.3118
#> W7Q14 1.3482 -4.0635 -2.9796 -2.3001 -0.6203 1.1962
#> W7Q15 0.8870 -4.8531 -3.4243 -2.6827 -1.0241 1.0880
#> W7Q17 1.5464 -4.3273 -3.6387 -3.0862 -1.7074 0.0272
#> W7Q18 1.3316 -2.7738 -1.5398 -1.0424 -0.1383 1.4559
#> W7Q19 1.6471 -2.7749 -1.6178 -1.0062 -0.2035 1.2243
#> W7Q20 1.9471 -2.3408 -1.1902 -0.6403  0.1760 1.4673
#>
#>       D.Cos X D.Cos Y D.Cos Z Theta Phi
#> W7Q1        1       0       0     0  90
#> W7Q2        1       0       0     0  90
#> W7Q3        1       0       0     0  90
#> W7Q4        1       0       0     0  90
#> W7Q5        1       0       0     0  90
#> W7Q6        1       0       0     0  90
#> W7Q7        1       0       0     0  90
#> W7Q8        1       0       0     0  90
#> W7Q9        1       0       0     0  90
#> W7Q10       1       0       0     0  90
#> W7Q11       0       0       1    90  90
#> W7Q12       0       0       1    90  90
#> W7Q13       0       0       1    90  90
#> W7Q14       0       0       1    90  90
#> W7Q15       0       1       0   NaN   0
#> W7Q17       0       1       0   NaN   0
#> W7Q18       0       1       0   NaN   0
#> W7Q19       0       1       0   NaN   0
#> W7Q20       0       1       0   NaN   0

In the output, the simple loading structure can be seen. Aldo note that $$\theta$$ reports NaN when both $$\theta$$ and $$\phi$$ is zero. This is because when converting to spherical coordinates, items oriented parallel to the $$y$$-axis will have $$\cos (0)$$ in the denominator of the $$\arctan$$ function, resulting in undefined values.

# 3. Plotting

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 appearing 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(mod3, 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 is 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, and Conformity were 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 Fairness seems not to be.

The orthogonal model is displayed below rotated rotated $$15^{\circ}$$ clockwise. In the graphical output the perpendicular orientation of the items forms a cross pattern in which each axis holds a unidimensional model indicated by the items parallel to it. The orthogonal model is foremost useful for studying the items under the assumption that the items are unidimensional and that the unidimensions are uncorrelated. Note that the orthogonal model will change the location of respondents in the latent space compared to the compensatory model. This can have an effect when studying profiles (see examples below), such that profile tendencies can emerge or disappear depending on the choice of model.

# Plot the orthogonal model
plot(mod2)

Figure 3: The orthogonal model plotted with the model rotated $$15^{\circ}$$ clockwise.

## 3.1. 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(mod3, constructs = TRUE,
items = c(15,17,18,19,20),
construct.lab = c("Compassion", "Fairness", "Conformity"))

Figure 4: The items from the Conformity construct plotted with the model rotated $$15^{\circ}$$ clockwise.

Figure 5: 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 does not seem to belong to any construct explicitly. Typing $$16$$ 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(mod3, constructs = TRUE,
items = 16,
construct.lab = c("Compassion", "Fairness", "Conformity"))

Figure 6: The item W7Q16 plotted with the three constructs and with the model rotated $$15^{\circ}$$ clockwise.

Figure 7: 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 above is 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. We can also note that item W7Q15, $$MDISC = .923$$, $$MDIFF_{range} = [-4.680, 1.051]$$) has the second lowest global discrimination score. Compared to W7Q16, however, item W7Q15 does seem to belong to the Conformity construct, observable when comparing angle orientation ($$\theta = -19.432^{\circ}, \phi = 31.515^{\circ}$$) and direction discrimination ($$DDISC_1 = .502$$, $$DDISC_2 = .332$$, $$DDISC_3 = .912$$). In other words, even if item W7Q15 shows signs of statistical violations in the model, the item still informs us of the content of the Conformity construct.

## 3.2. 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 that 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(mod3, diff.level = 5)

Figure 8: All items plotted on difficulty level 5 and with the model rotated $$15^{\circ}$$ clockwise.

Figure 9: All items plotted on difficulty level 5 and with the model rotated $$90^{\circ}$$ clockwise.

## 3.3 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(mod3, scale = TRUE,
constructs = TRUE,
construct.lab = c("Compassion", "Fairness", "Conformity"))

Figure 10: All items scaled to uniform length and plotted with the model rotated $$15^{\circ}$$ clockwise.

Figure 11: All items scaled to uniform length and plotted with the model rotated $$90^{\circ}$$ clockwise.

## 3.4. D3mirt Profile Analysis

The plot() function can also display respondents in the three-dimensional model represented as spheres whose coordinates are derived from the respondent’s trait scores. This allows for a profile analysis in which respondents can be separated and plotted as subsets conditioned on single or multiple criteria. The resulting output shows where the respondents are located in the model and, accordingly, what model profile best describes them. To do this, the user can either plot all respondents by setting ind.scoresto TRUE, or plot a subsection of respondents by creating a separate data frame and use it in the profiles argument when calling plot().

In the example below, the first option is illustrated also using the levels argument. Regarding the latter, the plot() function uses as.factor() to count the number of factor levels in the data imputed in the levels argument. This information is then used to assign colors for the spheres representing respondents when plotting. 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. By separating respondents using color coding, it is sometimes possible to display group-level effects.

Generally, it can be helpful to hide vector arrows with hide = TRUE when plotting respondent profiles to avoid visual cluttering. The example below displays all respondents using the gender variable included in the built-in data set. The gender variable is also color-coded so that males are displayed as blue spheres and females as red spheres.

# Load the data set to be used in levels
# Use as.matrix to remove any attributes
data("anes0809offwaves")
x <- as.matrix(anes0809offwaves)

Call plot() and use the gender data, contained in column two in data frame $$x$$, in the levels argument. 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 that the model axes represent unidimensional singular structures or traits.

# Plot profiles with item vector arrows hidden with hide = TRUE
# Score levels: 1 = Blue ("male") and 2 = Red ("female")
# Plot profiles with item vector arrows hidden
# Score levels: 1 = Blue ("male") and 2 = Red ("female")
plot(mod3, hide = TRUE, ind.scores = TRUE,
levels = x[, 2],
sphere.col = c("blue", "red"),
x.lab = "Compassion",
y.lab="Conformity",
z.lab="Fairness")

Figure 12: Gender profile for the anes0809offwaves data set plotted with the model rotated $$15^{\circ}$$ clockwise.

Figure 13: Gender profile for the anes0809offwaves data set plotted with the model rotated $$90^{\circ}$$ clockwise.

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

In the figure, a simple gender profile can be observed, showing that more women tend to have higher levels of compassion. When rotating the model $$90^{\circ}$$ clockwise, there seems to be no obvious gender difference related to Conformity or Fairness.

## 3.5. Plotting Confidence Intervals

It is also possible to plot a confidence interval in the shape of an ellipse surrounding the spheres in the latent space. In the example below, the younger individuals ($$\leq30$$) are selected and plotted with a $$95\%$$ CI. To subset respondents, create a new data frame by combining respondents’ factor scores from mod3$fscores with age data from column one in data frame $$x$$. Then assign respondent data conditioned on age to a new data frame using the subset() function. # Column bind trait scores in mod3 with the age variable W3Xage from data frame x z <- data.frame(cbind(mod3$fscores, x[, 1]))

# Subset data frame z conditioned on age <= 30
z1 <- subset(z, z[, 4] <= 30)

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 illustrated below.

# Check number of factor levels with nlevels() and as.factor()
nlevels(as.factor(z1[, 4]))

# Use rep() to create a color vector to color groups based on the nlevels() output
# z1 has 14 factor levels
colvec <- c(rep("red", 14))

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 that the CI limit can be adjusted with the ci.level argument.

# Call plot() with profile data on age with item vector arrows hidden
plot(mod3, 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 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 $$15^{\circ}$$ clockwise.

Figure 15: 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 is as follows.

In Figures 7 and 8 we can see a tendency for an age profile 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.

# 4. Exporting the RGL Device

Some options for exporting the RGL device are shown below. In addition, it is also possible to export graphical devices in R Markdown documents with rgl::hook_webgl() 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(mod3, constructs = TRUE)
s <- rgl::scene3d()
rgl::rglwidget(s,
width = 1040,
height = 1040)

# Export a snap shoot of an open RGL device directly to file
plot(mod3, constructs = TRUE)
rgl::rgl.snapshot('RGLdevice.png',
fmt = 'png')

# Getting Help, Feedback, and Questions

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

# References

Adler, D., & Murdoch, D. (2023). Rgl: 3d Visualization Using OpenGL [Computer software]. https://dmurdoch.github.io/rgl/index.html

Chalmers, R., P. (2012). mirt: A Multidimensional Item Response Theory Package for the R Environment. Journal of Statistical Software, 48(6), 1-29. https://doi.org/10.18637/jss.v048.i06

DeBell, M., Krosnick, J. A., & Lupia, A. (2010). Methodology Report and User’s Guide for the 2008-2009 ANES Panel Study. Palo Alto, CA, and Ann Arbor, MI: Stanford University and the University of Michigan.

McKinley, R. L., & Reckase, M. D. (1983). An Extension of the Two-parameter Logistic Model to the multidimensional latent space, Report ONR83-2. Iowa City, IA, American College Testing Program.

Muraki, E., & Carlson, J. E. (1995). Full-Information Factor Analysis for Polytomous Item Responses. Applied Psychological Measurement, 19(1), 73-90. https://doi.org/10.1177/014662169501900109

Reckase, M. D. (2009). Multidimensional Item Response Theory. Springer. https://doi.org/10.1007/978-0-387-89976-3

Reckase, M. D. (1985). The Difficulty of Test Items That Measure More Than One Ability. Applied Psychological Measurement, 9(4),401-412. https://doi.org/10.1177/014662168500900409

Reckase, M. D., & McKinley, R. L. (1991). The Discriminating Power of Items That Measure More Than One Dimension. Applied Psychological Measurement, 15(4), 361-373. https://doi.org/10.1177/014662169101500407