# RMX/PIccc: An Extended Person–Item Map and a Unified IRT Output for eRm, psychotools, ltm, mirt, and TAM

^{*}

^{†}

## Abstract

**:**

`RMX::plotPIccc()`function, which creates an extended version of the classical PI Map, termed “PIccc”. It juxtaposes the person parameter distribution to various item-related functions, like category and item characteristic curves and category, item, and test information curves. The function supports many item response models and processes the return objects of five major R packages for IRT analysis. It returns the used parameters in a unified form, thus allowing for their further processing. The R package RMX is freely available at osf.io/n9c5r.

## 1. Introduction

But Rasch would have wondered about what happened to the use of graphs. And I think he would have been quite justified in this. Could it be that we have used computers in a wrong way? Since Rasch retired from active duty, have we emphasized the power of computers to do complicated calculations and solving complicated equations over the power of the computers to make nice and illustrative graphs?([2]; p. 388)

`plotPImap()`function, psychotools [15] with the

`piplot()`function (Figure 1), or TAM [16] with the

`IRT.WrightMap()`wrapper to the WrightMap package.

- −
- Only the item difficulty/category threshold parameters are drawn, which is only partial information for models involving discrimination, guessing, or laziness parameters.
- −
- Although the threshold parameters are drawn for polytomous items, it is difficult to recognize which categories are likely to be chosen across the latent scale. Especially, the effects of threshold disorder are difficult to deduce.
- −
- Beyond item/threshold difficulty parameters, we may also learn a lot about our items in terms of information. The category and item information curves may tell us a lot if set into relation to the person parameter distribution.
- −
- Current implementations do not easily support flexibly arranging the items according to their characteristics (beyond difficulty) or, in the multidimensional case, dimensions.
- −
- Current implementations do not allow for varying the area proportions used for the person parameter histogram and the item parameter part. In Figure 1, it would be advantageous if we could increase the upper part at the expense of the item part.

`RMX::plotPIccc()`, which overcomes the restrictions of the “classical” PI Map in several respects. We term this modified diagram “PIccc” for it shows the Person–Item confrontation using category characteristic curves CCCs) and many other functions. Note that, although the package carries “Rasch” in its name, it does not only refer to the “Rasch Family of Models” (cf. [1]) but also to extensions covered by the term “Item Response Theory” (which is indicated by the “X”).

## 2. The RMX Package and the **plotPIccc()** Function

**plotPIccc()**

#### 2.1. Functionality Overview

`type=`argument, the options of which are listed in Table 1.

`classical=TRUE`option, which draws the PI-Map in its traditional form (see Section 3).

`plotPIccc()`function supports two modi, either

- −
- drawing one type of curve for a set of items (default); or
- −
- drawing several types for one item (by providing a vector of types).

`isel=`option (default: all items). For multidimensional models, the

`dsel=`options allows item selection according to dimensions. The items may be sorted according to various item characteristics with the

`isort=`option (see Table 2).

`isort="disc"/"guess"/"lazy"`for the RM, the RSM, and the PCM. Note that selecting the

`isort="disc"`option for the NRM will result in sorting by the average of the category discrimination parameters per item. Alternatively, the user may achieve any sorting by specifying the order of items in the

`isel=`option. Additionally, the logical

`gsort=`option switches for multidimensional models between sorting items within each dimension (

`FALSE`) vs. globally (

`TRUE`), i.e., across all dimensions.

- −
- the test information function (TIF) for the entire set of items (
`TIF=TRUE`); - −
- the TIF of the selected items (
`sTIF=TRUE`); - −
- the standard error (SE) for all items (
`SE=TRUE`); - −
- the SE of the selected items (
`sSE=TRUE`); and - −
- the kernel density estimate (
`dens=TRUE`)

`funwprop=`and

`funhprop=`) and the range of the latent continuum to plot.

`funcol=`, which will also take precedence over the

`infcol=`option. For multidimensional models, all options of the upper (person parameter) part of the PIccc accept color vectors. If the standard palette is used (default) and there are items with more than 8 categories, the colors will be recycled.

`RMX::plotPIccc()`contains a list with the parameters used for plotting, thus fostering further processing, e.g., in a results table (see Section 3).

#### 2.2. Some Technical Details

`RMX::plotPIccc()`function automatically detects the package used for parameter estimation and unifies the various outputs.

#### 2.2.1. Person Parameters

`person.parameter()`in eRm,

`personpar()`in psychotools,

`factor.scores()`in ltm,

`fscores()`in mirt, and

`IRT.factor.scores.tam()`or

`tam.wle()`in TAM), some of which support several estimation variants. As the return objects of the model parameter estimation routines contain only the item parameters (except for TAM),

`RMX::plotPIccc()`applies the appropriate PP estimation routine from the originating package with the default options. If a non-default PP estimation method is desired, one may use the

`pp=`option and provide the return object of the respective package. For TAM,

`RMX::plotPIccc()`uses the PPs provided already contained in the return object of the estimation routine.

`pp=NULL`), the

`RMX::plotPIccc()`function estimates the person parameters internally. This may take a considerable amount of time for some models. Now, creating a PIccc likely takes several rounds with fine-tuning the graphical options until the optimal result is obtained. Letting the function recalculate the PPs each time can make the fine-tuning unnecessarily cumbersome. Therefore,

`RMX::plotPIccc()`returns the PP object (as an attribute), which is automatically detected and could then be re-used in the

`pp=`option:

`# first run:`

`plotresult = RMX::plotPIccc(object, ... options ...)`

`# subsequent runs:`

`RMX::plotPIccc(object, pp=plotresult, ... better options ...)`

#### 2.2.2. Model Formulations

`rasch()`,

`ltm()`,

`tpm()`,

`gpcm()`, and

`grm()`) support an

`IRT.param=TRUE`option in their function calls to switch between the two formulations (mirt also supports an

`IRTparams=TRUE`in its

`coef()`function and TAM has a set of “

`IRT.`”-prefixed output functions, but these cannot be used here). Either way,

`RMX::plotPIccc()`transforms all parameters into the slope threshold formulation, if necessary, and uses these values for plotting and in the return object.

#### 2.2.3. Threshold Definitions

`RMX::plotPIccc()`function uses the Andrich/Masters formulation and transforms the input, if necessary. For

`type="CCC"`or

`type="TCC"`,

`RMX::plotPIccc`indicates threshold disordering with an asterisk (or the symbol set with

`disind=`).

`ak`in mirt), the latter restricted to equality across items (notation adapted; see [34], Equation (3.32), for details). The advantage of the TCB variant is that it allows for formulating a multidimensional NRM with ${\alpha}_{i}^{*}$, which allows for an analogue interpretation as the loadings of a factor analysis. The

`RMX::plotPIccc()`function uses the Bock parametrization [27], transforming the parameters following Thissen et al. [34], Equation (3.32), if necessary. Such a transformation is also applied for multidimensional models for we then obtain the category slope parameters for each item required for drawing. If the user requests

`type="TCC"`for an NRM, the slopes of the lines are the category boundary discrimination ($CBD$) parameters

#### 2.2.4. Multidimensional Models

`RMX::plotPIccc()`detects a multidimensional model automatically from the return object of the originating package. It plots (in the one type/several items mode) a diagram for each selected item and each selected dimension appearing in the model. Hence, an item appearing in more than one dimension (within-item multidimensionality) will be plotted more than once. The package supports both between- and within-item multidimensional compensatory models (The mirt package also supports non-compensatory models (sometimes referred to as partially compensatory (e.g., [36], Chapter 4), but only for dichotomous data. “[P]artially compensatory polytomous MIRT modes are yet to be developed.” ([37], p. 47)). It is important to note the following (dimensions indexed by $\ell =1\dots m$): we obtain each item’s parameters in the slope intercept (“regression”) formulation per dimension, i.e., a vector of length m of discrimination parameters ${a}_{i\ell}$ and the item’s intercept ${d}_{i}$. A slope threshold (“IRT”) formulation equivalent to the multidimensional slope for the 2PL (M2PL; cf. [36], Equation (4.5), p. 86) is the MDISC index

`Error in MDIFF(…):Item 1 is not of class "graded" or "dich"`), thus limiting the support of RMX to these models. Multidimensional PCMs, RSMs, and GPCMs could not be drawn that way.

`RMX::plotPIccc()`, we have to keep in mind that the parameters used here “give the relative difficulty of the item related to the corresponding coordinate dimension” ([36], p. 89).

#### 2.2.5. Information Functions

`RMX::plotPIccc()`function provides, therefore, the

`infomax=`option, which takes either the keywords

`infomax="auto"`and

`infomax="equal"`or a numeric value indicating the maximum value to plot. With

`"auto"`, each diagram is zoomed to its individual maximum, whereas

`"equal"`uses a common scale for all visible information diagrams (i.e., the common maximum across items in the multiple items/one function mode or the common maximum across the chosen information functions in the one-item/multiple-functions mode).

#### 2.2.6. The Internal Structure and the Return Object

`RMX::plotPIccc()`function, but users may directly call the five extractor functions (i.e.,

`ext_erm()`,

`ext_psy()`,

`ext_ltm()`,

`ext_mirt()`, and

`ext_tam()`) using the triple colon (

`:::`) operator. Each of them expects the return object of the originating package and determines the dimensionality of the model, extracts the item parameters (per dimension, if applicable), calculates/extracts the person parameters (also per dimension, if applicable), counts the response frequencies of all categories of each item, and returns a list. Note that all item parameters will be returned in the slope threshold (“IRT”) formulation. The

`cleaner()`function selects the required items and (if admissible) dimensions (

`isel=`,

`dsel=`) and sorts them (

`isort=`). The return object of the

`cleaner()`is then used by the

`drawer()`.

- −
- matrices with the location estimates in the slope-threshold formulation;
- −
- discrimination, guessing, and laziness parameters;
- −
- a vector indicating each item’s model (as mirt allows for varying models across items);
- −
- a vector of length n with the person parameter estimates of this dimension;
- −
- vectors of length ${n}^{*}=1001$ (
`tmin`to`tmax`) with TIF, sTIF, SE, and sSE; and - −
- matrices (${n}^{*}\times k$) with CCCs, TCCs, CIFs, and IIFs,

`RMX::plotPIccc()`function may also serve as a tool for further processing the results of an analysis (e.g., as a table; see the last listing 4), even if no diagram is required (option

`plot=FALSE`).

## 3. Worked Examples

`RMX::plotPIccc()`, we use the example dataset

`big5`delivered with the RMX package. Mimicking a students’ survey, it comprises 21 items covering the Big Five (i.e., Openness, Conscientiousness, Extraversion, Agreeableness, and Neuroticism; [40]) and 1076 simulated respondents. The response format of all items was Likert-type, with the categories “very inapplicable”, “rather inapplicable”, “neither-nor”, “rather true”, and “very true” (translated from the German original).

`RMX::plotPIccc()`:

Listing 1. Example of a GPCM with psychotools using the Agreeableness items. |

library(RMX) |

library(psychotools) |

mod1 = psychotools::gpcmodel(big5[,c(2,7,12,17)]) |

RMX::plotPIccc(mod1) |

`pplab="abs"`for absolute frequencies (alternatively

`pplab="rel"`for percentages or

`pplab="dens"`for the kernel density estimates). The green line is the test information function (TIF) and the red line the standard error (SE). Additionally, if only a subset of items is used, dashed lines indicate the TIF and the SE of the selected items in the respective colors. The top right segment holds the legend for the one-function/several-items mode and the category frequency barchart for the one-item/several-functions mode (see Figure 4 for the latter). The lower left segment shows the item-related functions, i.e., the CCCs by default in the one-function/several-items case or the selected functions in the one-item/several-functions case. The lower right segment shows the category response frequency barcharts of each item in the one-function/several-items mode and the legend of the respective function in the one-item/several-functions mode. The upper : lower and left : right proportions can be adapted with the

`funhprop=`and the

`funwprop=`option, each taking decimal values between zero and one. Values of zero or one for either option will switch on/off the entire regions so that each of the four segments can be drawn alone.

Listing 2. Example of a GPCM using TAM. |

library(TAM) |

mod2a = TAM::tam.mml( big5[,c(4,9,14,19)],verbose=FALSE) # Neuroticism |

mod2b = TAM::tam.mml.2pl(big5[,c(4,9,14,19)],verbose=FALSE) |

RMX::plotPIccc(mod2a,type=c("CCC","TCC","BIF"),isel=2,infomax=1.05) |

RMX::plotPIccc(mod2b,type=c("CCC","TCC","BIF"),isel=2,infomax=1.05) |

`Q9R`of the Neuroticism subscale. In this mode, the barchart with the category frequencies is shifted to the top and the legends are now placed to the right of each diagram. In the top left area, we now see not only the TIF and SE lines (solid) but also the respective dashed lines for the selected item, as mentioned above. The latter allow for comparing the item’s contribution to the test information. Note that the arrangement of the three functions follows the ordering in the

`type=`option.

`infomax=`option to equalize the scales of the two information functions). Here, we see clearly that the improvement in fit due to varying slopes comes at the cost of information. The item shows a threshold disordering for both models, which is indicated by the red asterisks. Thus, the weaknesses of the item become visible at a glance. Moreover, the comparison of the sSE curves of this item (dotted red lines) shows that the GPCM-based standard errors are remarkably larger than those based on the PCM, which is a result of the lower discrimination parameter of this item.

Listing 3. Example of a multidimensional GPCM using mirt and the Big Five example dataset. |

big5mod = "O = 5,10,15,20,21 |

C = 3,8,13,18 |

E = 1,6,11,16 |

A = 2,7,12,17 |

N = 4,9,14,19 |

COV=O*C*E*A*N" |

big5res = mirt::mirt(big5,big5mod,itemtype="gpcm",method="MHRM") |

big5est = RMX::plotPIccc(big5res,classical=TRUE, lmar=3, ylas=2, |

funhprop=0.6,dencol=NA,usedimcol=TRUE, |

dimcol=c("#bef7ff", "#a0dcff", "#82c2ff", "#63a7ff", "#458cff"), |

tifcol=grey(0.6), secol="dodgerblue4") |

`lmar=3`and

`ylas=2`options allowed for printing the item labels, which have been automatically extended by the dimension labels. With the option

`usedimcol=TRUE`, we colored the items’ dots according to their respective latent dimension. Note further that, in the

`classical=TRUE`variant, threshold disordering is indicated with dotted lines.

`pdf()`or

`png()`function of R (the former yielding scalable images). That way, the user may choose the optimal window proportions (

`width=`,

`height=`), which is the reason why the plot opens by default in an external window (RStudio/posit [41] users may set

`extwin=FALSE`to use the internal graphics viewer). Internally,

`RMX::plotPIccc()`uses for the external graphics window the generic

`dev.new()`function of R with the

`noRStudioGD=TRUE`option set. The

`extwin=FALSE`option is also required if one uses

`RMX::plotPIccc()`for compiling a markdown output, which is readily supported by RStudio/posit. Additionally, the option

`resetpar=`(default:

`TRUE`) controls, whether the graphic parameters (set with

`par()`) are restored after the drawing has finished. Setting to

`FALSE`allows for further refinements of the diagram (e.g., additional text, arrows, etc.).

`RMX::plotPIccc()`function returns (invisibly) a list with all values used for plotting, which may be useful for publishing the results. Listing 4 shows, exemplarily, how to build a table for

`LATEX`using the

`xtable`package [42] of R using the return object

`big5est`from Listing 3:

Listing 4. Processing the return object of the analysis of Listing 3. |

xtable::xtable(big5est$N$thresholds) |

\begin{table}[H] |

\centering |

\begin{tabular}{rrrrr}\hline |

& Q4 & Q9R & Q14 & Q19 \\\hline |

1 & -0.97 & -2.79 & -2.09 & -1.51 \\ |

2 & 0.49 & 0.90 & -0.32 & 0.07 \\ |

3 & 0.29 & -0.32 & -0.22 & 0.27 \\ |

4 & 2.20 & 2.99 & 1.51 & 2.09 \\\hline |

\end{tabular} |

\end{table} |

## 4. Discussion

`RMX::plotPIccc()`function may also be used to draw simple diagrams of a single item’s CCC, TCC, CIF, IIF, or the category frequencies barplot only by using the

`isel=`,

`funhprop=`, and

`funwprop=`options. Thus, the package offers enormous flexibility by covering functionality, which may require more programming effort in the other packages, if supported at all. It further improves some of the other packages’ functions in terms of graphical options.

`classical=TRUE`). This may require to split items across multiple diagrams, e.g., according to sub-scales or other substantive criteria. However, the alternative (so far) was to plot each, say, CCC separately per item (possibly gathering them in a plot matrix), which makes comparisons more difficult, not to mention the limited comparability to the person parameter distribution. Therefore, the presented solution seems to be a major step forward in this respect.

#### 4.1. Threshold (Dis)Ordering in the NRM

`type="TCC"`, thus allowing for easily detecting threshold disordering. Importantly, the

`type="CCC"`will not allow for detecting threshold disordering for the NRM as a category could indeed lack a range on the latent continuum, along which it has a larger probability to be chosen than any other category, although thresholds are ordered. This is in contrast to the (G)PCM, where threshold disordering is always associated with categories “vanishing” behind others. To our knowledge,

`RMX::plotPIccc()`is the first program directly implementing a graphical disorder detection feature for the NRM.

#### 4.2. Sorting Items in the NRM

`RMX::plotPIccc()`function allows for sorting polytomous items according to several criteria, including the discrimination parameters. For the NRM, this is not possible in a straightforward manner because it estimates a discrimination parameter for each category of an item. Therefore, we use the mean of the discrimination parameters per item for sorting. Alternatively, sorting could also be achieved by using the item-wise (i.e., single-indexed) discrimination parameter as defined by Thissen et al. (2010; [34]), which is planned for a further release of RMX.

#### 4.3. Estimating the Person Parameters

`extrapolate=TRUE`option in the

`coef()`function. In contrast, the psychotools package will just return

`NA`for respondents with zero or perfect scores.

`NA`if a response vector contains missing values. Therefore, the

`N=`shown in the person parameter distribution area may differ from the actual sample size if the originating package was psychotools. In contrast, the MML-based packages can handle zero and perfect scores.

## 5. Conclusions and Outlook

`RMX::plotPIccc()`. The current version has been developed with

- −
- eRm 1.0-2 (https://cran.r-project.org/package=eRm; accessed on 28 August 2023),
- −
- ltm 1.2-0 (https://cran.r-project.org/package=ltm; accessed on 28 August 2023),
- −
- psychotools 0.7-3 (https://cran.r-project.org/package=psychotools; accessed on 28 August 2023),
- −
- mirt 1.40 (https://cran.r-project.org/package=mirt; accessed on 28 August 2023), and
- −
- TAM 4.1-4 (https://cran.r-project.org/package=TAM; accessed on 28 August 2023).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

General Terms | |

IRT | Item Response Theory |

PP | Person Parameter |

JML | Joint Maximum Likelihood Estimation |

CML | Conditional Maximum Likelihood Estimation |

MML | Marginal Maximum Likelihood Estimation |

eRm | extended Rasch modeling |

ltm | latent trait models |

mirt | multidimensional IRT models |

TAM | Test Assessment Module |

Models | |

RM | Rasch Model |

PCM | Partial Credit Model |

RSM | Rating Scale Model |

2/3/4PL | (Birnbaum) 2PL/3PL/4PL |

GPCM | Generalized Partial Credit Model |

GRM | Graded Response Model |

NRM | Nominal Response Model |

Functions and Curves | |

ICC | Item Characteristic Curve |

IRF | Item Response Function (= ICC) |

CCC | Category Characteristic Curve |

ICRF | Item Category Response Function |

CRF | Category Response Function |

OCC | Operating Characteristic Curves |

ORF | Operating Response Function |

CBD | Category Boundary Discrimination |

TCC | Threshold Characteristic Curve |

CBRF | Category Boundary Response Function |

CIF | Category Information Function |

IIF | Item Information Function |

TIF | Test Information Function |

sTIF | Test Information Function based on the selected items |

SE | Standard Error |

sSE | Standard Error based on the selected items |

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**Figure 1.**Example of a “classical” PI Map of the example dataset (delivered with RMX; see Section 3) drawn with

`psychotools::piplot()`.

**Figure 3.**PIccc example for the same result object used in Figure 1 with all options at their default values. For details see text. * indicate threshold disordering, but the symbol can be changed with the

`disind=`option.

**Figure 4.**The one-item/multiple-functions mode. * indicate threshold disordering, but the symbol can be changed with the

`disind=`option.

Option | Curve Type |
---|---|

type="CCC" | the Category Characteristic Curve, a.k.a, Item Category Response Functions (ICRF), Category Response Functions (CRF), Operating Characteristic Curves (OCC), or Option Response Functions (ORF). The CCCs describe the probability of responding in a certain category given the location on the latent trait. |

type="TCC" | the Threshold Characteristic Curve, a.k.a, Category Boundary Response Function (CBRF), Category Boundary Curves, Cumulative Probability Curves, or Boundary Characteristic Curves (cf. [6], p. 329). They describe “the probability of responding positively rather than negatively at a given boundary between two categories” (Ostini and Nering, 2006, [17], p. 9). |

type="IIF" | the Item Information Function |

type="CIF" | the Category Information Function, a.k.a Item Response Information Function (e.g., [18]) |

type="BIF" | both the CIF and the IIF (“Both Information Functions”) |

Sort Option | Sort Criterion | Applicable to |
---|---|---|

isort="mean" | the mean difficulty | all models |

isort="median" | the median difficulty | all models |

isort="var" | the variance of the thresholds | polytomous models |

isort="min" | the minimum threshold | polytomous models |

isort="max" | the maximum threshold | polytomous models |

isort="range" | the threshold range | polytomous models |

isort="disc" | the discrimination parameter | 2/3/4PL, GPCM, GRM, and NRM |

isort="guess" | the guessing parameter | 3/4PL |

isort="lazy" | the laziness parameter | 4PL |

isort="none" | keeping the original ordering | default |

Option | Color of … |
---|---|

Person Parameter Area: | |

dimcol= | …the PP histogram(s) |

dencol= | …the PP density line(s) |

tifcol= | …the TIF line(s) |

secol= | …the S.E. line(s) |

Item Parameter Area: | |

funcol= | …the function lines, i.e., CCC, TCC, CIF; (see usedimcol=) |

infcol= | …the IIF lines |

discol= | …the disordered threshold indicator |

bgcol= | …the background of the function plots |

gridcol= | …the grid in the function plots |

usedimcol= | Use dimension colors (dimcol=) for thresholds ( classical=TRUE only; overrides funcol=) |

Model | eRm | Ltm | TAM | Mirt | Psychotools |
---|---|---|---|---|---|

Rasch Model (RM; [19]) | √ | √ | √ | √ | √ |

2PL model [20] | √ | √ | √ | √ | |

3PL model [20] | √ | √ | √ | √ | |

4PL model [21,22] | √ | √ | |||

Partial Credit Model (PCM; [23]) | √ | √ | √ | √ | √ |

Rating Scale Model (RSM; [24]) | √ | √ | √ | √ | |

Generalized Partial Credit Model (GPCM; [25]) | √ | √ | √ | √ | |

Graded Response Model (GRM; [18]) | √ | √ | |||

Graded Rating Scale Model (GRSM; [26]) | √ | ||||

Nominal Response Model (NRM; [27,28,29]) | (√) ${}^{a}$ | √ |

Q4 | Q9R | Q14 | Q19 | |
---|---|---|---|---|

1 | −0.97 | −2.79 | −2.09 | −1.51 |

2 | 0.49 | 0.90 | −0.32 | 0.07 |

3 | 0.29 | −0.32 | −0.22 | 0.27 |

4 | 2.20 | 2.99 | 1.51 | 2.09 |

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## Share and Cite

**MDPI and ACS Style**

Kabic, M.; Alexandrowicz, R.W.
RMX/PIccc: An Extended Person–Item Map and a Unified IRT Output for eRm, psychotools, ltm, mirt, and TAM. *Psych* **2023**, *5*, 948-965.
https://doi.org/10.3390/psych5030062

**AMA Style**

Kabic M, Alexandrowicz RW.
RMX/PIccc: An Extended Person–Item Map and a Unified IRT Output for eRm, psychotools, ltm, mirt, and TAM. *Psych*. 2023; 5(3):948-965.
https://doi.org/10.3390/psych5030062

**Chicago/Turabian Style**

Kabic, Milica, and Rainer W. Alexandrowicz.
2023. "RMX/PIccc: An Extended Person–Item Map and a Unified IRT Output for eRm, psychotools, ltm, mirt, and TAM" *Psych* 5, no. 3: 948-965.
https://doi.org/10.3390/psych5030062