# Estimating the Multidimensional Generalized Graded Unfolding Model with Covariates Using a Bayesian Approach

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## Abstract

**:**

## 1. Introduction

#### 1.1. The GGUM and Its Multidimensional Extension

_{j}is the latent trait level of person j, $\mathsf{\alpha}$

_{i}is the discrimination parameter of item i, $\mathsf{\delta}$

_{i}is the location parameter of item i, $\mathsf{\tau}$

_{ik}is the kth subjective response category threshold for item i, C is the number of response options minus 1, and M = 2C + 1.

_{jd}is the latent trait level of person j on the d

^{th}dimension, and

**θ**

_{j}= (θ

_{j1}, θ

_{j2}, …, θ

_{jD}) are assumed to follow a multivariate normal distribution. $\mathsf{\alpha}$

_{id}is the discrimination parameter of item i on the d

^{th}dimension, $\mathsf{\delta}$

_{id}is the location parameter of item i on the d

^{th}dimension, $\mathsf{\psi}$

_{ik}is the threshold parameter of the k

^{th}multidimensional subjective response category for item i, and $\mathsf{\tau}$

_{ik}is the k

^{th}subjective response category threshold for item i. D is the number of dimensions. C is the number of response options minus 1 and M = 2C + 1. The MGGUM estimated in the package bmggum only considers between-item multidimensionality (simple structure), which means that each item measures only a single trait, as shown in Figure 1. Therefore, $\mathsf{\alpha}$

_{id}, $\mathsf{\delta}$

_{id}, and $\mathsf{\tau}$

_{ik}= 0 for all d except one. Within each dimension, the unidimensional GGUM still applies. Users will need to rely on theories to decide which item loads on which factor, just like in confirmatory factor analysis. Note that the MGGUM defined in Equation (2) resembles the confirmatory multidimensional generalized graded unfolding model (CMGGUM) proposed in Wang and Wu (2016). The CMGGUM was proposed to handle both between and within-item multidimensionality, which means that it allows cross-loadings. In the case of between-item dimensionality, MGGUM is equivalent to CMGGUM.

#### 1.2. Estimating the MGGUM

#### 1.3. Incorporation of Covariates

_{jd}is the latent trait level of person j on the d

^{th}dimension, ${\mathrm{X}}_{\mathrm{j}\mathrm{p}}$ is the p

^{th}covariate of person j, ${\mathsf{\beta}}_{\mathrm{p}\mathrm{d}}$ is the regression coefficient of the relationship between the d

^{th}dimension and the p

^{th}covariates, p is the number of covariates, ${\mathrm{\u2107}}_{\mathrm{j}\mathrm{d}}$ is the residual and is assumed to be normally distributed with a mean of 0 and variance of ${\mathsf{\sigma}}^{2}$.

#### 1.4. Bayesian Model Fit Diagnostics

## 2. Study 1. Model Estimation Accuracy

#### 2.1. Method

**θ**~ MVN($\mathsf{\beta}$

**X**, $\omega $), $\mathsf{\beta}$ ~ MVN(0, 1), $\omega $ ~ lkj_corr_cholesky(1). Random initial values generated by bmggum were used. Based on the preliminary trials, 2000 iterations with 2 chains were sufficient to achieve convergence. Therefore, in this study, 2000 iterations with 2 chains were performed and the first 1000 iterations were discarded as burn-in. Model convergence was assessed using the Gelman–Rubin diagnostic index (Gelman and Rubin 1992), which compares the variability of samples after burn-in within parallel chains with the variability between parallel chains. If the ratio of variability between parallel chains to variability within parallel chains is less than 1.05, we considered it as evidence for model convergence. If a certain replication failed to converge, it was discarded, and an additional replication was conducted until 100 valid replications were obtained per condition. Overall, model nonconvergence was rare, specifically less than 4%, in this study.

^{th}item, $\widehat{\mathsf{\alpha}}$ is the parameter estimate, and $\mathsf{\alpha}$ is the true parameter. To have a single value of each estimation accuracy index for each condition, the obtained Cor, bias, and Ae values for item and person parameter estimates were averaged across replications and dimensions. Larger Cor and smaller bias and Ae indicate more accurate parameter estimation. The power/Type I error rates for detecting the correlations between traits and covariates were also computed by examining whether the 95% confidence interval of the posterior distribution of the regression coefficients included zero. If zero was included in the 95% confidence interval, it was considered statistically non-significant; if zero was not included in the 95% confidence interval, it was considered statistically significant.

#### 2.2. Results

## 3. Study 2. Model Selection Accuracy

#### 3.1. Method

#### 3.2. Results

## 4. Study 3. Empirical Illustration

#### 4.1. Method

#### 4.2. Results

_{GGUM2004}= 0.34, SE

_{GGUM2004}= 0.10; r

_{GGUM}= 0.34, SE

_{GGUM}= 0.10; r

_{mirt}= 0.32, SE

_{mirt}= 0.10), and no significant gender difference was found for assertiveness (r

_{GGUM2004}= −0.18, SE

_{GGUM2004}= 0.10; r

_{GGUM}= −0.18, SE

_{GGUM}= 0.10; r

_{mirt}= −0.14, SE

_{mirt}= 0.11). Age differences in orderliness (r

_{GGUM2004}= 0.01, SE

_{GGUM2004}= 0.04; r

_{GGUM}= 0.01, SE

_{GGUM}= 0.04; r

_{mirt}= 0.01, SE

_{mirt}= 0.04) and assertiveness (r

_{GGUM2004}= 0.03, SE

_{GGUM2004}= 0.04; r

_{GGUM}= 0.03, SE

_{GGUM}= 0.04; r

_{mirt}= 0.04, SE

_{mirt}= 0.04) were not significant.

## 5. Discussion

#### 5.1. The Benefit of Multidimensional Estimation and the Incorporation of Covariates

#### 5.2. Model Selection

#### 5.3. Implications

#### 5.4. Limitations and Future Directions

#### 5.5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Missing | r | Traits | Beta | Cor | Bias | Ae | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Items = 5 | Items = 10 | Items = 5 | Items = 10 | Items = 5 | Items = 10 | ||||||||||

RO = 2 | RO = 4 | RO = 2 | RO = 4 | RO = 2 | RO = 4 | RO = 2 | RO = 4 | RO = 2 | RO = 4 | RO = 2 | RO = 4 | ||||

0 | 0 | 2 | 0.00 | 0.60 | 0.84 | 0.75 | 0.92 | 0.00 | 0.00 | 0.00 | 0.00 | 0.62 | 0.41 | 0.50 | 0.29 |

0.25 | 0.65 | 0.85 | 0.78 | 0.93 | 0.00 | 0.00 | 0.00 | 0.00 | 0.59 | 0.40 | 0.48 | 0.30 | |||

5 | 0.00 | 0.59 | 0.84 | 0.75 | 0.92 | 0.00 | 0.00 | 0.00 | 0.00 | 0.62 | 0.41 | 0.50 | 0.29 | ||

0.25 | 0.65 | 0.86 | 0.78 | 0.93 | 0.00 | 0.00 | 0.00 | 0.00 | 0.59 | 0.40 | 0.48 | 0.29 | |||

0.5 | 2 | 0.00 | 0.64 | 0.86 | 0.78 | 0.93 | 0.00 | 0.00 | 0.00 | 0.00 | 0.59 | 0.39 | 0.48 | 0.29 | |

0.25 | 0.67 | 0.86 | 0.79 | 0.93 | 0.00 | 0.00 | 0.00 | 0.00 | 0.58 | 0.39 | 0.47 | 0.29 | |||

5 | 0.00 | 0.69 | 0.87 | 0.81 | 0.93 | 0.00 | 0.00 | 0.00 | 0.00 | 0.56 | 0.38 | 0.46 | 0.28 | ||

0.25 | 0.70 | 0.87 | 0.81 | 0.93 | 0.00 | 0.00 | 0.00 | 0.00 | 0.56 | 0.38 | 0.46 | 0.29 | |||

0.2 | 0 | 2 | 0.00 | 0.54 | 0.79 | 0.69 | 0.90 | 0.00 | 0.00 | 0.00 | 0.00 | 0.65 | 0.46 | 0.55 | 0.33 |

0.25 | 0.61 | 0.81 | 0.73 | 0.90 | 0.00 | 0.00 | 0.00 | 0.00 | 0.62 | 0.45 | 0.52 | 0.33 | |||

5 | 0.00 | 0.52 | 0.78 | 0.69 | 0.89 | 0.00 | 0.00 | 0.00 | 0.00 | 0.66 | 0.46 | 0.55 | 0.33 | ||

0.25 | 0.60 | 0.81 | 0.73 | 0.91 | 0.00 | 0.00 | 0.00 | 0.00 | 0.63 | 0.45 | 0.52 | 0.33 | |||

0.5 | 2 | 0.00 | 0.58 | 0.81 | 0.73 | 0.91 | 0.00 | 0.00 | 0.00 | 0.00 | 0.63 | 0.44 | 0.52 | 0.32 | |

0.25 | 0.63 | 0.82 | 0.75 | 0.91 | 0.00 | 0.00 | 0.00 | 0.00 | 0.61 | 0.44 | 0.51 | 0.33 | |||

5 | 0.00 | 0.63 | 0.83 | 0.77 | 0.91 | 0.00 | 0.00 | 0.00 | 0.00 | 0.60 | 0.42 | 0.50 | 0.32 | ||

0.25 | 0.66 | 0.83 | 0.77 | 0.91 | 0.00 | 0.00 | 0.00 | 0.00 | 0.59 | 0.43 | 0.49 | 0.32 |

Sample Size | Missing | r | WAIC | LOO | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Items = 5 | Items = 10 | Items = 5 | Items = 10 | |||||||

RO = 2 | RO = 4 | RO = 2 | RO = 4 | RO = 2 | RO = 4 | RO = 2 | RO = 4 | |||

200 | 0.00 | 0.30 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

0.60 | 0.98 | 0.99 | 1.00 | 1.00 | 0.88 | 0.97 | 1.00 | 1.00 | ||

0.90 | 0.62 | 0.96 | 0.77 | 1.00 | 0.39 | 0.92 | 0.64 | 1.00 | ||

1.00 | 0.54 | 0.53 | 0.63 | 0.58 | 0.66 | 0.71 | 0.73 | 0.68 | ||

0.20 | 0.30 | 0.97 | 1.00 | 1.00 | 1.00 | 0.94 | 1.00 | 1.00 | 1.00 | |

0.60 | 0.89 | 1.00 | 0.99 | 1.00 | 0.76 | 1.00 | 0.99 | 0.99 | ||

0.90 | 0.71 | 0.95 | 0.71 | 1.00 | 0.49 | 0.83 | 0.63 | 0.98 | ||

1.00 | 0.55 | 0.36 | 0.59 | 0.61 | 0.75 | 0.61 | 0.72 | 0.73 | ||

500 | 0.00 | 0.30 | 1.00 | 1.00 | 1.00 | 1.00 | 0.99 | 1.00 | 1.00 | 1.00 |

0.60 | 0.94 | 1.00 | 0.98 | 1.00 | 0.92 | 0.99 | 0.98 | 1.00 | ||

0.90 | 0.85 | 0.99 | 0.86 | 1.00 | 0.59 | 0.97 | 0.76 | 1.00 | ||

1.00 | 0.56 | 0.33 | 0.62 | 0.59 | 0.77 | 0.55 | 0.70 | 0.72 | ||

0.20 | 0.30 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | |

0.60 | 0.99 | 1.00 | 0.99 | 1.00 | 0.95 | 1.00 | 0.99 | 1.00 | ||

0.90 | 0.82 | 0.97 | 0.90 | 1.00 | 0.50 | 0.92 | 0.82 | 1.00 | ||

1.00 | 0.29 | 0.33 | 0.55 | 0.50 | 0.67 | 0.62 | 0.71 | 0.57 | ||

1000 | 0.00 | 0.30 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

0.60 | 0.97 | 0.99 | 1.00 | 1.00 | 0.96 | 0.99 | 1.00 | 1.00 | ||

0.90 | 0.89 | 1.00 | 0.97 | 1.00 | 0.68 | 1.00 | 0.95 | 1.00 | ||

1.00 | 0.49 | 0.34 | 0.55 | 0.51 | 0.71 | 0.58 | 0.59 | 0.64 | ||

0.20 | 0.30 | 1.00 | 1.00 | 1.00 | 1.00 | 0.99 | 1.00 | 1.00 | 1.00 | |

0.60 | 0.99 | 1.00 | 1.00 | 1.00 | 0.98 | 1.00 | 1.00 | 1.00 | ||

0.90 | 0.93 | 0.98 | 0.95 | 1.00 | 0.69 | 0.95 | 0.85 | 1.00 | ||

1.00 | 0.32 | 0.28 | 0.56 | 0.37 | 0.63 | 0.55 | 0.70 | 0.45 |

Items | Alpha | Delta | Tau | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

bmggum | ggum 2004 | ggum | mirt | bmggum | ggum 2004 | ggum | mirt | bmggum | ggum 2004 | ggum | mirt | |

Order1 | 1.57 (0.25) | 1.95 (0.37) | 1.96 (0.37) | 1.79 (0.30) | 1.96 (0.48) | 1.53 (0.30) | 1.52 (0.29) | 1.45 (0.25) | −1.98 (0.49) | −1.57 (0.29) | −1.56 (0.29) | 1.47 (0.24) |

Order2 | 1.10 (0.17) | 1.23 (0.67) | 1.23 (0.67) | 1.15 (0.19) | 2.15 (0.49) | 2.69 (17.50) | 2.59 (15.90) | 1.71 (0.35) | −3.18 (0.53) | −3.67 (17.77) | −3.57 (16.18) | 2.67 (0.40) |

Order3 | 1.49 (0.23) | 1.56 (0.37) | 1.57 (0.50) | 1.55 (0.26) | 2.11 (0.50) | 2.37 (6.27) | 2.58 (10.65) | 1.53 (0.35) | −2.29 (0.51) | −2.57 (6.44) | −2.78 (10.89) | 1.69 (0.35) |

Order4 | 1.10 (0.20) | 1.23 (0.23) | 1.22 (0.23) | 1.24 (0.24) | 1.58 (0.45) | 1.24 (0.25) | 1.24 (0.25) | 1.16 (0.23) | −1.21 (0.42) | −0.94 (0.20) | −0.94 (0.20) | 0.86 (0.18) |

Order5 | 1.26 (0.22) | 1.24 (0.54) | 1.24 (0.56) | 1.41 (0.40) | −3.21 (0.49) | −4.53 (22.29) | −4.42 (20.45) | −1.60 (0.34) | −0.46 (0.41) | −1.84 (23.33) | −1.72 (21.55) | −0.84 (0.44) |

Order6 | 1.11 (0.18) | 1.25 (0.66) | 1.26 (0.63) | 1.15 (0.21) | −2.48 (0.54) | −3.53 (14.42) | −3.72 (16.77) | −1.91 (0.41) | −1.18 (0.58) | −2.41 (15.22) | −2.62 (17.49) | 0.65 (0.39) |

Order7 | 1.94 (0.32) | 2.61 (0.60) | 2.61 (0.57) | 2.25 (0.39) | −2.54 (0.44) | −2.74 (9.90) | −2.68 (7.82) | −2.06 (0.28) | −1.43 (0.46) | −1.79 (10.03) | −1.74 (7.95) | 1.05 (0.28) |

Order8 | 1.96 (0.33) | 2.71 (0.55) | 2.71 (0.55) | 2.34 (0.44) | −2.29 (0.45) | −1.82 (0.27) | −1.82 (0.27) | −1.79 (0.25) | −1.19 (0.45) | −0.90 (0.24) | −0.89 (0.24) | 0.80 (0.21) |

Order9 | 0.68 (0.14) | 0.55 (0.19) | 0.55 (0.19) | 0.63 (0.16) | −0.74 (0.33) | −0.82 (0.46) | −0.82 (0.45) | −0.68 (0.30) | −0.81 (0.25) | −0.73 (0.28) | −0.73 (0.28) | 0.70 (0.22) |

Order10 | 1.69 (0.33) | 2.45 (0.47) | 2.45 (0.48) | 2.21 (0.45) | −0.01 (0.10) | −0.03 (0.07) | −0.03 (0.07) | −0.03 (0.09) | −1.29 (0.10) | −1.15 (0.07) | −1.15 (0.07) | 1.18 (0.08) |

Order11 | 1.57 (0.29) | 2.20 (0.43) | 2.20 (0.43) | 1.91 (0.37) | 0.26 (0.12) | 0.22 (0.08) | 0.22 (0.08) | 0.21 (0.10) | −1.23 (0.11) | −1.11 (0.08) | −1.10 (0.08) | 1.13 (0.09) |

Assertiveness1 | 2.04 (0.29) | 2.81 (0.44) | 2.82 (0.45) | 2.46 (0.37) | 1.70 (0.48) | 1.29 (0.26) | 1.28 (0.26) | 1.20 (0.22) | −2.51 (0.49) | −2.01 (0.27) | −2.00 (0.27) | 1.93 (0.23) |

Assertiveness2 | 1.21 (0.22) | 1.56 (0.35) | 1.31 (0.40) | 1.46 (0.31) | 1.06 (0.44) | 0.84 (0.34) | 3.58 (26.53) | .63 (0.27) | −3.67 (0.54) | −2.97 (0.47) | −5.94 (26.36) | 2.89 (0.40) |

Assertiveness3 | 2.65 (0.39) | 4.13 (0.77) | 4.10 (0.80) | 3.36 (0.54) | 1.81 (0.53) | 2.09 (314.35) | 2.25 (686.92) | 1.21 (0.31) | −2.85 (0.54) | −2.98 (314.36) | −3.13 (686.93) | 2.14 (0.31) |

Assertiveness4 | 0.74 (0.12) | 0.71 (0.64) | 0.74 (0.43) | 0.77 (0.15) | 2.09 (0.61) | 4.07 (25.83) | 5.49 (39.59) | 1.44 (0.42) | −1.60 (0.62) | −3.74 (27.82) | −5.23 (40.62) | 0.98 (0.37) |

Assertiveness5 | 2.47 (0.38) | 4.17 (0.86) | 4.19 (1.01) | 3.20 (0.57) | −2.73 (0.41) | −2.92 (28.05) | −2.95 (54.15) | −2.11 (0.24) | −1.23 (0.42) | −1.72 (28.09) | −1.75 (54.24) | 0.80 (0.22) |

Assertiveness6 | 2.32 (0.35) | 3.74 (0.56) | 4.20 (0.74) | 3.23 (0.65) | −2.37 (0.44) | −2.59 (3.22) | −1.81 (0.17) | −1.73 (0.18) | −1.39 (0.44) | −1.77 (3.22) | −1.02 (0.16) | 0.91 (0.15) |

Assertiveness7 | 3.23 (0.41) | 7.42 (2.23) | 7.68 (2.34) | 4.80 (0.96) | −2.61 (0.43) | −2.65 (82.96) | −2.64 (201.70) | −1.97 (0.23) | −1.65 (0.44) | −1.84 (82.96) | −1.85 (201.70) | 1.14 (0.21) |

Assertiveness8 | 2.55 (0.40) | 4.28 (0.88) | 4.44 (0.95) | 3.39 (0.60) | −2.38 (0.40) | −2.00 (0.24) | −1.86 (0.16) | −1.85 (0.17) | −1.29 (0.40) | −1.10 (0.23) | −0.97 (0.16) | 0.90 (0.15) |

Assertiveness9 | 1.13 (0.22) | 1.32 (0.28) | 1.34 (0.28) | 1.34 (0.28) | −0.06 (0.14) | −0.02 (0.11) | −0.02 (0.11) | −0.06 (0.12) | −1.37 (0.14) | −1.20 (0.11) | −1.20 (0.11) | 1.23 (0.11) |

Assertiveness10 | 1.09 (0.20) | 1.30 (0.28) | 1.30 (0.28) | 1.25 (0.26) | −0.05 (0.15) | −0.01 (0.11) | −0.01 (0.11) | −0.05 (0.13) | −1.23 (0.13) | −1.08 (0.11) | −1.08 (0.11) | 1.11 (0.11) |

Assertiveness11 | 1.44 (0.29) | 1.88 (0.40) | 1.88 (0.39) | 1.54 (0.35) | −0.41 (0.12) | −0.34 (0.08) | −0.34 (0.08) | −0.37 (0.11) | −1.36 (0.12) | −1.15 (0.09) | −1.15 (0.09) | 1.24 (0.11) |

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**MDPI and ACS Style**

Tu, N.; Zhang, B.; Angrave, L.; Sun, T.; Neuman, M.
Estimating the Multidimensional Generalized Graded Unfolding Model with Covariates Using a Bayesian Approach. *J. Intell.* **2023**, *11*, 163.
https://doi.org/10.3390/jintelligence11080163

**AMA Style**

Tu N, Zhang B, Angrave L, Sun T, Neuman M.
Estimating the Multidimensional Generalized Graded Unfolding Model with Covariates Using a Bayesian Approach. *Journal of Intelligence*. 2023; 11(8):163.
https://doi.org/10.3390/jintelligence11080163

**Chicago/Turabian Style**

Tu, Naidan, Bo Zhang, Lawrence Angrave, Tianjun Sun, and Mathew Neuman.
2023. "Estimating the Multidimensional Generalized Graded Unfolding Model with Covariates Using a Bayesian Approach" *Journal of Intelligence* 11, no. 8: 163.
https://doi.org/10.3390/jintelligence11080163