# Applying SEM, Exploratory SEM, and Bayesian SEM to Personality Assessments

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## Abstract

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## 1. Introduction

## 2. An Overview of Factor Analytic Techniques

#### 2.1. Exploratory Factor Analysis (EFA)

#### 2.2. Confirmatory Factor Analysis (CFA)

#### 2.3. Exploratory Structural Equation Modeling

#### 2.4. Estimation Methods

#### Maximum Likelihood vs. Weighted Least Squares

#### 2.5. Bayesian Structural Equation Modeling

#### 2.5.1. Types of Priors

^{10})) or weakly informative priors if prior information is presented from past analyses. Cross-loadings are estimated in an exploratory way and penalized by being assigned shrinkage priors with a mean of zero and a small variance ([35] Liang, 2020). For example, utilizing a normal prior of N (0, 0.01) suggests “the prior belief that a 95% chance the true cross-loading falls between −0.196 and 0.196” ([38] Liang et al., 2020, p. 876). In comparison to the independent clusters model CFA, which allows each item to load on one factor and sets all the cross-loadings to zero ([13] Marsh et al., 2009), Bayesian estimation is flexible in estimating models by regulating the variability in cross-loadings through controlling prior distributions. This enables models that are not computed by frequentist methods such as ML to be estimated ([38] Liang et al., 2020).

#### 2.5.2. Bayesian Model Fit Evaluation

## 3. Research on IPIP-NEO-120

#### Reliability and Validity Evidence of the IPIP-NEO-120

## 4. Significance of the Current Study

## 5. Illustrations of SEM, ESEM, and BSEM Techniques Using IPIP-NEO-120 Agreeableness Scale

**RQ1**.- To what extent do parameter estimation methods (SEM ML, WLSMV, and Bayesian SEM) affect model fit for correlated-factor, bifactor, and hierarchical factor models?
**RQ2**.- To what extent do factor loading constraints (allowing vs. restricting weak off-target loadings; ESEM vs. CFA) affect model fit for correlated-factor, bifactor, and hierarchical factor models?
**RQ3**.- To what extent will the use of different priors in BSEM affect model fit for correlated-factor, bifactor, and hierarchical factor models?

## 6. Methods

#### 6.1. Data and Measure

#### IPIP-NEO-120

#### 6.2. Sample

#### 6.3. Descriptive Statistics

#### 6.4. Data Analyses

## 7. Results

#### 7.1. The Effects of Estimation Methods

#### 7.1.1. Correlated-Factor Models

#### 7.1.2. Bifactor Models

#### 7.1.3. Hierarchical Models

#### 7.1.4. Fit Differences for Estimation Procedures

#### 7.1.5. ML vs. WLSMV

#### 7.1.6. ML vs. Bayesian Informative Priors

#### 7.1.7. WLSMV vs. Bayesian Informative Priors

#### 7.2. The Effects of Allowing and Restricting Off-Target Loadings (ESEM vs. CFA)

#### 7.2.1. Correlated-Factor Models

#### 7.2.2. Bifactor Models

#### 7.2.3. Hierarchical Models

#### 7.2.4. Fit Differences When Allowing Off-Target Loadings

#### 7.3. The Effects of Different Priors in BSEM on Model Fits

**Correlated-Factor Models.**The results in Table 7 for the correlated-factor model reveal that the Bayesian-IP models on average provided noticeably better fits ($\overline{BCFI}$ = 0.983, $\overline{BTLI}$ = 0.983, and $\overline{BRMSEA}$ = 0.022) than Bayesian-NIP models ($\overline{BCFI}$ = 0.906, $\overline{BTLI}$ = 0.891, and $\overline{BRMSEA}$ = 0.055). Improvements in fit when using different informative priors were not noticeable, but as variance increased, BTLI was to some extent increased (BTLI = 0.980 vs. 0.984) and BRASEA was slightly decreased (BRMSEA = 0.024 vs. 0.021). In keeping with the cutoff criteria of CFIs and TLIs ≥ 0.95 and RMSEAs ≤ 0.06, all Bayesian-IP models yielded exceptional fits.

**Bifactor.**Consistent with correlated-factor model results, Bayesian-IP models produced better model fit indices ($\overline{BCFI}$ = 0.990, $\overline{BTLI}$ = 1, and $\overline{BRMSEA}$ = 0.001) than did Bayesian-NIP models ($\overline{BCFI}$ = 0.929, $\overline{BTLI}$ = 0.915, and $\overline{BRMSEA}$ = 0.048). Using different informative priors did not improve model fit. In accordance with cutoff criteria for CFIs and TLIs ≥ 0.95 and RMSEAs ≤ 0.06, all Bayesian models with informative priors yielded excellent fits.

**Hierarchical Models.**Similar to correlated-factor and bifactor models, Bayesian-IP models produced noticeably better model fit indices ($\overline{BCFI}$ = 0.983, $\overline{BTLI}$ = 0.978, and $\overline{BRMSEA}$ = 0.025) than did Bayesian-NIP ($\overline{BCFI}$ = 0.876, $\overline{BTLI}$ = 0.861, and $\overline{BRMSEA}$ = 0.061). Model fit when using informative priors with N (0, 0.005) was better than those with large variance. According to the cutoff rules of CFIs and TLIs ≥ 0.95 and RMSEAs ≤ 0.06, all Bayesian-IP models yielded excellent fits.

#### 7.4. Fit Differences for ESEM WLSMV Models versus Bayesian Models with Informative Priors

## 8. Discussion

#### 8.1. The Effects of Parameter Estimation Methods on Model Fit

#### 8.2. The Effects of Factor Loading Constraints on Model Fit

#### 8.3. Bayesian Structural Equation Models and Exploratory Structural Equation Models

#### 8.4. Implications

#### 8.5. Recommendations for Future Research

#### 8.6. Limitations

## 9. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Examples of correlated-factor, hierarchical, and bifactor measurement models. Note: F = factor; FG = general factor; HO = hierarchical factor. In (

**a**), the correlated-factor model, covariation in the observed indicators is explained by six interrelated factors; in (

**b**), the hierarchical model, covariation in the observed indicators is captured by lower-order factors whose interrelationship is in turn captured by the hierarchical factor; in (

**c**), the bifactor model, covariation in the observed indicators is primarily captured by a general factor with the specific factors capturing additional covariation not captured by the general factor. For simplicity, residual variances and intercepts are omitted.

**Figure 2.**Examples of exploratory SEM (

**a**) correlated-factors, (

**b**) hierarchical, and (

**c**) bifactor models. Note: F: factor; FG: general factor; HO: higher-order factor. For simplicity and illustrative purposes, the figures above only contain a subset of cross-loadings (represented with the dashed lines).

**Table 1.**Descriptive Statistics and Conventional Reliability Estimates for IPIP-NEO-120 Agreeableness Facet Scores (N = 447,500).

Domain | Facet | Mean: Scale (Item) | SD: Scale (Item) | Alpha | Omega |
---|---|---|---|---|---|

Agreeableness | 87.95 (3.66) | 12.54 (0.52) | 0.85 | 0.90 | |

Trust | 13.36 (3.34) | 3.57(0.89) | 0.84 | 0.85 | |

Morality | 16.22 (4.06) | 3.05 (0.76) | 0.71 | 0.72 | |

Altruism | 16.52 (4.13) | 2.69 (0.67) | 0.70 | 0.70 | |

Cooperation | 14.51 (3.63) | 3.75 (0.94) | 0.71 | 0.72 | |

Modesty | 12.41 (3.10) | 3.43 (0.86) | 0.71 | 0.75 | |

Sympathy | 14.92 (3.73) | 3.16 (0.79) | 0.70 | 0.71 | |

Facet Means | 14.66 (3.67) | 3.27 (0.82) | 0.73 | 0.74 |

Estimation Method(s) | Models |
---|---|

SEM/CFA (ML) | 1. 6 correlated factors |

2. Bifactor | |

3. Hierarchical model | |

SEM/CFA (WLSMV) | 4. 6 correlated factors |

5. Bifactor | |

6. Hierarchical model | |

ESEM (ML) | 7. 6 correlated factors |

8. Bifactor | |

9. Hierarchical model | |

ESEM (WLSMV) | 10. 6 correlated factors |

11. Bifactor | |

12. Hierarchical model | |

BSEM (informative priors) | 13. 6 correlated factors |

14. Bifactor | |

15. Hierarchical Model |

Models | CFI | ||||||||

CFA ML | ESEM ML | MLMean | CFA WLSMV | ESEM WLSMV | WLSMVMean | Bayesian NIP | Bayesian IP | Bayesian Mean | |

6 Correlated factor | 0.906 | 0.983 | 0.945 | 0.894 | 0.987 | 0.941 | 0.906 | 0.983 | 0.951 |

Bifactor | 0.929 | 0.990 | 0.96 | 0.935 | 0.993 | 0.964 | 0.929 | 0.99 | 0.967 |

Hierarchical | 0.876 | 0.977 | 0.927 | 0.867 | 0.986 | 0.927 | 0.876 | 0.983 | 0.938 |

Mean | 0.904 | 0.983 | 0.944 | 0.899 | 0.989 | 0.944 | 0.904 | 0.985 | 0.952 |

Models | TLI | ||||||||

CFA ML | ESEM ML | MLMean | CFA WLSMV | ESEM WLSMV | WLSMVMean | Bayesian NIP | Bayesian IP | Bayesian Mean | |

6 Correlated factor | 0.891 | 0.968 | 0.93 | 0.877 | 0.976 | 0.927 | 0.891 | 0.983 | 0.939 |

Bifactor | 0.915 | 0.978 | 0.947 | 0.921 | 0.985 | 0.953 | 0.915 | 1.00 | 0.960 |

Hierarchical | 0.861 | 0.962 | 0.912 | 0.851 | 0.977 | 0.914 | 0.861 | 0.978 | 0.926 |

Mean | 0.889 | 0.969 | 0.930 | 0.883 | 0.979 | 0.931 | 0.889 | 0.987 | 0.942 |

Models | RMSEA | ||||||||

CFA ML | ESEM ML | MLMean | CFA WLSMV | ESEM WLSMV | WLSMVMean | Bayesian NIP | Bayesian IP | Bayesian Mean | |

6 Correlated factor | 0.055 | 0.029 | 0.042 | 0.09 | 0.039 | 0.065 | 0.055 | 0.022 | 0.047 |

Bifactor | 0.048 | 0.025 | 0.037 | 0.072 | 0.031 | 0.052 | 0.048 | 0.001 | 0.035 |

Hierarchical | 0.061 | 0.032 | 0.047 | 0.099 | 0.039 | 0.069 | 0.061 | 0.025 | 0.051 |

Mean | 0.055 | 0.029 | 0.042 | 0.087 | 0.036 | 0.062 | 0.055 | 0.016 | 0.044 |

CFI | TLI | RMSEA | |||||||

ML | WLSMV | Diff | ML | WLSMV | Diff | ML | WLSMV | Diff | |

Correlated factor | 0.945 | 0.941 | −0.004 | 0.930 | 0.927 | −0.003 | 0.042 | 0.065 | 0.023 |

Bifactor | 0.960 | 0.964 | 0.004 | 0.947 | 0.953 | 0.006 | 0.037 | 0.052 | 0.015 |

Hierarchical | 0.927 | 0.927 | 0.000 | 0.912 | 0.914 | 0.002 | 0.047 | 0.069 | 0.022 |

Mean | 0.944 | 0.944 | 0.000 | 0.930 | 0.931 | 0.001 | 0.042 | 0.062 | 0.020 |

ML | Bayesian-IP | Diff | ML | Bayesian IP | Diff | ML | Bayesian IP | Diff | |

Correlated factor | 0.945 | 0.983 | 0.038 | 0.930 | 0.983 | 0.053 | 0.042 | 0.022 | −0.020 |

Bifactor | 0.960 | 0.990 | 0.030 | 0.947 | 1.000 | 0.053 | 0.037 | 0.001 | −0.036 |

Hierarchical | 0.927 | 0.983 | 0.056 | 0.912 | 0.978 | 0.066 | 0.047 | 0.025 | −0.022 |

Mean | 0.944 | 0.985 | 0.041 | 0.930 | 0.987 | 0.057 | 0.042 | 0.016 | −0.026 |

WLSMV | Bayesian-IP | Diff | WLSMV | Bayesian IP | Diff | WLSMV | Bayesian IP | Diff | |

Correlated factor | 0.941 | 0.983 | 0.042 | 0.927 | 0.983 | 0.056 | 0.065 | 0.022 | −0.043 |

Bifactor | 0.964 | 0.990 | 0.026 | 0.953 | 1.000 | 0.047 | 0.052 | 0.001 | −0.051 |

Hierarchical | 0.927 | 0.983 | 0.056 | 0.914 | 0.978 | 0.064 | 0.069 | 0.025 | −0.044 |

Mean | 0.944 | 0.985 | 0.041 | 0.931 | 0.987 | 0.056 | 0.062 | 0.016 | −0.046 |

**Table 5.**Goodness-of-Fit Statistics for the Factor Models Allowing vs. Restricting Weak Off-Target Loadings.

Model | Mean CFI | Mean TLI | Mean RMSEA | |||
---|---|---|---|---|---|---|

CFA | ESEM | CFA | ESEM | CFA | ESEM | |

Correlated-Factor | 0.900 | 0.985 | 0.884 | 0.972 | 0.073 | 0.034 |

Bifactor | 0.932 | 0.983 | 0.918 | 0.982 | 0.060 | 0.028 |

Hierarchical | 0.872 | 0.982 | 0.856 | 0.970 | 0.080 | 0.036 |

Grand Mean | 0.922 | 0.984 | 0.910 | 0.970 | 0.061 | 0.037 |

Models | CFI | TLI | RMSEA | ||||||
---|---|---|---|---|---|---|---|---|---|

CFA ML | ESEM ML | Diff | CFA ML | ESEM ML | Diff | CFA ML | ESEM ML | Diff | |

Correlated-factor | 0.906 | 0.983 | 0.077 | 0.891 | 0.968 | 0.077 | 0.055 | 0.029 | −0.026 |

Bifactor | 0.929 | 0.99 | 0.061 | 0.915 | 0.978 | 0.063 | 0.048 | 0.025 | −0.023 |

Hierarchical | 0.876 | 0.977 | 0.101 | 0.861 | 0.962 | 0.101 | 0.061 | 0.032 | −0.029 |

Mean | 0.904 | 0.983 | 0.080 | 0.889 | 0.969 | 0.080 | 0.055 | 0.029 | −0.026 |

Models | CFA WLSMV | ESEM WLSMV | Diff | CFA WLSMV | ESEM WLSMV | Diff | CFA WLSMV | ESEM WLSMV | Diff |

Correlated-factor | 0.894 | 0.987 | 0.093 | 0.877 | 0.976 | 0.099 | 0.09 | 0.039 | −0.051 |

Bifactor | 0.935 | 0.993 | 0.058 | 0.921 | 0.985 | 0.064 | 0.072 | 0.031 | −0.041 |

Hierarchical | 0.867 | 0.986 | 0.119 | 0.851 | 0.977 | 0.126 | 0.099 | 0.039 | −0.06 |

Mean | 0.899 | 0.989 | 0.090 | 0.883 | 0.979 | 0.096 | 0.087 | 0.036 | −0.051 |

6 Correlated Factors | Parameters | PPP | PPPP | BIC | DIC | BCFI | BTLI | BRMSEA |
---|---|---|---|---|---|---|---|---|

BSEM with default priors | 87 | 0 | 29,257,462.5 | 29,256,504.9 | 0.906 | 0.891 | 0.055 | |

BSEM with informative priors | ||||||||

BSEM-Cross loadings (CL) priors: N (0, 0.005) | 207 | 0 | 0.039 | 29,001,466.6 | 28,997,563.2 | 0.983 | 0.980 | 0.024 |

BSEM-Cross loadings (CL) priors: N (0, 0.01) | 207 | 0 | 0.875 | 29,001,791.6 | 28,997,523.5 | 0.983 | 0.983 | 0.022 |

BSEM-Cross loadings (CL) priors: N (0, 0.02) | 207 | 0 | 0.999 | 29,001,550.1 | 28,997,509.9 | 0.983 | 0.983 | 0.021 |

BSEM-Cross loadings (CL) priors: N (0, 0.03) | 207 | 0 | 0.999 | 29,001,401.1 | 28,997,502.3 | 0.983 | 0.984 | 0.021 |

Mean of BSEM with informative priors | 207 | 0 | 0.728 | 29,001,552.4 | 28,997,524.7 | 0.983 | 0.983 | 0.022 |

Bifactor | Parameters | PPP | PPPP | BIC | DIC | BCFI | BTLI | BRMSEA |

BSEM with default priors | 96 | 0 | 29,179,630.6 | 29,178,573.2 | 0.929 | 0.915 | 0.048 | |

BSEM with informative priors | ||||||||

BSEM-Cross loadings (CL) priors: N (0, 0.005) | 214 | 0 | 0 | 29,027,629 | 28,924,943.8 | 0.99 | 1 | 0.001 |

BSEM-Cross loadings (CL) priors: N (0, 0.01) | 214 | 0 | 0 | 29,024,892.2 | 28,927,616.1 | 0.99 | 1 | 0.001 |

BSEM-Cross loadings (CL) priors: N (0, 0.02) | 214 | 0 | 0.001 | 29,031,978.6 | 28,918,152 | 0.99 | 1 | 0.001 |

BSEM-Cross loadings (CL) priors: N (0, 0.03) | 214 | 0 | 0.001 | 29,036,419.5 | 28,912,666.6 | 0.99 | 1 | 0.001 |

Mean of BSEM with informative priors | 214 | 0 | 0.001 | 29,030,229.8 | 28,920,844.6 | 0.99 | 1 | 0.001 |

Hierarchical | Parameters | PPP | PPPP | BIC | DIC | BCFI | BTLI | BRMSEA |

BSEM with default priors | 78 | 0 | 29,357,897.2 | 29,357,037.8 | 0.876 | 0.861 | 0.061 | |

BSEM with informative priors | ||||||||

BSEM-Cross loadings (CL) priors: N (0, 0.005) | 196 | 0 | 0 | 29,000,254.9 | 28,997,540.7 | 0.983 | 0.983 | 0.022 |

BSEM-Cross loadings (CL) priors: N (0, 0.01) | 196 | 0 | 0 | 29,000,200.2 | 28,997,593.1 | 0.983 | 0.977 | 0.025 |

BSEM-Cross loadings (CL) priors: N (0, 0.02) | 196 | 0 | 0.002 | 29,000,235.2 | 28,997,613.5 | 0.983 | 0.974 | 0.027 |

BSEM-Cross loadings (CL) priors: N (0, 0.03) | 196 | 0 | 0.004 | 29,000,206.8 | 28,997,583.1 | 0.983 | 0.978 | 0.025 |

Mean of BSEM with informative priors | 196 | 0 | 0.002 | 29,000,224.3 | 28,997,582.6 | 0.983 | 0.978 | 0.025 |

CFI | TLI | RMSEA | |||||||
---|---|---|---|---|---|---|---|---|---|

ESEM WLSMV | Bayesian-IP | Diff | ESEM WLSMV | Bayesian-IP | Diff | ESEM WLSMV | Bayesian-IP | Diff | |

Correlated factor | 0.987 | 0.983 | −0.004 | 0.976 | 0.983 | 0.007 | 0.039 | 0.022 | −0.017 |

Bifactor | 0.993 | 0.99 | −0.003 | 0.985 | 1 | 0.015 | 0.031 | 0.001 | −0.03 |

Hierarchical | 0.986 | 0.983 | −0.003 | 0.977 | 0.978 | 0.001 | 0.039 | 0.025 | −0.014 |

Mean | 0.989 | 0.985 | −0.004 | 0.979 | 0.987 | 0.008 | 0.036 | 0.016 | −0.020 |

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**MDPI and ACS Style**

Hong, H.; Vispoel, W.P.; Martinez, A.J.
Applying SEM, Exploratory SEM, and Bayesian SEM to Personality Assessments. *Psych* **2024**, *6*, 111-134.
https://doi.org/10.3390/psych6010007

**AMA Style**

Hong H, Vispoel WP, Martinez AJ.
Applying SEM, Exploratory SEM, and Bayesian SEM to Personality Assessments. *Psych*. 2024; 6(1):111-134.
https://doi.org/10.3390/psych6010007

**Chicago/Turabian Style**

Hong, Hyeri, Walter P. Vispoel, and Alfonso J. Martinez.
2024. "Applying SEM, Exploratory SEM, and Bayesian SEM to Personality Assessments" *Psych* 6, no. 1: 111-134.
https://doi.org/10.3390/psych6010007