# Comparing Bayesian and Maximum Likelihood Predictors in Structural Equation Modeling of Children’s Lifestyle Index

^{*}

## Abstract

**:**

## 1. Introduction

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_{3}:**H**

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_{5}:**H**

_{6}:- First moment properties of raw individual observations are mainly used in statistical techniques, thus making the techniques much simpler than second moment properties of the sample covariance matrix. Hence, B-SEM is easier to apply in more complex states.
- Direct latent variable estimation is possible, which simplifies the process of obtaining factor score estimates compared to classical regression methods.
- As manifest variables are directly modeled with their latent variables using familiar regression functions, B-SEM provides a more direct interpretation. It can also use common methods of regression modeling, such as residual and outlier analyses in conducting statistical analysis.

## 2. Theoretical Background of Maximum Likelihood-Structural Equation Modeling (ML-SEM) and Bayesian-SEM (B-SEM)

#### 2.1. ML-SEM

- (a)
- $\mathit{\mu}$ is a $\left(\mathit{m}\times \mathit{n}\right)$ matrix that represents factor loadings from modeling the regressions of ${\mathit{y}}_{\mathit{i}}$ on ${\Omega}_{\mathit{i}}$.
- (b)
- ${\Omega}_{\mathit{i}}$ is a $\left(\mathit{n}\times \mathbf{1}\right)$ vector with normal distribution $\mathit{N}\left(\mathbf{0},\mathit{\Phi}\right)$ and is representative of the constructs (latent variables). ${\Omega}_{\mathit{i}}$ $\mathit{i}=\mathbf{1},\dots ,\mathit{n},$ are identically independent, have no correlation with ${\mathit{\epsilon}}_{\mathit{i}}$, and have normal distribution $\mathit{N}\left(\mathbf{0},\mathit{\Phi}\right)$. To modify the exogenous and endogenous latent variables’ association, ${\Omega}_{\mathit{i}}$ is partitioned into $\left({\mathit{\lambda}}_{\mathit{i}},{\mathit{\omega}}_{\mathit{i}}\right)$, where ${\mathit{\lambda}}_{\mathit{i}}$ and ${\mathit{\omega}}_{\mathit{i}}$ are $\mathit{r}\times \mathbf{1}$ and $\mathit{s}\times \mathbf{1}$ vector variables, respectively, with latent structures.
- (c)
- ${\mathit{\epsilon}}_{\mathit{i}}$ is a $\left(\mathit{m}\times \mathbf{1}\right)$ random vector with $\mathit{N}\left(\mathbf{0},{\mathit{\psi}}_{\mathit{\epsilon}}\right)$ distribution that represents the error measurement.

- (a)
- $\mathit{\Sigma}$ is an $\mathit{r}\times \mathit{r}$ matrix of structural parameters representing the relationships among endogenous latent variables. This matrix is assumed to have zeroes in the diagonal elements.
- (b)
- $\mathit{\gamma}$ is an $\mathit{r}\times \mathit{s}$ matrix of regression parameters relating both exogenous and endogenous latent variables, and ${\mathit{\pi}}_{\mathit{i}}$ is a $\mathit{r}\times \mathbf{1}$ vector of disturbances.
- (c)
- ${\mathit{\pi}}_{\mathit{i}}$ is an error term presumed to have $\mathit{N}\left(\mathbf{0},{\mathit{\psi}}_{\mathit{\pi}}\right)$ distribution, where ${\mathit{\psi}}_{\mathit{\pi}}$ is a diagonal covariance matrix and this vector is uncorrelated with ${\mathit{\omega}}_{\mathit{i}}$.

#### 2.2. B-SEM

- $c$ is the number of categories for ${\mathit{x}}_{\mathbf{1}}$
**;** - ${\tau}_{c}-1$ and ${\tau}_{c}$ represent the threshold levels associated with ${\mathit{y}}_{\mathbf{1}}$.

- ${\mathit{\Phi}}^{-\mathbf{1}}(\cdot )$ is the inverse standardized normal distribution;
- $N$ is the total number of cases;
- ${N}_{r}$ is the number of cases in the $r$th category.

#### 2.3. Modeling Description

## 3. Materials and Methods

#### 3.1. Data Structure

#### 3.2. Ethics Statement

#### 3.3. Sampling

^{2}, and the size and direction of parameter estimates (see Table 1). Therefore, (483 − 22 − 9 = 452) 452 observations were considered as the final data of the study.

## 4. Results

- Prior I: Unknown loadings in $\mathit{\Lambda}$ are all made equal to 0.35, and the measures corresponding to $\left\{{\theta}_{1},{\theta}_{2},{\theta}_{3}\right\}$ are $\left\{0.6,0.5,0.2\right\}$.
- Prior II: The hyperparameter values are considered half of the values in prior I.
- Prior III: The hyperparameter values are considered a quarter of the values in prior I.
- Prior IV: The hyperparameter values are considered double the values in prior I.

- ${\theta}_{1}$ is the coefficient of parental socioeconomic status indicator;
- ${\theta}_{2}$ is the coefficient of household food security indicator;
- ${\theta}_{3}$ is the coefficient of parental lifestyle indicator.

^{2}), mean absolute error (MSE), and mean absolute percentage error (MAPE) are the most familiar statistical indices for comparison purposes. Table 7 presents the formulas of these indices and outputs of the ML and Bayesian approaches.

^{2}value for the B-SEM model was greater than for the ML-SEM model, and the RMSE, MSE, and MAPE values for the B-SEM model were lower than for ML-SEM. Therefore, according to the performance indices, B-SEM predicted children’s lifestyle better than the ML-SEM model. The main reason B-SEM performed better is the ML framework defined, which permits simultaneous self-adjustment of parameters and effective learning of the association between inputs and outputs in causal and complex models.

## 5. Discussion

^{2}, RMSE, MSE, and MPEA indices, SEM with the Bayesian approach was more effective at predicting children’s lifestyle with the dataset obtained from Urumqi, Xinjiang, China.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Likelihood, posterior, and prior for a parameter (source: [30]).

Observation Number | Mahalanobis D-Squared | p1 | p2 |
---|---|---|---|

36 | 22.56 | 0.0016 | 0.0084 |

88 | 20.31 | 0.0067 | 0.0091 |

92 | 18.92 | 0.0092 | 0.0104 |

134 | 36.58 | 0.0116 | 0.0124 |

228 | 32.71 | 0.0231 | 0.0178 |

256 | 30.08 | 0.0854 | 0.0364 |

372 | 28.19 | 0.0932 | 0.0392 |

411 | 25.44 | 0.1589 | 0.0421 |

444 | 19.76 | 0.2876 | 0.0482 |

Characteristics | Percentage | Characteristics | Percentage |
---|---|---|---|

Gender: | Average hours per day of using technology: | ||

Boy | 45.60% | Less than one hour per day | 13.20% |

Girl | 54.40% | 1 to 2 h per day | 15.70% |

School grades: | 3 to 4 h per day | 40.70% | |

Grade 1 | 14.20% | More than 4 h per day | 30.70% |

Grade 2 | 16.80% | Physical activities in a week: | |

Grade 3 | 16.60% | None | 44.20% |

Grade 4 | 16.30% | 1 or 2 times per week | 28.40% |

Grade 5 | 17.30% | 3 or 4 times per week | 19.70% |

Grade 6 | 18.80% | More than 4 times per week | 7.70% |

Study at home: | Average sleeping hours in a day: | ||

Less than one hour per day | 21.10% | Less than 7 h per day | 5.80% |

1 to 2 h per day | 29.40% | Between 7 and 8 h per day | 22.20% |

3 to 4 h per day | 33.10% | Between 8 and 9 h per day | 56.30% |

More than 4 h per day | 16.40% | More than 9 h per day | 15.70% |

Characteristics | Father (%) | Mother (%) | Characteristics | Father (%) | Mother (%) |
---|---|---|---|---|---|

Age: | Smoking Habit: | ||||

Less than or equal 30 years old | 18.5% | 21.6% | Smoker | 66.6% | 23.8% |

Between 31and 40 years old | 36.2% | 25.1% | Quitted | 15.7% | 13.7% |

Between 41 and 50 years old | 22.1% | 28.4% | Non-smoker | 17.7% | 62.5% |

More than 50 years old | 23.2% | 24.9% | Physical exercise: | ||

Education: | None | 54.4% | 33.6% | ||

Less than High School | 11.3% | 9.5% | 1 or 2 times in a week | 27.7% | 38.7% |

High school | 19.8% | 6.7% | 3 or 4 times per week | 14.8% | 11.2% |

Diploma | 37.7% | 41.9% | More than 4 times in a week | 3.1% | 16.5% |

Bachelor | 29.1% | 33.1% | Working hours in a day: | ||

Master or PhD | 2.1% | 8.8% | More than 14 hours per day | 26.7% | 8.2% |

Income: | 9–14 hours per day | 62.8% | 73.5% | ||

Less than RMB2000 per month | 11.7% | 20.6% | Less than 9 hours per day | 10.5% | 18.3% |

RMB2001-RMB3000 per month | 22.6% | 24.5% | Average sleeping hours in a day: | ||

RMB3001-RMB4000 per month | 33.9% | 22.1% | Less than 7 hours per day | 55.4% | 61.9% |

RMB4001-RMB5000 per month | 19.9% | 17.3% | Between 7 to 8 hours per day | 27.9% | 30.0% |

More than RMB5000 per month | 11.9% | 15.5% | More than 8 hours per day | 16.7% | 8.1% |

Work experience: | Drinking Alcohol Habit: | ||||

No work experience | 0.00% | 0.00% | Less than one time per month | 3.2% | 10.6% |

Less than 5 years | 7.4% | 19.2% | 1 time per month | 4.5% | 22.7% |

5-10 years | 12.9% | 21.7% | 2 to 3 times per month | 16.1% | 32.1% |

11-15 years | 36.6% | 26.6% | 1 time per week | 16.7% | 28.2% |

16-20 years | 32.8% | 23.6% | 2 to 3 times per week | 39.5% | 6.4% |

More than 20 years | 10.3% | 8.9% | 4 to 6 times per week | 18.7% | 0.00% |

Every day | 1.3% | 0.00% |

Parameter Description | Factor Loading |
---|---|

Parental Socioeconomic | |

Mother’s age | 0.43 |

Father’s age | 0.38 |

Mother’s education | 0.74 |

Father’s education | 0.39 |

Mother’s income | 0.43 |

Father’s income | 0.68 |

Mother’s work experience | 0.06 |

Father’s work experience | 0.05 |

Parents’ marriage length | 0.82 |

Parental Lifestyle | |

Mother’s drinking alcohol | 0.36 |

Father’s drinking alcohol | 0.73 |

Mother’s smoking habit | 0.48 |

Father’s smoking habit | 0.41 |

Mother’s physical exercises | 0.21 |

Father’s physical exercises | 0.09 |

Mother’s working hours | 0.76 |

Father’s working hours | 0.88 |

Mother’s average sleeping hours | 0.83 |

Father’s average sleeping hours | 0.71 |

Household Food Security | |

Worry about running out of food | 0.73 |

Do not have money: household | 0.82 |

Cannot afford to eat balanced meals: household | 0.93 |

Cut down food portions: household | 0.12 |

Do not eat the whole day: adults | 0.98 |

Do not have money: children | 0.04 |

Cannot afford to eat balanced meals: children | 0.25 |

Cannot afford enough food: children | 0.82 |

Skip a meal: children | 0.24 |

Children’s Lifestyle | |

Technology use | 0.92 |

Hours of study at home | 0.73 |

Child’s physical exercise | 0.49 |

Child’s sleep amount | 0.68 |

School grade | 0.46 |

Prior I | Prior II | Prior III | Prior IV | |||||
---|---|---|---|---|---|---|---|---|

Parameter | Estimate | STD | Estimate | STD | Estimate | STD | Estimate | STD |

θ_{1} | 0.561 | 0.021 | 0.555 | 0.033 | 0.549 | 0.069 | 0.584 | 0.121 |

θ_{2} | 0.493 | 0.088 | 0.461 | 0.097 | 0.452 | 0.102 | 0.503 | 0.201 |

θ_{3} | 0.203 | 0.096 | 0.192 | 0.051 | 0.180 | 0.091 | 0.221 | 0.138 |

θ_{13} | 0.739 | 0.108 | 0.721 | 0.101 | 0.598 | 0.027 | 0.751 | 0.102 |

θ_{16} | 0.683 | 0.112 | 0.677 | 0.109 | 0.655 | 0.111 | 0.686 | 0.138 |

θ_{19} | 0.822 | 0.087 | 0.816 | 0.078 | 0.801 | 0.098 | 0.852 | 0.203 |

θ_{22} | 0.733 | 0.039 | 0.730 | 0.035 | 0.722 | 0.069 | 0.763 | 0.093 |

θ_{27} | 0.763 | 0.109 | 0.755 | 0.099 | 0.743 | 0.106 | 0.771 | 0.126 |

θ_{28} | 0.883 | 0.119 | 0.844 | 0.081 | 0.822 | 0.077 | 0.896 | 0.119 |

θ_{29} | 0.827 | 0.044 | 0.814 | 0.041 | 0.759 | 0.036 | 0.834 | 0.66 |

θ_{210} | 0.711 | 0.066 | 0.697 | 0.057 | 0.666 | 0.051 | 0.723 | 0.107 |

θ_{31} | 0.734 | 0.029 | 0.726 | 0.026 | 0.669 | 0.039 | 0.742 | 0.127 |

θ_{32} | 0.822 | 0.071 | 0.816 | 0.064 | 0.798 | 0.061 | 0.831 | 0.104 |

θ_{33} | 0.928 | 0.191 | 0.909 | 0.161 | 0.852 | 0.170 | 0.832 | 0.206 |

θ_{35} | 0.981 | 0.058 | 0.921 | 0.052 | 0.832 | 0.048 | 0.883 | 0.067 |

θ_{38} | 0.816 | 0.161 | 0.799 | 0.152 | 0.764 | 0.143 | 0.802 | 0.188 |

Relation | Estimated Coefficients | |
---|---|---|

ML-SEM | B-SEM | |

Parental socioeconomic → Children’s life style | 0.549 * | 0.561 * |

Household food security → Children’s life style | 0.198 | 0.203 |

Parental lifestyle → Children’s life style | 0.488 * | 0.493 * |

Parental socioeconomic ↔ Parental lifestyle | 0.508 * | 0.513 * |

Parental socioeconomic ↔ Household food security | 0.519 * | 0.521 * |

Household food security ↔ Parental lifestyle | 0.611 * | 0.637 * |

Name of Index | Formula | ML-SEM Value | B-SEM Value |
---|---|---|---|

MAPE | $MAPE=\frac{1}{n}{{\displaystyle \sum}}_{i=1}^{n}\left|\frac{{y}_{i}^{\prime}-{y}_{i}}{{y}_{i}}\right|$ | 0.094 | 0.088 |

RMSE | $RMSE=\sqrt[2]{\frac{{\sum}_{i=1}^{n}{\left({y}_{i}^{\prime}-{y}_{i}\right)}^{2}}{n}}$ | 0.091 | 0.051 |

MSE | $MSE=\frac{{\sum}_{i=1}^{n}\left|{y}_{i}^{\prime}-{y}_{i}\right|}{n}$ | 0.128 | 0.105 |

R^{2} | ${R}^{2}=\frac{{[{\sum}_{i=1}^{n}\left({y}_{i}^{\prime}-{\overline{y}}_{i}^{\prime}\right).\left({y}_{i}-{\overline{y}}_{i}\right)]}^{2}}{{\sum}_{i=1}^{n}\left({y}_{i}^{\prime}-{\overline{y}}_{i}^{\prime}\right).{\sum}_{i=1}^{n}\left({y}_{i}-{\overline{y}}_{i}\right)}$ | 0.601 | 0.761 |

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**MDPI and ACS Style**

Radzi, C.W.J.b.W.M.; Hui, H.; Salarzadeh Jenatabadi, H.
Comparing Bayesian and Maximum Likelihood Predictors in Structural Equation Modeling of Children’s Lifestyle Index. *Symmetry* **2016**, *8*, 141.
https://doi.org/10.3390/sym8120141

**AMA Style**

Radzi CWJbWM, Hui H, Salarzadeh Jenatabadi H.
Comparing Bayesian and Maximum Likelihood Predictors in Structural Equation Modeling of Children’s Lifestyle Index. *Symmetry*. 2016; 8(12):141.
https://doi.org/10.3390/sym8120141

**Chicago/Turabian Style**

Radzi, Che Wan Jasimah bt Wan Mohamed, Huang Hui, and Hashem Salarzadeh Jenatabadi.
2016. "Comparing Bayesian and Maximum Likelihood Predictors in Structural Equation Modeling of Children’s Lifestyle Index" *Symmetry* 8, no. 12: 141.
https://doi.org/10.3390/sym8120141