# Accounting for Attribute Non-Attendance and Common-Metric Aggregation in the Choice of Seat Belt Use, a Latent Class Model with Preference Heterogeneity

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## Abstract

**:**

## 1. Introduction

## 2. Case Study and Data

## 3. Method

- The first process would be related to a mixed part of the model:
- $\omega $ estimation: assume R = 10 for 3 variables and 100 observations: there would be a matrix with 10 columns and 3 rows, and 100 values: the values would be filled by $\sum}_{i=1}^{n=3}\left(bet{a}_{i}+{\sigma}_{i}\times {\omega}_{i}\right)$, where $\omega $ would be estimated based on pseudo-random numbers or Halton sequences, and ${\sigma}_{i}$, or the SD of random parameters are values that would be estimated by maximum likelihood, and their initial values would be set by the investigator. Additionally, the initial value of beta would be set by investigator $\to \left({\beta}_{q}+{\omega}_{i}\right)$
- The multiplication of the above value by the vectors of observed coefficients would be saved as XR $\to \left({\beta}_{q}+{\omega}_{i}\right){x}_{ij}$
- The resultant would be multiplied by response$\to {\displaystyle \sum}_{j=1}^{J}{y}_{i,j}\left({\beta}_{q}+{\omega}_{i}\right){x}_{ij}]$
- The exponential of the above values in c would be calculated and would be summed up across the number of J or classes $\to \mathrm{exp}[{\displaystyle \sum}_{j=1}^{J}{y}_{i,j}\left({\beta}_{q}+{\omega}_{i}\right){x}_{ij}]$
- To have a probability based on the Multinomial logit model, the value in d would be divided by value in c. $\to \frac{\mathrm{exp}[{{\displaystyle \sum}}_{j=1}^{J}{y}_{i,j}\left({\beta}_{q}+{\omega}_{i}\right){x}_{ij}]}{{{\displaystyle \sum}}_{j=1}^{J}\mathrm{exp}[({\beta}_{q}+{\omega}_{i}){x}_{ij}]}$
- There are 10 observations (draws), along with Q columns, related to classes. Thus, the means of each class would be estimated by reducing the dimension of the random draws (R): the average of all the draws over each observation.
- Up to the above steps are related to $\to \frac{1}{R}{\displaystyle \sum}_{r=1}^{R}\frac{\mathrm{exp}[{{\displaystyle \sum}}_{j=1}^{J}{y}_{i,j}\left({\beta}_{q}+{\omega}_{i}\right){x}_{ij}]}{{{\displaystyle \sum}}_{j=1}^{J}\mathrm{exp}[({\beta}_{q}+{\omega}_{i}){x}_{ij}]}$.

- For latent class parts, the steps would be taken as follows:
- Create a vector of $\gamma $: this constitutes the initial value of a constant or the heterogeneity point related to a covariate T, which the class allocation is based on. For the first class the value would be set as 0 based on the literature review for model identification.
- Getting the exponential of T, times $\gamma $, which discussed in the above,
- The above would be transformed into a probability by dividing the values by the sum of all the components or classes.
- The above are related to the ${\pi}_{n,s}$ which is equal to $\frac{{e}^{{\beta}_{n|s}{x}_{nj|}}}{{{\displaystyle \sum}}_{k}{e}^{{\beta}_{n|s}{x}_{nj|}}}$ part of the above equation. It should be noted that ${\pi}_{n,s}$ acts as a constraint with $\sum}_{s}{\pi}_{n,s}=1$, so the sum of the probability of each observation across classes would be added up to 1.

- Now the resultants of item 1 and 2 would be multiplied. This would be the resultant of Equation (6).
- In order to transform the probability of Equation (6) into a likelihood, the sum of the probability of the individuals would be calculated and set as the likelihood.
- The log of 4 would be set as log likelihood and would be estimated by maximum likelihood.
- Now to come up with ANA and ACMA, we put constraints on the means and the SDs of the parameters. For coming up with ANA, we impose restrictions on some attributes’ means and for ACMA, we constrain two or more parameter means to be equal.
- Maximum likelihood would be estimated by the finite-difference method and with the help of Hessian and Gradient.

## 4. Results

## 5. Conclusions

## 6. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Attributes | Mean | Variance | Min | Max |
---|---|---|---|---|

Front-seat passenger belt status, belted versus not belted * | 0.32 | 0.217 | 0 | 1 |

Driver belt condition, not belted versus belted * | 0.140 | 0.120 | 0 | 1 |

Sunny weather condition, sunny as 1, versus others * | 0.682 | 0.216 | 0 | 1 |

Number of lanes, 2 lane as versus a single lane * | 1.415 | 0.243 | 1 | 2 |

Vehicle license, Non-Wyoming residence versus others * | 1.552 | 0.247 | 1 | 2 |

Time of observation: 11:30–1:30 versus others * | 0.229 | 0.176 | 0 | 1 |

Time of observation: 1:30–3:30 versus others * | 0.176 | 0.145 | 0 | 1 |

Latent class, no ANA, no ACMA (A) | ||

Attributes | FAA 1 | FAA 1 |

Weather: Sunny | −0.22 (−0.703) | 0.19 (0.846) |

Day of a week | 0.21 (0.679) | 0.63 (2.14) |

Number of lanes | 0.47 (1.033) | 0.85 (1.43) |

Driver belt status | −0.51 (−1.501) | 0.13 (0.49) |

Vehicle license plate registration | −0.75 (−2.826) | −0.46 (−2.54) |

Time of a day (11:30–1:30) | 1.60 (1.408) | −2.66 (−0.88) |

Time of a day (1:30–3:30) | −1.41 (−0.658) | 0.28 (0.526) |

Class probability, $\gamma $ | −0.16 (−0.134) | |

Classes share | 54% | 46% |

Log-likelihood | −3964 | |

AIC | 7959 | |

Mixed-mixed, no ANA, no ACMA (B) | ||

Attributes | FAA 1 | FAA 2 |

Weather: Sunny | 84.79 (1.422) | 0.06 (0.301) |

Day of a week | −67.71 (−1.33) | −0.85 (−8.168) |

Number of lanes | −106.19 (−1.36) | 0.44 (2.844) |

Driver belt status | −27.23 (−1.33) | 0.33 (2.542) |

Vehicle license plate registration | 42.60 (1.43) | 0.51 (3.534) |

Time of a day (11:30–1:30) | −31.37 (1.35) | 0.29 (2.587) |

Time of a day (1:30–3:30) | 133.77 (0.301) | −15.46 (−0.0089) |

SD. Driver belt status | 91.60 (1.30) | - |

SD. Vehicle license plate registration | 105.1 (1.37) | - |

SD. Time of a day (1:30–3:30) | 32.2 (1.39) | - |

Class probability, $\gamma $ | −0.089 (−1.050) | |

Classes share | 52% | 48% |

Log-likelihood | −3958.7 | |

AIC | 7954 | |

Latent class, ANA, ACMA (C) | ||

Attributes | ANA + ACMA | ANA |

Weather: Sunny | --- | 0.033 (4.666) |

Day of a week | --- | 0.366 (4.666) |

Number of lanes | −5.483 (−0.429) | −0.103 (−2.238) |

Driver belt status | −4.181 (−0.112) | --- |

Vehicle license plate registration | −1.239 (−0.461) | --- |

Time of a day (11:30–1:30) | 5.202 (0.414) | −0.298 (−7.104) |

Time of a day (1:30–3:30) | 5.202 (0.408) | --- |

Class probability, $\gamma $ | 2.186 (3.871) | |

Classes share | 10% | 90% |

Log-likelihood | −3969 | |

AIC | 7958 | |

Mixed-mixed, ANA, ACMA (with similar signs) (D) | ||

Attributes | ANA | ANA + ACMA |

Weather: Sunny | --- | 9.549 (0.140) |

Day of a week | --- | 46.419 (0.033) |

Number of lanes | −0.133 (−2.789) | −22.760 (−0.032) |

Driver belt status | 0.479 (4.516) | --- |

Vehicle license plate registration | −0.566 (−9.547) | --- |

Time of a day (11:30–1:30) | −0.271 (−3.647) | −23.192 (−0.033) |

Time of a day (1:30–3:30) | −0.271 (−2.85) | --- |

SD. Driver belt status | 0.655 (0.817) | |

SD. Vehicle license plate registration | 0.130 (0.620) | |

SD. Time of a day (1:30–3:30) | 0.933 (2.058) | |

Class probability, $\gamma $ | −3.542 (−6.791) | |

Classes share | 3% | 97% |

Log-likelihood | −3966 | |

AIC | 7958 | |

Mixed-mixed, ANA, ACMA (with reverse signs) (E) | ||

Attributes | ANA + ACMA | ANA |

Weather: Sunny | --- | 0.101 (0.58) |

Day of a week | --- | 0.705 (2.57) |

Number of lanes | −0.494 (−2.44) | 0.345 (1.24) |

Driver belt status | 0.703 (2.35) | --- |

Vehicle license plate registration | −0.889 (−3.11) | --- |

Time of a day (11:30–1:30) | 1.442 (1.40) | −6.742 (−0.42) |

Time of a day (1:30–3:30) | −1.442 (−1.42) | --- |

SD. Driver belt status | 1.079 (1.42) | |

SD. Vehicle license plate registration | 0.518 (1.37) | |

SD. Time of a day (1:30–3:30) | 2.586 (2.04) | |

Class probability, $\gamma $ | −0.598 (−0.83) | |

Classes share | 65% | 35% |

Log-likelihood | −3962 | |

AIC | 7950 |

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**MDPI and ACS Style**

Rezapour, M.; Ksaibati, K. Accounting for Attribute Non-Attendance and Common-Metric Aggregation in the Choice of Seat Belt Use, a Latent Class Model with Preference Heterogeneity. *Algorithms* **2021**, *14*, 84.
https://doi.org/10.3390/a14030084

**AMA Style**

Rezapour M, Ksaibati K. Accounting for Attribute Non-Attendance and Common-Metric Aggregation in the Choice of Seat Belt Use, a Latent Class Model with Preference Heterogeneity. *Algorithms*. 2021; 14(3):84.
https://doi.org/10.3390/a14030084

**Chicago/Turabian Style**

Rezapour, Mahdi, and Khaled Ksaibati. 2021. "Accounting for Attribute Non-Attendance and Common-Metric Aggregation in the Choice of Seat Belt Use, a Latent Class Model with Preference Heterogeneity" *Algorithms* 14, no. 3: 84.
https://doi.org/10.3390/a14030084