# Using the Multilevel Random Effect Model to Analyze the Behavior of Carpool Users in Different Cities

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Model Methodology

_{0j}is the intercept, β

_{1j}is the regression coefficient associated with predictor β

_{ij}, and the slope coefficient β

_{1j}is assumed to vary across districts depending on their effects at the city level. γ

_{ij}is the residual accounting for the level 1 random effects. The formulation is similar to a traditional regression model; however, there is an important difference, in that both the intercept and regression coefficients have subscript j, indicating that intercept β

_{0j}and slope coefficient β

_{1j}are permitted to vary across the level 2 administrative area.

_{ij}denotes the possibility of carpool users, and P

_{ij}= (0,1). To be more precise, to predict the possibility of carpool users, considering a binomial Y

_{ij}= (0,1) outcome, and P

_{ij}= [exp(Y

_{ij})/(1+exp(Y

_{ij}))], substitute this with Formulas (4) and (5), as shown below:

_{ij}is the logit prediction for the $i\mathrm{th}$ subject at level 1 and the jth unit at level 2, γ

_{00}is the intercept denoting the grand mean, W

_{j}, the regional (city) level characteristic, X

_{ij}the individual (carpool user) level characteristic, and γ

_{q}

_{0}is the regression coefficient associated with the regional level characteristic and individual level characteristic. In other words, the fixed effects determined by the regression coefficients of γ

_{qs}, which are associated with the slope variance for each variable at the individual level, are explained by a variable at the city level. μ

_{qj}is a random effect accounting for the random variation at level 2, where μ

_{j}~(0, τ

_{00}), and γ

_{ij}is the individual-level random effect, where γ

_{ij}~(0, σ

^{2}).

^{2}is the within-group variance in level 1 and ${\sigma}_{{u}_{0}}^{2}$ is the between-group variance. In the applied logistic model, the level 1 residuals are assumed to follow the standard logistic distribution, which has a mean of 0 and a variance of σ

^{2}= π/3 = 3.29. If the ICC is sufficiently close to zero, then there is effectively no variation in the violation carpool user between the level 2 regions, suggesting that the standard subject level models are adequate for such data. However, a significant value of ρ implies that the multilevel random effect logistic model should be used.

## 4. Data

#### 4.1. Data Description

#### 4.2. Data Analysis

## 5. Model Estimation Results

^{2}were 0.328 and 0.370 at two models calculated as a standard measure of the model fit. Next, it can be seen that, the AIC value of the multilevel logistic model is 1194.909, much smaller than that of the single-level model. This means that models including the city feature as a higher-level predictor via multilevel logistic regression have better goodness of fit than the standard logistic model, which set city features as an individual-level parameter. At last, the random-effect logistic regression model was chosen as the final model.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The DUO driver/passenger experience. The Metropia driver will be paired and identified as a driver by comparing the us-er’s GPS trajectories in real-time. The other passengers are also paired and identified as a DUO passenger, as shown in (

**a**). When a DUO passenger arrives at the destination (if the trip is valid), the DUO passenger will earn a base trekpoint incentive and a chance to win additional points by spinning a “prize wheel” in the app, as shown in (

**b**). In other words, the pairings of DUO passengers and drivers are performed automatically. Once the pairing is complete for a specific DUO passenger, the driver app will give an audio cue to the driver that the pairing with a specific DUO passenger has been successful. Finally, the passenger is given an additional bonus of half from the prize wheel by each driver receives, and the driver gets an email record, as shown in (

**c**).

Variables | No Carpool Experience | More Than One-Time Carpool Experience |
---|---|---|

The average number of total trips in one month | 42 | 59 |

The average of travel speed (mile/h) | 30.7 | 28.6 |

The percentage of peak hour trips (6 a.m.–10 a.m.) | 33.3 | 31.9 |

The average frequency of using APP per day | 1.4 | 2.0 |

The average number of carpooling as the driver | 1.7 | 9.9 |

The average frequency of using APP per day | 2.1 | 13.6 |

The average points during every trip for being a carpool driver | 12.0 | 37.1 |

The average points during every trip for being a carpool passenger | 21.7 | 68.0 |

Variables | No Carpool Experience | More than One-Time Carpool Experience | Total | |||
---|---|---|---|---|---|---|

Number | % | Number | % | Number | % | |

gender | ||||||

Male | 190 | 48.3 | 928 | 59.9 | 1118 | 57.5 |

Female | 195 | 49.6 | 584 | 37.7 | 779 | 40.1 |

Other | 8 | 2.0 | 38 | 2.5 | 46 | 2.4 |

Age | ||||||

18–25 | 24 | 6.1 | 144 | 9.3 | 168 | 8.6 |

26–34 | 72 | 18.3 | 434 | 28.0 | 506 | 26.0 |

35–45 | 148 | 37.7 | 570 | 36.8 | 718 | 37.0 |

46–55 | 75 | 19.1 | 264 | 17.0 | 339 | 17.4 |

56–65 | 74 | 18.8 | 138 | 8.9 | 212 | 10.9 |

Living situation | ||||||

Own house | 307 | 78.1 | 1,1250 | 72.6 | 1432 | 73.7 |

Rental house | 73 | 18.5 | 319 | 20.6 | 392 | 20.1 |

Live with friends/family | 13 | 3.3 | 106 | 6.8 | 119 | 6.1 |

How much flexibility to change the departure time | ||||||

No flexibility | 203 | 51.7 | 706 | 45.5 | 909 | 46.8 |

1–15 min | 190 | 48.3 | 844 | 54.5 | 1034 | 53.2 |

How long to find a parking | ||||||

Less than 5 min | 376 | 95.7 | 1416 | 91.4 | 1792 | 92.2 |

5–15 min | 16 | 4.1 | 120 | 7.7 | 136 | 7.0 |

Over 15 min | 1 | 0.3 | 14 | 0.9 | 15 | 0.8 |

Variable | 2016 Austin | 2017 Austin | 2016 El Paso | 2017 El Paso | 2016 Tucson | 2017 Tucson |
---|---|---|---|---|---|---|

GOVTPT | 4875 | 4670 | 580 | 461 | 1186 | 1258 |

LTVEH | 161,222 | 164,358 | 69,242 | 69,033 | 82,880 | 82,389 |

GTVEH | 109,150 | 113,315 | 102,865 | 105,427 | 54,636 | 56,966 |

DRALONE | 199,626 | 204,468 | 126,921 | 128,741 | 88,348 | 90,617 |

CHCPLHH | 43,636 | 42,179 | 86,020 | 85,287 | 91,469 | 94,680 |

CHSGLHH | 38,405 | 39,268 | 52,735 | 52,377 | 46,762 | 44,247 |

SALAINCO | 36,782 | 38,867 | 26,676 | 27,380 | 25,275 | 26,224 |

MEDAGE | 36.9 | 36.9 | 40.0 | 39.9 | 38.5 | 37.4 |

Variable | Description and Unit | Mean | S.E. | Min. | Max. |
---|---|---|---|---|---|

Dependent variable | |||||

REPUSER | Carpool user = 1; Non-carpool user = 0 | ||||

Area level characteristics (Level 2) | |||||

lnGOVTPT | ln (The total worker population of government workers’ principal mode to get from home to work is public transportation in each city) | 7.28 | 0.85 | 6.13 | 8.50 |

lnLTVEH | ln (The total worker population that the number of available vehicles at home and household members’ use is less than one vehicle in each city) | 11.45 | 0.33 | 11.14 | 12.01 |

lnDRALONE | ln (The total worker population of the commuter worker is male, and the primary travel type of working is driving alone in each city) | 11.09 | 0.35 | 10.73 | 11.62 |

lnCHSGLHH | ln (The total worker population that single-parent household that has children between the age of three and 17 who are enrolled in school in each city) | 10.72 | 0.11 | 10.56 | 10.87 |

Individual level characteristics (Level 1) | |||||

lnTotalTrip | ln (The number of total trips per month) | 3.58 | 1.09 | 0 | 5.67 |

Morning | The percentage of users starting a trip in the morning (6 a.m.–10 a.m.) during the observation month. | 0.32 | 0.18 | 0 | 1 |

DuoTimes | The number of times the user takes a trip as either a carpool passenger or driver during the observation month. | 19.51 | 29.88 | 0 | 264 |

lnTravelTime | ln (The average travel time of the user during the observation month) | 2.42 | 0.42 | 0.90 | 3.74 |

lnTravelSpeed | ln (The average travel speed of the user during the observation month) | 3.29 | 0.43 | 0.03 | 4.24 |

lnFreq | ln (The average frequency of using the Metropia app per day) | 0.17 | 1.09 | −3.43 | 2.27 |

lnDuodreward | ln (The average earned points per trip as a DUO driver during the observation month) | 0.03 | 5.87 | −9.21 | 5.16 |

DuoPct | The percentage of average earned points as either a DUO driver or passenger during the observation month. | 0.34 | 0.32 | 0 | 1 |

Male | Male = 1; others = 0 | 0.58 | 0.49 | 0 | 1 |

Age | Between 26 and 45 years old = 1; others = 0 | 0.46 | 0.50 | 0 | 1 |

Live | Living with friends/family = 1; others = 0 | 0.06 | 0.24 | 0 | 1 |

Flexibility | Do you have flexibility to change your departure time? 1 = Yes; 0 = No | 0.53 | 0.50 | 0 | 1 |

Parking | How long does it typically take you to find a parking? 1 = More than 5 min; 0 = Less than 5 min. | 0.08 | 0.27 | 0 | 1 |

Variables | Logistic Model | Multilevel Logistic Random-Effect Model | |||||
---|---|---|---|---|---|---|---|

Coef. | p-Value | S.E. | Coef. | p-Value | S.E. | ||

Aggregate level | lnGOVTPT | 0.275 | 0.076 * | 0.155 | 0.280 | 0.075 * | 0.158 |

lnLTVEH | 2.624 | 0.053 * | 1.356 | 2.627 | 0.057 * | 1.381 | |

lnDRALONE | −1.045 | 0.074 * | 0.584 | −0.990 | 0.096 * | 0.595 | |

lnCHSGLHH | 8.926 | 0.003 *** | 3.006 | 8.879 | 0.004 *** | 3.058 | |

Individual level | lnTotalTrip | 4.685 | 0.066 * | 2.548 | 5.209 | 0.046 ** | 2.614 |

Morning | 0.815 | 0.030 ** | 0.375 | 0.666 | 0.089 * | 0.391 | |

DuoTimes | 0.015 | 0.059 * | 0.008 | 0.023 | 0.009 ** | 0.009 | |

lnTravelTime | −0.402 | 0.023 ** | 0.177 | −0.500 | 0.006 ** | 0.182 | |

lnTravelSpeed | 0.952 | 0.000 *** | 0.237 | 1.439 | 0.000 *** | 0.257 | |

lnFreq | −4.674 | 0.067 * | 2.550 | −5.166 | 0.048 ** | 2.616 | |

lnDuodreward | 0.136 | 0.000 *** | 0.014 | 0.119 | 0.000 *** | 0.014 | |

DuoPct | 2.554 | 0.000 *** | 0.538 | 3.486 | 0.000 *** | 0.601 | |

Male | 0.219 | 0.135 | 0.146 | 0.320 | 0.033 ** | 0.150 | |

Age | 0.330 | 0.024 ** | 0.146 | 0.291 | 0.054 * | 0.151 | |

Live | 0.965 | 0.011 ** | 0.379 | 1.326 | 0.001 *** | 0.388 | |

Flexibility | 0.243 | 0.091 * | 0.144 | 0.299 | 0.053 * | 0.155 | |

Parking | 1.449 | 0.000 *** | 0.315 | 1.487 | 0.000 *** | 0.322 | |

Intercept | −13.424 | 0.001 *** | 4.200 | −13.810 | 0.001 *** | 4.280 | |

sigma_u | - | 0.634 | 0.027 ** | 0.276 | |||

rho | - | 0.109 | 0.002 ** | 0.045 | |||

N | 1943 | 1943 | |||||

AIC | 1492.879 | 1194.909 | |||||

BIC | 1693.204 | 1300.777 | |||||

Log-likelihood | −658.564 | −679.234 | |||||

McFadden’s pseudo-R^{2} | 0.328 | 0.370 |

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**MDPI and ACS Style**

Chen, T.-Y.; Jou, R.-C.; Chiu, Y.-C.
Using the Multilevel Random Effect Model to Analyze the Behavior of Carpool Users in Different Cities. *Sustainability* **2021**, *13*, 937.
https://doi.org/10.3390/su13020937

**AMA Style**

Chen T-Y, Jou R-C, Chiu Y-C.
Using the Multilevel Random Effect Model to Analyze the Behavior of Carpool Users in Different Cities. *Sustainability*. 2021; 13(2):937.
https://doi.org/10.3390/su13020937

**Chicago/Turabian Style**

Chen, Tzu-Ying, Rong-Chang Jou, and Yi-Chang Chiu.
2021. "Using the Multilevel Random Effect Model to Analyze the Behavior of Carpool Users in Different Cities" *Sustainability* 13, no. 2: 937.
https://doi.org/10.3390/su13020937