Error distributions assumptions in random regret choice models: towards error Frechit specifications

Figures & data

Figure 1. Location of Noord Brabant Province in the Netherlands. Source: https://nl.wikipedia.org/wiki/Noord-Brabant.

Table 1. Estimation results – first case study.

	Utility				Regret (RRsum)
	Logit	Logit + EC	Weibit	Weibit + EC	Logit	Logit + EC	Frechit	Frechit + EC
Travel Time	−0.0572	−0.0457	−0.0407	−0.0310	−0.0614	−0.0427	−0.0433	−0.0296
(t-value)	(−14.27)	(−10.93)	(−15.36)	(−12.11)	(−14.10)	(−11.77)	(−14.85)	(−12.23)
Travel Distance	−0.114	−0.105	−0.102	−0.0924	−0.325	−0.159	−0.305	−0.167
(t-value)	(−15.91)	(−11.26)	(−15.79)	(−10.20)	(−12.56)	(−10.14)	(−12.40)	(−11.77)
EC_CarBike_Mean		0.0131		0.0126		0.870		0.837
(t-value)		(3.06)		(2.31)		(3.31)		(3.02)
EC_CarBike_Std.^a		0.0275		0.0314		0.0552		0.0531
(t-value)		(1.44)		(1.26)		(2.12)		(1.51)
EC_CarWalk_Mean		0.00		0.00		0.147		0.158
(t-value)		(fixed)		(fixed)		(1.32)		(1.65)
EC_CarWalk_Std.		0.00		0.00		0.0303		0.0266
(t-value)		(fixed)		(fixed)		(3.29)		(4.24)
EC_BikeWalk_Mean		0.0309		0.0301		1.490		1.463
(t-value)		(3.20)		(4.15)		(5.23)		(4.77)
EC_ BikeWalk_ Std.		0.0314		0.0297		0.0134		0.0154
(t-value)		(1.52)		(1.07)		(1.93)		(2.89)
Final log-likelihood	−991.977	−986.825	−983.715	−979.528	−1012.287	−973.283	−985.401	−951.488
Rho-squared	0.220	0.224	0.227	0.230	0.204	0.235	0.225	0.252
Adj. Rho-squared	0.219	0.220	0.225	0.225	0.203	0.229	0.224	0.246
BIC	1998.063	2015.977	1981.539	2001.383	2038.683	2003.002	1984.911	1959.412

^aStandard deviation.

Table 2. Market shares – first case study.

		Prediction (%)
		Utility				Regret (RRsum)
	Observation (%)	Logit	Logit + EC	Weibit	Weibit + EC	Logit	Logit + EC	Frechit	Frechit + EC
Car	40.59	41.55	42.65	39.29	37.87	41.76	42.37	39.04	39.45
Bike	32.64	37.83	37.05	28.13	29.36	38.08	36.63	28.35	29.11
Walk	26.77	20.62	20.30	32.58	32.77	20.16	21.00	32.61	31.44
RMSE	–	8.10	8.10	7.47	7.36	8.64	7.24	7.41	5.96

Table 3. Validation – first case study.

	Utility				Regret (RRsum)
Average of 6 validations	Logit	Logit + EC	Weibit	Weibit + EC	Logit	Logit + EC	Frechit	Frechit + EC
Adj. Rho-square	0.162	0.162	0.167	0.168	0.149	0.172	0.168	0.187
BIC	2141.435	2161.395	2128.663	2146.429	2174.893	2147.008	2127.935	2108.466

Figure 2. Location of Beijing in China. Source: https://en.wikipedia.org/wiki/File:China-Beijing.png.

Table 4. Attribute level^a – second case study.

		E-Bike	Car	Bus
Travel Time	Main Time	(50, 70, 90)	(20, 30, 40)	(20, 35, 50)
	Possible Delay	(0: fixed)	(0, 30, 60)	(0, 30, 60)
	Access Time	(0: fixed)	(0: fixed)	(5, 10, 15)
	Egress Time	(0: fixed)	(5: fixed)	(5, 10, 15)
	Parking Search Time	(0: fixed)	(5: fixed)	(0: fixed)
	Transfer Time	(0: fixed)	(0: fixed)	(0, 10, 20)
Travel Cost	Main Cost	(1, 2, 3)	(10, 20, 30)	(2, 6, 10)
	Parking Cost	(0: fixed)	(0, 15, 30)	(0: fixed)

^aUnit: Time (minute) / Cost (Yuan).

Table 5. Estimation results – second case study.

	Utility		Regret (RRsum)
	Logit	Weibit	Logit	Logit + EC	Frechit	Frechit + EC
Travel Time	−0.0318	−0.0243	−0.0152	−0.0104	−0.0115	−0.0077
(t-value)	(−28.43)	(−25.91)	(−23.07)	(−20.52)	(−25.12)	(−19.72)
Travel Cost	−0.0356	−0.0330	−0.0222	−0.0129	−0.0196	−0.0108
(t-value)	(−21.59)	(−24.16)	(−26.90)	(−18.79)	(−22.79)	(−15.15)
EC_EBikeCar_Mean				0.00		0.00
(t-value)				(fixed)		(fixed)
EC_ EBikeCar _Std.				0.00		0.00
(t-value)				(fixed)		(fixed)
EC_CarBus_Mean				0.228		0.261
(t-value)				(1.93)		(1.67)
EC_ CarBus _Std.				0.110		0.103
(t-value)				(4.15)		(4.41)
EC_EBikeBus_Mean				0.0153		0.0145
(t-value)				(2.61)		(2.18)
EC_ EBikeBus_ Std.				0.0242		0.0204
(t-value)				(1.26)		(1.05)
Final log-likelihood	−2237.516	−2219.928	−2253.941	−2221.913	−2234.109	−2189.923
Rho-squared	0.238	0.244	0.232	0.243	0.239	0.254
Adj. Rho-squared	0.237	0.243	0.232	0.241	0.239	0.252
BIC	4490.814	4455.638	4523.664	4491.172	4484.000	4427.192

Table 6. Market shares – second case study.

		Prediction (%)
		Utility		Regret (RRsum)
	Observation (%)	Logit	Weibit	Logit	Logit + EC	Frechit	Frechit + EC
E-bike	22.67	26.56	24.11	39.84	34.15	35.31	30.06
Car	27.42	38.42	36.03	16.20	20.13	19.27	22.60
Bus	49.91	35.01	39.86	43.96	45.72	45.42	47.34
RMSE	–	18.92	13.31	21.36	14.23	15.70	9.19

Table 7. Validation – second case study.

	Utility		Regret (RRsum)
Average of 9 validations	Logit	Weibit	Logit	Logit + EC	Frechit	Frechit + EC
Adj. Rho-square	0.186	0.191	0.172	0.183	0.179	0.195
BIC	4792.795	4754.265	4875.153	4830.199	4833.643	4760.039

Table A1. First cast study.

	RRmax				RRlog
	Logit	Logit + EC	Frechit	Frechit + EC	Logit	Logit + EC	Frechit	Frechit + EC
Final log-likelihood	−1015.838	−1000.751	−995.976	−981.294	−1034.352	−985.305	−1013.502	−970.229
Adj. Rho-squared	0.200	0.209	0.215	0.224	0.185	0.221	0.202	0.233
BIC	2045.785	2043.829	2006.061	2004.915	2082.813	2012.937	2041.113	1982.785

Table A2. Second cast study.

	RRmax				RRlog
	Logit	Logit + EC	Frechit	Frechit + EC	Logit	Logit + EC	Frechit	Frechit + EC
Final log-likelihood	−2245.027	−2213.615	−2239.913	−2209.973	−2273.499	−2197.551	−2265.052	−2187.772
Adj. Rho-squared	0.235	0.244	0.237	0.245	0.225	0.250	0.229	0.253
BIC	4505.836	4474.576	4495.608	4467.292	4562.780	4442.448	4545.886	4422.890

Table A3. First cast study.

	RRmax				RRlog
	Logit	Logit + EC	Frechit	Frechit + EC	Logit	Logit + EC	Frechit	Frechit + EC
RMSE	8.70	8.12	7.88	7.52	9.15	7.79	8.74	7.29

Table A4. Second cast study.

	RRmax				RRlog
	Logit	Logit + EC	Frechit	Frechit + EC	Logit	Logit + EC	Frechit	Frechit + EC
RMSE	21.25	14.50	19.36	14.11	23.39	11.77	22.05	9.43

Table A5. First cast study.

	RRmax				RRlog
Average of 5 validations	Logit	Logit + EC	Weibit	Weibit + EC	Logit	Logit + EC	Frechit	Frechit + EC
Adj. Rho-square	0.145	0.154	0.162	0.168	0.129	0.160	0.144	0.173
BIC	2184.477	2183.713	2142.159	2146.193	2226.689	2176.784	2187.399	2144.564

Table A6. Second cast study.

	RRmax				RRlog
Average of 9 validations	Logit	Logit + EC	Weibit	Weibit + EC	Logit	Logit + EC	Frechit	Frechit + EC
Adj. Rho-square	0.175	0.187	0.178	0.189	0.163	0.192	0.167	0.195
BIC	4856.223	4806.179	4837.319	4790.895	4925.457	4774.673	4903.971	4757.275