Table 4.2 presents the results from the four models. Model 1 is an RPL model with no consideration given for ANA. All parameter estimates are highly significant and of expected sign.^{[1]} Model 1 represents a clear improvement over an MNL model, which has a log likelihood of 4832.78.
The stated nonattendance rate for free flow time is low but not trivial, at 12.05 per cent. The rate for uncertainty is higher at 36.52 per cent. What is notable about Model 2 however is the large decrease in model fit, from a log likelihood of 4211.83 to 4241.81. Whilst the precise reason for this deterioration is not certain, it may be due to reporting error in the stated nonattendance responses. Any improvement in model fit that may be expected from incorporating correctly revealed stated non attendance responses may be overwhelmed by a decrease in model fit stemming from incorrect responses. For example, Collins et al. (2013) found evidence of reporting error. When they included the stated nonattendance response as a covariate in the nonattendance component of the model, the inference for one of the attributes was that only 52 per cent of those who stated that they ignored the attribute actually did so. Applying this technique in the present context,^{[2]} we infer that 15.37 per cent of stated attenders to free flow time do not attend, while fully 72.61 per cent of stated nonattenders do in fact attend. For uncertainty, 52.66 per cent of stated attenders do not attend, while only 9.55 per cent of stated nonattenders do in fact attend. Thus for free flow time, stated attendance responses appear to be more accurate than nonattendance responses, and for uncertainty these findings are reversed. It must be noted that as with stated responses, the RPANA model cannot be assumed to be free from error. Nonetheless, the discrepancy in findings further calls into question the accuracy of stated responses, and motivates the present research.
The nonattendance rates inferred in Model 3 are consistently higher than the rates retrieved from the stated responses, at 15.99 and 59.84 per cent for free flow time and uncertainty, respectively. Since the RPL model nests within the RPANA model, a likelihood ratio test can be performed. The null hypothesis that Model 3 is equivalent to the pure RPL specification of Model 1 can be rejected, as with two degrees of freedom, the test statistic of 18.92 exceeds the chisquared critical value of 5.99 at the 95 per cent confidence level. Clearly, Model 3 is also a statistically significant improvement in model fit over Model 2.
Model 4, which introduces seven additional parameters for the nonattendance covariates, represents a further improvement in model fit over Model 3 (29.41; ). After accounting for the influences of varying information load, the mean ANA rates are slightly lower than in Model 3. For each of the covariates, the associated base level under the effects coding is reported, as well as the estimated
Table 4.2: Model results.
Taste parameters
Model 1
Model 2
Model 3
Model 4
RPL
RPL stated ANA
RPANA
RPANA with co variates
Parameter
tratio
Parameter
tratio
Parameter
tratio
Parameter
tratio
Total time
μ
1.499
10.30
1.520
9.68
1.497
9.57
1.490
9.46
σ
1.332
8.67
1.327
8.46
1.325
8.39
1.331
8.41
Free flow time
μ
1.851
15.34
1.836
14.01
1.469
8.81
1.310
8.47
σ
1.409
14.20
1.455
13.04
1.178
10.36
1.095
10.19
Stop/start time
μ
2.372
8.97
2.428
8.47
2.331
8.72
2.318
8.22
σ
1.590
7.21
1.632
6.98
1.650
7.47
1.639
6.84
Slowed down time
μ
2.284
12.85
2.348
12.38
2.289
11.34
2.321
11.09
σ
1.038
6.76
1.043
6.92
1.084
6.37
1.079
6.21
Congestion time
μ
1.726
12.12
1.708
12.63
1.729
11.40
1.733
10.93
σ
1.319
10.74
1.321
11.52
1.346
10.85
1.363
9.82
Uncertainty
μ
4.015
14.89
3.696
14.28
2.197
6.85
2.262
7.06
σ
1.685
9.17
1.817
8.52
0.807
4.07
0.835
4.31
Total cost
μ
1.222
36.49
1.201
35.55
1.252
37.20
1.267
35.39
Running cost
μ
1.094
10.05
1.058
9.39
1.102
9.98
1.083
9.53
Toll cost
μ
1.999
53.71
1.942
54.03
2.008
78.41
1.998
74.90
Attribute nonattendance
Parameter
iratio3
Parameter
iratio3
Free flow time
γ
—
—
—
—
1.659
9.17
2.057
6.72
ANA rate
—
—
12.05%
—
15.99%
—
11.34%
—
Uncertainty
γ




0.399
19.24
0.304
13.45
ANA rate


36.52%

59.84%

57.54%

Table 4.2: (Continued)
Model 1
Model 2
Model 3
Model 4
Free flow time nonattendance co variates
RPL
RPL stated ANA
RPANA
RPANA with covariates
Parameter tratiob
Number of levels 2
— —
— —
— —
0.820
—
Number of levels 3 or 4
— —
— —
— —
0.820
3.77
Number of alternative 3 or 4
— —
— —
— —
0.313
—
Number of alternatives 5
— —
— —
— —
0.313
1.64
Number of attributes 3 or 4
— —
— —
— —
1.066
—
Number of attributes 5 or 6
— —
— —
— —
1.066
3.25
Uncertainty nonattendance co variates
Parameter
tratiob
Number of alternative 3 or 4
— —
— —
— —
0.437
—
Number of alternatives 5
— —
— —
— —
0.437
2.77
Number of attributes 3 or 4
— —
— —
— —
0.604
—
Number of attributes 5 or 6
— —
— —
— —
0.604
3.92
Number of choice tasks 6 or 15
— —
— —
— —
0.392
—
Number of choice tasks 9
— —
— —
— —
0.692
0.56
Number of choice tasks 12
— —
— —
— —
1.084
2.44
Model fit
LL
4211.83
4241.81
4202.37
4187.67
K (number of parameters)
15
15
17
24
AIC
1.841
1.854
1.837
1.834
atratio for difference to 0% ANA.
bM'atio for difference to base level under effects coding.
parameters, and the tratio for the difference between the parameter and the base level.
The number of attribute levels was only found to have a significant influence on inferred nonattendance to free flow time. If three or four levels are used in place of two, the nonattendance rate increases (i.e. attendance to free flow time decreases). This is consistent with Hensher (2006), who detected this phenomenon in one of four models estimated. In terms of this dimension of information load, a high load does result in fewer attributes being processed, where the effect seems to be selective in terms of which attributes.
The number of alternatives has a significant influence for both attributes, albeit only marginally so for free flow time. As the number of alternatives increases from three or four to five, ANA rates decrease. This is further evidence for the same finding in Hensher (2006), who found evidence in three out of four models. The present evidence also suggests that the influence may be consistent across a number of different attributes. That is, the effect does not influence just some attributes out of the full set. The suggestion that preserving attributes makes it easier to differentiate between an increasing number of alternatives (Hensher, 2006) is plausible, and suggests that as one dimension of information load increases (i.e. the number of alternatives), reducing information load on another dimension (the number of attributes processed) might make the decision harder.
Since nonattendance is inferred for specific attributes with the RPANA model, it is possible to test for the influence of the total number of attributes on the probability of nonattendance to these specific attributes. This is aided by the design, in which uncertainty was present for all subdesigns, and free flow time for 12 out of 16 subdesigns. Indeed, this influence is significant for both attributes. As the number of attributes increases from three or four to five or six, the probability of not attending to free flow time and uncertainty increases. Put another way, as additional attributes are introduced, attention may be diverted away from these attributes that are frequently or always present. There may be some specific features of the survey design in this study which is having some influence on this finding. Nonattendance to free flow time may be influenced by the unbundling of the congestion time into slowed down time and stop/start time, which occurs only when there are five or six attributes. Yet for uncertainty, any potential link between the attribute and the nature of the additional attributes is less clear. The influence of the number of attributes on nonattendance to these two specific attributes was not examined in Hensher (2006), as such an analysis was precluded by the estimation of an ordered heterogeneous logit model for each of the four possible numbers of attributes presented. In those models, the dependent variable was the number of attributes ignored. Use of a single RPANA model using all subdesigns allows for the influence of the number of attributes on nonattendance to specific attributes to be identified.
Whilst the number of choice tasks was found to have some influence on non attendance to uncertainty, the findings were inconsistent. Whilst there is more non attendance to 12 tasks than six or nine, this does not extend to 15 tasks, where this would have suggested that more choice tasks results in more serial nonattendance.
The final dimension of the design that was varied in the study was the range of the attribute levels. Whereas Hensher (2006) and Louviere and Hensher (2001) found that nonattendance increased as the range decreased, no evidence was found in the present analysis.
Since the dimensions of the choice tasks systematically vary over the 16 subdesigns, under Model 4 the inferred nonattendance rates will also vary across these subdesigns. Table 4.3 reports both the nonattendance rates and dimensions of each subdesign. The nonattendance rates have a large range for both attributes, from 1.40 to 53.55 per cent for free flow time, and from 19.32 to 91.90 per cent for uncertainty. Consider now the dimensions of the subdesigns that generate the most extreme nonattendance probabilities. For both attributes, the lowest non attendance rates occur in subdesign 12, which has nine choice tasks, five alternatives (the largest number possible), four attributes and just two attribute levels. Broadly, nonattendance is reduced (i.e. attendance is increased) with more alternatives, fewer attributes and fewer attribute levels. Nonattendance to free flow time is maximised in subgroup 11, with four alternatives, five attributes and four attribute levels. Nonattendance to uncertainty is maximised in subgroup 13, with 12 choice tasks, four alternatives, six attributes and two attribute levels.
Next we consider the impact of ANA on the WTP distributions for the two attributes for which nonattendance is handled. In this analysis, the distributions were not conditioned on the observed choices. The fixed total cost attribute was used,^{[3]} lessening the occurrence of extreme WTP values, and resulting in a lognormal WTP distribution. Since the structural parameters of this distribution can be difficult to interpret, we report its mean, median, mode and standard deviation in Table 4.4. These measures are of the continuous component of the distribution, which represents the valuations of those respondents that attend to the attribute, and ignores the point mass at zero, which represents the nonattenders. Additionally, the unconditional mean is reported, which includes the influence of nonattendance.
Accounting for nonattendance by any means results in a larger mean WTP. The relative increase is larger for uncertainty (which can readily be discerned by examining the ratio measures), which is to be expected given the higher nonattendance rates for this attribute. Whereas there is little difference in WTP for free flow time between stated ANA (Model 2) and inferred ANA (Models 3 and 4), inferred WTP is a little higher for uncertainty.
The unconditional mean WTPs, in which the nonattendance conditions are allowed to influence the mean WTP of attenders, align much more closely with the mean of the base RPL model (Model 1). All of the ratios lie between 0.81 and 1.12, with the closest to one being the comparison of free flow time between the RPL model and the RPANA model which handles varying information load. This suggests that the RPL model may perform a reasonable job of approximating the unconditional mean WTP, even if some do not attend to the attribute. The
Table 4.3: Inferred ANA rates for each subdesign.
Sub
design
Number of choice tasks
Number of alternatives
Number of attributes
Number of levels of attributes
Range of attribute levels
Free flow time ANA rate
Uncertainty ANA rate
1
15
4
4
3
Base
12.03%
43.67%
2
12
4
4
4
Wide
12.03%
77.21%
3
15
3
5
2
Wide
18.28%
72.18%
4
9
3
5
4
Base
53.55%
65.77%
5
6
3
3
3
Wide
a
43.67%
6
15
3
3
4
Narrow
a
43.67%
7
6
4
6
2
Narrow
18.28%
72.18%
8
9
5
3
4
Wide
a
19.32%
9
15
5
6
4
Base
38.13%
51.98%
10
6
5
6
3
Wide
38.13%
51.98%
11
6
4
5
4
Narrow
53.55%
72.18%
12
9
5
4
2
Narrow
1.40%
19.32%
13
12
4
6
2
Base
18.28%
91.90%
14
12
3
3
3
Narrow
a
77.21%
15
9
3
4
2
Base
2.58%
36.47%
16
12
5
5
3
Narrow
38.13%
82.55%
aFree flow time not present in subdesigns with three attributes.
Table 4.4: Willingness to pay with respect to total cost.
Model 1
Model 2
Model 3
Model 4
RPL
RPL stated ANA
RPANA
RPANA with covariates
WTP
WTP
Ratio M2/M1
WTP
Ratio M3/M1
WTP
Ratio M4/M1
Free flow time Mean
$20.81
$22.96
1.10
$22.06
1.06
$23.28
1.12
Mean (unconditional)
—
$20.20
0.97
$18.53
0.89
$20.64
0.99
Median
$7.72
$7.97
1.03
$11.02
1.43
$12.78
1.66
Mode
$1.06
$0.96
0.90
$2.75
2.60
$3.85
3.63
Standard deviation
$52.12
$62.09
1.19
$38.24
0.73
$35.46
0.68
Uncertainty
Mean
$3.67
$6.46
1.76
$7.37
2.01
$6.99
1.91
Mean (unconditional)
—
$4.10
1.12
$2.96
0.81
$2.97
0.81
Median
$0.89
$1.24
1.40
$5.32
6.01
$4.93
5.57
Mode
$0.05
$0.05
0.88
$2.77
53.60
$2.46
47.49
Standard deviation
$14.72
$33.08
2.25
$7.07
0.48
$7.01
0.48
distribution itself may be dramatically different, and to investigate this, we now turn to the other measures of the lognormal distribution.
Whereas the median and mode under stated ANA vary only a small amount from Model 1, the inference of ANA results in a dramatic increase in these values. The very large increase in the mode is particularly notable. With uncertainty, for example, the ratios are 53.6 and 47.49 for Models 3 and 4, respectively. Such ratios are due to the very small mode in the RPL model (i.e. the denominator of the ratio). A mode WTP of $0.05 is very close to zero, and suggests that the many values close to zero in the lognormal distribution in the RPL model is a result of the large number of respondents not attending to the attribute. Accommodating nonattendance with stated responses makes little noticeable difference to the mode. In contrast, the two RPANA models capture the nonattendance with a point mass at zero, and allow the mode of the lognormal distribution (i.e. its peak) to assume much more reasonable values, that are more likely to be representative of those that attend to the attribute. This same phenomenon likely also explains the large decrease in standard deviation for Models 3 and 4, as the distribution no longer needs to approximate the coefficients of both nonattenders (near zero), and nonattenders (some way out from zero). This decrease in the standard deviation under the RPANA models is consistent with Hensher (2007), who found that employing stated ANA decreased the standard deviation of the normal distribution. Interestingly though in this instance, the use of stated ANA in Model 2 increases the standard deviation.
The analysis of the WTP will conclude with two observations. First, comparing the two RPANA models, there is little consistent difference in the measures of the lognormal distribution as the information load covariates are introduced. However, with the nonattendance rates varying dramatically as the information load varies, the full distribution including the point mass at zero will change to a large extent. Second, whilst the RPANA models might not have dramatically different unconditional means to the conventional RPL model, again the distribution of coefficients is very different, where this might have particular implications for valuation. When the valuations are applied, it is common to segment the distribution of valuations using quantiles. It may be appropriate to choose the number of such segments based on the ANA rate.
[1] Note that the moments of the lognormal distribution are associated with the underlying normal distribution, and recall that the sign of the time attributes was reversed (i.e. forced to be negative).
[2] Full model results are not presented in the interests of brevity.
[3] To be precise, this means that the WTP values are applicable for 12 of the 16 subdesigns.
Found a mistake? Please highlight the word and press Shift + Enter