A single versus a two-part model

There are several pieces of evidence for assessing whether the two-part model is worthwhile.

First, I consider the estimated coefficients for the explanatory variables — including their sign, magnitude, and significance levels. I compare the coefficients from the single BL and NB1+

models (models 2 and 3, Table 30 and Table 31), which capture the binary (zero versus any) and count (volume of trips 1 and above) in isolation. There are some differences in which

explanatory variables have significant coefficients, in addition to some with significant coefficients in both models but with opposite signs. (The estimated coefficients for the binary and count portions of the H models (model 4, Table 32) are almost identical to those for the separate BL and NB1+ models and suggest the same conclusions.) This suggests that there may be some differences in what influences making any trips versus making successively higher numbers of trips. For instance, there is an opposite influence on making any trips versus a higher volume of trips for the following variables.

• Female versus male (─ any, + volume, meaning an apparent negative impact on making any trips but a positive impact on the volume of trips). In addition, note the interaction effects involving gender: in each model, the observed direction of influence of being female is amplified for those with kids (for volume) and for older females (for both any and volume).

The effect of female and older (─) seems to have a stronger effect on trip volume than the probability of any trips.

• Employment, in general, vs. not working (+ any , ─ volume).

• Employment part-time or with multiple jobs, vs. full-time or not working (─ any, + volume).

In both models, this serves to somewhat attenuate the effect of employment in general, among this sub-group.

• Recentness of immigration (+ any , ─ volume). Furthermore, note that foreign-born status only matters for making any trips once recentness is taken into account; by contrast a foreign-born dummy variable has a significant additional negative effect for volume only.

• Weekend days of the week, among those who are employed (─ any, + volume). This serves to attenuate the otherwise positive influence of employment on the chances of any travel, and the otherwise negative effect on trip volume, among the employed on weekdays.

In addition, there is apparently more effect, that is, coefficients that are only significant, or more significant, in only one of the models, either the BL (model 2, modeling the probability of “any”) or NB1+ (model 3, modeling only the volume of trips), for the following variables: Household size (+ , more impact on volume); Single-parent status (+, more impact on volume); Females with kids (+, more impact on volume); Older females (─, more on volume, perhaps); Suburban and urban built environments, vs. town/country (+, more impact on volume); Friday (+, more impact on volume); Number of cars (while having any cars, versus none, has a significant positive effect in both models, the number of cars is not significant in the BL (any) model, and has a seemingly complicated relationship with volume of trips in the NB models, discussed more below).

Comparing the coefficients from the separate BL and NB1+ models (models 2 and 3 in Table 30 and Table 31), which account for “any” and “volume” separately, to the single NB0+

model (model 1, Table 29), which treats all counts together, we see that estimates for all the above variables show a weaker effect and in some cases are not significant in the single model (model 1). For instance, while models 2 and 3 suggest that employment increases the chance of any trips but decreases trip volume, the coefficient in the single model is negative but not quite significant (with p=0.139), for the model with the entire sample), is even less significant for the low-access segment (p=0.512), and is significant but with opposite signs in the medium- and high-access segments, respectively. Separating the effect on any trips versus on the volume of

trips offers more clarity on the influence of employment on activity. These sorts of differences would seem a justification for handling the phenomena of any trips versus successively higher numbers of trips separately, especially for the purposes of drawing conclusions about the role of particular explanatory variables.

Another consideration is whether separating the “any” and “volume” portions of the model improves the overall performance of the model. Considering various goodness of fit measures as well as the accuracy of the predicted values, the hurdle model is no worse than the single model, and may be slightly better, but it is unclear if the improvement is substantial. In particular, the information criteria statistics (AIC and BIC) are lower using the two-part hurdle model (model 4) than the single NB0+ model (model 1) for all of the segments, but only at the first decimal place (for instance reduced from 3.897 to 3.849 in model 4a versus 1a; see a consolidated side-by-side comparison in Table 35). The pseudo-R² values for the hurdle models (model 4) are also higher than for the single NB0+ models (model 1), but only by about 0.01 unit.

However, with such low pseudo-R² values overall, these increases are substantial on a percentage basis. For instance, the increase from 0.028 to 0.040 between models 1d and 4d, respectively, comprises a 42-percent improvement (see Table 35).

Table 35 also shows some measures based on the accuracy of the predicted values generated by the models. These are also barely improved in the hurdle model compared to the ordinary negative binomial model (models 1 and 4, respectively). For instance, for the versions estimated using the high-access segment, the average absolute difference between the actual and predicted value decreases just 0.001 units (or 0.03%), from 1.569 to 1.568 trips difference.

The percent of cases for which the average distance is less than 1 trip increases slightly from 37.8% to 38.0%. However, additional gains in accuracy, though still small, are achieved in model 5, which uses the final specification (discussed in 4.6 and shown in Table 37). Perhaps an

important aspect to this final specification is that the explanatory variables for the binary and count components of the model differ somewhat. Forcing them to be identical, as done in models 1 through 4 for the purposes of comparison may have undermined the main theoretical advantage of the hurdle model framework.

I conclude that there are probably differences in the factors influencing “any” versus

“volume” of trips, but that we are not able to model either that well. As a result, the hurdle model is only slightly better than the alternative, but it is theoretically superior, as well as measurably superior, even if just barely. I use it for the remainder of the analysis.

Table 35. Comparison of model performance of the single negative binomial model versus the joint hurdle model Desired

value for measure

Model estimated using:

Entire sample (d) High-access segment only (c) Model 1d

* Uses a different specification. See section 4.6 and Table 37.

4.5.3 Differences in model fit and the accuracy of predictions within different