Zonal-Based Multiple Regression

TRANSPL: Urban Transportation Planning and Engineering

ZONAL-BASED MULTIPLE REGRESSION

 An attempt is made to find a linear relationship between the number of trips produced or attracted by zone and average socioeconomic characteristics of the households in each zone.  The following are some interesting considerations:

1. Zonal models can only explain the variation in trip making behavior between zones. - for this to happen, it would be necessary that zones not only had an homogeneous socioeconomic composition, but represented as wide as possible a range of conditions - a major problem is that the main variations in person trip data occur at the intra-zonal level

2. Role of the intercept - one would expect the estimated regression line to pass through the origin; however, large intercept values have often been obtained - if this happens the equation may be rejected, in on the contrary, the intercept is not significantly different from zero, it might be informative to re-estimate the line, forcing to pass through the origin

3. Null zones - it is possible that certain zones do not offer information about certain dependent variables (e.g. there can be no HB trips generated in non-residential zones) - Null zones must be excluded from analysis; although their inclusion should not greatly affect the coefficient estimates (because the equation should pass through the origin)

4. Zonal totals versus zonal means - When formulating the model, there is a choice between using aggregate or total variables, such as trips per zone and cars per zone, or rates (zonal means), such as trips per household per zone and cars per household per zone. - In the first case, the regression model would be

Yi = 0 + 1X1i + 2X2i + … + kXki + Ei

- Whereas the model using rates would be

yi = 0 + 1x1i + 2x2i + … + kxki + ei

with yi = Yi/Hi; xi = Xi/Hi; ei = Ei/Hi and Hi the number of households in zone i

TRANSPL Notes of AM Fillone, DLSU-Manila Reference: Modelling Transport by Ortuzar and Willumsen TRANSPL: Urban Transportation Planning and Engineering

HOUSEHOLD-BASED REGRESSION

 Decreasing zone size, especially if zones are homogeneous may reduce intra-zonal variation. However, smaller zones imply a greater number of them and this has two consequences: o More expensive models in terms of data collection, calibration and operation; o Greater sampling errors, which are assumed non-existent by the multiple linear regression model

 In a household-based application each home is taken as an input data vector in order to bring into the model all the range of observed variability about the characteristics of the household and its travel behavior.  The calibration process, as in the case of zonal models, proceeds stepwise, testing each variable in turn until the best model (in terms of some summary statistics for a given confidence level) is obtained.

Example 4.3: Consider the variables trips per household (Y), number of workers (X1) and number of cars (X2). Table 4.3 presents the results of successive steps of a step-wise model estimation; the last row also shows (in parenthesis) values for the t-ratio (equation 4.9). Assuming large sample size, the appropriate number of degrees of freedom (n – 2) is also a large number so the t-values may be compared with the critical value of 1.645 for a 95 % significant level on a one- tailed test.

Step Equation R2

1 Y = 2.36 X1 0.203

2 Y = 1.80 X1 + 1.31 X2 0.325

3 Y = 0.91 + 1.44 X1 + 1.07 X2 0.384 (3.7) (8.2) (4.2)

Comments:  The third model is a good equation in spite of its low R 2. The intercept 0.91 is not large (compare it with 1.44 times the number of workers, for example) and the regression coefficients are significantly different from zero (H0 is rejected in all cases).

The Problem on Non-linearities

 Linear regression model assumes that each independent variable exerts a linear influence on the dependent variable  Figure 4.9 presents data for households stratified by car ownership and number of workers. It can be seen that travel behavior is non-linear with respect to family size  There is a class of variables, those of a qualitative nature, which usually shows non-linear behavior (e.g. type of dwelling, occupation of the head of the household, age, sex).  In general, there are two methods to incorporate non-linear variables into the model: 1. Transform the variables in order to linearize their effect (e.g take logarithms, raise to a power) 2. Use dummy variables. In this case the independent variable under consideration is divided into several discrete intervals and each of them is treated separately in the model. - In this form it is not necessary to assume that the variable has a linear effect, because each of its portions is considered separately in terms of its effect on travel behavior. - For example, if car ownership was treated in this way, appropriate intervals could be 0, 1, and 2 or more cars per household. As each sampled household can only belong to one of the intervals, the corresponding dummy variable takes a value of 1 in that class and 0 in the others.

Example 4.4: Consider the model of Example 4.3 and assume that variable X2 is replaced by the following dummies

Z1, which takes the value of 1 for households with one car and 0 in other cases, Z2, which takes the value of 1 for households with two or more cars and 0 in other cases.

 It is easy to see that non-car-owning households correspond to the case where both Z1 and Z2 are 0. The model of the third step in Table 4.3 would now be:

2 Y = 0.84 + 1.41 X1 + 0.75 Z1 + 3.14 Z2 R = 0.387

 Even without the better R2 value, this model is preferable to the previous one just because the non-linear effect of X2 (or Z1 and Z2) is clearly evident and cannot be ignored.  Note that if the coefficients of the dummy variables were for example, 1 and 2, and if the sample never contained more than two cars per household, the effect would be clearly linear.  The model is graphically depicted in Figure 4.10.

OBTAINING ZONAL TOTALS

 In the case of zonal-based regression models, this is not a problem as the model is estimated precisely at this level  In the case of household-based models, though, an aggregation stage is required  Precisely because the model is linear the aggregation problem is trivially solved by replacing the average zonal values of each independent variable in the model equation and then multiplying it by the number of households in each zone  Thus, for the third model of Table 4.3, we would have

Ti = Hi (0.91 + 1.44X1i + 1.07X2i) where Ti is the total number of HB trips in the zone, Hi is the total number of households in it, and Xji is the average value of variable Xj for the zone  On the other hand, when dummy variables are used, it is also necessary to know the number of households in each class for each zone; for instance, in the model of Example 4.4 we require:

Ti = Hi(0.84 + 1.41X1i) + 0.75 H1i + 3.14 H2i

where Hji is the number of households of class j in zone i

CROSS-CLASSIFICATION OR CATEGORY ANALYSIS

The Classical Model

Category Analysis (UK) or Cross-classification (USA)  The method is based on estimating the response (e.g. the number of trip production per household for a given purpose) as a function of household attributes  Its basic assumption is that trip generation rates are relatively stable over time for certain household stratifications  The method finds these rates empirically and for this it typically needs a large amount of data

Variable Definition and Model Specification

Mathematical form: tp(h) = Tp(h) / H(h) where tp(h) is the average number of trips with purpose p (and at a certain time period) made by members of households of type h Tp(h) is total number of trips in cell h, by purpose group H(h) is the number of households in the group

. The categories are chosen such that the standard deviations of the frequency distributions are minimized (Figure 4.11)

Figure 4.11 Trip-rate distribution for household type . The method has, in principle, the following advantages: a. Cross-classification groupings are independent of the zone system of the study area b. No prior assumptions about the shape of the relationship are required c. Relationships can differ in form from class to class (e.g. the effect of changes in household size for one or two car-owning households may be different

. The traditional cross-classification method has also several disadvantages:

a. The model does not permit extrapolation beyond its calibration strata, although the lowest or highest class of a variable may be open-ended b. There are no statistical goodness-of-fit measures for the model, so only aggregate closeness to the calibration data can be ascertained c. Unduly large samples are required, otherwise cell values will vary in reliability because of differences in the numbers of households being available for calibration at each one d. There is no effective way to choose among variables for classification, or to choose best groupings of a given variable e. If it is required to increase the number of stratifying variables, it might be necessary to increase the sample enormously

IMPROVEMENTS TO THE BASIC MODEL Multiple Classification Analysis (MCA)

MCA is an alternative method to define classes and test the resulting cross- classification which provides a statistically powerful procedure for variable selection and classification.

Example: Table 4.7 presents data collected in a study area and classified by three car-ownership and four household-size levels. The table presents the number of households observed in each cell (category) and the mean number of trips calculated over rows, cells and the grand average.

Household 0 car 1 car 2+ cars Total Mean trip rate size 1 person 28 21 0 49 (0.47) 2 or 3 persons 150 201 93 444 (1.28) 4 persons 61 90 75 226 (1.86) 5 persons 37 142 90 269 (1.90) Total 276 454 258 988 Mean trip rate (0.73) (1.53) (2.44) (1.54)

. The values range from 0 to 201 . There are four cells with less than the conventional minimum number (50) of observations required to estimate mean trip rate and variance with some reliability

. Using the mean row and column values to estimate average trip rates for each cell, including that without observations in the sample

. Deviation from the grand mean For zero cars: 0.73 – 1.54 = -0.81 For one car: 1.53 – 1.54 = - 0.01 For two cars or more: 2.44 – 1.54 = 0.90 . Deviations for each of the four household size groups For 1 person: 0.47 – 1.54 = - 1.07 For 2 or 3 persons: 1.28 – 1.54 = - 0.26 For 4 persons: 1.86 – 1.54 = 0.32 For 5 persons: 1.90 – 1.54 = 0.36

. Table 4.8 depicts the full trip-rate table together with its row and column deviations

Table 4.8 Trip rates calculated by multiple classification HH size Car ownership level Deviations 0 car 1 car 2 + cars 1 person 0.00 0.46 1.37 - 1.07 2 or 3 persons 0.46 1.27 2.18 - 0.26 4 persons 1.05 1.85 2.76 0.32 5 persons 1.09 1.89 2.80 0.36 Deviations - 0.81 - 0.01 0.90

The most important statistical goodness-of-fit measures associated with MCA are . An F-statistic to assess the entire cross-classification scheme; . A correlation ratio statistic for assessing the contribution of each classification variable; and . An R2 measure for the complete cross-classification model

TRANSPL Notes of AM Fillone, DLSU-Manila Reference: Modelling Transport by Ortuzar and Willumsen