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 onetailed test.
Step
1
2
3
Equation
Y = 2.36 X1
Y = 1.80 X1 + 1.31 X2
Y = 0.91 + 1.44 X1 + 1.07 X2
(3.7)
(8.2)
(4.2)
R2
0.203
0.325
0.384
Comments:
 The third model is a good equation in spite of its low R2. 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).
TRANSPL Notes of AM Fillone, DLSU-Manila
Reference: Modelling Transport by Ortuzar and Willumsen
TRANSPL: Urban Transportation Planning and Engineering
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 X 2 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:
Y = 0.84 + 1.41 X1 + 0.75 Z1 + 3.14 Z2



R2 = 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.
TRANSPL Notes of AM Fillone, DLSU-Manila
Reference: Modelling Transport by Ortuzar and Willumsen
TRANSPL: Urban Transportation Planning and Engineering
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
TRANSPL Notes of AM Fillone, DLSU-Manila
Reference: Modelling Transport by Ortuzar and Willumsen
TRANSPL: Urban Transportation Planning and Engineering
TRANSPL Notes of AM Fillone, DLSU-Manila
Reference: Modelling Transport by Ortuzar and Willumsen
TRANSPL: Urban Transportation Planning and Engineering
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
p
T (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)
tp(h)
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
TRANSPL Notes of AM Fillone, DLSU-Manila
Reference: Modelling Transport by Ortuzar and Willumsen
TRANSPL: Urban Transportation Planning and Engineering

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 crossclassification 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
size
1 person
2 or 3 persons
4 persons
5 persons
Total
Mean trip rate
0 car
1 car
2+ cars
Total
Mean trip rate
28
150
61
37
276
(0.73)
21
201
90
142
454
(1.53)
0
93
75
90
258
(2.44)
49
444
226
269
988
(0.47)
(1.28)
(1.86)
(1.90)
(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

TRANSPL Notes of AM Fillone, DLSU-Manila
Reference: Modelling Transport by Ortuzar and Willumsen
TRANSPL: Urban Transportation Planning and Engineering
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
0 car
1 car
2 + cars
1 person
0.00
0.46
1.37
2 or 3 persons
0.46
1.27
2.18
4 persons
1.05
1.85
2.76
5 persons
1.09
1.89
2.80
Deviations
- 0.81
- 0.01
0.90
Deviations
- 1.07
- 0.26
0.32
0.36
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