Question 3 of 7: Trip Generation — Cross-Classification and Regression

Question 3: Trip Generation — Cross-Classification and Regression (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A cross-classification table of observed trip rates and a matching table of forecast household counts, both stratified by number of cars (rows) and number of workers (columns); plus a fitted linear regression for the same trip rate. One cell — two or more cars with no worker — is empty in both tables, so it contributes nothing.

Find. The forecast trip production of every household type by the cross-classification method and by the regression method, the marginal effect of an extra worker and an extra car, and a comparison of the assumptions and limitations of the two approaches.

Approach. Multiply each cross-classification rate by the matching household count and sum; then evaluate the regression at every cell of the same stratification, applying the stated caps, multiply by the same household counts and compare the two totals.

1. Part (a) — apply the cross-classification rates cell by cell. The forecast production of a household type is the product of its observed rate and its forecast count, $$T_{ij}=r_{ij}\,H_{ij}$$ where $i$ indexes cars and $j$ indexes workers. Working across the first row, $1.1\times130=143$, $2.3\times90=207$, $3.6\times70=252$ and $5.5\times40=220$ trips per day.
2. Complete the table and total it. Repeating the same product for the remaining two rows gives the full matrix below.

   Part (a) — forecast trips by household type (cross-classification)

   Cars	0 workers	1 worker	2 workers	3 or more	Row total
   0	143.0	207.0	252.0	220.0	822.0
   1	138.0	792.0	1,064.0	1,900.0	3,894.0
   2 or more	—	200.0	368.0	1,680.0	2,248.0

   Summing the three row totals, $$\boxed{T_{(a)}=822+3{,}894+2{,}248=6{,}964 \text{ trips per day}}$$ Across the 1,470 forecast households this is an average of 4.74 trips per household, and the two largest single contributions — 1,900 and 1,680 trips — both come from the three-or-more-worker column, which is where the rate and the household count are simultaneously high.
3. Part (b) — describe what the regression coefficients mean. The fitted equation is $$\text{Trip rate}=0.79+1.62\,\text{WORKER}+0.88\,\text{CAR}$$ so each additional worker in the household adds $\boxed{1.62}$ trips per day and each additional car adds $\boxed{0.88}$ trips per day, with a base of 0.79 trips for a household with neither. Workers are therefore the stronger driver, by a factor of about 1.8, which is what one expects when the dominant purpose in a daily trip total is the journey to work — each worker generates a return commute plus associated linked trips. The car coefficient is positive and smaller: car availability does not create the activity, it removes the constraint that would otherwise suppress or shorten it. Both effects are linear and additive within the fitted range, and both are truncated by the stated caps, WORKER at 3 and CAR at 2, because the survey contained too few households beyond those values to estimate a slope reliably.
4. Evaluate the regression at each cell. Applying the caps, the expected rate for a household with $w$ workers and $c$ cars is $0.79+1.62\min(w,3)+0.88\min(c,2)$. For a household with two workers and one car, for example, $$r = 0.79+1.62(2)+0.88(1)=0.79+3.24+0.88=4.91 \text{ trips/day}$$ Repeating for all twelve strata gives the regression rate table.

   Part (b) — expected trip rate from the regression

   Cars	0 workers	1 worker	2 workers	3 or more
   0	0.79	2.41	4.03	5.65
   1	1.67	3.29	4.91	6.53
   2 or more	2.55	4.17	5.79	7.41

5. Multiply by the same household counts and total. Using the identical forecast household matrix keeps the two methods strictly comparable.

   Part (b) — forecast trips by household type (regression)

   Cars	0 workers	1 worker	2 workers	3 or more	Row total
   0	102.7	216.9	282.1	226.0	827.7
   1	100.2	723.8	1,374.8	1,632.5	3,831.3
   2 or more	—	208.5	463.2	1,482.0	2,153.7

   $$\boxed{T_{(b)}=827.7+3{,}831.3+2{,}153.7=6{,}812.7 \text{ trips per day}}$$
6. Part (c) — compare the two estimates before comparing the methods. The regression forecast is 151.3 trips per day lower, a difference of only 2.17 per cent on a zone total near 7,000 trips. Agreement that close at the zonal level is expected, because the regression was fitted to the same survey the cross-classification cells came from, so both reproduce the sample mean. The differences are concentrated in individual cells rather than in the total: the regression is well above the observed rate for two-worker one-car households (4.91 against 3.80) and well below it for the three-or-more-worker one-car and two-or-more-car households (6.53 against 7.60, and 7.41 against 8.40). Those are precisely the cells where the true relationship is not linear, and the linear model cannot follow it.
7. Set out the assumptions and limitations of each method. Cross-classification assumes only that households with the same characteristics make the same average number of trips, and that those rates are stable over the forecast horizon. It is non-parametric, imposes no functional form, and captures interaction between workers and cars automatically — the jump from 3.8 to 8.4 trips between the two-worker one-car cell and the three-or-more-worker two-or-more-car cell is inherited directly from the data. Its weaknesses are sample-driven: each cell needs enough surveyed households to give a stable mean, the number of cells grows multiplicatively with each additional stratifying variable, an empty cell (such as the two-or-more-cars, zero-workers cell here) cannot be filled at all, and there is no way to interpolate to a household type not surveyed. It also gives no measure of statistical fit and no way to test whether a variable matters.
8. State when each is the appropriate tool. Regression assumes a specific functional form — here linear and additive, with no interaction term — and assumes that the fitted coefficients remain valid in the target year. Its advantages are exactly complementary: it is parsimonious, it can be evaluated for any combination of inputs including ones never surveyed, it yields significance tests and a coefficient of determination, and it uses the whole sample to estimate each coefficient rather than partitioning the sample among cells. Its limitations are the imposed linearity, sensitivity to correlation between the explanatory variables (household size, workers and car ownership are strongly collinear in practice, so the individual coefficients are less reliable than the fitted total), the arbitrariness of the truncation rules, and the risk that a household-level equation applied to zonal averages produces an ecological fallacy. Sound practice in a Canadian regional model is to use cross-classification where the survey supports well-populated cells — typically the common household types — and to use the regression to fill sparse or empty cells and to extrapolate to household types the survey did not reach, which is precisely the role it plays in this question.