Question 3 of 7: Trip Production by Cross-Classification and by Regression

Question 3: Trip Production by Cross-Classification and by 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. Two 5 × 3 matrices, indexed by persons per household (1, 2, 3, 4, 5 or more) down the rows and vehicles per household (0, 1, 2 or more) across the columns, transcribed from page 3 of the paper:

Household inventory total: 1870 households.

The household inventory totals 1870 households. The regression alternative supplied in part (b) is $$\text{Trip rate} = -0.85 + 2.63\,\text{NPERSON} + 2.01\,\text{NVEH}$$ with NPERSON capped at 5 and NVEH capped at 2, matching the top row and right-hand column of the cross-classification table.

Find. The forecast trips generated by each of the fifteen household types, first from the observed cross-classification rates and then from the fitted regression rates, together with an interpretation of the regression coefficients and a comparison of the two methods.

Approach. Trip production by cross-classification is a pure multiplication: trips in a cell equal the cell trip rate times the number of households in that cell, and the zone total is the sum over cells. Part (b) repeats the exercise with the rates replaced by regression predictions, so the only new work is evaluating the fitted equation at each of the fifteen (NPERSON, NVEH) combinations.

1. Part (a) — apply the cross-classification rates cell by cell. For every household type, $$T_{pv} = R_{pv}\times H_{pv}$$ where $R_{pv}$ is the trip rate for households with $p$ persons and $v$ vehicles and $H_{pv}$ is the forecast number of such households. Taking the first cell as the pattern, a one-person household with no vehicle makes 2.6 trips/day and there are 100 such households, so $T_{1,0} = 2.6 \times 100 = 260$ trips. Repeating over all fifteen cells:

   Question 3(a) — forecast trips per day by household type, cross-classification method

   Persons per household	0 vehicles	1 vehicle	2 or more vehicles	Row total
   1	260	1200	600	2060
   2	528	1675	410	2613
   3	666	2300	560	3526
   4	1380	2415	882	4677
   5 or more	224	685	516	1425
   Column total	3058	8275	2968	14301

   $$\boxed{\text{Total forecast trips (cross-classification)} = 14\,301\ \text{trips/day}}$$ Two structural features of the answer are worth noting before moving on. Households with one vehicle produce 8275 trips, 58 per cent of the total, simply because that is the largest column of the household inventory. And the largest single cell is not the highest trip rate but the four-person, one-vehicle cell at 2415 trips, because trip production is a product of rate and frequency — the highest-rate cell (5 or more persons, 2 or more vehicles, at 17.2 trips/household) contributes only 516 trips because it contains just 30 households.
2. Part (b) — evaluate the fitted regression rate for each household type. Substituting each (NPERSON, NVEH) pair into the fitted equation, and remembering both caps, gives the fifteen predicted rates. For example, for a three-person household with one vehicle, $$R = -0.85 + 2.63(3) + 2.01(1) = -0.85 + 7.89 + 2.01 = 9.05\ \text{trips/household}$$ against the 9.20 trips/household in the observed table. The full set of predicted rates is:

   Question 3(b) — trip rates predicted by the fitted regression (trips per household per day)

   Persons per household	0 vehicles	1 vehicle	2 or more vehicles
   1	1.8	3.8	5.8
   2	4.4	6.4	8.4
   3	7.0	9.1	11.1
   4	9.7	11.7	13.7
   5 or more	12.3	14.3	16.3

   Compare with the observed rates in the Given block: the regression under-predicts small vehicle-poor households and over-predicts small vehicle-rich ones.

3. Multiply the regression rates by the same household inventory. The households are unchanged, so only the rates differ from Step 1:

   Question 3(b) — forecast trips per day by household type, regression method

   Persons per household	0 vehicles	1 vehicle	2 or more vehicles	Row total
   1	178	1137	870	2185
   2	485.1	1605	421.5	2511.6
   3	633.6	2262.5	553	3449.1
   4	1450.5	2452.8	821.4	4724.7
   5 or more	246	715.5	489.6	1451.1
   Column total	2993.2	8172.8	3155.5	14321.5

   $$\boxed{\text{Total forecast trips (regression)} = 14\,321.5\ \text{trips/day}}$$
4. Compare the two forecasts. The totals agree closely: $$\frac{14\,321.5 - 14\,301}{14\,301}\times 100 = +0.14\ \text{per cent}$$ That is expected rather than lucky. Least-squares regression fitted to the same underlying survey reproduces the weighted mean of the observations, so an aggregate over a household distribution resembling the calibration sample must come out close. The agreement within cells is much weaker, and the disagreement is systematic: the regression under-predicts the small, vehicle-poor households (the one-person, no-vehicle cell drops from 2.6 to 1.78 trips, a 32 per cent shortfall, and its cell forecast falls from 260 to 178 trips) and over-predicts the small, vehicle-rich households (the one-person, two-vehicle cell rises from 4.0 to 5.80 trips, and its forecast rises from 600 to 870 trips). Aggregation hides the errors; a corridor study focused on a single household type would not be able to.
5. Part (c) — interpret the coefficients. Because the fitted equation is linear and additive, each coefficient is a constant marginal effect: $$\frac{\partial R}{\partial \text{NPERSON}} = +2.63\ \text{trips/household per additional person},\qquad \frac{\partial R}{\partial \text{NVEH}} = +2.01\ \text{trips/household per additional vehicle}$$ Both signs are positive and both make clear intuitive sense. An additional person is an additional traveller with their own activity programme — work, school, shopping, social — so household trip production must rise with household size, and the estimated 2.63 trips per person per day is a plausible individual trip rate for an urban area. An additional vehicle raises production because it relieves the vehicle-availability constraint: in a one-vehicle household, two adults with conflicting schedules must chain, share or forego trips, and the second vehicle converts suppressed travel into realised travel. The magnitudes are also sensible relative to one another, with the person effect somewhat larger than the vehicle effect, since a person is a traveller whereas a vehicle only makes travelling easier.
6. State the two things the regression gets wrong, which is where the marks are. First, the intercept of $-0.85$ trips is physically meaningless: an empty household with no vehicle cannot make a negative number of trips. It is a fitting artefact with no interpretation outside the calibration range of one to five persons, and it is the reason the equation must never be extrapolated toward zero. Second, and more important for forecasting, the additive form imposes no interaction between persons and vehicles — the value of a second vehicle is assumed to be 2.01 trips whether the household contains one person or five. The observed table flatly contradicts this. For a one-person household the observed rate is 4.0 trips at both one vehicle and two or more vehicles, a saturation effect (one person cannot drive two cars at once), whereas the regression insists on adding another 2.01 trips. For a five-person household the observed step from one to two vehicles is $17.2 - 13.7 = 3.5$ trips, well above 2.01. The interaction is real and the linear model cannot represent it; a multiplicative form, or an interaction term $\text{NPERSON}\times\text{NVEH}$, would be needed.
7. Part (d) — compare the two methods. The comparison is best made on three axes.

   Assumptions. Cross-classification is non-parametric: it assumes only that households within a cell are homogeneous and that the cell rates are stable over the forecast horizon. It therefore reproduces any pattern present in the data, including the non-linearity and the person–vehicle interaction identified above, without being told to look for it. Regression imposes a functional form — linearity, additivity, constant marginal effects, and homoscedastic normally distributed residuals if any inference is to be drawn — and everything the data does that the form cannot represent becomes residual error.

   Data requirements. This is where the trade runs the other way. Cross-classification needs a reliable trip rate in every occupied cell, so the household travel survey must contain enough sampled households in each of the fifteen cells to give a stable mean; sparse cells (here, five-or-more-person households with no vehicle, of which there are only 20) yield noisy rates, and empty cells yield none at all. The regression estimates only three parameters from the whole sample, so it is far more economical of data, it borrows strength across cells, and it can predict a cell for which no household was surveyed.

   Application and transferability. Cross-classification cannot extrapolate: a forecast of six-person households or three-vehicle households has no cell to fall in, which is exactly why the source table caps its categories at "5 or more" and "2 or more". The regression extrapolates freely, though the caps written into part (b) show that the analyst does not trust it to. Cross-classification also produces a table that a lay committee can read and audit, which has real value in a public planning process, whereas the regression produces coefficients whose standard errors and goodness of fit can be reported and tested — a genuine advantage for defending the model technically.

   In practice the two are complementary: calibrate by cross-classification where the survey supports it, fill sparse or empty cells with the fitted regression, and check the aggregate agreement — here 0.14 per cent — as evidence that the two are telling the same story.