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. A base-year survey of 1,144 households making 8,074 trips, cross-classified by household size and income, together with a target-year forecast of 1,770 households in the same 15 cells.

Find. The target-year trips in each of the 15 cells by cross-classification (a) and by the fitted regression equation (b), and a comparison of the assumptions and limitations of the two methods (c).

Approach. Both methods produce a trip rate per household for every cell and then multiply by the forecast household count; they differ only in where the rate comes from — the observed cell mean in (a), a fitted linear function in (b).

(a) Cross-classification (category analysis)

1. Convert the survey into an observed trip rate for every cell. The rate is the cell’s trips divided by its households, \[r_{mn} = \frac{\text{trips}_{mn}}{\text{households}_{mn}},\qquad \text{e.g.}\quad r_{4,\text{low}} = \frac{1010}{120} = 8.42\ \text{trips/household}.\]

The rate surface behaves exactly as trip-generation theory predicts: it rises strongly with household size in every income column and rises with income at every household size, and it flattens at the top of the size range because the fifth and subsequent members of a household are disproportionately children.

2. Apply each observed rate to the forecast households in the same cell. \[T_{mn} = r_{mn} \times H_{mn},\qquad \text{e.g.}\quad T_{4,\text{low}} = 8.4167 \times 150 = 1{,}262.5\ \text{trips}.\]

3. Sum the 15 cells for the zonal production. \[\boxed{T_{(a)} = 2{,}905.6 + 6{,}775.1 + 2{,}961.8 = 12{,}642.5 \approx 12{,}643\ \text{trips/day}}\] The implied average is \(12{,}642.5 / 1770 = 7.14\) trips per household, up from the base-year 7.06, because the forecast shifts households toward the larger and higher-income cells.

(b) Linear regression rate

4. Evaluate the fitted equation for each of the 15 household types. With NPERSON capped at 5 and HINCOME coded 0, 1, 2, \[r = 0.46 + 1.96\,\text{NPERSON} + 1.66\,\text{HINCOME},\qquad \text{e.g.}\quad r_{4,\text{low}} = 0.46 + 1.96(4) + 1.66(0) = 8.30.\] Every additional person adds a constant 1.96 trips and every step up the income ladder adds a constant 1.66 trips, which is the substantive assumption the model makes.

5. Apply the fitted rates to the same forecast households and sum.

6. Total the regression forecast. \[\boxed{T_{(b)} = 2{,}851.8 + 6{,}499.2 + 3{,}165.4 = 12{,}516.4 \approx 12{,}516\ \text{trips/day}}\] The two methods differ by only 126.1 trips, about 1.0 per cent of the total, which is the expected outcome when a well-specified regression is fitted to the same data that the cross-classification tabulates.

(c) Assumptions and limitations compared

Cross-classification assumes that the observed trip rate in each cell is stable over time and transferable to the forecast year, and that household size and income between them capture everything that matters — but it imposes no functional form at all. That is its strength: the rate surface can be as irregular as the data, and the saturation at large household sizes visible in the table is reproduced automatically. Its limitations are equally clear. It has no statistical machinery, so there is no confidence interval and no significance test. It is extremely sensitive to thin cells: the five-or-more, low-income cell rests on 13 households, and the two-person, high-income cell on 27, so a handful of unusual households sets the rate for a whole forecast category. It cannot fill an empty cell at all, and it cannot extrapolate to a household type outside the sampled range. Adding a third explanatory variable multiplies the number of cells and makes the thin-cell problem worse.

Regression assumes a specific functional form — additive, linear and with no interaction between size and income — in exchange for statistical efficiency. Because it pools all 1,144 households to estimate three coefficients, it is far more stable in thin cells, it can fill an empty cell, and it comes with standard errors, t-statistics and an \(R^2\). Its limitations are the mirror image of those advantages. The linear form is imposed rather than tested, so the saturation the data actually show is lost: the equation keeps adding 1.96 trips for every person, and the analyst has to patch this by hand with the “NPERSON = 5 for 5 or more” cap the question supplies. The no-interaction assumption is visibly violated here — in the survey the income effect is much larger for small households than for large ones. The two worst cells make the point: for one-person high-income households the regression gives 5.74 against an observed 3.75, an error of 1.99 trips per household, while for five-or-more medium-income households it gives 11.92 against an observed 12.35. Treating the ordinal income code as a cardinal variable is a further imposition, since it forces the low-to-medium and medium-to-high steps to be identical.

In practice the two are complements rather than rivals, and the standard Canadian regional practice is to use them together: cross-classification for the well-populated cells where the data speak for themselves, and a fitted equation to fill sparse or empty cells and to extrapolate to household types the survey did not reach. The one per cent agreement between the two totals here is reassurance that neither is badly misspecified.