16-Civ-A6 Highway Design, Construction, and Maintenance · December 2015

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

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format: 98-Civ-A6 Transportation Planning & Engineering, National Examination, December 2015. Seven questions, each of equal value (20 marks); any five constitute a complete examination. Closed book, one two-sided aid sheet permitted. Three hours. All seven questions are solved below as a study resource.

Reference texts. Mannering, Washburn & Kilareski, Principles of Highway Engineering and Traffic Analysis (Wiley); Papacostas & Prevedouros, Transportation Engineering and Planning (Prentice Hall); Roess, Prassas & McShane, Traffic Engineering (Pearson); Ortúzar & Willumsen, Modelling Transport (Wiley) for the demand-model chapters (trip generation, distribution, mode choice, assignment).

Question 3: Trip Generation — Cross-Classification vs 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 matrix of households and survey trip rates by auto ownership (rows) and income (columns):

Given data — No. of households / survey trip rate (trips/HH)
Vehicles/HH	Low: HH	Low: rate	Med: HH	Med: rate	High: HH	High: rate
0	11	2	11	3	0	7
1	37	7	240	8	10	13
2	7	12	120	13	41	18
3 or more	0	17	4	18	19	23

Find. Zonal trips by the cross-classification rates (a) and by the regression rates (b), and a comparison of the two methods (c).

Approach. For each cell, trips $=$ (number of households) $\times$ (trip rate); sum over all twelve cells. Part (a) uses the tabulated survey rates directly; part (b) substitutes the regression-predicted rate for each (AUTO, INC) combination.

(a) Cross-classification — apply the survey rate cell by cell. Multiplying and summing down each income column: $$T_{\text{Low}}=11(2)+37(7)+7(12)+0(17)=365,$$ $$T_{\text{Med}}=11(3)+240(8)+120(13)+4(18)=3585,$$ $$T_{\text{High}}=0(7)+10(13)+41(18)+19(23)=1305.$$ The zonal total is $$\boxed{T_{(a)}=365+3585+1305=5255\ \text{trips}}$$
(b) Regression rates. Evaluate $r=1.225+5.1(\text{AUTO})+2.875(\text{INC})$ for each type, capping AUTO at 3. For example the 1-vehicle Low cell gives $r=1.225+5.1(1)+0=6.325$; the 2-vehicle Medium cell $r=1.225+5.1(2)+2.875=14.30$; the 3+-vehicle High cell $r=1.225+5.1(3)+2.875(2)=22.275$.
(b) Apply the regression rates cell by cell. Weighting each predicted rate by its household count: $$T_{\text{Low}}=11(1.225)+37(6.325)+7(11.425)+0=327.48,$$ $$T_{\text{Med}}=11(4.10)+240(9.20)+120(14.30)+4(19.40)=4046.70,$$ $$T_{\text{High}}=0+10(12.075)+41(17.175)+19(22.275)=1248.15.$$ The zonal total is $$\boxed{T_{(b)}=327.48+4046.70+1248.15\approx 5622\ \text{trips}}$$

Trips for each household type. The table below lists every one of the twelve cells (households × rate) under both methods.

Forecast trips by household type — (a) cross-classification / (b) regression
Vehicles/HH	Low (a)	Low (b)	Medium (a)	Medium (b)	High (a)	High (b)
0	22	13.48	33	45.10	0	0
1	259	234.03	1920	2208.00	130	120.75
2	84	79.98	1560	1716.00	738	704.18
3 or more	0	0	72	77.60	437	423.23
Column total	365	327.48	3585	4046.70	1305	1248.15

The regression rates used in (b) are 1.225 / 6.325 / 11.425 / 16.525 (Low), 4.10 / 9.20 / 14.30 / 19.40 (Medium) and 6.975 / 12.075 / 17.175 / 22.275 (High) trips/HH for 0 / 1 / 2 / 3+ vehicles.

(c) Comparison. The two totals differ by about 7 % (5255 vs 5622). The regression smooths the empirical rates onto a plane linear in AUTO and INC, so it does not reproduce the survey rates exactly — it raises the rate for the dominant 1-vehicle Medium cell (8.0 → 9.2) and thereby lifts the total.

Assumptions and limitations. The cross-classification method (a) assumes each category's observed rate is stable and transferable; it makes no functional-form assumption and reproduces the data exactly, but it needs a surveyed rate for every category (any sparsely surveyed cell gives a noisy rate) and cannot interpolate or extrapolate beyond the surveyed categories. The regression method (b) assumes trip rate is a linear, additive function of AUTO and INC; it fills empty or sparse cells and smooths sampling noise, and it extrapolates gracefully, but it imposes a rigid form (equal marginal effect of each additional vehicle, no AUTO×INC interaction) and can predict implausible rates outside the fitted range — which is exactly why AUTO is capped at 3. In practice cross-classification is preferred when the survey is large and the cells are well populated, and regression when data are thin.

Question 3 — forecast trips by method
Household group	(a) Cross-classification	(b) Regression
Low income	365	327.5
Medium income	3585	4046.7
High income	1305	1248.2
Zone total	5255	≈ 5622

Topic: Trip generation by cross-classification (category analysis) versus multiple linear regression.

Key relations: Trips $=\sum_{\text{cells}}(\text{HH}_{ij}\times r_{ij})$; regression rate $r=a+b\cdot\text{AUTO}+c\cdot\text{INC}$.

Why this works: Both methods multiply a per-household rate by the number of households in each category and sum. Cross-classification stratifies households into homogeneous cells and uses the empirical mean of each; regression fits a smooth surface through those same cells, trading exact fit for the ability to fill and extrapolate. They agree closely only when the regression plane happens to pass through the dominant cells.

Common pitfall: Forgetting to cap AUTO at 3 in the regression (the equation is only calibrated to AUTO ≤ 3), or weighting the regression rate by anything other than the actual household counts.

Source: Papacostas & Prevedouros, Transportation Engineering and Planning, Ch. 12 (trip generation); Ortúzar & Willumsen, Modelling Transport, Ch. 4.

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