THE UNIVERSITY OF LETHBRIDGE
DEPARTMENT OF GEOGRAPHY
GEOGRAPHY 3235: Quantitative Models for Geographic Analysis
Fall 2006
Assignment 7 – Gravity Model
Goal
To apply the gravity model outlined by Taaffe and Gauthier (1973) to intermetropolitan
interaction in Canada using air passenger flows.
Instructions
1. Open the gravity model template in Excel. This file contains all necessary data
required to calibrate the gravity model. The data for each of five destinations have been
formatted and organized for use based on three sources: 1. Statistics Canada 1996 Census
of Population, Statistics Canada 1996 Air Passenger Origin and Destination Domestic
Report (Catalogue 51-204-XIB),1 and a third party source for distance between airports
(in kilometers) (http://www.webflyer.com/travel/milemarker/). The destinations lists for
each of the five cities were compared, and the 46 destinations that were common to each
city were selected for use2.
2. Beginning with the Calgary sheet, calculate the population product (PiPj) for Calgary
(i = Calgary) and all other cities (j= 1 to 46, except Calgary itself). How does the
population product relate to interaction between communities?
3. Calculate expected interaction (Iij) using the gravity model:
I ij  k
Pi Pj
d ijb
Set b=1 and set k to a value of 1/100,000. This value of k will scale down the interaction
to a more convenient magnitude and avoid the possibility of negative logarithms in
further analysis.
4. Following the model provided in Figure 3.2 of Taaffe and Gauthier (1973), plot the
expected interaction on the X-axis of an XY graph (scatter plot) against total number of
1
The air passenger origin and destination data indicate final destinations only, irrespective of the number of
flight segments or intermediate stops to change planes.
2
For example, Edmonton is listed in the data as a destination for all five cities thus it is included among the
list of 46 destinations. Fort McMurray, on the other hand, was on the origin/destination list for only four
cities (Calgary, Toronto, Vancouver, and Winnipeg) and so was not included among the destinations to
make the results comparable across the five metropolitan areas.
THE UNIVERSITY OF LETHBRIDGE
DEPARTMENT OF GEOGRAPHY
GEOGRAPHY 3235: Quantitative Models for Geographic Analysis
Fall 2006
passengers for each center on the Y-axis. (Don’t print it yet!) Describe the scatter plot
and compare your results to those in Figure 3.2 of Taaffe and Gauthier (1973).
5. Following the model in Figure 3.3 of Taaffe and Gauthier (1973), apply a trend line to
the scatter graph produced above (Chart > Add Trendline). In the dialog box under Type,
select Linear; and under Options, select ‘Display R-squared value.’ Print this graph and
include it in your report.
6. Discuss the R2 value in relation to your response to 4a above. What can you conclude
about the trend line in light of this R2? Identify the most prominent outliers and speculate
on the reasons for the large error.
These results give us cause to reassess our assumptions. Perhaps the relationship
between expected gravity model interaction and observed passenger flows is geometric
rather than arithmetic. Let’s digress for a moment and explore this hypothesis.
7. Calculate passengers as a proportion of the population product by dividing total
passengers for each city by the population product – we shall call this new measure
“passengers per capita.”3 Calculate log of distance and log of passengers per capita.
8. You will notice that the values produced are very small and appear as zeroes until the
6th or 7th decimal place. To remedy this, scale passengers per capita by 109 and find its
log. Following Fig. 3.4 of Taaffe and Gauthier (1973) as a model, plot the log of distance
against the log passengers per capita. Print this graph and include it in your report.
What do you conclude from this graph?
Steps 7 and 8 give us good reason to reject the naïve assumptions we made in the early
analysis. Maybe if we take logs of the measures we used earlier, we can get a better fit!
Shall we see for ourselves?
9. Calculate the log of Iij and log of total passengers. Following the model provided in
Figure 3.5 of Taaffe and Gauthier (1973), plot the log of the expected interaction on the
X-axis of an XY graph (scatter plot) against the log of the total number of passengers for
each center on the Y-axis. Fit a trend line to the scatter as you did in question 5 above.
Print this graph and include it in your report. Compare the R2 value with the results of
#5/6.
Step 9 does seem to fit better, doesn’t it? Now that we need to determine the friction
of distance coefficient to a model built of logs.
3
Passengers per capita is a convenient and short misnomer used in Taaffe and Gauthier (1973: 77). We are
really calculating total passenger flows per population product of the metropolitan O-D pair.
THE UNIVERSITY OF LETHBRIDGE
DEPARTMENT OF GEOGRAPHY
GEOGRAPHY 3235: Quantitative Models for Geographic Analysis
Fall 2006
10. Save your data file under a new file name. In a new worksheet, convert all cell entries
to values (Copy > Paste Special >Values). Remove the top row leaving a header row that
can be used for SPSS variable names. Start SPSS and open your ‘new’ Calgary
worksheet. Just to test the SPSS Regression procedure, recompute the regressions
performed in questions 5 and 8 using the trend line fitting procedure in Excel. You should
get the same values.
Now run a multiple regression with log observed passenger flows as the dependent Y
variable, log population product as x1 and log distance as x2.
Note that the b2 term is negative, why is this?
11. Repeat the multiple regression analysis for Montreal, Toronto, Vancouver and
Winnipeg to obtain friction of distance coefficients and R2 for each city. Enter these
values in the Summary Table in your workbook.
12. In general, what do the friction of distance coefficients and R2 values tell us about the
air traffic interaction between these five metropolitan areas?
13. How can you account for the individual differences in friction of distance coefficients
and R2 values?
14. What have you learned about the gravity model as a predictor of interaction between
these Canadian cities?