Urban Rank Size Hierarchy: Page 8

Traditional Rank-Size Relationships

	We have seen that the rank-size distribution can be described by 
the mathematical formula with which most geographers are familiar

	                 r_i ((p_i)^q) = K

where q and K are constants, r is the rank of the ith city, and p_i is the 
population of that city (Berry and Garrison 1958). The simple or 
restrictive rank size relationship, which, as we saw, assumes the value 
of q to be l, the value of K to be the population of the largest city 
(P), and is frequently expressed in terms of the value of P_1. (Carroll 
1982).

	                  P_1 = (P)/(r_i).

	The debate about the nature of the urban rank size relationship 
continues for a variety of reasons. First, an epistemological concern: a 
mathematical description of a phenomenon, regardless of the precision 
with which the formula describes the data, is just a description, not an 
explanation. Several researchers thus take care to point out that the 
labeling of the rank size relationship as a law or a rule is incorrect.  
(Chorley and Haggett 1967, Stewart 1958, 222,244).  Szymanski and Agnew 
(1981) disparage the rank size relationship as one of three examples of 
poor scientific study in geography. They argue that a statistical 
relationship, found only occasionally, and without theoretical 
foundation, has been elevated to the status of law. Secondly, there are 
concerns about the nature of the "straight line" itself. These concerns 
arise from several considerations. First is a concern for the margin of 
error which can be described graphically. Log log graph paper affords 
wide latitude in plotting data. Even when the plotted data exhibit a 
"good fit", it is striking how far off the mark are predictions of the 
ranks of specific urban areas. Anyone who has had the experience of 
plotting a rank size curve knows how slight a difference even several 
thousands of population can make in the placement of a dot on the graph 
in the high and middle ranges of many sets of urban data. Rapoport 
(1978,847) reminds us that some monotonically decreasing curve will 
describe just about any group of objects arranged according to size. 
Secondly, the existence of a straight line on a graph also depends, to an 
extent, on the portion of the urban distribution selected for analysis    
the lowest levels of the urban hierarchy frequently deviate markedly from 
the projected straight line (and the mathematical formula) and thus are 
frequently omitted from analysis because attention is given to the 
largest urban areas. The largest cities, as well, usually deviate 
significantly from the values predicted by the formula or the straight 
line on a graph. This is the phenomenon known as the primate city. 
Attempts to accommodate the lower portion of the curve often displace the 
upper portion and vice versa. Sahal (1981,294) has discussed this as a 
problem of results being sensitive to the origin of the independent 
variable. Applied to the simple rank size rule this means, for example, 
that instead of attempting to predict the size of the 100th largest city 
by dividing the population of the largest city by l00, we could, with 
equal logic, but with different results, predict the size of the largest 
city by multiplying the population of the city of rank 100 by 100. This 
follows, of course, from equation (7) above in which the product of these 
two factors is a constant. Rosing (1966) relates how Zipf accomplished a 
similar end by determining the population of the largest city (New York), 
not by census data, but by the computation of the y intercept of a 
regression line through the ranking of the 100 largest cities on double 
log paper.

	There is evidence that a full urban distribution may be "S" shaped, 
reflecting a growth or logistics curve (Stewart 1958, 245), or "J" 
shaped, and that the linear orientation of the data exists only in 
portions of the distribution. Parr and Jones (1983, 284 85) for example, 
describe the rank size distribution as a lognormal distribution if 
truncated at a sufficiently high level. Carroll (1979) offers the 
greatest clarity on these issues. He points out that the rank size 
distribution is more properly classified as a relation between two 
variables than as a probability distribution. The rank-size distribution 
can be derived from three kinds of the class of skew probability 
distributions. All of these probability distributions ---  the lognormal, 
the Pareto and the Yule, are J shaped and highly skewed in the upper 
tail. Each of the distributions is unique although the upper tails are 
quite similar, therefore, if the examination of a set of urban data 
excludes the smaller urban centers, any or all three distributions might 
apply. Carroll states

	We have seen that the law of proportionate effect results in the 
	lognormal distribution. This law with a lower threshold results in the 
	Pareto. And, this law with a lower threshold at which new units enter 
	in a constant rate gives the Yule distribution. 

Nader (1984) suggests that a non-logarithmic approach to rank-size data 
yields results superior to that of a logarithmic model. Perhaps such non 
linear distributions are polymodal and reflect mixtures of two or more 
rank size distributions, as has been reported for some distributions in 
geology (Krombein and Graybill 1965,108,126). Another perspective on this 
issue is that of Sahal (1981) who states quite simply that it is a 
general characteristic of simple laws that they do not hold over the 
entire range of variables. Clearly the rank size relationship is still 
open to interpretation, and it is the intent of this research paper 
to suggest even another method which can generate a distribution similar 
to the rank-size rule.