CHAPTER 13

# Erosion of the Size-Density Relation

In Chapter 12 it appeared that slopes more negative than -2/3 were the result of including cities in the analysis. In this chapter we examine the reverse condition: slopes less negative than -2/3, slopes tending toward zero (which is to say tending toward no slope or relation at all).

At the conclusion of the article in which we presented the results of the last chapter, we added the findings shown in the next table, comparing our latest slopes with the earliest ones for which we could obtain data.[1] We were attempting to see if other countries might have experienced the kind of "erosion" of a -2/3 size-density slope which we saw earlier in England (Chapter 7).

 ``` Nation/Yr n b p Belgium 1816 9 -0.27 .035 1961 9 -0.13 <.001 France 1872 90 -0.14 <.001 1962 90 -0.29 <.001 Italy 1861 13 -0.14 .087 1961 19 0.32 <.001 Netherlands 1849 11 -0.12 .067 1960 11 -0.09 .009 Portugal 1878 19 -0.64 .832 1960 18 -0.44 .002 Spain 1833 48 -0.68 .931 1960 50 -0.42 .002```

It is evident that in every case the earlier slope is closer to -2/3. The earlier slopes for Italy, Netherlands, Portugal and Spain, in fact, conform to the hypothesis by the p > .05 criterion. Even in the case of France, the p-value improved from .00000005 to .00008, suggesting that, if we had still earlier data, départements  which were intended to be uniformly small still might have shown some conformity to the size-density law.

We developed a tentative explanation for the tendency of slopes to erode toward zero. If boundaries remain fixed while the population experiences redistribution, then any relation which once existed will, obviously erode; you can't change the independent variable while holding the dependent variable constant and expect the original relation to persist.

More particularly, suppose we have a set of territorial divisions which do fit the size-density law. Now fix the boundaries of the territorial divisions once and for all. This could happen through historical inertia (e.g., England) or technological change (e.g., the automobile and U.S. counties). Under normal processes of population growth high-density places will tend to become even more dense. This is the law of proportionate effect: nothing succeeds like success, the rich get richer, big population centers draw more people than small ones (gravity model). Ordinarily such growth would lead to further subdivision (down to some technologically/economically minimal size), but here no further subdivision occurs. The result is erosion of the slope.

This is easily simulated An initial group of 20 data points, scattered around a regression line (exact-fit points were computed, then a random number was added or subtracted from the vertical coordinate).

Next we move data points to the right. The arrow points have the same vertical coordinates as the original data points (no change in area), but the horizontal coordinates have been increased. The amount of increase is proportional to the original value, plus a positive or negative random number. The new data points resulting from the rightward moves are shown in green.

Finally, we compute a new regression line for the generated points. The slope has eroded has clearly eroded toward zero.

As originally put forward, the erosion hypothesis was only suggested, along with the data in Table 13-1 confirming that earlier slopes tended to be more negative. In 1982, with the help of my wife Karen,[2] I conducted a direct test of the erosion hypothesis. The question was: Does the slope become less negative as population becomes more concentrated?

We needed area and population data for stable sets of territorial divisions over significant periods of time. Ideally, stability would mean no change whatever in the boundaries of divisions throughout the period of study. We found such data[3] for

Austria (9 Bundesländer:  1910-1961)
Belgium (9 Provinces: 1816-1961)
England and Wales (54 Counties: 1801-1961)
France (90 Départements:   1872- 1962)
Netherlands (11 Provinces: 1830-1960)
Switzerland (24 Cantons: 1850-1960)
United States (48 States: 1900- 1970)
We also included studies[4] of three nations where the number of changes in territorial divisions was small relative to the number of divisions
Italy (14 Regioni:  1861-1911; 15: 1921-1951)
Portugal (19 Distritos:  1878-1920; 20: 1930-1960)
Spain (48 Provincías:   1838-1897; 50: 1900-1960)
We computed size-density slopes for each of the years indicated in the accompanying table:
.
 ```NATION/yr S.D. Slope NATION/yr S.D. Slope AUSTRIA NETHERLANDS 1910 1.593 -0.633 1830 0.521 -0.123 1923 1.562 -0.630 1840 0.510 -0.125 1934 1.537 -0.648 1849 0.491 -0.122 1951 1.422 -0.709 1859 0.487 -0.129 1961 1.411 -0.724 1869 0.490 -0.134 1879 0.511 -0.128 BELGIUM 1889 0.551 -0.123 1816 0.393 -0.272 1899 0.577 -0.123 1831 0.469 -0.219 1909 0.597 -0.124 1846 0.611 -0.099 1920 0.620 -0.131 1856 0.599 -0.094 1930 0.660 -0.115 1866 0.611 -0.084 1940 0.679 -0.114 1876 0.653 -0.083 1950 0.687 -0.109 1880 0.658 -0.083 1960 0.815 -0.098 1890 0.681 -0.077 1900 0.710 -0.078 PORTUGAL 1910 0.731 -0.084 1878 0.795 -0.644 1920 0.739 -0.094 1890 0.780 -0.638 1930 0.763 -0.104 1900 0.784 -0.634 1947 0.787 -0.121 1911 0.776 -0.631 1961 0.803 -0.133 1920 0.783 -0.620 1930 0.830 -0.604 ENGLAND & WALES 1940 0.837 -0.600 1801 0.144 -0.604 1950 0.855 -0.585 1811 0.149 -0.560 1960 0.898 -0.548 1821 0.151 -0.534 1831 0.157 -0.475 SPAIN 1841 0.164 -0.436 1833 0.533 -0.676 1851 0.173 -0.392 1850 0.559 -0.671 1861 0.187 -0.337 1857 0.564 -0.659 1871 0.198 -0.299 1877 0.583 -0.624 1881 0.216 -0.261 1887 0.589 -0.614 1891 0.230 -0.245 1897 0.610 -0.594 1901 0.231 -0.249 1900 0.608 -0.606 1911 0.243 -0.244 1910 0.618 -0.596 1921 0.249 -0.234 1920 0.647 -0.548 1931 0.253 -0.229 1930 0.682 -0.507 1951 0.258 -0.243 1940 0.711 -0.478 1961 0.251 -0.288 1960 0.815 -0.419 FRANCE SWITZERLAND 1872 0.609 -0.426 1850 0.763 -0.615 1881 0.634 -0.423 1870 0.792 -0.610 1891 0.649 -0.418 1880 0.778 -0.639 1901 0.692 -0.386 1888 0.818 -0.586 1911 0.725 -0.367 1900 0.843 -0.525 1921 0.755 -0.348 1910 0.855 -0.554 1931 0.794 -0.326 1920 0.857 -0.514 1946 0.792 -0.320 1930 0.860 -0.492 1954 0.838 -0.308 1941 0.847 -0.487 1962 0.848 -0.292 1950 0.864 -0.470 1960 0.917 -0.413 ITALY 1861 0.540 -0.142 UNITED STATES 1871 0.520 -0.177 1900 1.607 -0.477 1881 0.518 -0.159 1910 1.474 -0.526 1901 0.526 -0.125 1920 1.454 -0.531 1911 0.543 -0.101 1930 1.470 -0.518 1921 0.593 -0.042 1940 1.438 -0.526 1931 0.579 -0.043 1950 1.422 -0.533 1936 0.577 -0.039 1960 1.393 -0.541 1951 0.573 0.005 1970 1.403 -0.539```

There are a variety of concentration measures which might have been computed[5] We chose the standard deviation of log D (referred to in the table as S.D.). It seems to have escaped the attention of most authors[6] that any measure of concentration can in the end be shown to be based upon this mundane statistic. A standard deviation of zero indicates no concentration (even densities), with increasing values indicating increasing concentration. The standard deviation suffers from the fact that we cannot appropriately compare it across sets with different numbers of divisions. But it is perfectly suitable for our purposes.

Figs. 13-1 through 13-10 show historical values for slopes and standard deviations. To the left of each historical graphs is a scatter diagram showing the relation between the two. The size-density slopes are shown in green; the standard deviations (concentration measures) are shown in red. It will be obvious that the two are strongly related (increasing concentration leading to erosion of the slope). A following statistical table summarizes the results of our analysis.

FIG. 13-1. AUSTRIA     (see commentary below)

FIG. 13-2. BELGIUM

FIG. 13-3. ENGLAND & WALES

FIG. 13-4. FRANCE

FIG. 13-5. ITALY

FIG. 13-6. NETHERLANDS

FIG. 13-7. PORTUGAL

FIG. 13-8. SPAIN

FIG. 13-9. SWITZERLAND

FIG. 13-10. UNITED STATES

The following table summarizes several statistics which confirm what the visual presentation has already shown.

 ```Nation r2 p{r2=0} rho p{ρ=0} tau p{t=0} Austria .97 .0024 .90 .0719 .80 .0500 Belgium .50 .0040 .09 .7423 .03 .8695 England & Wales .86 .0001 .94 .0003 .87 .0001 France .99 .0001 .99 .0030 .96 .0001 Italy .83 .0007 .82 .0209 .61 .0218 Netherlands .74 .0001 .71 .0110 .55 .0062 Portugal .96 .0001 .75 .0339 .61 .0218 Spain .97 .0001 .99 .0010 .97 .0001 Switzerland .88 .0001 .92 .0037 .82 .0005 United States .98 .0001 .92 .0146 .82 .0044```

The first is a test of significance, against the null hypothesis, for the linear correlation coefficient[7]. In some cases, e.g. Belgium, the relation is clearly not linear. Nothing in our erosion hypothesis suggests that it should be; we only expect a monotonic increase of the one with the other, i.e., an ordinal relation.

For this reason two additional test statistics are provided, Spearman's rho, and Kendall's tau. Hays and Winkler[8] review the advantages and disadvantages of each as an alternative to r-square, concluding

...each index supplies information of importance about the general monotone relation that may exist between variables, quite irrespective of the true value of the correlation coefficient.... For moderately large samples, ... tau seems to provide a better test of the hypothesis of no association....
Austria, Fig. 13-1, interestingly, shows support for the hypothesis but in the reverse historical order expected. Over time the population has become less concentrated, and the size-density slope has become steadily more negative, even passing the theoretically expected - 2/3 value. A similar pattern shows up in the United States, Fig. 13-10: here the decreasing concentration reflects increased settlement in large states such as California. This may be more a problem of the level of measurement of area and density, i.e., the state vs. the county.

The historical charts for the other nations show remarkable conformity to the notion that size-density tends to erode toward zero over time. The chart for France, incidentally, projected back another century, would produce a slope slightly more negative than -1/2, supporting our earlier conjecture that — even when a deliberate effort is made to produce uniformly size divisions — the size-density relation tends to surface.

NOTES:

[1] B. R. Mitchell, European Historical Statistics, New York: Columbia University, 1975.

[2] The study was published as G. Edward Stephan and Karen H. Stephan, "Population Redistribution and changes in the Size- Density Slope", Demography, 21:35:40, 1984

[3] The data for Austria, Belgium, France, Netherlands and Switzerland came from Mitchell, Op. Cit.; England and Wales data were from B. R. Mitchell and H. C. Jones, Second Abstract of British Historical Statistics, Cambridge: Cambridge University Press, 1971; United States data was from the census.

[4] Mitchell, Op. Cit.

[5] I have reviewed a number of these measures and shown that each can be given a probability interpretation: G. Edward Stephan, "Measures of Population Distribution: A Probability Interpretation", Western Sociological Review, 7:3-10, 1976.

[6] except notably Maurice G. Kendall and Alan Stuart, The Advanced Theory of Statistics (3 vols.; 2nd ed.), New York: Hafner, 1963.

[7] (this will be in the methodological appendix)

[8] William L. Hays and Robert L. Winkler, Statistics: Probability, Inference and Decision, 849, New York: Holt, Rinehart and Winston, Inc., 1971.