Barriers to the Spatial Diffusion for the
Demographic Transition in Western Europe
Jean-Pierre BOCQUET-APPEL* and Lucienne JAKOBI*
Introduction
The zones exhibiting a major discontinuity, an abrupt change of the
values taken by spatialized variable(s) generally delineate patterns and
probably correspond to the zones of sharp changes for the causal variable(s)
underlying the observed surface relief, or to barriers in dynamic processes.
An example of this can be two contiguous areas of mortality generated by
ecological and/or socio-economic factors. A rapid change of the factors on
the map corresponds to a sharp change of the mortality. Such a change is
observed today on the borders separating Western Europe from the Federation of Russia. The underlying causal variable(s), summarized by the ‘sanitary system’, changes sharply on the border zone (Labbé, 1993). Another
example is the Trieste-St Petersburg line (Hajnal, 1965) which separates
two huge zones of nuptiality, high in the east, low in the west.
To detect the zones of major discontinuity for one or several superimposed variables, in the same geographic frame, i.e. for surfaces of variables, this is the goal of a technique of descriptive statistics called
Wombling (Womble, 1951; Adams, 1970; Barbujani, Oden, Sokal, 1989;
Bocquet-Appel, 1993). One can summarize the Wombling analysis in the
following way: when the studied area is divided into quadrats, of (preferentially small and) equal size, an estimator quantifies the rate of change
in each quadrat for all the variables studied. To this end, the variables of
interest are represented as quasi-continuous surfaces, whose local slopes
measure the local rate of change. For each surface-variable, the rate of
change is expressed by a vector currently known in analytical geometry
* CNRS, UMR 152 Musée de l'Homme, Paris
118
SPATIAL ANALYSIS OF BIODEMOGRAPHIC DATA
as the gradient vector. The information contained in the different surfacevariables is combined by summing (or averaging) these gradient vectors at
the center of each quadrat. That gives an estimator which summarizes the
overall rate of change in each quadrat. A mapping of the estimator-values
gives a representation of the magnitude of the change in all points for the
set of surface-variables. The zone of major discontinuity are those contiguous zones where the overall rate of change is maximum relatively to
the other zones of the map. Wombling analysis has been extended to qualitative data (Oden, Sokal, Fortin and Geobl, 1993) and generalized for quantitative data, to t a k e into account the information redundancy of the
surface’s reliefs, due to the functional correlations between the variables
(Bocquet-Appel and Bacro, 1994). A test for the null hypothesis of no barriers has been proposed. It takes into account the functional correlation
between the variables and the spatial autocorrelation within surfaces, under
the hypothesis that the spatial autocorrelation is isotropic (Bocquet-Appel,
1994). The location of the zones of abrupt changes does not indicate, by
itself, change direction (positive or negative). That direction is of relative
value, depending on the observer’s location on the map, as to go up or
down a hill slope will depend of the walker’s position. When the zones
of abrupt changes have been localized through the wombling technique, it
is necessary to look again at the data on the map.
In this paper we will use the Wombling analysis to identify, at the
scale of western Europe, the zone of abrupt changes which delineate the
main patterns brought about by the demographic transition, for four dates.
The data are those of the Princeton project (Coale and Treadway, 1986).
They represent indexes of fertility and nuptiality gathered at the district
or department level.
Statistical method
Generalized wombling: a recall
From the vectors of data on the map, where each vector element represents the value taken by a variable at a point, each surface of variable
is reconstituted by means of a spatial interpolator (for instance Akima,
1978; for a critical review see Briggs, 1992). Each surface is subdivided
into quadrats, each quadrat is approximated by a first order polynomial
fi(X), where the subscript i designs the i-th surface-variable. If there is no
value structurally possible at a location (i.e. in a lake or an ocean) the
interpolated value is eliminated from calculation. An estimate of the overall
rate of change, for Q surface-variables, located at point X= (x,y), the center
of a quadrat, where x and y design respectively the longitude and the latitude, can be:
W’(X) = a’R-‘a + b’R-‘b
(1)
BARRIERS TO THE SPATIAL DIFFUSION
119
where a = (a1, a *,..., , ai ,..., aQ)t, b = (b,, b, ,..., b ,,..., bQ) t, R-l represents respectively the vector (1 x Q) of the partial derivatives extracted from the
gradient vector Vf;= 2 (X), $(X)] = (ai, bi) of the polynomial approximating
t
the quadrats and the information matrix. This latter is the inverse of the
correlation matrix expressing a functional relationship between the variables (Bocquet-Appel and Bacro, 1994; Bocquet-Appel, 1994). By a direct
analogy with the usual definition of the variance (a sum of squared deviations from the mean), the quantity w(X) represents, in the univariate case,
a deviation (squared) between the rate of change of the surface-variable
observed at the point X, and the rate of the mean surface. This reference
surface being flat in every direction, its rate of change is null. Therefore,
one may call wZ(X”) the local spatial variance (Bacro and Bocquet-Appel,
1994), which is simply, in univariate data, the sum (squared) of the elements of the gradient vectors at point X, and in multivariate ones, a sum
of univariate variances weighted by their information redundancy.
The detection of a zone of major discontinuity can be done by applying Barbujani, Oden and Sokal’s algorithm (op. cit.) on those W 2 (X)values. This algorithm links together pairs of quadrats: i) for which the
w(X)-values are located in the zone α = 5% of the upper part of p(X)distribution of the map; ii) which are separated by an another quadrat (a
‘bridge’) located in the zone β = 10% of the v(X)-distribution. These contiguous quadrats, as opposed to single value, represent zones of major spatial discontinuity in the continuum of surface-variables. They are also called
‘barriers’. The individual influence of the variables in the barriers can be
estimated (Bocquet-Appel and Bacro, 1994). The barriers detected by Wombling analysis are not absolute values, but are relative to the area under
consideration. The higher house of a village is certainly not the highest
of the region.
From the four indexes gathered for the Princeton project, two will be
analyzed here: a standardized index of fertility for married women, $, and
the proportion of married women within the childbearing span (15-50), I,,,.
Both indexes vary between zero (no children; no married women) and one
(maximum fertility of the Hutterite; all women married). If the rate of illigitimate birth is negligeable, the overall fertility index is simply:
Z,= Ig x I,,,. In fact, I, is hardly usable because, in mixing the causes for the
overall fertility, it darkens the direction of the underlying processes. As
noticed by Coale and Treadway (1986) the same value for If, can be produced by two radically different causes: high nuptiality and low fertility
or conversely. We have decided to analyze, the indexes of marital fertility,
I, - the witness of the demographic transition -, and of nuptiality, Im, separately and together. The avalaible data did not start at the same date for
all the nations. The pre-transition data for missing dates within a nation,
were estimated by taking the first (the oldest) pre-transition value observed
120
SPATIAL ANALYSIS OF BIODEMOGRAPHlC DATA
in those data. Because the missing data for the Federation of Russia and
Yougoslavia cover the whole span of the time period analyzed in this study,
they were excluded. For each of the four years retained: 1870, 1900, 1930
and 1960, the number of data points on the map are respectively of 692,
692, 678 and 634. Each quadrat, subdividing the European map, was of
size 0.72° x 0.72°.
The demographic transition in Western Europe revisited
We recall here the main conclusions of the Princeton project. The
demographic transition took place on ancient regional patterns of nuptiality,
born long before the XIXth century and which remained unchanged until
the 1930’s. They disappeared in the 1960’s, due to the marriage boom.
Superimposed on those patterns of nuptiality, fertility declined in discrete
cultural areas (Watkins, 1986), typically delimited by linguistic boundaries,
observed in Germany (Knodel, 1974), Belgium (Lesthaeghe, 1977), and in
Switzerland (Van de Walle: cited by Anderson, 1986). Underlying to those
cultural areas, the hypotheses for the generating mechanism of the transition
that have been set forth, are the following: i) the diffusion of a cultural
innovation in the line of the work of Hagerstrand (1967), more rapid in
an homogenous area than between two areas separated by a boundary (Knodel, 1974; Lesthaeghe, 1977); ii) the conformism of the individuals to the
regional fertility norms and attitudes (Lesthaeghe, 1977) eventually generated by or covarying with; iii) the social structure of the geographical unit.
The data of the Princeton project do not permit to test these hypotheses
(not alternative anyway). Only less aggregated data might give the answer
which, generally, can be obtained from studies at the local (or even microlocal) scale. Such is the case with the data from Taiwan, on which the
diffusion and its dynamic are currently tested (Montgomery and Casterline,
1993), or from Scotland for which the influence of the economic structure
on immigration and the marital fertility and nuptiality indexes were
measured (Anderson and Morse, 1993a, 1993b).
One can perform a wombling analysis in order to answer (or to confirm the answers) of the following issues: from the observed patterns of
marital fertility (Ig)g) at the four dates, what were the geographic direction(s)
taken by the big-Lang of the transition, at the European scale? How long
do, the nuptiality patterns remain stable? Finally, in the overall fall of fertility, what was the relative influence in the patterns, of the change due
to the fertility, compared to that due to nuptiality?
The changes in the patterns of marital fertility (Id
Figure 1 shows the zones of major discontinuities at the four dates.
We will briefly describe their main features.
BARRIERS TO THE SPATIAL DIFFUSION
I
I
i
In 1870, a large area of controlled fertility (in short: ACF), broadly
outlining two thirds of France, was defined, in the east, by a long zone
of change, from Belgium running down the 1870 political boundaries of
France, i.e. excluding Alsace-Lorraine. Around Lyon the zone of abrupt
changes splits into two branches. One goes into a south-west direction,
he southern length of the Massif central, the Garonne Valley,
- -.I--. . .-..I I
a region hors &de
m discontinuit6
m
r&gion hors &ude
I I
Figure 1. Zone of abrupt changes observed in Western Europe
for the index of marital fertility (LJ) at four dates
121
122
S P A T I A L A N A L Y S I S O F BIODEMOGRAPHIC
DATA
and running above the Bearn and the Basque countries to Bordeaux. This
branch delineates the ACF in the south-west. The other branch follows the
west side of the Alps, to Nice. The large ACF is delineated in the northwest, by a zone running along the coast of Normandy, separating England
and Brittany from the continent. Within the ACF, the average value for Z,
is 4.45. Directly outside the ACF, I, takes the values, respectively, of 7.6-6.
in the north-east and south-east (Holland to Savoy), 6.8-6.2 in the northwest (England and Brittany) and 5.5-5.3 in the south and south-west, Another somewhat erratic zone of change starts from a Strasbourg-Stuggart
line in Germany, down the Swiss-Italian border, and covers a large part
of Switzerland. Outside that zone (Alsace, west of Switzerland) $ takes
the values of 7. and 2 7.5 on the zone. Zones of Ix’s abrupt variations,
which do not correspond to any ACF, can be observed in Sweden (I, = 7.6
vs 7.3), in the west of central Poland (I,= 7.1 vs 6.5) and the south of
Italy (/,=6.4 vs 7 in Sicilia). Other minor I, variations can be observed
within the major zones of change, delineating secondary patterns, such as
in the Massif Central, in the south-west (I,= 5.18 vs 4.45 in its north and
5.5 in its south), or in Gironde Valley (I,= 3.2). By 1900 (figure lb), this
long flanking zone in the east of France, disappeared in Lorraine and in
southern Garonne. The north-west boundary, which separated England from
the continent had vanished too, indicating that the transition was operating
on this side of the Channel. Other zones appeared in the east, in the Netherlands vs Germany ($=7 vs 6.2), in the north of Bavaria close to Thuringy,
and Germany vs Tcheki and also in the very east, along Poland’s borders.
By 1930, the flanking zone of France did not exist any more. Instead,
a vast zone of homogeneity lay from the west coast of the Atlantic to the
east border of Germany, except in Bavaria, and in the north-west of England. Yet three pillars of the past flanking zone still remain, located respectively, in the north (Belgium and Holland), in the center of Switzerland
and northern Italy, in the south, across the Basque Country. Far in the
east, a zone of low fertility appears vertically (from Poland to Bulgaria).
On this zone, from north to south, $ takes the values, in the west, of: 2.7,
3.3, 4.1, and in the east, of: 4.4, 4.0, 4.5. Another zone separates Greece
(1,=5.3) from Italy (6-6.5). By 1960, the north and central pillars of the
1870 major zone of transition were still persistent. Other minor zones appeared, vertically in Finland, Ireland (with an exceptionnally high Ig-value
of 6. I vs 3 in England), the center of Italy (I,= 3.1 in the north vs 4.1 in
the south), in Albania, and along Portugal.
At the continental scale, if the transition was a diffusion process even
through the linguistic boundaries, from 1870 to 1900, its directions were
triple: north-westward, towards England; north-eastward, towards Germany
passing through Lorraine; and southwards, througth Garonne Valley and
Lyon, towards Provence and the Mediterranean side of Catalunya in Spain.
From 1900 to 1960 the process went farther and farther from its starting area.
However two ancient pillars at high Ig values still exist, respectively located
at the Holland-Belgian borders and in Switzerland. By 1960, transition zones
BARRIERS TO THE SPATIAL DIFFUSION
123
existed in Ireland, Portugal and Albania. Linguistic boundaries (Flamish,
German, Italian and, except in Garonne Valley, the Oc language) may explain the observed patterns.
The changes in the patterns of nuptiality
(I,,,)
Figure 2 shows a long zone of change, laid horizontally on western
Europe. One end of this zone starts from eastern Austria, goes on to the
Switzerland and Italian border, discontinuously crosses France, goes down
to Portugal through central Spain via the Basque country, This long zone
of change stayed at the center of Europe until 1900 and then, by 1930
remained at state of trace. It later disappeared. A sampling pair of pointvalues drawn on each side of the zone gives, for 1870 and 1900, respectively, the average /,-value of 5.10 (N= 11) and 5.28 (N=7) above the
zone and of 5.54 and 5.47 below. A first pattern could be observed at the
continental scale, until 1900, which divided Europe from east to west horizontally with a nuptiality index in the north lower than in the south. Other
regional zones of abrupt changes were, from north to south, the following:
l
In Scotland, in 1870, 1900 and 1930. The average Im-values were
respectively: 3.7 (N=2), 3.75 (N =2) et 3.6 (N = 1) in the north, and 4.35,
4.1, 4.0 in the south. This zone delineates a known pattern at low nuptiality
in Scotland compared to that of England (Anderson and Morse, 1993a,
1993b);
l
In Belgium, from 1870 to 1960. There was a persistancy in the pattern of nuptiality, through the whole time-period, from north to south across
the country, with I m -values of the order of 4.9-5.7, 4.5-5.4, 5.0-6.9, i.e.
producing alternating patterns of mean, low and high nuptiality;
l
In the basque Country. The average Im values, for the years 1870
and 1900, were respectively of 5.4, 5.5, in the north and 6.0, 5.7 in the
south, i.e. higher in the south than in the north. Then, for the years 1930
and 1960, the trend was reversed. The average I m -values were higher in
the north (6.0, 6.2) than in the south (5.6, 5.3);
l
In Italy, for the years 1930 and 1960. It was located along the west
side of the country, perhaps separating the Italian boot, at high nuptiality
(5.-5.6), from Sardaigne (4.4-4.8).
A demographic cause for the variation of nuptiality patterns, is the
imbalance for the sex-ratio of the spoused canditates, whatever the underlying cause of that imbalance (migration due to market labor for males,
war, etc.). The cause identification depends on more local studies, at the
regional scale, such as in Scotland, where it has been demonstrated that
the very low nuptiality index, comparatively to England, was brought about
by a high out-migration for males (Anderson and Morse, 1993a, 1993b).
Contrary to the zone of changes delineating the patterns of marital fertility,
which seems to be clearly connected and forming parts of a common demographic phenomenon, those of nuptiality are more scattered, more regional,
124
SPATIAL ANALYSIS OF BIODEMOGRAPHIC DATA
showing alternating zones at high and low
Im-values, except for the long
zone of changes across Europe, which indicates a broad north-south trend.
The geographic setting of these regional patterns, as identified by their
barriers, was approximately the same in 1870 and in 1900. However it
changed in 1930 and again in 1960, becoming less and less differentiated.
Figure 2.1 Zone of abrupt changes observed in Western Europe for the
index of nuptiality (Im) at four dates
BARRIERS TO THE SPATIAL DIFFUSION
125
The changes in overall fertility, viewed as the changes in 1, and Im simultaneously
It is shown in figure 3. From 1870 to 1960, the zones of discontinuities
which delineate the patterns, are globally more similar to those exhibited by
the index of marital fertility, than by nuptiality. This simultaneous analysis
II
@
discontinuit6
\
-
m discontinuit6
r’? région hors Etude
,:
1
:,
‘:
II m region hors BtUde
Figure 3 . - Zone of abrupt changes observed in Western Europe for the
index of marital fertility (/9) and nuptiality (Im) taken simultaneously, at
four dates. The change in marital fertility represents, respectively, 72%,
87%, 80% and 72% of the overall change in fertility at each date during
the demographic transition
126
SPATIAL ANALYSIS OFBJODEMOGRAPHJCDATA
TABLE I.- CONTRIBUTION
OF MARITAL FERTILITY
(IR)
AND NUPTIALITY
(Im)
IN THE
GLOBAL CHANGE FOR OVERALL FERTILITY DURING THE TRANSITION
%
before 1789 (?)
1870
1900
1930
1960
.250 (?)
.719
.869
.804
.721
Im
.750 (?)
.281
.131
.196
.278
of I g and I m as generating causes for I f, permits to show their relative
influence on I f, which would be unobservable if I f were directly used instead. This was, for example, the case in 1 870 While the patterns of fertility were mostly differentiated by the long zone of change in marital
fertility, which separated 3/4 of France from the rest of Europe, within
France other zones of abrupt differentiation existed, but due to nuptiality,
for instance in the west (Brittany). Other zones could be observed in Scotland and in Poland, in this last country due to fertility. The influence of
each of these two causes, marital fertility and nuptiality in the global
change in fertility during the transition, as measured by the fraction represented by each of these variables in the barriers, is given in table 1.
At the western European scale, in 1870, 72% of the change in the overall
fertility was due to the decline of marital fertility, and 28% of the pattern
variation of nuptiality. By 1900, all of a sudden, in 30 years at most, the fall
of marital fertility represented 87% of the global change of the overall fertility,
while nuptiality represented 13% of the change. In 1930, the decline of nuptiality still represented 80% of the demographic change, showing that this process maintained its velocity. By 1960, its contribution disminished at 72%,
while 28% was due to the change in the nuptiality patterns.
At the continental scale, the fertility side of the demographic transition can be analyzed as the superposition of two demographic processes,
running at two different time scales. The first one is short term, and is
concerned with the spatial diffusion of birth control, the true master of the
transition. The second one might be at a longer term and is not concerned
with diffusion. It is the nuptiality pattern, which seems to divide Europe
into two broad regions, from an unknown date in the past until 1900, plus
local variations. This pattern is described by a line running obliquely
through Europe from east (Poland) to west (Portugal). The nuptiality index
is lower above this line (in the north) than below (in the south).
Concluding remarks
In the big bang scenario posited here, the direction taken by the transition was three folds: north-west, north-east and south. The immobility of
a long wave of change, along the east border of France, perhaps until 1900
BARRIERS TO THE SPATIAL DIFFUSION
127
and even WWI, is striking. This zone corresponds to several boundaries
(linguistic, geographic - the Alps - and political). From 1870 to 1900, in
a single generation, one witnessed a fulgurant increase, an outbreak, in the
fall of marital fertility in the pattern variation of the overall fertility. Although modest, nuptiality was again playing a role, from the 1960’s only.
The hypothesis assumed here is that the demographic transition was,
basically, a continuous diffusion process, from one or several starting
points, located not too far from each other. It excludes a multiple seed
scenario like pollen diffused by the wind. But such scenarios might be
easily introduced, using a hierarchical diffusion model (Robson, 1973). In
this last approach, cultural innovation diffusion is not a continuous spatial
process over the area, but a discontinuous one, which follows the stream
determined by the numerical hierarchy of urban entities and to which a
local diffusion term is added. The main technical difference between both
scenarios to fit the data - the Big-bang and the hierarchical -, is that while
the first corresponds to only one equation diffusion system (Okubo, 1980;
Murrray, 1989), the other corresponds to several.
The goal of wombling analysis is to identify the zones of major discontinuities. They typically indicate pattern boundaries. It is not to identify
their causes. The generating causes for the change, in the surface response
of variables, are most probably varying in the same localities. They deserve
all the researchers’ concern. More local studies, on less aggregated (causes)
data, should be preferentially carried out there.
Wombling analysis by itself cannot say to what kind of process the
change could be attributed. If the hypothesis is, for exemple, diffusion,
then the technique will localize its main obstacles, will help to identify
its directions. Yet, Wombling analysis will give a measure of the contribution of the different variables to the observed spatial change. A point is
in order here concerning the sampling of the data. If they are vectors (i.e.
a set of values), they should be sampled on the map approximately equally
spaced. Fortunately, this is generally the case for the biodemographic data,
which are gathered for administrative purposes. Otherwise, grave distorsions may result, in particular the identification of rupture zones in regions
where, simply, the sampled localities are more numerous than in the others.
Several ways exist to bypass this drawback by reaggregating the data on
a regular grid or, if the variables are numerous (say > lo), by sampling
the variables individually, each on a different set of localities in the same
region.
Acknowledgments
The manuscript has been prepared by Marie-France Leroy. Parts of
the artwork is by Danièle Fouchier. Documentary work on Germany has
been done by Karl-Ludwig Liebs.
128
S P A T I A L A N A L Y S I S O F BIODEMOGRAPHIC D A T A
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