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• if [1 2
p 1 p m] 0.750 then the relation between the indicators i and j is
ij 9 i 9 j
regarded as a dominance relation if min hp , p j a and maxhp , p j . a;
ij 9 i 9 j
ij 9 i 9 j
• if [1 2
p 1 p m] , 0.750 then the relation between the indicators i and j cannot
ij 9 i 9 j
be specified if min hp , p j . a.
ij 9 i 9 j
This first step determines the interval in which a is situated. If the dichotomization
threshold a is based solely on the relation between two indicators, it may occur that for
different pairs of indicators different respondents are dropped, such that the total number of respondents dropped exceeds 11.1 in case of equivalence or 25 in case of
dominance. If such is the case, the pair of indicators on which the decision about a is
based will not be declared an equivalence or dominance relation and the dichotomization threshold will be based on another pair of indicators.
In other words: ;i, j [I: iRj is declared equivalent iff [1 2 p 1 p m] 0.889,
ij 9 i 9 j
and a results in a solution that holds for at least 88.9 of the respondents. And ;i, j [I:
iRj is declared dominant iff [1 2 p 1 p m] 0.750, and a results in a solution that
ij 9 i 9 j
holds for at least 75 of the respondents. Still there is a problem when no equivalent relations are found. In that case, one needs to derive a maximum dichotomization
threshold a that decides about dominance. This value can also be derived, based on the
theorem of Chebyshev. If a is the maximum threshold, and one wants to obtain a
solution that holds for at least 75 of the population, the theorem can be written as 1 2 2
a m 0.75. As a consequence, a m 8. Because m is known, a can always be calculated.
The procedure, as described above results in a set of PCUs that can be presented in an implication scheme. A corresponding digraph indicates the possible stages between the
traditional and integrated states, while the orderings reflect the possible integration paths between both states.
4. Boolean analysis and the integration process: some examples
These examples discuss only one aspect of the integration process, namely the
4
changing gender relations. The respondents were confronted with the following
statements: a A woman always has to obey her husband.
b A woman can talk to another man, even when she does not know him. c When there are visitors, women preferably retire into a separate room.
d Apart from her domestic role, a woman has to play a role in religious, political
and social life also. e A woman has the right to work outside the house.
4
A more detailed analysis of these data can be found in Janssens 1995, 1997 and Janssens and Migratie 1995. Here, the discussion is restricted to the construction of the group norm and some considerations based
on the formal structure of this norm.
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287
The answers are recoded such that a 1 refers to an integrated and a 0 to a traditional attitude.
Here, two examples of the use of Boolean analysis are presented. The first example shows how the group norm of the female Turkish migrant population is determined. In a
second example, a comparison between structures is made. 4.1. Women and gender relations: the Turkish case
Table 4 presents the set R containing response patterns of 773 Turkish migrants,
based on a 5-item battery.
Table 4 Response patterns of Turkish respondents concerning gender relations
Pattern Frequency
Pattern Frequency
Pattern Frequency
00000 17
01010 5
10101 4
00001 39
10001 2
10110 1
00010 30
10010 2
11010 1
00100 5
11000 3
11100 1
01000 3
00111 170
01111 77
00011 185
01011 44
10111 42
00101 20
01101 5
11011 6
00110 8
01110 2
11110 1
01001 3
10011 17
11111 80
First, one evaluates the existence of an equivalence relation. For this purpose, the D -values, indicating the difference between both indicators, are calculated. If the
p
relation between the indicators i and j is considered as equivalent, this difference must be minimal. So the pair of items with a min D
is evaluated first.
p
According to the above Table 5, the relation between the indicators d and e results in a min D of 23. For this particular pair of items, [1 2
p 1
p m] 5 0.84. Because
p de 9
d 9e
0.84,0.889, the relation between the indicators i and j is regarded as non-equivalent. Not only their difference, but also their sum is the smallest. Because
a m 8 i.e. a 96.625, no equivalence relation is found and the dichotomization threshold a ,
min hp , p j, a ,50. The largest possible threshold that does not exceed 50 is 35.
de 9 d 9e
Consequently, the set of PCU9 is hac9, ad9, ae9, bd9, be9, cd9, ce9j. Fig. 4 presents the
Table 5 Possible PCUs and D -calculation based on the responses of the Turkish respondents
p
Pairs p
p D
S Pairs
p p
D S
ij 9 i 9 j
p p
ij 9 i 9 j
p p
ab 68
139 71
207 bd
21 455
434 476
ac 31
287 256
318 be
16 479
463 495
ad 10
521 511
531 cd
35 290
255 325
ae 9
543 534
552 ce
18 296
278 314
bc 65
250 185
315 de
50 73
23 123
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Fig. 4. Implication scheme representing the integration process of the Turkish respondents.
implication scheme based on the selection of PCUs. From the set of declared PCUs, R
x
can be derived. According to the definition of the integration process, Fig. 5 illustrates the stages between the traditional and integrated position. Fig. 6 shows the group norm
of the Turkish respondents R together with the set of the response patterns of the
x
group as a whole R . In this figure, all possible shifts between the traditional and integrated attitude are illustrated. The arrows printed in bold represent the unique
solution resulting from the application of Boolean analysis. Some statistics can be developed to describe this structure. The fit between the
obtained structure and the total population can be expressed in terms of the proportion of the population situated on the structure. In this case, the solution holds for 88.7 of the
population. As a result, a is accepted as the dichotomization threshold. A next
evaluation criterion is the number of response patterns involved. When R contains 27
different response patterns, nine are selected to describe the structure of the integration process. However, the number of respondents and the number of response patterns are
not the only bases for comparison. The obtained structure is a reproduction of a process at a given moment in time. One of the key problems behind this research is how the
Fig. 5. Digraph representation of the integration process of the Turkish respondents.
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289
Fig. 6. Comparison between R and R
among the Turkish respondents.
x
evolution from a traditional to an integrated attitude can be described. The answer is given by the number of possible paths between the terminal nodes of the graphical
representation. First, the number of paths can be used as an indication of the coherence of the group. Afterwards, the characteristics of the respondents situated on the different
positions and paths can be compared.
4.2. The generational evolution of gender relations In a second example, a comparison between different structures is made. Here, we
will restrict ourselves to a comparison based on group coherence. Group coherence refers to the variety of possible attitudes its members show. As a measure of
comparison, the number of paths between source and sink of the digraph representation of the integration process is used. Because the way a structure is derived is already
discussed in the previous example, only the results are compared.
For this purpose, three generations of Turkish and Moroccan Islamic women are selected and confronted with the same set of items: women migrated as adults during the
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Fig. 7. Comparison of gender relations between Turks left and Moroccans right ‘First generation adults’.
1960s and 1970s, women migrated as children during the 1960s and 1970s, and finally, women born and raised in Belgium.
In Figs. 7–9, the structure of the integration process, as obtained applying the Boolean analysis as it is described in the second part of this article, is presented. Between
brackets, the percentage of the population corresponding to that specific state of the integration process, is given. Where this structure is considered as the group norm, the
Fig. 8. Comparison of gender relations between Turks left and Moroccans right ‘First generation children’.
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Fig. 9. Comparison of gender relations between Turks left and Moroccans right ‘Born in Belgium’.
number of paths determines the tolerance or rigidity of that group. A substantial number of paths may also refer to a situation of uncertainty of a group trying to reach a new
balance within a changing environment. A comparison of these structures shows that the first generation that migrated as adults produces the most complex scheme among the
Moroccan respondents. Afterwards, the number of possible paths reduces. For the Turkish population, a similar evolution is detected, although the most complex structure
is found among the respondents who migrated as children. The diversity within both groups is remarkable, compared to the structures of the other categories.
The structural diversity, based on a comparison of the different generations, leads to the ‘spearhead’-hypothesis. From the obtained structures, one may deduce that the first
Moroccan generation who migrated as adults absorbed the shock of migration, where for the Turkish migrants it was the case for the generation who migrated as children.
According to the hierarchical representations, one may assume that the following generations achieve a balance, which is reflected in a more restricted number of possible
paths. These findings run parallel to the conclusions of previous research see Lesthaeghe and Surkyn 1994, who also refer to Bastenier and Dassetto 1993, that
describes a slowing down of the secularization process due to the pioneering work of the generation that made the transition from the native society to a society where new
networks and structures need to be developed. The retarded evolution of the Turkish minority can be explained by the spearhead function of the generation that migrated as
children. This proves that this pioneering work is not necessarily performed by the eldest segment of the first generation of migrants. The difference between both minorities may
partly be explained by the knowledge of the French language for those Moroccans living in Brussels, and the more isolated and protected environment in which the Turkish
women live. One may also expect that the diversity of the local migrant community and
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the networks developed by men who migrated earlier might play an important role as well.
5. Conclusions and discussion
