A Boolean Approachto the Measurement ofGroup Processes and Attitudes.doc

A Boolean Approach to the Measurement of Group Processes and Attitudes- subject: comparing group attitudes- problem: creating standard for comparison- method: Boolean analysis- illustration: process of integrationINTEGRATION PROCESS1. Individual evolutiontraditionalintegrateddepending on flexibility not predictable2. Within a groupsum of individual behaviourdepending on coherence of the group (norm)Integration is a one-way learning process, whereby somebody gradually closes the gap between the native culture of the land of origin and the culture of the society he or she lives in, depending on the adaptability of the individual and the group he or she belongs to.Formalisation of integrationPossible attitude changes on individual levelLet I = i1, i2, .inbe the set of n indicators,with ij I: ij 0, 1,Suppose:a person p1, characterised by (i1, i2, .in ),a person p2, characterised by (i1, i2, .in ),When we are interested in the behaviour of p12 possible reactions- p1 does not change his attitude since he only agrees with the common attitudes the attitude of p1 can be described as (i1, i2, .in ) (i1, i2, .in ),- p1 does change his attitude influenced by the attitude of p2 the attitude of p1 can be described as (i1, i2, .in ) (i1, i2, .in ),Ip1p2i11111i20101i30101i40000Table 1. Range of possible attitude change of respondent p1As a result: p1 may show the same pattern 1000,or p1 may evolve towards pattern 1110.step 1: p1 can be represented by 1000step 2: p1 can be represented by 1100p1 can be represented by 1010step 3: p1 can be represented by 11101100100011101010Figure 5.1. Possible evolution attitude p1 000010000001110010101110111100000000100000011100101011101111100000001000100111001010111011110001000000000001110110111111111111000000100000001100111011101111101000001000000010001010111011111110000010000000110010101110111111110000100000011100101011101111Table 2. Completing the structure1111101111011110100110101100000110000000Figure 2. Digraph integration processBoolean analysis1. Boolean minimisationSuppose a set of Boolean expressions C, each expression being a complete fundamental product.If two complete fundamental products c1 and c2, differ according to one generator, this generator can be removed to create a simpler expression q.This new expression q is included in the two complete fundamental products: q c1 = c1 and q c2 = c2.abcd and abcd yields abcabcd and abcd yields abc abc and abc yields abMinimisation step: Canonical projection (Flament, 1976)If no other Boolean expression r, for which r q = q and r c1 = c1 and r c2 = c2 can be found, q is called a prime implicant or ultimate canonical projection (PCU).Every set of Boolean expressions C can be reduced to a set of prime implicants or PCUs.Boolean analysis2. Boolean implicationSuppose a set of response patterns (Boolean expressions) R,R = pi | pi = pattern i with a frequency fi R0 = pi | pi = pattern i with a frequency fi = 0 R* = pi | pi = pattern i, with a frequency fi 0 If one takes R0 as the starting set for Boolean minimisation,the set of PCUs covers all non-existing patterns.If ab is a PCU, then the subpatterns ab, ab and ab are found as subpatterns of some complete fundamental products of R* a implies b or: a bBoolean implication introduces partial order between indicatorsBoolean analysis enables to represent a dataset by a set of partially ordered elementsBoolean analysis and integrationStructure obtained by Boolean implication Structure of integration processFor the aggregation of knowledge, Doignon and Falmagne (1985) are able to generalise Birkhoffs Theorem and prove that if a knowledge structure is closed under union and intersection, it is well-graded. This condition conforms the gradual aspect of the process of integration.Theuns (1991) proved that there is a one-to-one correspondence between the set of PCUs consisting each of one positive and one negative response and the family of knowledge spaces that are closed under union and intersection. The set of PCUs, consisting each of one positive and one negative response, enables us to obtain a structure that meets the conditions structuring the process of integration.Dichotomisation and quasi-equivalencejjip ijp ijp iip ijpijp ip jp jpTable 3. i j contingency tableIf for two indicators i and j, both referring to the same concept, p ij approximates p ij (pij p ij), and both exceeds a given dichotomisation threshold a, the relation between pij and p ij is defined as quasi-equivalent.1. If either p ij or p ij a and p ij p ij dominance relation2. If both p ij and p ij a and p ij p ij equivalence relation3. If both p ij and p ij a and p ij p ij quasi-equivalence relation4. If both p ij and p ij a and p ij p ij undefined relationObtaining a unique solutionProblem:Dichotomisation of R into R* and R0 - items versus patterns - minimal versus maximal approach no unique dichotomisation threshold (no norm)If the relation between two indicators i and j is declared equivalent, this decision results in the rejection of a minimal proportion of (p ij + p ij) / p of the original population.allowed deviation threshold b is selected such that 1 - (p ij + p ij) / p b.b considered as the group normall indicators must be included in the obtained solutionTheorem of Chebyshevs:for any set of observations (sample or population), the minimum proportion of the values that lie within k standard deviations of the mean is at least 1 - 1 / k2, where k is any constant greater than 1.Two allowed deviation thresholds b1 and b2 can be defined:b1 considering 3 SD if 1 - (p ij + p ij) / p b1 88.9% of the population b2 - considering 2 SD if 1 - (p ij + p ij) / p b2 75% of the population.The margins of the interval determining the quasi-equivalence relation between two indicators i and j.if 1 - (p ij + p ij) / p : b1 equivalence a maxp ij, p ij. b1 and b2 quasi-equivalence aminp ij, pij. b2 dominance relation, or, not specified.considering: a m / 8 all items must be includedPatternFreqPatternFreqPatternFreqPatternFreq1111119710111020111001001101141111103101101601101129001011611110111210101190110101001010311100121010101011001300100191110001101001101100010010007110111310011150101114000111191101103100101101010110001103110100210001190100113600010161100116100010301001070001002110010110000120100015000011116110001201111140010000500001036110000101111030011117200000131101111330111012001110600000021Table 4. Response patternsbba406138544a72131203478269747Table 5. ab contingency tableIndicator pairsp ijp ijDpab7213866ac36252217ad50173123ae24436412af15448433bc74224150bd11216957be2610074bf27394367cd1404793ce47249202cf29246217de35330295df23333310ef708515Table 6. Dp -calculation1 - (p ij + p ij) / p = .7925, and b1 .7925 b2 quasi-equivalent relationdichotomisation threshold a minp ij, p ij so that a 70 a = 50.a ea ca da fb fb ed cc ec fd fd ebeadcfFigure 3. Implication schemeabcdefbcdefacdefcdefbcefbefcefefefFigure 4. Graphical representationFigure 5. Comparison R* versus R*a. (a = 50)- 720 possible paths- 274 actual paths- 8 selected as group norm- grade of cohesion of 76.58Distance between patternsagrade of cohesionrestrained patternspossible paths5076.581184779.3913163682.4715243586.7420522989.412258Table 7. Comparison varying thresholdsIf a = 50 is the threshold on which a unique solution is based, than the distance of a response pattern that does not belong to the solution equals the number of dichotomisation thresholds one has to add successively to the origional threshold before the pattern is contained in the set of response patterns determined by the new thresholdIntegration process Turkish women11111 (10.3)10111 (5.5)01111 (10.0)00111 (22.0)01011 (5.7)00011 (24.0)00001 (5.1)00010 (3.9)00000 (2.2)- grade of cohesion: 88.7- restrained patterns: 9- possible paths: 6Integration process Moroccan women11111 (27.4)10111 (16.2)11011 (3.3)01111 (4.2)10011 (8.8)01011(1.8)00111 (10.0)00011 (12.5)00001 (4.4)00010 (1.4)00000 (2.8)- grade of cohesion: 92.8- restrained patterns: 11- possible paths: 10First generation adults11111 (6.0)11111 (13.7)01111 (9.6)10111 (15.3)11011 (2.9)01111 (3.4)01011 (7.2)00111 (17.9)10011 (9.9)01011 (2.9)00111 (14.9)00011 (31.8)00011 (17.9)00010 (5.9)00001 (5.4)00010 (3.2)00001 (4.3)00000 (2.4)00000 (4.6)Figure 8. Gender relations Turks (left) and Moroccans (right)First generation children11111 (14.2)11111 (32.7)10111 (7.3)11011 (1.8)01111 (11.0)10111 (21.1)10011 (3.3)01011 (3.7)00111 (3.7)10011 (11.3)00011 (4.3)00101 (17.4)00011 (8.4)00010 (3.2)00001 (5.2)00010 (0.0)00001 (3.7)00000 (0.7)00000 (0.4)Figure 9. Gender relations Turks (left) and Moroccans (right)Not migrated11111 (25.4)11111 (55.6)10111 (5.1)01111 (10.3)10111 (16.0)01111 (6.4)01011 (8.0)00111 (23.7)10011 (6.5)00111 (0.6)00011 (10.8)00011 (5.8)00010 (2.3)00001 (2.7)00010 (0.0)00001 (0.0)00000 (0.0)00000 (0.0)Figure 10. Gender relations Turks (left) and Moroccans (right)Recently migrated11111 (5.5)11111 (17.4)10111 (8.3)01111 (8.5)10111 (7.9)01111 (3.5)00111 (21.5)01011 (6.3)00111 (16.6)00011 (27.9)00101 (4.3)00011 (16.1)00010 (3.0)00001 (5.4)00001 (11.7)00000 (5.2)00000 (5.0)Figure 11. Gender relations Turks (left) and Moroccans (right)Turkish women - rural background11111 (2.5)11111 (3.7)01111 (2.6)01111 (5.5)01011 (7.2)00111 (14.7)01011 (7.2)00111 (9.1)00011 (14.7)00011 (38.2)00010 (6.9)00001 (9.1)00010 (5.4)00001 (6.8)00000 (5.6)00000 (10.6)Figure 12. Gender relations eldest gen. (left) and recently migr (right)Turkish women - urban background11111 (8.0)11111 (10.4)01111 (13.9)10111 (6.0)01111 (12.0)01011 (6.3)01101 (3.3)00111 (20.5)01011 (6.6)00111 (33.1)01001 (0.9)00101 (1.8)00011 (32.2)00101 (3.4)00011 (18.0)00010 (4.8)00001 (1.8)00010 (2.7)00001 (2.6)00000 (0.0)00000 (1.7)Figure 13. Gender relations eldest gen. (left) and recently migr (right)Morrocan women - ethnic background11111 (33.1)10111 (20.8)00111 (10.8)00011 (7.6)00010 (0.6)00001 (2.1)00000 (1.2)11111 (20.0)10111 (10.8)110