The Method of Purging Applied to Repeated Cross-Sectional Data

Abstract

In cross-sectional survey research, it is quite common to estimate the (standardized) effect of independent variable(s) on a dependent variable. However, if repeated cross-sectional data are available, much is to be gained if the consequences of these effects on longitudinal social change are considered. To assess these consequences, we describe a type of simulation in which longitudinal shifts in the independent variable’s distribution, and longitudinal variation in effect on the dependent variable are ‘purged’ from the data. Although the method of purging is known for many years, we add new practical features by relating the method to logistic and linear regression analysis. Because both logistic and linear regression analysis can be found in all major statistical packages, the method of purging is made available to a wider group of social scientists. With the use of repeated crosssectional data, gathered in the Netherlands between 1970 and 1998, the new practical features of the purging method are shown, using the SPSS packag

Full text

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Quality & Quantity 38: 1–16, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
1
The Method of Purging Applied to Repeated
Cross-Sectional Data
Practical applications using logistic and linear regression analysis
MANFRED TE GROTENHUIS?, ROB EISINGA and PEER SCHEEPERS
Department of Social Science Research Methods, University of Nijmegen, PO Box 9104, 65000 HE
Nijmegen, The Netherlands
Abstract. In cross-sectional survey research, it is quite common to estimate the (standardized) effect
of independent variable(s) on a dependent variable. However, if repeated cross-sectional data are
available, much is to be gained if the consequences of these effects on longitudinal social change are
considered.
To assess these consequences, we describe a type of simulation in which longitudinal shifts in
the independent variable’s distribution, and longitudinal variation in effect on the dependent variable
are ‘purged’ from the data. Although the method of purging is known for many years, we add new
practical features by relating the method to logistic and linear regression analysis. Because both
logistic and linear regression analysis can be found in all major statistical packages, the method
of purging is made available to a wider group of social scientists. With the use of repeated cross-
sectional data, gathered in the Netherlands between 1970 and 1998, the new practical features of the
purging method are shown, using the SPSS package.
Key words: purging, simulation, counter factual analysis, repeated cross-sectional survey, logistic
and linear regression analysis
Introduction
There is a long tradition in social sciences of collecting cross-sectional survey data.
As a result, a massive quantity of longitudinal data is available nowadays. These
data are in many events more or less suitable to test causal models. Besides, these
data are often tailor-made to test to what extent parameters in these models vary
over time. However, to explain longitudinal social change, one has to go beyond
the causal modeling of effects and their over-time variation. To answer the question
on the causes of social change, we have to asses the consequences of shifting
distribution(s) and of varying effect(s). One way to assess these consequences,
is to simulate a counter factual situation in which both the distribution of the
independent variable(s) and its effect(s) remain constant over time. This kind of
simulation is often labelled ‘purging’ (for an overview see: Clogg, 1978; Clogg &
?E-mail: m.tegrotenhuis@maw.kun.nl

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MANFRED TE GROTENHUIS ET AL.
Table I. Relationship between church membership and confessional party voting in
1970 and 1998, frequencies and (percentages)
Vote for
confessional
party
Church membership:
1970
Non-
member
1998
Non-
member
Church
member
Total Church
member
Total
No 472
(95.7)
21
(4.3)
493
(39.5)
286
(37.9)
469
(62.1)
755
(60.5)
758
(60.7)
490
(39.3)
1248
(100)
860
(96.1)
35
(3.9)
895
(59.9)
351
(58.5)
249
(41.5)
600
(40.1)
1211
(81.0)
284
(19.0)
1495
(100)
Yes
Total
Data derived from the ‘Cultural Changes’ data-set collected by the Social and Cultural
Planbureau (SCP), The Netherlands: author’s calculations.
Eliasen, 1988; Clogg et al., 1990) because one eliminates or purges the impact of
shifting distribution(s) and varying effect(s).1
For example, in the Netherlands, the mainstream confessional political party
CDA suffered from a massive loss of votes in the past decades. Also, there is
abundance evidence that religious affiliation has a strong effect on voting beha-
vior and over time this effect declined (Nieuwbeerta, 1995). Finally, there was a
profound distributional shift, i.e., church membership rates were declining during
the past century in the Netherlands.(Eisinga & Felling, 1990). These two facts,
i.e., declined church membership rates and a declined effect of religious affiliation,
may explain the over-time loss of confessional votes (Eisinga et al., 1997). In the
next section we will use this example to explain and demonstrate the practical
applications of purging using logistic and linear regression analysis.
Before wedo so,wefirstintroduce a2×2table based ontwopoints intime,just
to outline the basic idea behind the purging method applied to longitudinal data.
Next, we will show how purging can be done using logistic regression analysis
and a few constraints. Moreover, we will introduce more independent variables
and a decomposition of the overall impact. Finally, we will show how the method
of purging can be applied to a continuous dependent variable using a linear re-
gression model. These practical extensions, i.e., using (multiple) logistic and linear
regression analysis to purge distributional shift(s) and varying effect(s), are the core
contribution of this article.
1. The Method of Purging Exemplified by Means of a 2 × 2 Table
In this section we will give a example of purging on a 2 × 2 table with data from
repeated cross-sectional surveys. Tosave space, werestricted our data to twopoints
in time, i.e., 1970 and 1998. The data from the two samples are shown in Table I.

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THE METHOD OF PURGING
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Table II. Relationship between church membership and confessional party voting in
1998, constant association and constant distribution, frequencies and (percentages)
Vote for
confessional
party
Church membership:
Constant association in 1998
Non- Church
membermember
Constant association in 1998
Non- Church
member member
TotalTotal
No 857
(95.7)
38
(4.3)
895
(59.9)
227
(37.9)
373
(62.1)
600
(40.1)
1084
(72.5)
411
(275)
1495
(100)
568
(96.1)
23
(3.9)
591
(39.5)
329
(58.5)
375
(41.5)
904
(60.5)
1097
(73.4)
398
(26.6)
1495
(100)
Yes
Total
According to the row totals in Table I, (mainstream) confessional parties
suffered from heavy losses between 1970 and 1998.2In 1970 39.3% of the Dutch
voted for a confessional party, whereas in 1998 the percentage was down to 19.0.
Thispolitical turnover ispartly duetochanges inchurch members’ voting behavior,
as can be seen from the interior (shaded) cells in Table I. According to these cells,
62.1% of all church members voted for a confessional party in 1970 but it declined
to 41.5% in 1998. Confessional party voting among non-members was quite rare:
only 4.3% of them did so in 1970 and 3.9% in 1998. The change in voting behavior
of especially church members is reflected in the odds ratios as a measure of asso-
ciation: 36.9 in 1970 and 17.43 in 1998. Next to this change in association, we can
also ascertain a profound shift in church membership rates. In 1970, 60.5% of the
Dutch population considered themselves to be church members, whereas in 1998
this was 40.1%.
So, two processes may be responsible for the loss of confessional votes between
1970 and 1998. First, a change in the association between church membership
and confessional party voting as church members changed their voting behavior.
Second, a distributional shift in the (independent) variable church membership as
the percentage non-members increased between 1970 and 1998.
To assess the impact of each of these two processes, we performed two simula-
tions, one in which the association between church membership and confessional
party voting washeld constant, and one in which the church membership rates were
held constant. The outcomes are presented in Table II.
In Table II, the left 2 × 2 tabulation represents the simulation in which the
association between church membership and religious voting is held constant, i.e.,
the odds ratio is held on the 1970 level. To archieve this, we used the 1998 dis-
tribution of church membership (895/600) and the conditional percentages from
1970 (cf. Table I). On the basis of this information, we recalculated the cell

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MANFRED TE GROTENHUIS ET AL.
frequencies for 1998. For instance in cell 0.0 (non-member, no vote for confes-
sional party) the new frequency is 895 × 0.957 = 857. On the basis of the new
frequencies, the percentage that did vote for confessional party is recalculated:
(38 + 373)/1495 × 100 = 27.5%). So, if a constant association is simulated, the
percentage of votes for confessional parties in 1998 is 27.5 instead of the observed
19.0 (cf. Table I) and lies well outside the 95% confidence interval around 19.0.3
Practically, this finding implies that if the association between church membership
and religious voting not had changed over time, the percentage of votes for a
confessional party would have amounted to 27.5, which is significantly different
from the observed percentage.
In the right part of Table II, we simulated constant distribution. To keep the
distribution of the church membership variable in 1998 on the 1970 level, we use
the 1970 distribution (39.5/60.5), as new column total percentages in 1998. On the
basis of these percentages we recalculated column total frequencies for 1998. For
instance, the number of non-members is 0.395×1495 = 591. On the basis of these
frequencies we calculated new frequencies for the shaded cells. For example, the
number of non-members who did not vote for a confessional party is 591×96.1 =
568. The purged percentage of people who voted for confessional parties in 1998
is 23 + 375/1495 × 100 = 26.6%. So, if we simulate a constant distribution of
church membership, the percentage of votes for confessional parties is 26.6 instead
of the observed 19.0 and again lies outside the 95% confidence interval (cf. note
3). Practically, this finding implies that if the church membership’s distribution
not had changed over time, the percentage for a confessional party would have
amounted to 26.6 and is significant different from the observed percentage. As
these two simulations have an impact of respectively 8.5 and 7.6 (purged % in 1998
– observed % in 1998), the overall conclusion is that both the observed change in
association and the observed shift in contribution have contributed significantly
and almost equally to the decline of confessional party voting in the Netherlands
between 1970 and 1998.4
This example was only used to clarify the method of purging on longitudinal
data. However restricted to two sample points, itis quite a laborious way to produce
Table II. Needless to say that purging this way is close to impossible if we use a
large set of data with multiple data points and multiple causal factors. Fortunately,
we do not have to recalculate the marginals this way, as there is a relationship
between logistic regression analysis and the method of purging (Xie, 1989). We
elaborate upon this relationship and the resulting practical applications within the
field of longitudinal research in the next section.

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THE METHOD OF PURGING
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2. The Method of Purging Using Logistic Regression Analysis
Another, more easy way to produce all relevant information from Tables I and II,
is to perform a logistic regression analysis using the following equation:
logp1(conf.partyvoting)
P0(conf.partyvoting)
= b0+ b1church member + b2year + b3(church member × year)
(coding: confessional party voting: no = 0, yes = 1; church member: non-member
= 0, church member = 1; year: 1970 = 0, 1998 = 1).
Note that b0represents the logit among non-members in 1970, that b1denotes
the difference in logit between non-members and church members in 1970, that
b2is the difference in logit between 1970 and 1998 for non-members and b3is
the additional difference in logit between 1970 and 1998 for church members. The
resulting values of these parameters are presented in the second column Table III.
Next, we applied logistic regression analysis with the equation given in (1) to
the data in the left part of Table I (1970) together with the data in the left part of
Table II (constant association in 1998). Resulting parameters are again presented in
Table III. Note that b0and b1have the same values as for Table I, since we purge on
1998 while 1970 is our base-level. The parameters b2and b3however equal zero,
indicating that the association remained on the 1970 level.
Finally, a logistic regression analyses was performed on the data in the left part
of Table I (1970) together with the data in right part of Table II (constant distribu-
tion in 1998) again using Equation (1). The resulting parameters are the same as
for Table I, since only the marginal frequencies changed and logit parameters do
not depend on marginal distributions. From the resulting parameters, we calculated
the percentage that voted for a confessional party in 1998 using:
?
×fc
ftotal
(1)
ˆ p1(conf.partyvoting) =
c
1
e−(b0+b1churchmember+b2year+b3churchmember×year)+ 1
× 100(2)
(c denotes (shaded) cell, fcdenotes the cell frequency and ftotalis total frequency
of the 1998 sample).
The outcomes are presented in the last column of Table III.
As can be seen from Table III, logistic regression provides the same percentages
votes for a confessional party as in the example shown in Tables I and II. The
practical implication of it all is, that we only have to perform one single logistic
regression analysis on the data in Table I and let the computer calculate both
the percentage confessional votes while B2and b3equal zero (= constant asso-
ciation) and the percentage confessional votes using the 1970 weights (fc/ftotal),
(= constant distribution).

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MANFRED TE GROTENHUIS ET AL.
Table III. Percentage religious votes: (1) observed, (2) with constant association and (3) with constant distribution
Table
Parameters
Formula 2
% in 1998
b0
b1
b2
b3
1
−3.11
3.61
−0.09
−0.75
1/(e−(−3.11+−0.09)+ 1) × 860/1495 × 100+
1/(e−(−3.11+−0.09)+ 1) × 35/1495 × 100+
1/(e−(−3.11+3.61+−0.09+−0.75)+ 1) × 351/1495 × 100+
1/(e−(−3.11+3.61+−0.09+−0.75)+ 1) × 249/1495 × 100 =
19.0
Left
−3.11
3.61
0.00
0.00
1/(e−(−3.11)+ 1) × 860/1495 × 100+
part
1/(e−(−3.11)+ 1) × 35/1495 × 100+
of 2
1/(e−(−3.11+3.61)+ 1) × 351/1495 × 100+
1/(e−(−3.11+3.61)+ 1) × 249/1495 × 100 =
27.5
Right
−3.11
3.61
−0.09
−0.75
1/(e−(−3.11+−0.09)+ 1) × 568/1495 × 100+
part
1/(e−(−3.11+−0.09)+ 1) × 23/1495 × 100+
of 2
1/(e−(−3.11+3.61+−0.09+−0.75)+ 1) × 375/1495 × 100+
1/(e−(−3.11+3.61+−0.09+−0.75)+ 1) × 529/1495 × 100 =
26.6

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THE METHOD OF PURGING
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The exact procedure to run these simulations with the SPSS package, is shown
in Appendix 1.5
2.1.
EXTENSION OF THE MODEL: MULTIPLE CAUSAL FACTORS
The model thusfar used has only two independent variables (i.e., church member-
ship and year), whereas it may be necessary to take other variables into account
as well. In our example, the association in 1970 between church membership
and confessional party voting was much stronger compared to 1998. This differ-
ence mainly came about because church members turned away from confessional
parties. This behavioral change may be explained by declining church attendance
rates.6Therefore, we extend the previous model by including a variable measuring
church attendance and a cross-product term modeling the effect in 1998:
logp1(rel.partyvoting)
p0(rel.partyvoting)
= b0+ b1ch. mem. + b2year
+b3(ch. mem. × year)
+b4ch. attend.
+b5(ch. attend × year)(3)
(coding: confessional party voting: no = 0, yes = 1; church membership: non-
member = 0, church member = 1; year: 0 = 1970, 1 = 1998; church attendance:
0 =≤ once a month, 1 =≥ once a month).
The resulting parameters are: b0 = −3.12, b1 = 2.35, b2 = −0.10, b3 =
−0.09, b4= 2.00 and b5= −0.45. Compared with the previous results, b3is much
smaller this time, indicating that church attendance (partly) explains church mem-
bers’ behavioral change. Next to the longitudinal change in association between
church membership and religious voting (b2 and b3), the association between
church attendance and religious voting also changed between 1970 and 1998 (b2
and especially b5). To assess the impact of both changes simultaneously and seper-
ately, wecarried out three simulations. Inafirstsimulation, alllongitudinal changes
were purged (b2, b3and b5set to zero). In a second simulation, only the behavi-
oral changes of non-members and church members were purged (b2and b3set to
zero). Finally, in a third simulation, the behavioral changes of church attenders
and non-attenders were purged (b2and b5set to zero). The results are shown in
Table IV.
The impact of a changing association is small this time, as the purged percent-
age differs only slightly from the observed percentage (22.2 − 19.0 = 3.2). Of
this minor impact, only the changing association between church attendance and
religious voting has a significant impact.
Likewise, one may want to know whether the decline of church membership
rates or the decline of church attendance rates has had the strongest impact on

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Table IV. Percentage of religious votes while simulating constant association
% in 1998 95% confidence interval
(cf. note 3)
Observed
Constant association for both
church membership and church attendance
Constant association for church membership
Constant association for church attendance
19.017.0–21.0
22.2
20.8
21.5
ns
ns: lies within 95% confidence interval around 19%.
Table V. Percentage of religious votes while simulating constant distribution
% in 199895% confidence interval
(cf. note 3)
17.0—21.0 Observed
Constant distribution for both
church membership and church attendance
Constant distribution for church membership
Constant distribution for church attendance
19.0
33.1?
20.0
23.3
ns
ns: lies within 95% confidence interval around 19%.
the observed loss of confessional votes. To answer this question, we used a factor
decomposition method described by Liao (1989). Translated to our longitudinal
problem, this method implies that for finding the impact of the shift in church
membership’s distribution, purged percentages must be calculated within the two
categories of church attendance and multiplied by their 1998 weights of each of
the church attendance categories and finally summed. To find the impact of a chan-
ging distribution of church attendance, purged percentages have to be calculated
within the two categories of church membership, multiplied by their 1998 weights
and summed. The decomposition method for two x-variables is expressed in the
following formula:7
ˆ p1(y)=
?
k
ˆ pxk
fxk
ftotal
(4)
( ˆ pxkdenotes the purged percentage in category k of variable x, fxkdenotes the
frequency in category k of variable x and ftotalis total sample frequency).
With the use of (2), (3) and (4) the purged percentages are calculated and shown
in Table V.
From Table V we derive that the constant distribution of both church member-
ship and church attendance has a huge impact on the percentage of religious votes.
If both church membership rates and church attendance would have been stable

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THE METHOD OF PURGING
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between 1970 and 1998, the purged percentage is 33.1 instead of the observed
19.0. The separate impact of each variable is modest, with the impact of falling
church attendance rates being significant. The outcomes are quite logical as in the
Netherlands falling church membership rates go hand in hand with falling church
attendance rates (Te Grotenhuis & Scheepers, 2001). When Table V is compared
with Table IV, then the conclusion must be that the dramatic loss of religious votes
is primarily caused by falling church membership rates and church attendance
rates. In fact, if this process would not had occurred, then the loss of confessional
votes would have been 5.2% instead of the observed loss of 20.3% (cf. Table I).
The exact procedure to run the simulations from Tables IV and V with the use of
the SPSS package is shown in Appendix 2.
For heuristic and practical reasons the examples given were limited in data. For
demonstrations of some full-fledged purging analyses (i.e., multiple factors and
multiple datapoints) using logistic and linear regression analysis we like to refer
to (Eisinga et al., 1997; Gomulka & Stern, 1990; Te Grotenhuis et al., 1998; Ten
Have et al., 2002).
3. The Method of Purging Using Linear Regression Analysis
In the former section the dependent variable was nominal. However, the method
of purging can also be applied to dependent variables which are ratio or interval
scaled. As an example we choose a variable to measure respondents’ opinion about
abortion. From previous research it is known that church attendance and education
are relative strongly related to this opinion (Scheepers et al., 2002). We therefore
use the following linear regression equation with church attendance (5 categories),
education (7 categories) and year (2 categories):
ˆ yabortion = b0+ b1ch. attendance + b2edu. + b3year
+b4(ch. attendance × year)
+b5(edu. × year)
(coding: church attendance: 0 = never ...5 = weekly; education: 0 = lowest ...6 =
highest; year: 0 = 1970, 1 = 1998).
The resulting parameters are: b0= 2.60, b1= 0.27, b2= −0.10, b3= −0.15,
b4 = 0.07, b5 = 0.034. Note that the value of b0represents the mean score of
respondents who do not visit religious services and have the lowest educational
level in 1970, that b1is the effect of church attendance in 1970, that b2is the effect
of education in 1970, that b3is an additional constant for 1998 while b4and b5are
the additional effects of church attendance and education in 1998. Because we are
only interested in the mean score on y we use:
?
(¯ y denotes mean score, bois constant, ¯ x is mean score on variable p).
(5)
¯ y = bo+
p
bp¯ xp
(6)

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Table VI. Mean score on abortion variable while simulating constant associations
Mean in 1970Mean in 1998 95% confidence
interval8
Observed
Constant association for both
church attendance and education
Constant association for church attendance
Constant association for education
3.02 2.862.80–2.92
2.74
2.87
2.88
ns
ns
To hold all associations constant, b3, b4and b5have to be set to zero in (5) and a
new mean score on y is calculated. To hold all distributions constant, only the 1970
distributions are used in (5) to recalculate the new mean score on y. Following (6),
this can be achieved by taking the observed means from the 1970 distributions. We
can also decompose the impact of constant association and constant distribution.
For instance, if one likes to know the contribution of the shift in church attendance
rates, the 1998 means of year and education are substituted in (5) together with the
1970 mean of church attendance. In Table VI the outcomes of these simulations
are summarized.
From Table VI we derive that between 1970 and 1998 the mean score on the
abortion variable dropped from 3.02 to 2.86. This implies that, on average, Dutch
people gradually opposed less to the idea that women should have the right of
abortion. This decline would have been stronger if the association between church
attendance and education, on the one hand, and abortion, on the other, would have
been stable between 1970 and 1998. If only the association between church at-
tendance and abortion is held constant the impact is insignificantly lower (2.87)
and almost equal to a simulation in which the association between education
and abortion is held constant (also non-significant). Next, we simulated constant
distributions, the results can be found in Table VII.
As Table VII shows, constant distributions have quite different consequences.
Unlike Table VI, the simulated mean score in 1998 has significantly risen to 3.18.
This indicates that if church attendance rates would not have fallen and educational
level would not have risen, the opposition against abortion would have been higher,
instead of lower, compared to 1970. This impact is mainly attributable to the fall
of church attendance rates (2.86–3.13) and to a much lesser degree to rising levels
of education (2.86–2.91) which is inside the 95% confidence interval. The SPSS
syntax to run all simulations from Tables VI and VII is shown in Appendix 3.

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Table VII. Mean score on abortion variable while simulating constant distributions
Mean in 1970Mean in 1998 95% confidence
interval
(cf. note 8)
Observed
Constant distribution for both church
attendance and education
Constant distribution for church attendance
Constant distribution for education
3.022.86 2.80–2.92
3.18
3.13
2.91 ns
4. To Purge or not to Purge: Discussion and Conclusions
Thusfar we assumed that the observed percentages and mean scores between 1970
and 1998 differed significantly. In other words, we assumed that there was enough
statistical evidence that a social change indeed had taken place. Of course before
purging is applied, one must test whether the data indicate social change. If the test
fails to reject H0(= equal proportions or equal means) subsequent purging would
have no statistical meaning.
Likewise, we assumed that the observed shift in distributions and variations
of effects were significant. Of course, one should test this assumption before a
purging analysis is carried out. To test whether effects changed significantly over
time, it is best to include cross-product terms, modeling the interaction between the
predictor variables and time (often the years in which the surveys were conducted).
If these cross-product terms are not significant, purging these terms (i.e., setting
them to zero) is useless. The same argument applies to shifts in distributions of
predictor variables. In case there is lack of statistical evidence for this shift, one
should refrain from purging them. Practically this means that one should only purge
those changes and shifts that turned out to be significant.
A purging analysis is also questionable if one wants to assess the impact of
a certain shift in distribution while controlling for other causal factors. This may
result in a weak (controlled for) effect of the variable under investigation and hence
a low purging impact even when the shift in distribution is large. We suggest to
start with a simple regression analysis and assess the purging impact of a particular
variable. In subsequent analyses, one may look for other variables that may explain
the purging impact of that particular variable.
In this contribution we showed that the method of purging on longitudinal data
can be applied with the use of logistic and linear regression analysis. As these
analyses are widely used, the practical procedures we proposed may be used by a
wider group of social scientists who are interested in explaining longitudinal social
change. Especially in case of a large number of repeated cross-sectional surveys,

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MANFRED TE GROTENHUIS ET AL.
we feel that the method of purging could be a powerful tool to assess to what extent
distributional shifts and/or behavioural change explain longitudinal social change.
Appendix 1: SPSS Syntax for Simulation #1 and #2

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Appendix 2: SPSS Syntax for Simulation #3 and #4 and Decomposition

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MANFRED TE GROTENHUIS ET AL.
Appendix 3: SPSS Syntax for Simulation #5 and #6 and Decomposition

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Notes
1In the literature on purging this simulation is called marginal CG purging and three-factor CDG
purging respectively.
2We focused on the mainstream confessional parties, i.e., KVP, CHU, ARP and their successor
CDA. Small right-wing confessional parties were not taken into account because of their stable
electorate.
3To calculate the 95% confidence interval we used the following formula (assuming normal
distribution):
ˆ p ± zα/2
?
ˆ pˆ q
n
= 19 ± 1.96
?19 × 81
1495
= 19 ± 2 = [17%;21%].
4The total decline of confessional votes is 20.3% (39.3–19.0). So, there is a unexplained per-
centage of 4.2% (20.3–8.5–7.6). This remaining part is contributable to the joint effect of shifting
distribution and changing association.
5All Appendices inthis articlecan be downloaded as ASCII filesfrom the firstauthor’s homepage
(http://baserv.uci.kun.nl/∼mtgroten). The files are written as syntax for SPSS for windows, version
6.0 and higher.
6According to social science theory, one of the explanations for this behavioral change of church
members is a declining integration within a religious community (cf. Stark, 1994; Ultee et al., 1992).
We take church attendance as indicator for (the lack of) integration within a religious community.
7The decomposition method described by Liao (1989) is not limited to two variables. One only
has to calculate purged percentages within each combination of the left out variables which makes it
a bit more laborious.
8To calculate the 95% confidence interval we used the following formula (assuming normal
distribution):
¯ x ± zα/2
s
√n= 2.86 ± 1.96
1.3
√2097
= 2.86 ± 0.056 = [2.80;2.92].
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