Download Base models for Configural Frequency Analysis

Psychology Science, Volume 46, 2004 (1), p. 150-170
Base models for Configural Frequency Analysis
1
ALEXANDER VON EYE
Abstract
In this article, a grouping of base models for Configural Frequency Analysis (CFA) is
proposed. Four groups are discussed. The first involves log-linear models and estimates
expected cell frequencies using information provided by the observed cell frequencies. The
second group uses a priori probabilities. These probabilities can exist in the form of, for
instance, population parameters. The third group uses a priori probabilities that conform to
substantive models. The fourth group of base models represents hypotheses concerning the
multivariate distribution in cross-classifications. In this group, CFA methods are used to
determine whether and where deviations from distributional assumptions exist. Data examples are provided from the areas of politics, birth statistics, and adolescent aggressive behavior. In the discussion, the relation between log-linear modeling and CFA is specified. It is
stated that the foundation of CFA goes far beyond what one would assume if CFA were
equivalent to residual analysis in log-linear modeling.
Key words: Configural Frequency Analysis, base model, log-linear model, crossclassification
1
Alexander von Eye, Michigan State University, Department of Psychology, 119 Snyder Hall, East Lansing,
MI 48824-1117, USA; E-mail: [email protected]
Base models for Configural Frequency Analysis
151
Base models for Configural Frequency Analysis
In the garden of classification methods, Configural Frequency Analysis (CFA; Lienert,
1968; Lautsch & von Weber, 1995; von Eye, 2002a) plays a particular role. In contrast to
such methods as cluster analysis or latent class analysis, both of which create a priori unknown groups from raw data, CFA asks whether clusters of existing groups contain more or
fewer cases than expected. CFA shares the characteristic of analyzing existing groups with
discriminant analysis and with logistic regression. However, in contrast to these two methods, the typical application of CFA is exploratory in nature, as is the case for many other
methods of classification. There exist confirmatory methods of CFA. However, these have
found only few applications (later in this article, confirmatory applications are discussed).
CFA has been developed intensively since its first presentation in 1968 (see Lienert, 1969),
and is now among the more popular methods of data analysis.
One of the reasons for the increasing popularity of CFA is that the results of this method
of data analysis are deemed easy to interpret. Specifically, CFA allows researchers to explore
whether for a particular cell of a cross-classification, the null hypothesis H0: E[Ni] = Ei can be
rejected, where E indicates the expectancy, Ni is the observed frequency for Cell i, Ei is the
estimated expected frequency for Cell i, and i indexes all cells in a cross-classification. If a
significance test suggests that E[Ni] > Ei, the cell, also called Configuration i, is said to constitute a CFA type. If E[Ni] < Ei, Cell i is said to constitute a CFA antitype. If neither is the
case, that is, if E[Ni] = Ei, Cell i is said to follow the distribution proposed in the CFA base
model. In different words, CFA identifies those cells as type-constituting that contain more
cases than expected. Cells that contain fewer cases than expected constitute antitypes. Cells
with observed frequencies that do not differ from their expected counterparts beyond chance
are typically not the focus of interpretational efforts. Reference to the base model that was
used to estimate the expected cell frequencies is rarely made.
In this article, we discuss base models that can be used for CFA. Specifically, we discuss
four groups of base models. First, we discuss log-linear models. These are the most frequently used base models. Indeed, the original CFA base model (Lienert, 1969) is a loglinear main effect model. Second, we discuss models in which cell probabilities are determined based on population parameters. Third, we consider models that determine cell probabilities based on a priori considerations (see, for instance, von Eye, 2002a, Ch. 8). Fourth,
we discuss base models that reflect assumptions concerning the distributions of the variables
under study (von Eye & Gardiner, 2003). We also present examples and arguments that can
be used for the selection of base models. In the next section, we first present a brief overview
of the method of CFA. The following sections discuss the four groups of base models.
1. The five steps of Configural Frequency Analysis
CFA is performed in five steps (von Eye, 2002a). The first step involves selecting a base
model. This article is concerned with the nature of base models. The selection of base models
also requires that theoretical considerations concerning the scale characteristics of the variables in the cross-classification be taken into account. In addition, the sampling scheme for
data collection determines which kind of base model is admissible (von Eye & Schuster,
1998; von Eye, Schuster, & Gutiérrez-Peña, 2000). Neither of these two issues is of concern
152
A. von Eye
here. Instead, the characteristics and applications of four groups of base models are discussed.
The second step of CFA involves the selection of a concept of deviation from independence. This issue is relatively new in the discussion of CFA (von Eye, Spiel, & Rovine, 1995).
It is based on the idea that a number of definitions of deviations from some base model exist
that reflect different data characteristics (Goodman, 1991). The concept of deviation that is
used in routine CFA applications is that of the standardized residual. This concept is marginal-dependent. Other concepts are marginal-free, and measures of deviation can behave
differently, depending on whether they are marginal-free or marginal-dependent (von Eye &
Mun, 2003). At the moment of this writing, there exist no CFA applications that consider
these characteristics of measures and tests of deviation, and there exists no theoretical work
that discusses these characteristics in CFA models other than 2-sample CFA.
The third step involves the selection of a significance test. A large number of tests has
been proposed for use in CFA. Recent work has shown that these tests differ dramatically in
their characteristics. For example, von Eye (2002b) has shown that, depending on sample
size, some of the more popular tests of CFA tend to identify more types, whereas others tend
to identify more antitypes. von Weber, Lautsch, and von Eye (2003) showed that the $-error
of CFA tests can differ by large quantities.
The fourth step involves estimating (or determining) the expected cell frequencies, performing the significance tests, and identifying those configurations that constitute types or
antitypes. The fifth step involves interpreting types and antitypes. A complete example of
CFA follows.
Data example. In a study published by DiFrancisco and Critelman (1984; from Lawal,
2003, p. 251), respondents from 5 nations indicated their level of education and whether they
followed politics regularly. The five nations were 1 = USSR, 2 = USA, 3 = UK, 4 = Italy,
and 5 = Mexico. The three levels of education were 1 = primary education, 2 = secondary
education, and 3 = college education or higher. A score of 1 on the third variable indicates
that a respondent follows politics regularly, a 2 indicates that a respondent does not follow
politics regularly. The cross-classification of these three variables appears in Table 1, along
with CFA results. The standard normal z-test was used, and the significance level " was
Bonferroni-adjusted to be "* = 0.001667. For the present analyses, we used the original CFA
base model, that is, the log-frequency model
log Ei = λ + λ Country
+ λkEduc + λlPolit
j
where the subscripts to the 8 terms index the estimated parameters, and the superscripts
index the variables. Obviously, this is the model of independence among the three variables
under study. Types and antitypes emerge if these variables covary. This base model of CFA
is also known as first order CFA.
Table 1 shows that there is a large number of types and antitypes that describe patterns of
political interest that deviate significantly from the assumption that the three variables Country, Level of education, and Interest in politics are independent from each other. In the following paragraphs, we present sample interpretations of types and antitypes as well as patterns of types and antitypes.
Base models for Configural Frequency Analysis
153
Table 1:
CFA of the cross-classification of the three variables Country (C), Educational level (E),
and Interest in politics (I)
CEI
111
112
121
122
131
132
211
212
221
222
231
232
311
312
321
322
331
332
411
412
421
422
431
432
511
512
521
522
531
532
Ni
94
84
318
120
473
72
227
112
371
71
180
8
356
144
256
76
22
2
166
256
142
103
47
7
447
430
78
25
22
2
Êi
387.578
183.188
261.063
123.391
139.735
66.046
323.482
152.893
217.890
102.985
116.627
55.123
285.759
135.063
192.480
90.975
103.026
48.695
240.692
113.763
162.124
76.628
86.778
41.015
335.166
158.416
225.760
106.705
120.839
57.114
statistic
-14.912
-7.328
3.524
-.305
28.193
.733
-5.364
-3.307
10.373
-3.152
5.868
-6.347
4.155
.769
4.578
-1.570
-7.983
-6.692
-4.814
13.336
-1.581
3.013
-4.270
-5.311
6.109
21.578
-9.834
-7.910
-8.991
-7.293
p
.00000000
.00000000
.00021266
.38009021
.00000000
.23187817
.00000004
.00047128
.00000000
.00081139
.00000000
.00000000
.00001626
.22096045
.00000234
.05820057
.00000000
.00000000
.00000074
.00000000
.05699597
.00129473
.00000978
.00000005
.00000000
.00000000
.00000000
.00000000
.00000000
.00000000
Antitype
Antitype
Type
Type
Antitype
Antitype
Type
Antitype
Type
Antitype
Type
Type
Antitype
Antitype
Antitype
Type
Type
Antitype
Antitype
Type
Type
Antitype
Antitype
Antitype
Antitype
The first antitype in Table 1, constituted by Cell 111, suggests that fewer individuals with
primary education from the USSR than expected follow politics regularly. The second antitype, constituted by Cell 112, suggests that also fewer individuals with primary education
from the USSR than expected do not follow politics regularly. This result may sound implausible. However, these two antitypes, taken together, indicate that there are fewer individuals
with primary education from the USSR in this sample (for details on aggregation of results
from CFA see von Eye, 2002a, Ch. 10.8). Therefore, these antitypes reflect the association
between the variables Country and Level of Education rather than political interest.
These two antitypes, and the corresponding two antitypes in the individuals from the
USA, are counterbalanced by the types constituted by Cells 511 and 512. These types sug-
A. von Eye
154
gest that there are more Mexicans than expected from the assumption of variable independence who have primary education.
Other types and antitypes may be interpretable as indicating whether individuals do or do
not follow political events more or less regularly than expected. For example, the type constituted by Cell 311 suggests that more individuals with primary education from the UK than
expected indicate that they follow politics regularly. Accordingly, fewer individuals with
secondary education from the US indicate that they don’t follow politics regularly. Readers
are invited to interpret additional types and antitypes as well as patterns of types and antitypes.
The fact that some of the type and antitype patterns reflect particular variable associations can give rise to follow-up data analyses. In the present data example, researchers may
consider an alternative base model. This could be the base model in which the variables
Country and Level of Education are allowed to interact but are proposed to be independent of
the variable Political Interest. This is the base model of a Prediction CFA (P-CFA; Heilmann, Lienert, & Maly, 1979; cf. von Eye & Schuster, 1998). The types and antitypes that
result from P-CFA in the present data example can be interpreted such that particular patterns of Country and Level of Education allow one to predict Political Interest. The log-linear
representation of this base model is
log ei = λ + λ Country
+ λkEduc + λlPolit + λ CountryxEduc
.
j
jk
Alternatively, if the grouping of variables in predictors and criteria is not desired, one
may consider performing a second order CFA, that is, an analysis with a base model that
takes all pairwise associations into account. The log-linear representation of this base model
is
log Ei = λ + λ Country
+ λkEduc + λlPolit + λ CountryxEduc
+ λ CountryxPolit
+ λlkEducxPolit .
j
jk
jl
(This base model is employed in Section 2.1, below). Both of these base models as well
as the one used for the analysis of the data in Table 1 are log-linear models. In the following
sections, we describe four groups of base models, one of which includes log-linear models.
2. Four groups of base models for CFA
In this section, we present four groups of base models for CFA. These are (1) log-linear
models, (2) models based on population parameters, (3) models with a priori determined
probabilities, and (4) models based on distributional assumptions. In either case, a CFA base
model must meet the following three criteria (von Eye & Schuster, 1998):
1.
Uniqueness of interpretation of types and antitypes: there must be only one reason
for discrepancies between observed and expected cell frequencies. Sample cases of
such reasons include the presence of higher order interactions, main effects, and predictor-criterion relationships.
Base models for Configural Frequency Analysis
2.
3.
155
Parsimony: base models must be as simple as possible. That is, base models must
include as few terms as possible and terms of the lowest possible order. Methods
have been proposed to simplify CFA base models (Schuster & von Eye, 2000).
Consideration of sampling scheme: the sampling scheme of all variables must be
considered. Particular sampling schemes can have the effect that the selection of
base models is limited. For example, product-multinomial sampling precludes using
base models of zero order CFA.
From a more general perspective, CFA base models play the role of a superordinate null
hypothesis. The observed data are assumed to have been drawn from a population in which
the base model provides a valid description of the frequency distribution in the crosstabulation. Types and antitypes suggest that, at least in specific sectors of the table, this null
hypothesis must be rejected.
2.1 Log-linear base models for CFA
Log-linear models allow one to describe relationships among variables. For instance, loglinear models can be used to depict association structures, dependency structures, or joint
variable distributions (Goodman, 1984). In the context of CFA, the first two possibilities
have been used. The concept behind using log-linear models lies in the definition of a CFA
base model given above. The base model includes all relationships that researchers do not
wish to be the reason for the emergence of types and antitypes. Types and antitypes can then
emerge only if the relationships exist that are not part of the base model. Such relationships
can either be global, that is, involve the entire range of variable categories, or local, that is,
involve only a selection of variable categories (Havránek & Lienert, 1984). For example, if
researchers wish that types and antitypes reflect nothing but variable associations, the main
effects must be part of the base model. An example of such a base model is the original CFA
base model used by Lienert (1969).
In contrast to log-linear modeling, the base model of a CFA is not altered in the presence
of rejected null hypotheses of model fit. Rather, researchers interpret the types and antitypes
as indicated in the data example above, and attempt to explain the discrepancies from the
base model. As was illustrated in the first data example, this interpretation can focus on
individual types and antitypes as well as on groups of types and antitypes.
Log-linear base models can be classified in two groups (von Eye, 2002a). The first group
is that of global models. These are models in which all variables have the same status. There
is no distinction between, for example, predictors and criteria. Within the group of global
models, there exists a hierarchy. At the bottom of this hierarchy, there is the base model of
zero order CFA. This model proposes that the cross-classification under study reflects no
effects whatsoever. The resulting expected frequencies are thus uniformly distributed. Types
and antitypes can emerge if effects exist, any effects.
At the second level of this hierarchy, we find first order CFA. This is the base model that
proposes that main effects exist but no variable associations. This model is also called the
model of variable independence or the base model of classical CFA. Types and antitypes will
emerge if variable associations exist.
A. von Eye
156
At the subsequent higher levels, increasingly higher order interactions are taken into account. Types and antitypes emerge if associations exist at the levels not considered in the
base model. For d variables, the highest possible order of a base model takes the d-1st order
interactions into account. Types and antitypes can then emerge only if interactions of the dth
order exist. In different words, the highest possible order of CFA base model identifies types
and antitypes only if a saturated hierarchical log-linear model is needed to provide a satisfactory description of the observed frequency distribution.
The second group of log-linear CFA base models is that of regional models. These are
models that group variables. Examples of such models include Prediction CFA which distinguishes between predictors and criteria; Interaction Structure Analysis (ISA; Lienert &
Krauth, 1973) which distinguishes between two groups of variables of equal status; and ksample CFA which includes one or more classification variables and one or more variables
that are used to distinguish between these groups. Models of CFA that distinguish among
more than two groups of variables have been discussed (Lienert & von Eye, 1988), but have
found limited application.
The following paragraphs present examples of log-linear base models. For these examples, we assume that all variables have been observed under a multinomial sampling scheme
such that there are no constraints on the selection of base models. For the examples, we assume that the four variables, A, B, C, and D are included in the analyses.
Examples of global base models.
Zero order base model:
log Ei = λ
First order base model:
log Ei = λ + λ jA + λkB + λlC + λmD
Second order base model:
log Ei = λ + γ Aj + γ lB + γ lC + γ mD +
AB
AD
BC
BD
CD
+ γ jk
+ γ AC
jl + γ jm + γ kl + γ km + γ lm
Examples of regional base models.
ISA in which the first variable group contains variables A and B, and the second variable
group contains variables C and D:
log Ei = λ + γ Aj + γ lB + γ lC + γ mD +
CD
+ γ AB
jk + γ lm
Base models for Configural Frequency Analysis
157
P-CFA in which variable D is predicted from variables A, B, and C:
log Ei = λ + γ Aj + γ lB + γ lC + γ mD
AB
ABC
+ γ jk
+ γ jlAC + γ klBC + γ jkl
Data example. In the following paragraphs, we re-analyze the data from the first example. As was indicated in the discussion of the results of this example, some of the types and
antitypes reflect the association between Country and Level of Education or, in general, first
order associations. Thus, if researchers ask the question whether types and antitypes result at
the level of the remaining second order association, a second order CFA can be performed.
In the typical case, the number of types and antitypes in higher order CFA applications is
smaller than in lower order applications, because more effects are taken into account. The
Table 2:
Second order CFA of the data in Table 1
Configuration
111
112
121
122
131
132
211
212
221
222
231
232
311
312
321
322
331
332
411
412
421
422
431
432
511
512
521
522
531
532
Ni
94
84
318
120
473
72
227
112
371
71
180
8
356
144
256
76
22
2
166
256
142
103
47
7
447
430
78
25
22
2
Êi
94.261
83.739
310.282
127.718
480.457
64.543
235.001
103.999
366.791
75.209
176.208
11.792
339.334
160.666
272.270
59.730
22.396
1.604
167.386
254.614
143.713
101.287
43.902
10.098
454.019
422.981
71.944
31.056
21.036
2.964
statistic
-.027
.028
.438
-.683
-.340
.928
-.522
.785
.220
-.485
.286
-1.104
.905
-1.315
-.986
2.105
-.084
.313
-.107
.087
-.143
.170
.468
-.975
-.329
.341
.714
-1.087
.210
-.560
p
.48929071
.48863800
.33064370
.24733225
.36685148
.17665137
.30085750
.21634957
.41301778
.31370450
.38757386
.13474938
.18279871
.09427841
.16206149
.01763773
.46661597
.37710080
.45735818
.46540306
.44319895
.43243690
.32003912
.16479129
.37092086
.36644349
.23763608
.13859947
.41680103
.28783443
158
A. von Eye
following analyses will show whether this is the case here too. To make results comparable,
2
the following CFA also uses the z-test and the Bonferroni-adjusted "* = 0.0016667. Results
are displayed in Table 2.
The results in Table 2 show clearly that our earlier interpretations were correct. The types
and antitypes in Table 1 resulted because first order associations exist. At the level of the
second order association, not a single type or antitype emerges.
2.2 CFA base models that are based on a population parameter
Application of log-linear base models presupposes in the typical case that the cell probabilities are estimated from sample information. In particular instances, however, population
parameters are known or can be derived from population statistics. Examples of such parameters include gender distributions, cohort size, family characteristics such as number of
children in a family, or number of children raised in single parent households. An early example of a CFA based on population parameters can be found in Spiel and von Eye (1993).
In a similar fashion, the number of cases that are expected to fall in a particular range of such
characteristics as intelligence or extraversion can be derived from test norms that are supposed to be valid for entire populations.
When population parameters are known, the expected cell probabilities can be calculated
as specified in a base model. Consider the case in which the three variables A, B, and C are
observed, each having two categories. The univariate marginal probabilities for these three
variables are π 1A , π 2A , π 1B , π 2B , π 1C , and π 2C . The bivariate marginal probabilities are
A.C
.
. BC
π ijAB
. , π i.k , and π . jk . To illustrate, consider the base model of first order CFA which
implies only main effects, that is, variable independence. For this model the expected cell
probability is
ABC
π ijk
= π iAπ Bj π kC ,
with i, j , k = {1, 2} . The corresponding expected cell frequency is calculated to be
ABC
.
Eˆ ijk = N π ijk
A more complex base model proposes, for example that A and B are unrelated, but A and
C as well as B and C can be related. The expected cell probability for this base model can be
calculated by
π ijk =
2
π iA.k.Cπ ..BC
jk
π kC
Readers recalculating these results using SPSS or SYSTAT will notice that different expected cell frequencies result. Readers recalculating these results using Rindskopf’s (1987) log-linear modeling program will
notice that this program produces the same expected cell frequencies as reported here. The reasons for this
discrepancy need to be explored.
Base models for Configural Frequency Analysis
159
(Bishop, Fienberg, & Holland, 1975) and the expected cell frequency is as before. Most if
not all CFA base models are simple models, that is, the expected frequencies can be estimated from the marginals using closed form equations. Thus, for most models, equations of
the kind given here can be derived.
Data example. For the following example, we use data from a study by Spiel and von
Eye (1992). In this study, the authors asked whether Austrian students of psychology and
journalism are representative of families in Austria in regard to the size of their families of
origin. 244 students of psychology and 179 students of journalism indicated, among other
information, the number of children in their families of origin. Table 3 displays the relative
frequencies of numbers of children, pooled over student gender and major, and the population probabilities. The population probabilities were taken from the Austrian census.
Table 3:
Probabilities and relative frequencies of numbers of children in families in Austria
Number of children
Austria (Census)
Student sample
(Spiel & von Eye, 1992)
1
.4829
.1844
2
.3466
.4350
3
.1194
.2128
>3
.0510
.1678
As Table 3 indicates, the distribution of numbers of children is quite different in the
pooled sample of students of psychology and journalism than in the population. We now ask
whether particular patterns of Major (M; 1 = psychology, 2 = journalism), number of children (C; labels as in Table 3), and Gender (G; 1 = females, 2 = males) deviate from the population such that types or antitypes result. We use the model of variable independence as the
base model. That is, we propose that major, number of children in the family of origin, and
gender are unrelated.
We analyze the M x C x G cross-classification in two ways. First, we employ standard
first order CFA in which we estimate the expected cell frequencies from the observed univariate marginal frequencies.
The expected frequencies for this analysis can be estimated using the log-linear model
C
G
log Ei = λ + λ M
j + λk + λl .
2
This translates into the well known X formula for the estimation of expected cell frequencies,
N j.. N.k . N..l
.
Eˆ ijk =
N2
Second, we substitute the probabilities from Table 3 for the marginal relative frequencies
for numbers of children. This translates into the following equation for the estimation of
expected cell frequencies:
160
A. von Eye
Eˆ ijk = N j..π k N..l / N .
For both analyses, we use the z-test and the Bonferroni-adjusted "* = 0.003125. Table 4
displays results from both analyses.
The two panels in Table 4 display two configural analyses that differ only in the reference population that is used to estimate the expected cell frequencies. In the reference population of Austrian psychology and journalism majors, the three variables Major, Number of
Children, and Gender are independent and, consequently, not a single type or antitype
emerges. In contrast, when Austria’s general population is used as a reference, three antitypes and six types emerge. The three antitypes suggest that female students of psychology
and journalism are less likely than expected from general population parameters to be the
sole children of their families of origin. In addition, male psychology majors are also less
likely than expected children from one-child families. The types of the psychology majors
suggest that male students are more likely than expected to have either two siblings or more
than three. Female journalism majors stem with increased likelihood from families with two,
Table 4:
Two CFA analyses of the cross-classification of Major (M), Number of Children in the
family of origin (C), and Gender of student (G)
Patterns
Standard first order CFA
CFA taking a priori probabilities into
account
a
ˆ
ˆ
z
p(z)
z
p(z)
MCG
Ni
Type/
Ei
Ei
Antitype?
111
34
31.70
.42
.335
82.92
-5.73
< "*
A
112
14
13.30
.20
.422
34.78
-3.53
< "*
A
121
76
74.77
.16
.437
59.58
2.13
.017
122
23
31.36
-1.55
.060
24.99
-.40
.345
131
43
36.57
1.11
.133
20.52
4.96
< "*
T
132
10
15.34
-1.39
.082
8.61
.47
.319
141
37
28.85
1.57
.058
8.77
9.54
< "*
T
142
7
12.10
-1.49
.069
3.68
1.73
.042
211
15
23.25
-1.76
.039
60.84
-5.88
< "*
A
212
15
9.75
1.70
.045
25.52
-2.08
.019
221
52
54.85
-.41
.340
43.71
1.25
.106
222
33
23.01
2.14
.016
18.34
3.43
< "*
T
231
22
26.83
-.96
.168
15.06
1.79
.037
232
15
11.25
1.13
.129
6.32
3.46
< "*
T
241
19
21.17
-.48
.315
2.70
4.96
< "*
T
242
8
8.88
-.30
.383
2.70
3.23
< "*
T
a
”< "*” indicates that the tail probability is smaller than can be expressed with three decimals, and that a type or antitype exists
Base models for Configural Frequency Analysis
161
three, or more children. Male journalism majors are with increased likelihood children from
families with more than three children.
In all, with reference to the population of psychology and journalism majors, the three
variables Major in College, Number of children in family of origin, and Gender are not associated to the extent that types and antitypes emerge. In contrast, with reference to the general
Austrian population, psychology and journalism majors are less likely to have no siblings,
and they are more likely to have two or more siblings.
2.3 CFA base models based on a priori probabilities
A particular group of CFA base models results from theoretical models and assumptions.
Consider an experiment in which coins are tossed and dies are rolled. For coins, one assumes
that heads and tails appear at equal rates, that is, with probability 1/2. For dies, one assumes
that each of the six faces appears with probability 1/6. Similar assumptions can be made for
diamonds in urns, for cards in standard decks, and for the rates with which the numbers in
the lotto game 6 out of 49 are drawn. Using CFA under the usual base models, one can test
whether a coin is even, whether Donata is able to roll more 6es than Alex, both using the
same die, whether beginners have more luck playing poker, or whether alcohol consumption
affects the rates of heads and tails. The sample equations given in Section 2.2 can be used for
these purposes in a parallel fashion.
More complex situations exist. For example, von Eye (2002a; Ch. 8) showed that the
probabilities for positive first, second, and higher order differences can be described using a
probability model that suggests a probability pattern that differs greatly from a uniform distribution. Univariate deviations from such patterns suggest main effects that can lead to the
emergence of types and antitypes that need explanation. This applies accordingly when such
differences are studied in a multivariate context.
An additional group of base models may be considered. In mathematical Psychology, researchers derive mathematical descriptions of processes that culminate in testable models.
Alternative models are often formulated and plotted against each other. For example, Erdfelder and Buchner (1998) derived and tested several competing models of the hindsight
bias. Evidence of hindsight bias is typically observed in situations in which subjective probabilities are estimated in retrospect with outcome information available. Raters are asked to
indicate subjective probabilities as if the outcome information were unavailable. Hindsight is
observed if subjective probabilities are biased in the direction of the outcome. This has also
been called the knew-it-all-along effect (Wood, 1978). Erdfelder and Buchner (1998) derived
a multinomial model of the cognitive processes involved in the hindsight bias phenomenon
(see also Winman, Juslin, & Björkman, 1998). Sample equations appear, for instance in
Erdfelder and Buchner’s Table 3 (1998, p. 396).
Now, from the perspective of CFA applications, it is interesting to note that models as the
ones derived by Erdfelder and Buchner allow one to predict the probabilities of particular
outcome patterns. These probabilities can be compared with observed frequencies using
goodness-of-fit tests. It can thus be attempted to decide which of a number of competing
models describes the observed frequency distribution better. In addition, and this leads to an
application of CFA, one can ask which observed frequencies in particular deviate from the
predicted ones such that CFA types or antitypes result.
A. von Eye
162
In the present article, we give no example for this kind of model. Examples of analyses
that take the a priori probabilities of first and second differences into account can be found in
von Eye (2002a).
2.4 CFA models that are based on distributional assumptions
As was indicated in Section 2.1, log-linear models are typically used to explore (1) variable associations, (2) dependency structures, or (3) joint variable distributions. The CFA
models discussed in the first two groups of base models allow one to perform analyses that
reflect specific forms of variable associations and dependency structures. These are associations or dependency structures that result in types and antitypes and are not part of a base
model. In this section, we discuss a new field of application for CFA, that is, the examination
of distributional characteristics of cross-classifications (for more detail see von Eye & Gardiner, 2003; von Eye, 2003).
Consider a multivariate space that is spanned by d continuous variables. The random vector x (d x 1) follows a d-dimensional normal distribution if its density can be described by
f ( x) = (2π )− d / 2 | Σ |−0.5 e−0.5( x− µ ) / Σ
−1
( x−µ )
,
where µ is the mean and the covariance matrix Σ is positive definite and has rank d.
The characteristics of the multivariate normal distribution imply that if X follows this distribution, each subset of X will also follow this distribution. In particular, each univariate
distribution will be normal too. The reverse is known not to be true. Variables that are normally distributed individually, do not necessarily display a multivariate normal distribution
in general, unless, for example, independence is assumed.
A large number of tests has been proposed to determine whether a sample can be assumed to stem from a multinormal distribution. Among these tests are graphical methods
2
which first calculate the squared multivariate Mahalanobis distances r of the individual data
points, xi, from their means, then rank order these distances, and third plot them against the
χ12−αi ;2 distribution, with 1 - αi = (i - 0.5)/N (see Jobson, 1992). Most popular are Mardia’s
tests (1970, 1980) which use the concepts of multivariate skewness and kurtosis.
The currently known tests of multinormality propose hypotheses that are compatible with
predictions based on multinormality. However, they do not test multinormality directly. Von
Eye and Gardiner (2003) proposed a procedure that tests multinormality directly. This procedure can be seen as a multivariate extension of the well known χ 2 -test of univariate normality. Specifically, the procedure of von Eye and Gardiner proceeds in the following steps:
1.
Split the scales that are used to observe the variables under study in segments. Two
approaches have been proposed. One approach involves creating equal-length segments on the scales. The other approach involves creating equal-probability segments on the scales. Equal-length segments imply that the probability of segments
decreases as their distance from the mean increases. Equal-probability segments im-
Base models for Configural Frequency Analysis
2.
3.
163
ply that their length increases with their distance from the mean. Let the number of
segments of scale j be cj.
Cross the segmented scales. The crossed segmented scales create a d-dimensional
space with C = Π jc j sectors.
Calculate for each of the C sectors the probability under the assumption of a dvariate normal distribution. The methods proposed by Genz (1992) can be used to
calculate these probabilities. These probabilities are
2
z1i +1 z j +1
zkd+1
z1i z 2j
zkd
p( z1i − z1i +1 , z 2j − z 2j +1 ,..., zkd − zkd+1 ) = ∫ ∫ ... ∫ Ψ ( z1 , z 2 ,..., z d ) dz1dz 2 ...dz d ,
4.
5.
where the subscripts index the segments and the superscripts indicate the variables.
Calculate the expected frequency for Sector sij...,k as Êi,j, ...,k = Npi,j,...,k. The following
steps identify von Eye and Gardiner’s sector test as a CFA method.
Determine for each sector whether the observed frequency, Ni,j,...,k differs from the
expected frequency. The null hypothesis for these sector-specific tests is
E[ N i , j ,...,k ] = Eˆ i , j ,...,k If this comparison suggests that a sector contains significantly
more or fewer objects than expected based on the joint density function of the d variables under study, this sector evinces a violation of multivariate normality. Therefore, the assumption of multivariate normality must be rejected at least for this sector. The tests that can be used to examine the sectors are known from CFA.
In addition to the CFA-like sector tests, von Eye and Gardiner proposed an overall good2
ness-of-fit test that is based on the sum of the X - components.
Data example. In a study on the development of aggressive behaviors in adolescence,
Finkelstein, von Eye, and Preece (1994) presented 114 adolescents about 12 years of age
with a questionnaire that included questions about verbal aggression against adults (VAAA),
physical aggression against peers, (PAAP), and aggressive impulses (AI). The respondents
processed the instrument in 1983, 1985, and 1987. One condition that must be met for proper
application of parametric multivariate statistical procedures is that of multinormality. In the
present example, we examine the three variables from the wave in 1983, VAAA83, PAAP83,
and AI83.
Before testing multinormality, we ask whether these three variables can be considered
normally distributed when analyzed at the univariate level. Descriptive statistics appear in
Table 5.
Table 5 suggests that the univariate skewness and kurtosis values of the three variables
are not worrisome. The standard deviations are somewhat small, which can be explained by a
relatively large number of ties near the scale mean. Figure 1 displays the probability plots of
the three variables.
The probability plots of the three variables in Figure 1 show that the variables are, at the
level of univariate analysis, very near the normal distribution. We thus retain the hypothesis
that these three variables are, in the population, normally distributed.
We now ask whether these three variables can be assumed to stem from a multinormal
population. In a first step, we inspect the 3D scatterplot of these three variables. Figure 2
displays the scatterplot.
A. von Eye
164
Table 5:
Descriptive statistics for the three self report variables VAAA83, PAAP83, and AI83
VAAA83
114
7.000
36.000
19.208
5.661
0.417
0.226
0.222
0.449
N of cases
Minimum
Maximum
Mean
Standard Dev
Skewness(G1)
SE Skewness
Kurtosis(G2)
SE Kurtosis
PAAP83
114
8.000
44.000
21.291
8.254
0.606
0.226
-0.471
0.449
AI83
114
5.000
29.000
16.890
5.432
0.100
0.226
-0.442
0.449
3
2
1
0
-1
-2
-3
0
10
3
2
1
0
-1
-2
-3
0
20
VAAA83
Expected Value for Normal Distribution
Expected Value for Normal Distribution
Expected Value for Normal Distribution
Figure 1:
Probability plots of the variables VAAA83, PAAP83, and AI83
10
20
30
PAAP83
40
50
30
40
3
2
1
0
-1
-2
-3
0
10
20
AI83
30
Base models for Configural Frequency Analysis
165
Figure 3:
3D scatterplot of the variables VAAA83, PAAP83, and AI83
The visual inspection of the three-dimensional distribution of these three variables is certainly inconclusive. Obviously, the variables are strongly correlated (rVAAA83-PAAP83 = 0.606
rVAAA83-AI83 = 0.495, rAI83-PAAP83 = 0.372). However, it is hard to make out whether the deviations
from multinormality are so strong that the null hypothesis must be rejected. Employing
Mardia’s tests of multivariate skewness and kurtosis, we obtain skew = 0.163 with p =
0.9790 and kurtosis = 11.152 with p < 0.001, suggesting that the three variables do not conform with hypotheses that are compatible with the assumption of multinormality.
This information, although useful, does not provide us with hints at where in the threedimensional space deviations are strongest. Therefore, we perform a CFA of the threedimensional space. Specifically, we create three equi-probable segments for each of the three
indicators of aggression. Segment 1 of each variable contains the low-scoring adolescents,
Segment 2 contains the adolescents with scores about the mean, and Segment 3 contains the
high-scoring adolescents, that is, the adolescents with above average aggression scores.
Then, we cross the thus categorized variables to obtain a 3 x 3 x 3 cross-classification. To
calculate the expected cell frequencies, we use the method proposed by Genz (1992). This
calculation takes the correlations among the three variables into account. We then perform a
CFA of this cross-classification, using the z-test and the Bonferroni-adjusted "* = 0.00185.
Table 6 displays results of the configural analysis.
A. von Eye
166
Table 6:
CFA of the three-dimensional distribution of the variables VAAA83, PAAP83, and AI83,
performed with the goal of testing multinormality
Configuration
111
112
113
121
122
123
131
132
133
211
212
213
221
222
223
231
232
233
311
312
313
321
322
323
331
332
333
Êi
Ni
15
4
2
7
2
2
2
2
0
11
5
6
2
9
3
1
4
4
2
1
2
0
1
4
1
8
14
8.664
5.252
2.131
5.170
3.918
1.908
1.489
1.371
.792
3.657
4.005
2.690
4.803
6.148
4.815
2.874
4.342
4.040
.756
1.323
1.458
1.913
3.949
5.295
2.528
6.569
12.040
2
statistic
2.153
-.546
-.090
.805
-.969
.066
.419
.537
-.890
3.840
.497
2.018
-1.279
1.150
-.827
-1.105
-.164
-.020
1.431
-.281
.449
-1.383
-1.484
-.563
-.961
.558
.565
p
.01567217
.29237026
.46416812
.21052392
.16625234
.47357112
.33758449
.29551351
.18674901
.00006162
.30947955
.02180589
.10043393
.12507364
.20403187
.13451753
.43488734
.49200377
.07618921
.38929046
.32688650
.08329751
.06889284
.28677873
.16831158
.28832552
.28608148
Type/
Antitype?
Type
The overall goodness-of-fit X for the model in Table 6 is 40.05 (df = 17; p = 0.00127).
This significant value suggests that the null hypothesis of multinormality must be rejected.
The sector-specific CFA tests will indicate whether there are particular sectors with too many
or too few cases, compared to the expected frequencies that were calculated based on the
assumption of multinormality.
The sector-specific results in Table 6 suggest that there is one sector with more cases
than expected based on the hypothesis of multinormality. In Sector 211, we find 11 adolescents, but less than 4 were expected for this sector. These are respondents who are about
average in verbal aggression against adults, and below average both in physical aggression
against peers, and aggressive impulses.
Given this result, researchers can respond by discussing the assumption that the three aggression variables follow a multinormal distribution. There may be reasons why this assump-
Base models for Configural Frequency Analysis
167
tion fails to carry. Alternatively, or in addition to discussing this assumption, the researchers
may consider selective resampling, with the goal of completing the sample.
For reasons of comparison, we present in Table 7 results from standard, first order CFA,
that is, from a CFA of the cross-classification displayed in Table 6. In this CFA, only the
main effects of the three variables are taken into account.
2
The overall Pearson goodness-of-fit X for the first order CFA base model is 96.12 (df =
20; p < 0.01). This value suggests significant deviations from the assumption of variable
independence. Two types emerge which carry this result. The first type is constituted by
Configuration 111. It contains 9 adolescents. Only about 3 had been expected. These are
adolescents that score below average in all three aggression scales. The second type is constituted by Configuration 333. This type suggests that more adolescents with high scores on all
three variables than expected based on the hypothesis of variable independence were found.
Obviously, this result differs from the one presented in Table 6.
Table 7:
First order CFA of the variables VAAA83, PAAP83, and AI83, performed with the goal of
testing deviations from independence
Configuration
111
112
113
121
122
123
131
132
133
211
212
213
221
222
223
231
232
233
311
312
313
321
322
323
331
332
333
Ni
Êi
15
4
2
7
2
2
2
2
0
11
5
6
2
9
3
1
4
4
2
1
2
0
1
4
1
8
14
5.452
4.787
4.920
3.407
2.992
3.075
4.089
3.590
3.690
6.814
5.983
6.150
4.259
3.740
3.843
5.111
4.488
4.612
4.997
4.388
4.510
3.123
2.742
2.819
3.748
3.291
3.382
statistic
4.090
-.360
-1.316
1.946
-.573
-.613
-1.033
-.839
-1.921
1.603
-.402
-.060
-1.095
2.720
-.430
-1.818
-.230
-.285
-1.341
-1.617
-1.182
-1.767
-1.052
.704
-1.419
2.596
5.773
p
.00002162
.35958151
.09403134
.02580282
.28320486
.26995857
.15081630
.20068376
.02737383
.05442260
.34383465
.47595022
.13684194
.00326195
.33350828
.03450379
.40898962
.38780038
.08999708
.05290488
.11864076
.03859086
.14636441
.24080438
.07788918
.00471728
.00000000
Type/
Antitype?
Type
Type
A. von Eye
168
3. Discussion
Repeatedly, CFA has been labeled as nothing but a re-packaged residual analysis for loglinear models. Indeed, a good number of CFA applications uses the routine base model of
CFA which is the first order model, a log-linear model of variable independence. Lehmacher
(2000) and von Eye (2002) have raised the argument that even though many CFA base models use the same methods as log-linear modeling for the estimation of expected cell frequencies and for the analysis of the cell-wise deviances, there exist fundamental differences in the
goals pursued with these two methods. Log-linear modeling is used to devise a model that
describes the data well. If there are significant deviations from the frequencies proposed by a
model, researchers try to modify the model such that it describes the data without significant
deviations. In contrast, CFA is used to identify and explain configurations that constitute
types and antitypes, that is, significant deviations from some suitably specified base model.
The present article shows that the labeling of CFA as a method of log-linear residual
analysis can be disputed from an additional perspective. This perspective considers the base
models that can be specified for CFA application. Four groups of base models have been
identified in this article. Only the first group includes log-linear models. As was mentioned
above, the best known CFA base model in this group is the model of variable independence,
used in Lienert’s (1969) original CFA. As was indicated above, this group of base models is
very flexible. It allows one to consider a very large number of base models, including global
and regional models, and models with and without covariates.
The second group of base models uses population parameters instead of estimating the
expected cell frequencies from the data. The third group of models is fueled by a priori considerations. These considerations reflect, for instance, processes as described by mathematical psychological theories, or theoretical propositions such as the a priori probability that can
be determined for events as draws from urns or coin toss outcomes. The most frequently
discussed base model in this group specifies the a priori probability of sign patterns for differences between adjacent scores in, for instance, time series.
The fourth group of models is parametric in nature. This group allows one to test hypotheses concerning the multivariate distribution of categorized and cross-classified continuous variables. This group is new (von Eye & Gardiner, 2003). It uses Genz’ (1992) method
which allows one to estimate the probability of rectangular sectors in a multivariate space
under the assumption of multinormality. Methods of CFA are used in this context first to
specify these sectors, and second to determine whether individual sectors contain more or
fewer cases than expected based on the assumption of multinormality. Further developments
of this approach will allow researchers to test additional distributional assumptions. Alternative definitions of sectors will be proposed in the future, for example, convex sectors, identified by methods of, for example, cluster analysis.
The selection of a base model for a particular application of CFA is obviously not arbitrary. The following criteria can be used for selection:
1.
2.
A base model must conform to the specification of CFA base models given above
(von Eye & Schuster, 1998): uniqueness of interpretation, parsimony, and consideration of sampling scheme.
A base model must also take into account the available information about the phenomenon under study. For example, population parameters must be taken into ac-
Base models for Configural Frequency Analysis
3.
169
count if available. This applies accordingly to theoretical considerations concerning
a priori probabilities, or to models that describe the process under study.
A base model can be a hypothesis about a probability distribution. von Eye and Gardiner (2003) proposed using CFA methods to test hypotheses concerning multinormality.
We thus conclude that CFA is a method with a much broader foundation than considered
by arguments that reduce this method to a derivative of log-linear modeling. CFA is of interest whenever researchers ask questions concerning the frequencies in individual cells and
groups of cells.
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