Configurational Analysis Research Paper

1. Profiles And Configurations

William Stern introduced in 1911 the distinction between the notions of variability and psychography. Variability is the focus when many individuals are observed in one characteristic with the goal to describe the distribution of this characteristic in the population. Psychographic methods aim at describing individuals in many characteristics. Stern also states that these two methods can be combined.

When describing an individual, results are often presented in form of profiles. For example, test results of the (Minnesota Multiphasic Personality Inventory) personality test typically are presented in form of individual profiles, and individuals are compared to reference profiles. A profile describes the position of an individual on standardized, continuous scales. Thus, one can also compare the individual’s relative standing across several variables. Longitudinally, one can ask, whether an individual’s relative standing and/or the correlation with some reference change. For example, a profile may resemble the pattern typical of schizophrenics. Individuals can be grouped based on profile similarity.

In contrast to profiles, configurations are not based on continuous but on categorical variables. Specifically, the ensemble of categories that describes a cell of a cross-classification is called configuration (Lienert 1969). Consider, for example, the variables Gender (female–male), Marital Status (married–not married), and Happiness (yes–no). The cross-classification of these three variables has eight cells. The combinations of categories that describe these cells are the configurations. For example, the three categories female and married, and happy constitute the first configuration. The combination male and not married, and unhappy constitutes the last configuration.

CA investigates such configurations from several perspectives. First, CA identifies configurations. This involves creating cross-classifications, or, when variables are originally continuous, categorization and then creating cross-classifications. Second, CA asks, whether the number of times, a configuration was observed, could have been expected from some a priori specified model. Significant deviations will then be studied in more detail. Third, CA asks whether the cases found displaying different configurations also differ in their mean and covariance structures in variables not used for the cross-classification. This question concerns the external validity of configurational statements. Other questions are conceivable.

In the following section, a theoretical background for CA is provided. Specifically, CA will be embedded in differential psychology and the person-oriented approach. In Sect. 3, configural frequency analysis, a chief method for CA will be described. In Sect. 4, other methods such as cluster analysis and log-linear modeling will be described briefly.

2. Configurational Analysis: Theoretical Embedding

This section covers two roots of configurational analysis (CA), differential psychology, and the Person-Oriented Approach. The fundamental tenet of differential psychology is that ‘individual differences are worthy of study in their own right’ (Anastasi 1994, p. ix). This is often seen in contrast to General Psychology where it is the main goal to create statements that are valid for an entire population. General Psychology is thus chiefly interested in variables. The data carriers themselves, e.g., humans, play the role of replaceable random events. They are not of interest per se. In contrast, Differential Psychology considers the data carriers units of analysis. The smallest unit would be the individual at a given point in time. However, larger units are often considered, e.g., all individuals that meet the criteria of geniuses, alcoholics, and basketball players.

Differential psychology as both a scientific method and an applied concept presupposes that the data carriers’ characteristics are measurable. In addition, it must be assumed that the scales used for measurement have the same meaning for every data carrier. Third, it must be assumed that the differences between individuals are measurable. In other words, it must be assumed that data carriers are indeed different when they differ in location on some scale. It is well known that this is often not the case with questionnaires and tests in the social sciences.

The Person-Oriented Approach (see Magnusson 1998, Bergman and Magnusson 1991, 1997, von Eye et al. 2001) is a relative of Differential Psychology. It is based on five propositions (Bergman and Magnusson 1997, von Eye et al. 2000) that stress the individual nature and complexity of functioning, process, and development (FPD); the lawfulness and structure of FPD; and the organization of FPD in patterns. Most important from the present perspective is Proposition 5, which states that some patterns and configurations will be observed more frequently than others are, or more frequently than expected based on prior knowledge or assumptions. These patterns can be called common types. CFA defines types as configurations that occur more often than expected from some chance model. Accordingly, there will be patterns or configurations that are less frequently observed than expected from some chance model. CFA terms these the antitypes.

Thus, the person-oriented approach and CA meet where (a) patterns of scores or categories are investigated, and (b) where the tenet of differential psychology is employed that it is worth the effort to investigate individuals and groups of individuals. The methodology employed for studies within the frame- work of the person-oriented approach is typically that of CA. The following section describes configural frequency analysis, one of the central statistical methods of CA.

3. Configural Frequency Analysis (CFA)

Configural frequency analysis is introduced by using a data example. Consider the three variables, Gender (G) (1=Male, 2=Female); Motive for Suicide At-tempt (M) (1=llness, 2=Psychiatric Disorder, 3=Alcoholism); and Outcome of Suicide Attempt (1=Survived, 2 =Dead) (see Lienert 1978, von Eye et al. 1996). Crossed, these three variables form a 2×3×2 contingency table. This table appears in Table 1, along with results from CFA.

The first column of Table 1 lists the configurations. The first configuration is the combination of the categories, Male and Illness and Survived Suicide Attempt. The second configuration is the combination Male and Illness and Died, and so forth. The second column displays the frequency with which each of the configurations occurred. CA allows researchers to investigate cross-classifications of the kind presented in Table 1 from the perspectives listed above. The following paragraphs illustrate the first two perspectives using CFA as a sample method of analysis.

3.1 Identification Of Configurations

The first perspective involved identification of con-figurations. The 12 configurations in Table 1 are identified as the cells of a 2×3×2 cross-classification of three categorical variables. Each individual in the sample displays one of the 12 possible patterns and can therefore be unambiguously assigned to one of the cells.

3.2 Comparison Of Observed With Expected Cell Frequencies: CFA Types And Antitypes

When inspecting the observed frequencies in Table 1, it becomes immediately obvious that the 12 con-figurations were observed at different frequencies. Whereas seven alcoholic males were counted to have survived a suicide attempt, 86 ill females were counted to have survived an attempt. While it does make sense to compare raw frequencies, more often researchers try to explain the observed frequency distribution based on an underlying model. Such a model is said to explain the observed data if no significant discrepancies between the observed distribution and the distribution predicted by the model exists.

CFA goes the opposite route. It specifies a so-called base model that is not supposed to explain the data (examples of base models follow in Sect. 3.3). Discrepancies from this model indicate what is going on, that is, the effects that created the observed distribution. These effects manifest in the above mentioned common types and common antitypes. More specifically, let f_i be the observed frequency for Cell i, and e_i the expected frequency for Cell i. The base model is used to estimate the expected cell frequencies. Then, if statistically f_i > e_i, the configuration that describes Cell i constitutes a CFA type. If f_i < e_i, the configuration constitutes a CFA antitype. If f_i = e_i, the configuration constitutes neither a type nor an antitype.

3.3 CFA Base Models

Admissible CFA base models meet the following three criteria (von Eye and Schuster 1998). The first criterion is that there be only one way to deviate from the base model for clear-cut interpretation of types and anti-types. For example, first order CFA uses the log-linear main effect model for a base model. This model implies no interactions among variables. Deviations from this model are only possible if there are interactions.

The second criterion, introduced by von Eye and Schuster (1998), is that the sampling scheme be considered. Specifically, the multinomial and the product multinomial sampling schemes have major implications for the selection of base models and interpretation of results (von Eye et al. 1999).

Parsimony is a third criterion for the selection of base models. More specifically, a base model must take into account interactions of the lowest possible order.

Base models can be subdivided in two groups. The first group is that of global CFA models. Models in this group assign the same status to all variables in the analysis. There is a hierarchy of global models. It begins at the level where no effects are assumed at all. Thus, the expected cell frequency for Cell i is estimated as e_i = t/N, where t is the total number of cells in the cross-classification and N is the sample size. This model is termed zero-order CFA. At the next higher level, first-order CFA proposes that main effects can exist for each variable, but no interactions. First-order CFA is the original approach to CFA (Lienert 1969). The expected cell frequencies for Table 1 were estimated using the first-order CFA base model (this model will be explained in more detail later in this section). Accordingly, second order CFA proposes that first-order interactions exist between all pairs of variables, and so forth. In each of these models types and antitypes can emerge only if effects of orders higher than the ones assumed in the base model exist.

Regional CFA models distinguishes between two or more groups of variables. Examples of regional CFA models include Prediction CFA (P-CFA) which assumes that the variables are grouped into predictors and criteria. P-CFA types (antitypes) allow one to predict the occurrence (lack of occurrence) of criterion configurations from particular predictor configurations.

Most CFA base models can be expressed in form of simple log-linear models. Thus, expected cell frequencies can be estimated from the marginal frequencies of the variables involved in the cross-classification. A general expression for log-linear models and, thus, log-linear CFA base models, is log F=Xλ where F is the array of expected frequencies under study (the third column in Table 1), X is the design matrix (comparable to a design matrix in analysis of variance), and λ is a parameter vector. Estimation is typically performed using maximum likelihood methods. Many CFA base models lead to very simple computational equations.

Once the expected cell frequencies are estimated, they can be compared with the observed frequencies. A large number of statistical tests have been proposed for these comparisons. One of the best of these tests is the binomial approximation of the z-test. The test statistic is z_i = ( f_i – e_i)√(e_i(1-e_i /N)). Inserting the values from the data example in Table 1 yields for the first configuration the test statistic z₁₁₁= (64 -59.416)/√ 59.416(1-59.416/482)=0.635. This value is the same as the one for the test statistic for Configuration 111 in Table 1 . Other tests include the exact binomial test, the χ²-component test, Lehmacher’s (1981) hypergeometric tests, and the F-test (cf. von Eye and Rovine 1988).

3.4 Statistical Decisions Concerning Types And Antitypes

For each of the configuration specific test statistics a tail probability is then calculated. Recent results suggest that standard testing as described above tends to underestimate the probability of antitypes (Indurkhya and von Eye 2000). Future research will have to devise methods that eliminate this asymmetry. One very important characteristic of the tests performed in CFA is that they are not independent of each other. Rather, they can be dependent because (a) the avail-able information has already been exhausted or (b) capitalizing on chance. Therefore, measures must be taken to protect the significance level α. A number of adjustments have been proposed for protection of α. One of the most conservative yet most frequently employed adjustments is known as Bonferroni adjustment. This procedure creates an adjusted significance level, α*, that is α* =α/t, where α is the original, nominal significance level, and t is the total number of tests performed. In the example in Table 1 we test each cell. Therefore, t =12. For α=0.05 we obtain α*=0.05/12 = 0.0041667. Obviously, the adjustment makes the decision as to whether a type or antitype exists more conservative. Therefore, the Configurations, 211, 212, 221, and 231 for which the tail probabilities are below the usual α =0.05 fail to constitute types and antitypes. Still, even under this conservative decision rule, CFA manages to identify two types and one antitype in the data in Table 1. The first type is constituted by Configuration 122. It suggests that more men suffering from psychiatric disorders than expected from the base model of variable independence do not survive a suicide at-tempt. The second type is constituted by Configuration 232. It suggests that more alcoholic women than expected from the base model die in suicide attempts. The antitype, Configuration 131, indicates that fewer alcoholic men than expected from the base model survive a suicide attempt.

3.5 Interpretation Of Types And Antitypes

When interpreting results from CFA, researchers use two sources of information. The first source includes the identified types and antitypes. Most typically (see, for example, Table 1), only a subset of configurations deviates from the CFA base model to the extent that types or antitypes are constituted. If there exist only few significant deviations, the base model is contradicted locally, and in the other sectors of the data space spanned by the cross-classification under study the base model holds.

The second important source of information is the CFA base model. In the analyses for Table 1 the base model proposed that the three variables, Gender, Motive, and Outcome of Suicide Attempt be unrelated throughout. The two emerging types and the one antitype, however, suggest that there exist local associations such that particular categories of these variables, that is, particular configurations surface more often (and less often) than could be expected under the base model. Thus, the base model experiences local violations in three sectors, yet it holds in the remaining nine sectors. The base model for the CFA of the data in Table 1 can be violated by any relationship among the three variables.

It is important to realize that the application of CFA within the context of Configurational Analysis implies a perspective of variables that is different than the perspective taken when analyzing variables. With-in CA, variable interactions and main effects are of interest only in so far as they are part of a CFA base model. The main focus of interest, however, lies in the possible local deviations from the base models. Such deviations are always interpreted at the level of configurations, that is, locally. There have been only a small number of attempts to interpret groups of types and groups of antitypes (von Eye and Brandtstadter 1997, Gutierrez-Pena and von Eye 2000).

3.6 Variants Of CFA

By far most CFA base models are derivatives of log-frequency models. However, there exist circumstances under which the estimation of expected cell frequencies from the observed frequencies might be problematic. An example where the probabilities of configurations are determined by information other than the observed distribution can be found in longitudinal CFA (von Eye 1990). When change scores are calculated for ordinal variables, the top and the bottom ranks have fewer possibilities for change. Whereas the middle ranks can stay the same, move up, or move down, the top rank can only stay the same or move down. Accordingly, the bottom rank can only stay the same or move up. These differences must be considered when estimating expected cell frequencies (for details see von Eye 1990, Chap. 6).

A second domain of development of methods for CFA is that of Bayesian CFA. Bayesian methods differ from standard, frequentist methods in that conclusions about a parameter or about unobserved data are made in terms of probability statements that are conditional on the observed value of the dependent measure. Implicitly, statements are conditioned on known values of the independent variable. In contrast, standard, frequentist statistical inference is based on an evaluation of the procedure used to estimate the parameter for the distribution of possible values of the dependent measure, conditional on the true yet un-known value of the parameter (Gelman et al. 1995). One interesting and important characteristic of Bayesian statistics is that prior information can easily be taken into account.

Thus far, there have been only a small number of attempts to formulate a Bayesian CFA (Wood et al. 1994, Gutierrez-Pena and von Eye 2000, von Eye et al. 2000). Each of these approaches is (a) unrelated to loglinear modeling methods for estimation of expected cell frequencies, and (b) uses the so-called noninformative priors. The rationale for using this kind of priors is to let the data speak for themselves, that is, without consideration of information external to the data under study (Gelman et al. 1995). However, in many instances such information exists. Thus, future developments will discuss methods of incorporating prior information in the form of prior distributions.

A second route of development that has just begun to be traveled is that of defining types of types (and antitypes). Traditionally, CFA types and antitypes are defined in a way similar to residuals in log-linear models, that is, as significant discrepancies from some base model. In 1991, Goodman made explicit that there are many ways to deviate from expectancy. Three of these ways were used by von Eye, Spiel, and Rovine (1995) to define different concepts of types and antitypes. The first concept is a marginal-dependent relative of the correlation, ρ. The second is a marginal-free variant of the odds ratio, and the third concept is a marginal-dependent variant of the odds ratio. Results of simulation studies suggest that the patterns of types and antitypes identified under the three concepts can differ dramatically. In addition, these patterns can be different than the patterns identified using the original concept of CFA types and antitypes. Gonzales Deben (1998) proposed a fourth approach to defining types and antitypes that is based on a residual definition by Rudas et al. (1994).

3.7 Alternative Methods Of Configurational Analysis

Section 3 suggests that configuration frequency analysis is the chief method of data analysis in CA. However, other methods have been proposed and used. Two of these methods will be briefly reviewed here, log-linear modeling and cluster analysis.

Cluster analysis (see Hartigan 1975, von Eye et al. 1999) is a method that allows researchers to identify groups that were unknown to exist before analysis. Cluster analysis creates groups (clusters) that contain cases that are more similar to each other than to cases from other groups. The cluster formation process typically follows one of the following three strategies:

(a) an existing sample is subdivided until some optimality criterion is reached; (b) the individuals in a sample are agglomerated until some optimality criterion is reached; and (c) density centers are formed and individuals gravitate to the nearest center. For more detail on the decision making involved when searching for clusters see von Eye et al. (1999). Cluster analysis and CA meet where researchers hold the assumption that data analysis at the aggregate level can obscure interpretable differences between individuals or groups of individuals.

The typical application of log-linear modeling focuses on variables. Thus, this method finds more applications in standard, aggregate level analysis (Agresti 1990, Bishop et al. 1975). However, several variants of log-linear modeling can be fruitfully employed in CA. Two sample approaches will be briefly reviewed in this context. The first approach concerns models that specify effects that are specific to subtables. For instance, Clogg and Shockey (1988) proposed models that allow one to ask whether, in two-way cross-classifications, an association between the two variables that constitute the table is the same throughout. In an example with the variables Riding a Car for Fun and Drug Use, the authors showed that there is an association such that those respondents who ride a car often for fun also tend to consume stronger drugs. However, those respondents who never ride a car for fun tend to consume stronger drugs than those who ride a car a few times a year for fun do. This pattern suggests that the association between the two variables switch as one inspects different regions of a cross-classification.

The second approach is linked to nonstandard log-linear models (Rindskopf 1990). Standard log-linear models specify main effects and interactions in terms of coding schemes that are parallel to the ones used for analysis of variance (see, e.g., Christensen 1997). Nonstandard models include contrasts that cannot be expressed using these terms. Rather, these effects, comparable to planned mean comparisons, are added to the model to test specific hypotheses. These hypotheses typically concern effects that occur only in specific regions of a cross-classification. Von Eye and Brandtstadter (1998) proposed using nonstandard log-linear models to test hypotheses about causal effects. While such models may suffer from problems of parameter intercorrelations that can be viewed parallel to multicollinearity in analysis of covariance or multiple regression, they allow researchers to ask quest-ions that go beyond standard questions of variable associations.

These and other approaches to log-linear modeling meet CA where hypotheses are formulated and tested that do not apply to everybody in a population but only to distinct groups. It is assumed that the variable relationships in these groups differ from the ones found for the rest of the populations.

To conclude, configurational analysis (CA) represents a methodology that allows researchers to counter-balance the far more frequently used methodology that focuses on variables rather than on individuals. Its chief representatives, cluster analysis and con-figural frequency analysis are well-developed statistical techniques that allow researchers to specify statements about individuals and groups of individuals. Variables play in these statements the role of carriers of information.

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