types of change: application of configural frequency analysis in repeated measurement designs

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Types of change: Application of configuralfrequency analysis in repeated measurementdesignsAlexander von Eye a & John R. Nesselroade aa The Pennsylvania State UniversityPublished online: 27 Sep 2007.

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169

Experimental Aging Research, Volume 18, Number 4, 1992, ISSN 0734-0664 “1992 Beech Hill Enterprises Inc.

QUANTITATIVE TOPICS IN RESEARCH ON AGING

J.J. McArdle and S.A. Cohen, Eds.

Types of Change: Application of Configural Frequency Analysis in Repeated Measurement Designs

ALEXANDER VON EYE AND JOHN R. NESSELROADE The Pennsylvania State University

Many researchers are concerned both with intraindividual change patterns and interindividual differences and similarities in those change patterns. Configural Frequency Analysis (CFA) provides a way to identify overrepresentations (types) and under- representations (antitypes) in the frequencies of multiple variable classifications organized to reflect patterns of change. Three methods of CFA for analyzing repeated measures data are considered. To establish trends, two of them require at least ordinal data and the third requires interval data. Data analysis and the interpretation of results are illustrated. CFA is compared with residual analysis from log-linear modeling.

n recent years, several new repeated measurement data I analysis procedures, suitable for addressing developmental questions, have appeared. Examples include log- linear modeling (Bishop, Fienberg, & Holland, 1975), prediction analysis (PA; Hildebrand, Laing, & Rosenthal, 1977; cf. Froman & Hubert, 1980), information theory (Krippendorff, 1986), and configural frequency analysis (CFA; Krauth & Lienert, 1973; Lienert, 1969). Although it would be of interest to compare all of these methods in detail, the present paper focuses on CFA and its application to the study of change.

Log-linear, information theory, and prediction analyses can be used for exploratory analysis but most applications, in particular of prediction analysis, are confirmatory. Results obtained with these methods are usually expressed in general rather than in differential terms. CFA is typically used in exploratory, differential contexts although it can be used in a confirmatory fashion.

We will begin by comparing CFA to residual analysis in log-linear modeling (Haberman, 1973), an approach to which it is closely related. Configural frequency analysis is a statistical method for the detection of types in the analysis of discrete variables. When more individuals with a particular pattern of characteristics are found than expected, the pattern is called a type. Essentially, to identify types, CFA compares observed frequencies of a contingency table with expected frequencies estimated under some model. The method of log-linear modeling can be used for the estimation.

CFA and Residual Analysis

Lienert and collaborators (Krauth & Lienert, 1973; Lienert, 1%9) began developing CFA quite independently of log-linear modeling in 1969. CFA’s similarity to residual analysis in log-linear modeling is that it focuses on residuals, that is, the (standardized) differences between observed and expected frequencies. Both CFA and residual analysis typically examine all cells of a contingency table. Therefore, problems of multiple testing arise. Solu- tions for these problems have been discussed with respect to CFA (e.g., Perli, Hommel, & Lehmacher, 1985). Also, significance tests have been developed specifically with CFA in mind (e.g., Lehmacher, 1981; Lindner, 1984). It is obvious, however, that such tests may be applied in residual analysis also.

One way in which CFA differs from residual analysis is that whereas CFA is used to identify those cells in a contingency table that contradict a set of null hypotheses, residual analysis compares observed and expected cell frequencies “to obtain more insight into the failure of the model” (Haberman, 1973, p. 216), and to obtain indicators for the formulation of alternative models (Agresti, 1984). Results of CFA concern cells that are interpreted with reference to the meaning of the pattern of states of variables that constitute these cells.

In longitudinal research, change patterns are configurations with time-related characteristics. The present paper focuses on the application of CFA to repeated observa-

The authors dedicate this paper to G.A. Lienert. The authors are indebted to Lars R. Bergman, G.A. Lienert, Peter Metzler, and Mike J. Rovine for helpful comments on earlier versions of

Correspondence regarding this paper should be sent to Alexander von Eye, College of Health and Human Development, S-1 10 Henderson South, this paper. J.R. Nesselroade’s work is supported in part by the MacArthur Foundation Research Program on Successful Aging.

The Pennsylvania State University, University Park, PA 16802, U.S.A. J.R. Nesselroade is now at the University of Virginia.

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170 VON EYE/NESSELROADE

TABLE 1

Configural Frequency Analysis of the Variables Logorrhea, Running Away From Home, Fear of Dark, and Bedwetting

Configuration

1 1 1 1 1 1 1 2 1 1 2 1 1 1 1 1 1 2 2 2 2 2 2 2

1 2 2 2 2 1 1 1 1 2 2 2

2 1 1 2 2 1 1 2 2 1 1 2

2 1 2 1 2 1 2 1 2 1 2 1

23 3.332 10.800 3.5*10-”(T) 11 9.661 .434 0.6645 18 4.915 5.923 3.2*10-3 (TI 15 7

20 12 16 18 43 13 43 38

122 50

14.252 9.308

26.988 13.731 39.812 15.732 45.613 23.207 67.288 43.946

127.420 64.828

.200 - .762 - 1.372 - .472 - 3.886

.578 - .400

-2.155 -3.115 - .927 - .531 - 1.934

0.8413 0.4463 0.1701 0.637 1 0.0001 0.5630 0.6890 0.0312 0.0018 0.3541 0.5954 0.0531

2 2 2 2 249 187.966 5.208 1.9*10-* (T)

n = 698 l.. . = 122, 2... =576 .1.. = 184, .2.. = 514 ..1.=282, ..2.=416 ... 1=179, ... 2=519

T = type, A = antitype, rn = observed frequencies, rh =expected frequencies.

tions to identify types of change patterns. We will demon- strate the use of CFA in a differential context to identify types which indicate associations that apply only to selected subsets of the data space (Nesselroade, 1983, 1988). Examples of types of change patterns include temporal trajectories observed more often than expected by chance, and behavior problems observed at a given age only.

Preliminary Example of CFA

The first example of CFA involves a clinical data set (Krauth & Lienert, 1974). Six hundred and ninety-eight children were scored (1 = present, 2 = absent) on each of the following behavior disorders: L = logorrhea that is, uncontrolled, uninterrupted speaking, R = running away from home, F = fear of dark, and B = bedwetting. Table 1 gives the cross-tabulation of the four variables.

More than a third of the children did not show any of the symptoms [f(2222) = 2491. Only 23 children showed pattern 1111, that is, all four symptoms. The research question typically approached with log-linear modeling concerns the interaction structure of the variables. Resid- ual analysis can then be used to identify those configurations that lead to the rejection of a particular model, and to obtain hints for the formulation of better fitting models. The research question underlying CFA concerns single

configurations. For example, what syndrome is observed more often than expected by chance? Or, what syndrome is observed less often than expected by chance?

Next to the observed cell frequencies, Table 1 gives the expected frequencies estimated under the assumption of total independence of all four variables. In log-linear modeling this assumption is met by a main effect model in which categories of single variables can differ in their frequencies, and aU interaction terms are set equal to zero. Table 1 shows that the expected frequency of, for instance, the syndrome of all four behavior disorders is e = 3.33. However, f ( l l l1) = 23 children manifested this syndrome.

Subsequently, we will discuss significance tests. Syn- drome 11 11 will be shown to form a type. This type may be interpreted as a local association between L, R, F, and B, indicating that children suffer from all four symptoms far more often than expected by chance under the independence model. Syndrome 1222 was observed less often than expected. In CFA terms, this syndrome constitutes an antitype, indicating that logorrhea as a mono- symptomatic behavior disorder is very unlikely under the independence model. Table 1 contains three types (syndromes 11 11, 1121,2222) and two antitypes (syndromes 1222, and 2122).

This example illustrates how CFA compares observed

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TYPES OF CHANGE: CFA 171

frequencies m with expected frequencies fi estimated under some model. Types and antitypes emerge only if the difference between m and fi is larger than statistically tolerable. It is important to note that the absolute mag- nitude of both m and fi is not considered in CFA. Rather, CFA tests how large m is relative to fi.

The following sections of this paper will give a more detailed introduction to the rationale and methods of CFA. Then, three methods of applying CFA to repeated observations will be discussed. The first of these methods analyzes the pathways that subjects follow through time. These pathways are treated as states of a single variable. The second method considers each variable at each point in time as a separate variable. Thus, a contingency table is formed that has at least as many dimensions as observation points in time. The third method describes each univariate time series by an orthogonal polynomial. The coefficients of these polynomials, which can be interpreted as descriptive statistics, may be categorized, and analyzed using CFA.

Rationale and Methods of CFA

Let a d dimensional contingency table X by given, X = {m}, where m denotes the observed frequencies. Then, the probability of the observed frequency m,, d is p(m,, ,d). Utilizing methods for log-linear modeling (Bishop, Fien- berg, 8t Holland, 1975), the expected cell frequency (h,, ,d) may be estimated under certain assumptions that specify the effects used to explain the observed frequency distribution. One of the simplest assumptions negates the presence of any effects in X. Under this assumption ( f i i , ,d) can be estimated by

d d

where c, denotes the number of states that can be assumed by the jth variable. Within the framework of CFA, the assumption of absence of any effects was discussed by Lienert and von Eye (1989), under the label of configural cluster analysis.

Under the assumptions that only main effects exist, and that the variables under study are totally independent of each other, a given cell frequency may be estimated by

fi,, , d = m, m , ... m d/n(d-l) (2)

where x, denotes the marginal sum of the ith category of the first variable, and so on. In two-dimensional tables, Formula 2 is equivalent to the well-known rule, according to which “expected frequency = row sum * column sum / n,” or, in terms of Formula 2,

hi,= m,.m,,/n.

The assumption of total independence among the variables under study was made in the first conception of CFA (Lienert, 1%9). Any deviation from this assumption

is necessarily due to relations among variables. Therefore, types or antitypes can be interpreted as local indicators of the presence of a structure that contradicts the null hypothesis of a chance distribution.

In most applications CFA estimates expected cell frequencies using (2). The model underlying this formula takes into account that the states of the variables may be frequented differently. It generates the same expected frequencies as the model representing the absence of any effects if the marginal sums are equal.

Both models of CFA share the interpretation of types and antitypes as indicating that there is a structure in the data that can only be explained if assumptions are made that go beyond the ones in these CFA models. A hier- archy of CFA models may be generated by assuming in- creasingly complex relations among variables. The first step beyond the model of total independence is the assumption that the structure in a given data set can be explained by pairwise associations of variables. Within the framework of CFA, von Eye and Lienert (1984; cf. von Eye, 1988, 1990) defined second and higher order types and antitypes on the basis of this assumption. If this assumption is rejected, relations among triplets, quadru- plets, etc., rather than pairs of variables are needed to explain the variation in the data set.

Several statistical test procedures have been proposed to decide whether or not the null hypothesis

can be rejected. Krauth and Lienert (1973) proposed using the nonparametric binomial test. This test calculates the two-sided tail probability of m = fi or m assuming more extreme values as

where p: = fi/n and

and n if a=m,, ,d

mi, ,d else e = { If fi is large enough a normal approximation can be used for which the test statistic is

z = (m, ,d - fi)/(npq)1’2 ( 5 )

If H, holds, a test statistic distributed as chi-square can be used for which

Z2(alpha/2) = chi2(a). (6)

However, this approximation tends to suggest conservative decisions. More powerful tests have been proposed that can be applied only under certain assumptions

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172 VON EYEINESSELROADE

concerning population parameters (Lehmacher, 1981) or to dichotomized data (Lindner, 1984).

Irrespective of whether the generally applicable test procedures given in (4)-(6) or more specialized tests are used, it cannot be assumed that several tests performed on the same sample are independent of each other. There- fore, several procedures have been discussed to keep the factual alpha-level as close as possible to the nominal one. The most common procedure is the Bonferroni adjustment (see Miller, 1966), according to which for t tests the adjusted alpha becomes

alpha* = alpha/t. (7)

Alpha* guarantees that an a priori determined alpha is not exceeded for a family of tests. As an alternative, one can test sequentially using Holm’s (1979) general procedure which leads to the successive significance levels

alpha: = alpha/t, alpha: = alpha/(t - 1), ..., and, in more general terms, (8) alpha? = alpha/(t - k + l),

with k = 1,. .. ,t . Obviously, already the second significance test is less conservative when (8) is applied than when (7) is applied. (Note, that (8) is effective only if the tail probabilities are ordered in an ascending sequence.)

Recently, Shaffer (1986) proposed a modification of Holm’s procedure that may lead to even more powerful testing. Holland and Copenhaver (1987, 1988) showed that under certain dependence assumptions these new procedures can be improved even more.

Using the H, given in (3) and the test statistics given in (4), (3, and (a), CFA types and antitypes may be defined as follows:

(1) A type is defined by

(9)

The “greater than” relation is satisfied if the probability for (9) was observed to be, for instance, B < a*, if the binomial test was used, p(z) < a*, if the z-test was used, or ~(1’) < a*, if a chi-square test was used. (2) Accordingly, an antitype is defined by

The “less than” relation is satisfied if the probability for (10) was observed to be, for instance,

B < a*, p(z) < a*, or p(f) < a, respectively.

In the next sections CFA will be applied to longitudinal data. Special emphasis will be given to the specification of the model for the estimation of expected cell frequencies.

CFA in Longitudinal Research

The following sections of this paper will present three methods for analyzing repeated observations using CFA.

Two of these methods require transformations to deal with time-related dependencies of the autocorrelation type. These are the first method to be discussed, in which trajectories of Ss through time, that is, trends, are considered as states of a single variable, and the third, in which repeated observations are described with orthogonal polynomials. In the second method to be discussed, each variable at each point in time is considered a separate variable. The resulting contingency table enables one to focus on local time-related dependencies of the variable(@ under study.

The Investigation of Trends with CFA

Often in longitudinal research interest is in the time- related trend displayed by one or more variables. Here a trend is defined as an overall increase or decrease in a variable across time, irrespective of the level from where the change started. If, for instance, a researcher investigates age-related decline in cognitive functioning or the emergence of wisdom with age, interest might be only in change patterns rather than in the absolute level of performance. CFA can be used to determine whether or not there are change patterns that occur either more or less often than expected by chance.

The main properties of the approach include: 1. Change patterns are investigated irrespective of the

mean of a series of measures; 2. Degree and order of autocorrelations do not jeopar-

dize results; 3. Both uni- and multivariate approaches may be

realized; 4. Both discrete and ordinal discrete variables may be

analyzed, and that either separately or combined; 5 . Distributional assumptions do not need to be made; 6. In addition to the variables that are observed more

than once, classification variables such as group membership or personality characteristics, or independent, experimental variables may be used.

For the sake of simplicity, let one continuous random variable, A, be observed three times on each of a sample of subjects. Then a change pattern can be generated by assigning to the differences a,,l - a,

1 if a,+r > a, d,= [ 0 if a,,l = a, (1 1)

Using this transformation each person’s vector of three measures is transformed into a vector containing two discretized differences, d, and da. Because each of the d:s can assume three values, nine different change patterns are possible. In more general terms, if d variables are observed t times, and the change patterns are defined using (ll),

- I if a,,, < a, [i= 1,2].

different change patterns are possible.

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Sample size considerations. The sample size required for a CFA depends on the characteristics of the applied significance test. If, for instance, Pearson’s chi-square test is chosen the required sample size may be calculated following Cochran’s (1954) suggestions according to which the expected cell frequency should be at least 5 when df = 1. Wise (1963) considers Pearson’s chi-square test still valid if expected frequencies are as low as 2. In a com- parative simulation study, Larntz (1978) showed that Pearson’s chi-square test performs well even if the minimal cell expectations are 0.5 (cf. Koehler, 1986; Koehler & Larntz, 1980; Simonoff, 1985; Zelterman, 1987). Let the required expected cell frequency be f, then the required sample size is at least n = fc. It should be noted, however, that this is the sample size required for the approximation of the chi-square distribution to be suffi- ciently exact. In order to detect types and antitypes one needs much larger samples.

A considerable reduction in the required sample size can be obtained if instead of (1 1) a transformation is used that takes into consideration, for instance, only the signs of the differences d,:

+ if a,,, 2 ai

- if a,+, < ai. di= {

If (1 3) is applied the minimal sample size is reduced from fc to n’=f*2d(‘-1). If a classification variable that can assume v states is added to the design, n and n’ have to be multiplied by v to estimate the required sample size.

Categorization of continuous time series. Transforma- tions as described in (1 1) and (13) are useful under the following conditions.

(1) Measures are at the ordinal level, and the number of values is unknown. Examples of such measures include responses from psychiatric patients who say they “feel better” and responses from students who say they “like school this year better than last year.” For such measures, transformations (11) and (13) establish a metric for the change process. CFA allows the researcher to form a classification of individuals when this kind of weak data is given.

(2) A researcher is interested only in change that takes place between time-adjacent observation points.

(3) Conditions for the application of parametric tests (t-test, ANOVA) are violated. For instance, to apply repeated measurement ANOVA one needs independent, identically distributed random variables. Often, these conditions are not met.

The analysis of longitudinal data with CFA using the transformations (11) or (13) may be performed under several different assumptions concerning the autocorrelation of the difference measures and the relations among the variables under study. The different assumptions lead to different estimations of expected frequencies and, therefore, to different types and antitypes. Three of the most important of these assumptions will be discussed in the present context.

The first assumption is applicable only to designs in which there are no classification variables. Suppose d variables have been observed t times (with d 2 1 and t > 1) . Then, each variable can have 3(?’) or 2+l) states, if (1 1) or (13, respectively, is applied. The expected frequencies may be estimated under the assumption that the transformed variables are totally independent. Under this assumption, types or antitypes can emerge only if there are pairwise or higher order associations among the transformed variables under study. Main effects of these variables, which indicate first order autocorrelations, are taken into account in the estimation of expected frequencies. Therefore, first order autocorrelations do not affect the detection of types. These characteristics of the search process imply that the interpretation of the resulting types and antitypes can be based solely on local associations between the slopes of the variables under study.

Estimating expected frequencies. The expected values can be estimated using simple log-linear procedures. However, whereas in the application of log-linear methods global model fits are typically attempted, in the application of CFA simultaneous tests are performed to determine the structure of local associations that exist beyond the model under which the expected frequencies were estimated.

When k classification variables are included in a design in which d variables have been observed t times, two kinds of assumptions can be made. In each case, the d variables are transformed using either (11) or (13). The first model under which classification variables may be considered assumes total independence of all variables. Types and antitypes resulting under this model have the same characteristics as types and antitypes resulting from designs without classification variables. However, in addition to these characteristics each type and antitype must be interpreted as contingent upon particular states of the classification variables. Therefore, types and antitypes indicate local associations that become apparent only under certain conditions, or - to put it in more concrete terms - they indicate profile similarities between slopes of variables that apply only to certain groups of subjects.

The second model takes into account both the empirical associations among the slopes of the repeatedly observed variables and the associations among the classification variables. Therefore, types and antitypes can emerge only if the assumption of independence between the variables observed over time and the classification variables is invalid.

To illustrate the search for trend types two examples will be given. The first example involves a data set from Lienert (1978, p. 987). The data resulted from an experiment in which n = 150 adult subjects were assigned at random to one of two experimental groups. The first group performed a cognitive task in a relaxed atmo- sphere. (Sl). The second group performed the same task under stress (S2). Each of these groups was halved into a less intelligent group (11) and a more intelligent group (12) on the bases of their intelligence test scores. Perfor- mance was scored both with respect to the quantity of

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TABLE 2

Configural Frequency Analysis of the Variables Stress, Intelligence, Quantitative and Qualitative Test Performance

Configuration

P Q S I m fi Z P(Z)

1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2

1 1 1 1 2 2 2 2 3 3 3 3 1 1 1 1 2 2 2 2 3 3 3 3

1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2

N = 150 l . . . =82, 2...=68 .1.. =48, .2.. =47, .3.. =55 ..1.=70, ..2.=80 ... 1=76, ... 2=74

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

13 17

1 1 6

12 2 3 2 3

19 3

11 3 1 1 2 0

12 10 0 1 7

20

6.204 6.041 7.090 6.904 6.075 5.915 6.942 6.760 7.109 6.922 8.124 7.910 5.145 5.009 5.880 5.725 5.037 4.905 5.757 5.606 5.895 5.740 6.737 6.560

150

2.786 4.551

- 2.343 - 2.300 - 0.03 1

2.552 - 1.920 - 1.479 - 1.963 - 1.526

3.923 - 1.793

2.626 - .913 - 2.053 - 2.013 - 1.376 - 2.25 1

2.652 1.891

- 2.477 - 2.017

.103 5.365

0.005 0.000005 (T) 0.019 0.021 0.975 0.01 1 0.055 0.139 0.050 0.127 0.00009 (T) 0.073 0.009 0.361 0.040 0.044 0.169 0.024 0.008 0.059 0.013 0.044 0.918 0.00000008 (T)

T = type, A = antitype, m = observed frequencies, rfi = expected frequencies.

the solutions (P; number of items processed) and the correctness of the solutions (Q; number of items correct). Both measurements were determined twice, that is once before the stress treatment and once after in the experimental group, and once before and once after the same length of time in the control group. The correctness of the solutions was scored as either increasing from the first to the second observation (1). remaining unchanged (2), or decreasing (3), the quantitative performance was scored as either increasing (1) or not increasing (2).

Here the resulting data set will be analyzed under the assumption that all variables under study are totally independent of each other. The resulting table has 2(Pl,P2) x 3(Ql,Q2,Q3) x 2(Sl,S2) x 2(11,12) = 24cells. The nominal alpha was set a priori to alpha = 0.05. Using Bonferroni adjustment the critical alpha becomes alpha* = 0.002083. The data and results of this trend analysis are given in Table 2.

Configural frequency trend analysis reveals three change types in this experimental data set. The first type - read from the top of the table - is characterized by the configuration P1, Q1, S1,12. It consists of those Ss who are more intelligent (I2), did not have to work under stress (Sl), and improved their cognitive performance both in terms of the number of items processed (Pl) and the number of items correct (Ql). Empirically, this profile occurs almost three times as often as expected under the null hypothesis of total independence among the four variables under study. The second type is formed by the subjects who are less intelligent (11), worked under stress (S2), and increased the number of items processed (Pl). However, there are indicators that these subjects suffered from the stress. They showed a decrease in the number of items solved (43). Even worse are the stress effects in the third type. More subjects than expected under the null hypothesis were more intelligent subjects (12) who

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I75 TYPES OF CHANGE: CFA

responded to the experimentally induced stress (S2) by solving fewer items (Q3), and keeping the number of items processed constant (P2).

In the examples in Table 2 three types and no antitype emerged, showing that even under the conservative Bon- ferroni alpha adjustment local deviations from the assumption of total independence among variables can be identified. However, without further analysis using, for example, log-linear models (Bishop, Fienberg, & Holland, 1975), the structure of associations and interactions underlying these types cannot be determined. Therefore, in the following cxample CFA will estimate expected frequencies under a log-linear model that takes into account both any association among the repeatedly observed variables, and any association among the classification variables. Under these assumptions, types and antitypes emerge only if the null hypothesis of independence between these two groups of variables does not hold true.

The data that will be discussed were collected in the same experiment as were the data just analyzed (cf. Lienert & Krauth, 1974). In the present analysis n = 150 adult subjects were assigned at random to one of three experimental conditions (S): S1 = no treatment, S2 = alcohol, and S3 = sleep deprivation. Both before and after real- ization of these conditions a cognitive performance test was administered. The solutions provided by the subjects were scored with respect to their correctness (Q) and speed (P). The change in Q has three levels: Q1, Ql,Q3, where Q1 indicates improvement; the change in P has two levels: P1 and P2 where P1 indicates improvement. The contingency table resulting from these three variables has 2(Pl,P2) x 3(Ql,Q2,Q3) x 3(Sl,S2,S3) = 18 cells. When estimating expected frequencies we considered the association between Q and P, and assumed both Q and P to be independent of S. Holm adjustment was used with a starting alpha* = 0.05/18 = 0.00278. The results of the CFA are summarized in Table 3.

The application of CFA to the data in Table 3 reveals four types and one antitype. Reading from the top to the bottom of the table, the first type displays the pattern P1, Q1, S1. This pattern indicates that the subjects in this group belong to the control group (Sl), and improved from the first to the second observation both in the correctness and the speed of their solutions. Since the information on intelligence is not used in this analysis, this type can be interpreted only with reference to the experimental treatment. The two treatment conditions obviously impaired the performance of subjects exposed to them to an extent that leads to an agglomeration of subjects from the control group in pattern Q1, S1. This agglomeration was not expected under the assumption of independence of the two observables from the treatment variable. The result that improvement in both cognitive measures under the two treatment conditions is highly unlikely is in accordance with this interpretation. Pattern P1, Q1, S2 almost forms an antiytpe. The tail probability for this pattern is with p = 0.00695 only marginally greater than the critical value of alpha* = 0.00385.

Whereas the control group shows improvements in

both performance variables, it turns out to be unlikely that the alcohol group improves in speed. The type pattern Pl,Q3, S2 indicates that in this group there are lower correctness scores at the second point in time, that is under the influence of alcohol. Only 10 subects were expected for this group, however, 22 were observed.

The third experimental group forms two types. The first of these displays the pattern P2, Q2, S3 which indicates that sleep deprivation leads to lower scores in speed and to no change in the correctness of the solutions. The second type has the pattern P2, 43, S3. This pattern shows that under sleep deprivation it is likely that performance deteriorates. Together, the first and the second type of this experimental group can be interpreted as showing that sleep deprivation slows subjects down and excludes higher correctness scores.

The antitype is formed by pattern P2,43, Sl . This pattern indicates that without stress it is highly unlikely that subjects display poorer scores in both performance measures. Only one subject from the control group was found showing this pattern. However, thirteen were expected.

CFA of Time Representations of Variables

In the last paragraph, change patterns were analyzed under two premises. First, the level of scores was considered irrelevant. Therefore, raw data were transformed such that only slope information was retained. Second, relations among slope patterns were taken into account in the estimation of expected frequencies. Therefore, emerging types and antitypes reflect local associations beyond similarities among slope patterns. In the present paragraph, both premises will not be stated. Data will be used in raw form, if possible, or transformed such that level information is maintained. As a consequence, types and antitypes will reflect local associations among variables at given points in time.

To allow a clear-cut interpretation of types and antitypes as reflecting time related local relations, this approach is best applied if there are neither experimental nor classification variables. There are three possible reasons why types and anyitypes can emerge in using this kind of variable. The first reason is the presence of associations among classification variables. The second reason is the presence of associations among classification and repeatedly observed variables. The third reason is the presence of associations among the repeatedly observed variables. Because without further detailed investigation it cannot be decided what the reasons for the detection of types and antitypes are, it is recommended in the present context either not to include classification variables in an analysis, or to allow these variables to be associated with each other and with all other variables under study.

Let d variables be observed t times with d 2 1 and t > 1. Then two cases can be distinguished on the basis of the nature of these variables. If these variables are discrete in nature such as "employed vs. jobless," raw data, that is the counts at each point in time, can be analyzed directly by CFA. If, however, quantitative variables are used,

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176 VON EYEINESSELROADE

TABLE 3 Configural Frequency Analysis of the Variables Treatment,

Quantitative and Qualitative Test Performance

Configuration

P Q S m m Z P(Z) P <

1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2

1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3

N = 150 l . .=82 , 2..=68 . l .=48 , .2.=47, .3.=55 ..l =70, ..2=50, ..3=30

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

30 1 1

18 4 1 5

22 0

14 2 0 2 8

14 1

13 14

14.933 10.667 6.400

10.733 7.667 4.600

12.600 9.OOO 5.400 7.467 5.333 3.200

11.200 8.000 4.800

13.067 9.333 5.600

4.109 - 3.071 -2.182

- 1.360 - 1.705 - 2.237

- 2.367

- 1.470 - 1.808 - 2.858 O.OO0 4.268

- 3.494 1.239 3.618

2.302

4.470

2.453

O.ooOo4 (T) 0.00333“ 0.002 0.029 0.021 0.174 0.088 0.025 0.000006 (T) 0.00278 0.018 0.014 0.142 0.071 0.004 1 .Ooo O.ooOo2 (T) 0.00292 0.0005 (A) 0.00357 0.215 0.0003 (T) 0.003 13

T = type; A = antitype; m =observed frequencies; rR = expected frequencies. Tritical values after Holm (1979) adjustment.

the data must be categorized such that they can be compared across observations, and such that both level and slope information is conserved. Examples of transformations that preserve this information include the categorizing of variables at percentile points that are computed jointly for all observation points.

Let the ith variable have c discrete or categorized categories, with i = 1 ,..., d and d 2 1. Then, the matrix under study has

c = tqc, (14)

cells, with t > 1. A first model under which the expected frequencies

may be estimated assumes total independence among variables and observations. This assumption includes independence both within the d variables and within the t observations. Obviously, the problem connected with the assumption of total independence is that violations can be due to associations solely among variables, irrespective of whether or not there are associations of slopes. Therefore, a second model may be preferred, especially if the assumption of independence of variables cannot be justified. This second model assumes that there may

be any relation among the variables, and that there is no autocorrelation. In this model the associations among variables are accounted for, and, therefore, there is only one way in which types and antitypes can emerge. This way is given by the violation of the assumption of independence of variables over time.

The following example, which was taken from Lienert and von Eye (1984), investigates the time needed to solve “master mind” tasks. A sample of n = 1 18 children and adolescents from third, fourth, and seventh grades played master mind with a personal computer (see Funke & Hussy, 1979). Each subject was presented with eight problems in a sequence. The problems did not differ in their complexity. The time available for each problem was not restricted; however, the maximal number of trials per problem was restricted to eight. For the purposes of the present paper the time needed in the sixth and in the seventh trial of a master mind problem will be analyzed. Because of the wide range of time needed, the time measures were categorized. Three levels were used with T1 = less than 11 sec, T2 = between 11 and 20 sec, and T3 = more than 20 sec. The resulting cross-tabulation is given in Table 4. (Note that in Table 5, n = 106 because 12 subjects solved the given problem before trial 6.)

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TABLE 4

Configural Frequency Analysis of the Variables “Time Needed in Trial 6” and “Time Needed in Trial 7”

Configuration

Trial 6 Trial 7 m fi P(m) P <

1 1 1 2 1 3 2 1 2 2 2 3 3 1 3 2 3 3

N = 106 l .=39 , 2.=22, 3.=34 .1=39. .2=28. .3=39

27 10 2 8

13 12 4 5

25

14.349 10.301 14.349 12.141 8.716

12.141 12.509 8.981

12.509

106

0.0015 (T) 0.00714“ 0.913 O.ooOo5 (A) 0.00556 0.261 0.191 0.882 0.007 (A) 0.00833 0.213 0.001 (T) 0.00625

~~

T = type; A = antitype; rn = observed frequencies; fi = expected frequencies. .Critical values after Holm (1979) adjustment.

Table 4 is analyzed under the assumption that the time needed for a decision in trial 6 is independent of the time needed in trial 7. Alpha adjustment was done using the Holm method with a starting value of alpha* = 0.00556. CFA revealed two types and two antitypes. Read from the top to the bottom table, the first type turns out to be a stability type. The pattern TI , T1 represents subjects who consistently make quick decisions. The number of subjects showing this pattern was 2.5 times as many subjects as expected.

The next two patterns that significantly deviate from their expected frquencies are T1, T3, and T3, TI. The former pattern describes students who make a quick decision in trial 6 and use a relatively long decision time in trial 7. A subsample of 14 subjects was expected to display this pattern; only two subjects were found to show it. Hence, it can be concluded that a transition from a quick decision in trial 6 to a time consuming decision in trial 7 is highly unlikely. The transition in the inverse direc- tion, that is from long decision times in trial 6 to short decision times in trial 7 is also highly unlikely. The pattern T3, T1 forms the second antitype. (Note, that the detection of the second antitype is due to the application of Holm’s adjustment. If Bonferroni’s method had been used the local null hypothesis could not have been rejected for this pattern.)

The second type in this data set is also a type of stable response times. The pattern T3, T3 describes those Ss who consistently display long decision times. Twice as many subjects as expected were found in this cell. Altogether, the pattern of types and antitypes suggests the conclu- sion that subjects keep the speed with which they respond in a master mind game constant.

CFA of Coefficients of Orthogonal Polynomials

In the last two sections there was an emphasis on either the slope (Section on “Investigation of Trends”) or the location of subjects (Section on “Time Representations of Variables”). In the present section, information on both the location of subjects and on the change of this location across time will be analyzed simultaneously. This analysis will be done in two steps. First, each subject’s sequence of measures will be described using orthogonal polynomials. Second, the resulting descriptive polynomial coefficients will be categorized and configurations of states of coefficients will be analyzed using CFA (see Krauth, 1973; Krauth & Lienert, 1978).

A univariate series of t > 1 measures can be approximated by a polynomial using a general regression approach by

for each subject, where x represents the observation points, and the b, are estimators of regression coefficients, with i = 0,1, .... The estimators b, can be determined using, e.g., least square methods. Using (15), the estimated score for the jth observation is

9 = b, + b,j + b j z +..., (16)

for j = 1 , ..., t. The regression equation (15) can be re- formulated as

y = + alpl + alpz +..., (17)

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where the p, denote orthogonal polynomials of the ith degree, with i = 0, ..., and the a, denote the polynomial coefficients. For t points in time a polynomial of maxi- mally t-1st degree can be approximated. Of course, the t-1st degree polynomial fits these measures exactly. For reasons of parsimony, however, polynomials of degrees lower than t-1 are preferred, if there are no substantial deviations from the estimated values.

If (1 7) is used to describe a given sequence of measures both the coefficients a, and the polynomials 0, have characteristics that make them attractive for the present purposes. First, the coefficients a, may - as in linear regression analysis - be substantively interpreted. The coefficient a,, always gives the arithmetic mean of the time series under study; a, describes the linear trend, and therefore, is an estimator of the linear regression coefficient, az describes the quadratic trend, and so forth. Second, the estimators a, are independent of the degree of the eventually selected polynomial. As a consequence, if the researcher decides that a higher degree polynomial is needed for an adequate description of a time series, only the additional coefficient needs to be estimated, and the coefficients of the polynomials of lower degrees remain constant. This characteristic follows from the orthogonality of the polynomials.

Another characteristic that must be mentioned in the present context concerns the spacing of the observations. If the observations t are equidistant, that is equally spaced, the values of the orthogonal polynomials need not be calculated. For most instances, they are given in standard textbooks of analysis of variance (e.g., Kirk, 1982; Table E12). If observations are not equidistant the values of the orthogonal polynomials need to be calculated (cf. Bliss, 1970; von Eye & Hussy, 1980).

The orthogonal coefficients a, can be estimated by

po(j) = 1 , with j = 1 ,..., t, and

for i =0, 1 , ..., r, where r denotes the degree of the last coefficient to be estimated, i.e., the degree of the polynomial.

For the typal search with CFA, let d variables have been observed t times, with d 2 1 and t > 1 . Then, for each subject the observed measures of the ith variable can be approximated by a polynomial of degree r,, with 1 < r, < t-1 . The parameter r, must be the same for each subject. Each of the polynomial coefficients of each variable, i.e., a,,,, a,,, a,,, . . . , must be categorized to enable the researcher to apply CFA. Then, a contingency table results that has

be rl = 2, and r, = 3, and r3 = 2. Suppose cI1 = 2, cll = 2, cil = 2, cia = 3, c23 = 2, cII = 3, and c31 = 3. Then, the number of cells of the contingency table under study is c = 2*2* ... *3*3 = 432.

The method discussed in the present section is suitable especially if there is only a small number of variables that can be approximated by polynomials of low degrees. How- ever, compared to the methods discussed in the previous sections, CFA of polynomial coefXcients is independent of the number of observation points, and level and curvature can be simultaneously taken into account.

The following example analyzes a data set in which a sample of n = 42 college students was observed four times on state anxiety (see Nesselroade, Pruchno, & Jacobs, 1986; Wittmann, 1988). At each occasion, the state measures were obtained using two parallel forms of a state anxiety questionnaire. For the present purposes it was decided to smooth the time series for each parallel form using second degree polynomials. Thus, for each subject the parameters of the following equation were estimated:

y = bo + bit + bit’,

with t = {1,2,3,4} denoting the equidistant points in time. Resulting estimates have been calculated for both parallel forms. (These are available from the authors.)

TABLE 5

Configural Frequency Analysis of Linear and Quadratic Polynomical

Coefficients of Two Parallel Tests

A1 A2 B1 B2 m r i i P(m)

1 1 1 1 12 1 1 1 2 0 1 1 2 1 3 1 1 2 2 0 1 2 1 1 0 1 2 1 2 6 1 2 2 1 1 1 2 2 2 4 2 1 1 1 0 2 1 1 2 0 2 1 2 1 6 2 1 2 2 0 2 2 1 1 0 2 2 1 2 1 2 2 2 1 1 2 2 2 2 8

N=42

3.220 2.660 3.898 3.220 3.220 2.660 3.898 3.220 1.981 1.637 2.399 1.981 1.981 1.637 2.399 1.981

0.00008 (T) 0.128 0.889 0.070 0.070 0.095 0.177 0.807 0.263 0.377 0.062 0.263 0.263 0.982 0.599 0.001 (T)

l . . .=26 , 2 . . .= 16 cells, where cu denotes the number of states of the jth polynomial coefficient of the ith variable. If, for example, d = 4 variables have been observed at least t = 4 times, three polynomial coefficients have to be categorized. Let the degree of the first polynomial coefficients

. I . , = 21, .2.. = 21

. = 19, . -2. = 23

.., 1 = 23, . . .2 = 19

T = type; m = observed frequencies, fi = expected frequencies.

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I I I I I r I I I I I I 1 I

1 1.4 1.8 2.2 2.6 3 3.4 3.8

points in time FIGURE 1. Averaged slope of Type 1111 (mean set to zero).

To illustrate the use of CFA of orthogonal polynomials it was decided to analyze only the coefficients b, and bl. A natural cut-off for the first parameter is b, = 0. Posi- tive regression coefficients indicate an overall increase in state anxiety over the four points in time. Negative regression coefficients describe an overall decrease. In a similar manner, two groups of slope coefficients were consti- tuted. Positive bz coefficients indicate that the curvature of state anxiety development is concave, and negative b coefficients indicate a convex curvature. Since for each subject four dichotomous parameters were analyzed, a 2 x 2 x 2 x 2 contingency tabel resulted. Table 5 sum- marizes the results of the application of CFA.

The expected frequencies in Table 5 were estimated under the assumption of total independence among the four parameters under study. The binomial significance test given in (4) was applied. Table 5 shows that two types emerged. The first type displays pattern 11 11, that is it contains subjects who reported on both scales positively accelerated increasing state anxiety. This type strongly contradicts the assumption of statistical independence of the four polynomial parameters in the sample. Rather, it provides evidence supporting the parallelism of forms of the questionnaire, and that across time. Figure 1 depicts the averaged curvatures of the first type for both

parallel forms. For reasons of comparability, information on the mean of the mean state anxiety is not incor- porated in Figure 1.

The second type emerging in the present analysis displays pattern 2222. Subjects of this type show in both forms of the questionnaire an accelerated decrease in state anxiety over time. Therefore, the second type also sup- ports the paralleIism of the two forms. In Figure 2 the averaged slopes of the state anxeity characteristics for the second type are depicted. Again, information on the means is neglected.

Table 5 provides two additional indicators of the equiv- alence of the two forms of the state anxiety questionnaire. First, the two other configurations with like patterns in the two forms, that is configurations 1212 and 2121 tend toward constituting types. Both of these configurations contain more than twicc as many subjects as expected under the assumption of independence of the polynomial coefficients. Complementing this indicator, a second characteristic of the data is the tendency for subjects who respond differentially to the two forms to constitute antitypes. In all but one of these configurations the observed frequencies are smaller than the expected ones. Statistical- ly, configurations 1122 and 1211 come close to being antitypes.

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-1 0

-20

-30

-40

-50

-60

-70

-80

-90

-1 00

-110

-1 20

-130 I I 1 I I I 1 I I 1 1 1 I I I

1 1.4 1.8 2.2 2.6 3 3.4 3.8

points in time

FIGURE 2. Averaged slope of Type 2222 (mean set to zero).

Discussion

Three methods of Configural Frequency Analysis for the analysis of repeated measurement and longitudinal data have been presented and discussed. The first method enables the researcher to analyze trend patterns of ordinal or interval data where patterns are defined as sequences of categorized differences between time-adjacent measures. In these differences, the means of the time series are not considered. However, they may be used, for instance, as classification variables.

The second approach, which also requires at least ordinal data, involved analysis of time series by crossing variables observed at several points in time. To make the crossing possible, cut-off points for categorization of continuous variables must not vary over time. This conserves level information.

The third approach, which requires interval data, fits orthogonal polynomials to the time series. The coefficients of these polynomials can be categorized, and analyzed using CFA. With this approach, even long time series can be analyzed, and level and slope information can be taken into account simultaneously. All three CRA approaches can lead to a statistical class-

ification of change patterns into types and antitypes. Each

type and antitype constitutes a class of intraindividual change patterns that differs in the frequency of its oc- currance from the expected frequency under the hypothesis of no change over time. The different patterns, which represent interindividual differences in intraindividual change, are interpretable with respect to two characteristics: (1) substantive meaning; and (2) whether they represent a type or an antitype. In the data presented in the preceding section, for example, the first type included subjects whose anxiety level increased over time, and the second type included subjects with decreasing anxiety levels.

When using CFA to represent change or other developmental phenomena, one identifies extreme frequencies of configurations. Overrepresented configurations of variables reflecting change information are construed as types of change. Types of change patterns may be interpreted as local associations among the variables. These local associations are inferred to produce an increased relative frequency of certain change patterns. Complementing this increased prevalence rate, antitypes describe configurations that are less often observed than expected under the null hypothesis of a chance distribution.

In many instances types can be interpreted as sectors of the data space that provide empirical support for sub-

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181 TYPES OF CHANGE: CFA

stantive concepts. An example is given in the previous section in which the types supported the notion of two parallel test forms. Antitypes can typically be interpreted as sectors of the data space in which the concepts signified by particular configurations are not realized in relation to an assumed underlying frequency distribution. Thus, antitypes tell which multivariate patterns are less likely to occur than predicted from a given set of expectations.

Both types and antitypes have great potential for testing notions of developmental sequence. For example, a theory that predicts the sequence X+Y+Z derives support both from an abundance of cases that manifest the pattern XYZ and from a relative paucity of cases that manifest the alternative patterns of ZYX, ZXY, etc. When change patterns are cross-classified with an index of chrono- logical age, the patterning of types and antitypes across age classifications can support or negate a hypothesized stage model of development.

Advanced conceptions of development (e.g., Feather- man’s [ 19851 development as duration dependent process) can also be tested by CFA. For example, if individuals are cross-classified as leaving and remaining in a given state by length of time spent in that state, CFA’s types and antitypes would represent more-than-likely and less- than-likely change patterns. Short duration “leaver” and long duration “remainer” antitypes would support the duration dependence notion just as would short duration “remainer” and long duration “leaver” types. Short duration “remainer” and long duration “leaver” antitypes, by contrast, would indicate a situation in which fewer than expected individuals remain in a given state very long, but of those who do, fewer than expected eventually leave it. Duration by leaving or remaining by sequence (e.g., X4Y-Z) cross-classifications can provide rigorous tests of quite “strong” developmental models.

Finally, other kinds of outcomes offer ways of resolving familiar problems in measuring changes and differences. For instance, in the example with the two parallel forms of the state anxiety questionnaire, the relative frequency of change patterns that contradicted the parallelism of forms was decreased. The frequency of supporting patterns was increased. In the example in Table 4, the configural frequency analysis made it obvious that large discrepancies between quantitative and qualitative cognitive performance are unlikely.

Of particular concern in longitudinal aging research is the loss of subjects. Individuals no longer participate in a study because they die, they are no longer able to participate, or they move away to live with relatives. In either instance, the resulting time series are shorter than for individuals participating in the entire study. From a data analysis perspective there are two reasons why data in a longitudinal study may be missing. The first reason is that data are, in principle, available but the researchers were not able to collect them. This type of missing data occurs when subjects move away or miss a data collection wave. Here, methods of missing data estimation can be applied to impute the missing information (Little & Rubin, 1987; Rovine & Delaney, 1990).

The second reason why data may be missing is that they do not exist. An example of this type of reason is the death of subjects. Here, imputation of missing values would be questionable. In many instances, researchers eliminate cases with non-existing data, thus possibly bias- ing results. It is, therefore, desirable to include as much information from these subjects as possible. Two methods have been suggested to handle this problem (Lienert & von Eye, 1986; von Eye, 1989). The first method estimates parameters of time series that are independent of the length of the time series. Examples of such parameters include the linear trend or measures of change in performance relative to some criteria. Estimation of these parameters requires only two observation points. The second method introduces an additional category, the state “data not existing.” This category is added to the variables beginning with the point in time when subjects can no longer provide data.

The advantages of these methods over casewise or list- wise deletion are twofold. First, the sample size is not decreased when data are non-existing. Second, the entire sample can be analyzed rather than just a subsample. Thus, results will not be biased because of attrition.

It is important to consider the role of CFA among the methods of categorical analysis. As was emphasized throughout this paper, CFA leads to a classification of subjects. This classification results from comparing observed with expected cell frequencies. The interpretation of types and antitypes refers to the assumptions under which cell expectancies were estimated. To generate the classification, CFA uses methods for the estimation of expected frequencies as in log-linear analysis, and applies nonparametric statistics and simultaneous testing.

CFA uses the same algorithms as log-linear modeling to estimate expected cell frequencies. In most instances, CFA null hypotheses refer to simple models. To calculate expected cell frequencies for such models, closed forms can be given. CFA applies mostly nonparametric statistical tests. For instance, the binomial, chi-square and Lehmacher’s (198 1) more powerful hypergeometric tests are nonparametric tests, applicable under very liberal conditions. CFA uses methods from simultaneous testing to keep the factual alpha level constant.

These characteristics suggest that CFA is an eclectic method. It shares algorithms in common with statistical methods such as log-linear modeling, in particular residual analysis, but also information theory and prediction analysis (Hildebrand et al., 1977). Its contribution resides in its differential perspective which allows the researcher to identify types and antitypes from local associations among variables.

In longitudinal research, CFA attempts to identify types of change, that is, patterns that occur more often than expected under some models. With log-linear models, one attempts to find a model that accounts for the variation at each observation point. Examples of such models include first order Markov chain models which are fitted to a subset of observation points and, then, applied to the remaining observation points (Anderson & Goodman,

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1957; Bishop et al., 1975). Again, the goal of the application of log-linear models is the specification of a model. CFA attempts to identify those developmental patterns that contradict a local null hypothesis.

In spite of the differences between log-linear modeling and CFA we conclude that the two methods may be used in a complementary fashion. In thorough data analysis one is interested in the interaction structure of the variables. Therefore, after CFA one might want to specify a log-linear model. This may be done either under inclu- sion of all cells or blanking out type and antitype cells. The second approach usually leads to extremely simple models. The first approach leads to the specification of those interactions that are necessary to explain the presence of types and antitypes.

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Appendix Computer Programs for CFA

There are two ways to calculate CFA solutions. The first is to use a main frame program for log-linear modeling, e.g., SPSSX or SAS. Each of these programs allows the user to calculate the expected frequencies for any CFA model. For instance, the first order CFA model corre- sponds to a log-linear main effect model. The expected frequencies then can be compared to the observed frequencies using the tests discussed in the paper. Most of the programs provide standardized residuals. These can be used as test statistics for CFA tests. Alpha must be adjusted by hand calculation.

For PC users, there are program packages that perform log-linear modeling. Examples include SY STAT. This program allows one to calculate expected frequencies for virtually any CFA model. One major disadvantage of the SYSTAT expected frequencies is that they are printed with only two decimals. Thus, the resulting CFA tests are only crude approximations.

A CFA program for PCs is available from the authors. The program calculates a first order CFA, that is a CFA under the assumption of total independence of all variables. The following eight significance tests are available: (1) the chi-square test, (2) the z = SQRT (chi-square) test, (3) the binomial test, (4) the binomial test via the Stirling approximation, ( 5 ) Lehmacher’s asymptotic hypergeometric test, (6) Lehmacher’s test with Kuechenhoffs con- tinuity correction, (7) the z-test, and (8) Anscombe’s z- test.

The program is interactive. It allows the user to key in data or to read data from an ASCII file. The program sends results to a printer. Limitations of the program are as follows:

Number of variables < 11; Number of categories per variable < 10. The program runs under DOS 2.1 or higher on IBM

and compatible PCs. It was written and tested in interpreted MS GWBASIC. More features are being incor- porated.

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types of change: application of configural frequency analysis in repeated measurement designs

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