Using Mosaic Displays in Configural Frequency Analysis

Transcription

1 Methods of Psychological Research Online 2001, Vol.6, No.3 Internet: Institute for Science Education 2001 IPN Kiel Using Mosaic Displays in Configural Frequency Analysis Eun Young Mun, Alexander von Eye, Hiram E. Fitzgerald and Robert A. Zucker 1 Abstract The present study proposes using Mosaic displays to depict results of Configural Frequency Analysis (CFA). The Mosaic display is a graphical method to examine multiway cross-tabulated data. Its unique strength as a graphical tool for CFA lies in its capability to show more than one dimension of the data in a way that instant comparisons of proportions can be made without much effort. Mosaic displays allow one to illustrate not only cell frequencies but also patterns of types/antitypes in CFA. In the present study, we compare Mosaic displays using real data examples to existing graphical methods that examine cell frequencies or test statistics of types and antitypes. Keywords: Configural frequency analysis, mosaic display, graphics, categorical data 1 Authors note: We would like to thank Christof Schuster for helpful comments on earlier versions of this article. This research was supported, in part, by NIAAA grant #2 R01 AA07065 to Robert A. Zucker and Hiram E. Fitzgerald. Correspondence concerning to this article should be addressed to Eun Young Mun, MSU-UM Longitudinal Study, 4660 South Hagadorn Road, Suite 620, East Lansing, MI 48823; Phone: (517) , Fax: (517) , Electronic mail: muneunyo@msu.edu

2 E.Y. Mun, A. von Eye, H.E. Fitzgerald, & R.A.Zucker: Using Mosaic Displays in CFA Introduction Visual display serves a unique function in our communication of findings that tables or words won t do: Visual display adds impact to the intended message (Tukey, 1989, 1990). Although graphics can never replace the need for tables, computations, or words, a good graph can provide an immediate and inescapable shot at a phenomenon of interest (Tukey, 1989, 1990). The complementary relationship between a graphic display and analysis is fittingly described in Tukey s words. A picture may be worth a thousand words, but it may take a hundred words to do it (Tukey, 1986; cf. Wainer, 1990). Unfortunately, traditional graphs based on the two-dimensional space do not handle multivariate relationships well (Wainer & Velleman, 2001). It is awkward to display more than two variables, not to mention four. Therefore, there is a great demand for graphical methods that help understand multivariate relationships. The present study proposes using Mosaic displays (Friendly, 1992, 1994; Hartigan & Kleiner, 1981, 1984) to depict results of Configural Frequency Analysis (CFA; Lienert, 1969; von Eye, 1990, 2001, in prep; von Eye, Spiel, & Wood, 1996). The Mosaic display can accommodate the graphic needs of CFA. It simultaneously illustrates not only cell frequencies but also patterns of types and/or antitypes in CFA. The present article describes CFA and its needs for graphic displays, Mosaic displays, and it provides step-by-step demonstrations of the Mosaic displays of CFA results using data examples. 2. Configural Frequency Analysis (CFA) and Its Needs for Visual Display CFA is a multivariate method for typological research that involves categorical variables. CFA can be applied in both exploratory and confirmatory research. Using CFA, researchers ask whether cells contain fewer or more cases than expected from some chance model. Most of these models can be specified using log-linear models. A candidate for a chance model is virtually any log-linear model, including non-hierarchical and non-standard models. Most CFA models can be expressed in terms of the log-frequency model, log F = Xλ, (1)

3 166 MPR-Online 2001, No. 3 where F is the array of frequencies in the cross-tabulation, X is the indicator or design matrix that contains all vectors needed for the specified model including the intercept, main effects, interaction effects, and/or covariate effects, and λ is the parameter vector. In the Classical CFA (Lienert, 1969), expected frequencies are computed under the assumption of total independence of all variables under study (von Eye, 1990). However, virtually any set of assumptions can be incorporated into CFA, allowing more complex hypotheses to be tested. Accordingly, there exists a variety of ways to test whether a configuration constitutes a type and/or an antitype (von Eye, 2001). When a cell contains more cases than expected it is said to constitute a CFA type. When there are fewer cases than expected, a cell is said to constitute a CFA antitype. For all the statistical tests for types and antitypes of a given configuration in CFA, a general null hypothesis is expressed as H : F m = F, 0 ij ij (2) where F m ij is the model-based expected and F ij is the true expected frequency of cell ij of a given configuration (cf. DuMouchel, 1999). If F ij > F m ij, a cell is said to constitute a CFA type. If, in contrast, F ij < F m ij, a cell is said to constitute a CFA antitype. If, statistically, F ij = F m ij, a cell constitutes neither a type nor an antitype 2. Identification of types and antitypes serves two key roles in CFA. First, the presence of types and antitypes in a CFA model functions as a red flag to suggest that the hypothesized model is not a good representation of the data; second, it shows where variables in a cross-classification are associated. A graphic display of types and antitypes would facilitate these two key functions in CFA. However, few graphical techniques are available to display types and antitypes of CFA. One way to illustrate types and antitypes is to show test statistics of types and antitypes in a bar graph (e.g., von Eye & Niedermeier, 1999, p. 195; see also Figure 6 in this article for an example). Height of a 2 Throughout the present article, we identified types and antitypes in CFA base models assuming that types and antitypes do not exist in the population. This assumption, however, may lead to incorrect identification of types and antitypes when types/antitypes exist in a population. An alternative approach to identify types and antitypes suggested by Kieser and Victor (1999) utilizes an additional model for types/antitypes superimposed on the base model. Using this alternative approach may result in different numbers and/or configurations of types and antitypes. Since the focus of the present article was to demonstrate usefulness of mosaic displays for CFA, however, we identified types and antitypes using the classical CFA approach.

4 E.Y. Mun, A. von Eye, H.E. Fitzgerald, & R.A.Zucker: Using Mosaic Displays in CFA 167 bar represents the magnitude of test statistics of types and antitypes, and types and antitypes can easily be identified by drawing two lines across bars representing critical values. However, an ideal graphic display of CFA should feature not only types and antitypes but also proportionate or raw cell frequencies of a given configuration. Test statistics of types and antitypes result from discrepancies between the observed and expected frequencies. The latter depends on the hypotheses of the model tested under study. Therefore, magnitude of test statistics of types and antitypes alone does not disclose anything about observed frequencies of a cross-tabulation, which facilitate understanding with regard to where local associations are reflected in a cross-classification. Observed cell frequencies of a configuration or multi-way contingency table have typically been plotted in a bar graph by arraying multiple categorical dimensions into one dimension listing all possible cell indices (Hartigan & Kleiner, 1981, 1984; Wang, 1985; see Figure 5 in this article for an example). Height of a bar represents the magnitude of a cell frequency proportional to others. For example, Mahoney (2000) converted a three-way contingency table (4 2 2) to a two-way contingency table (4 4) by arraying the last two categorical variables into one dimension with cell frequencies shown in bars clustered by the first variable in a clustered bar chart. There are ways to improve this type of bar charts by adding a third dimension or adjusting the width of bars (see Clogg, Rudas, & Matthews, 1997). However, the Mosaic display can also be an alternative for traditional two-dimensional bar charts in graphics of multivariate relationships. Furthermore, to our knowledge, there is no graphical technique yet developed to achieve the two critical features needed for the graphical presentation of CFA simultaneously: display of types and antitypes and display of cell frequencies. To accommodate these two features for graphic presentation of CFA results, this article proposes using mosaic displays for CFA, and illustrates this technique in comparison to the two other techniques used previously mentioned (e.g., Mahoney, 2000; von Eye & Niedermeier, 1999). 3. Mosaic Displays The mosaic display, proposed by Hartigan and Kleiner (1981, 1984) is a graphical method for examining cross-tabulated data. A mosaic, defined as the collection of tiles or rectangles for the n-way contingency table is formed by dividing a square n times vertically and then horizontally (or vice versa) in a successive manner until all cell con-

5 168 MPR-Online 2001, No. 3 figurations are displayed. Each of the cell counts is represented in a mosaic display by a rectangular area proportional to the cell frequencies of other cell configurations so that relative size of a tile or rectangle becomes an indicator for whether the observed data deviate from the CFA base model. Relatively larger rectangles suggest large observed frequencies. Likewise, relatively smaller rectangles denote smaller observed frequencies. If main effects or associations are hypothesized in the CFA base model or when information other than cell frequencies is to be displayed, adding shade and color (Wang, 1985) or incorporating residuals or signs into the tiles (Friendly, 1994) helps determine whether the observed data deviate from the hypothesized model. Thus, in CFA applications, relative sizes of tiles still indicate cell frequency but additional components such as shading, color, numbers, or signs can address the deviation of the observed data from the specified model. Thus, the usefulness of a mosaic display goes beyond displaying just frequencies. In addition, mosaic displays can be especially helpful for illustrating multi-way contingency tables by examining successive mosaic displays sequentially as successive variables are brought into the cross-tabulation (Friendly, 1994). To summarize, there are two characteristics of a mosaic that suit the needs of CFA: display of cell frequencies and patterns of type/antitype. First, relative sizes of rectangles in a mosaic display do not change as a function of the hypotheses or models tested under study since rectangles or tiles reflect the observed frequencies of a crosstabulation. Therefore, regardless of the hypotheses or models tested, the size of a tile always corresponds to the magnitude of an observed frequency of a given crosstabulation. Second, incorporating color, shading, sign, or numbers to the mosaic display allows researchers to discriminate types and/or antitypes and determine whether the tested model is a good representation of the data or not. In the following section, several data examples are used to illustrate Mosaic displays in CFA with step-by-step descriptions. 4. Data Examples 4.1. CFA Base Model Consider the following data example. In a study on child behavior problems (Mun, Fitzgerald, von Eye, Puttler, & Zucker, 2001; Zucker et al., 2000), a sample of 215 boys was rated twice by parents using the Child Behavior Checklist for Ages 4-18 (CBCL; Achenbach, 1991). The first rating occurred when the boys were between three and five

6 E.Y. Mun, A. von Eye, H.E. Fitzgerald, & R.A.Zucker: Using Mosaic Displays in CFA 169 years old and the second rating occurred when they were six to eight years old. Following Achenbach (1991), a T-score of 60 was used as the clinical cut-off for externalizing and internalizing behavior problems in the clinical range. Based on the averaged parental ratings, boys were assigned to clinical levels of externalizing behavior problems at wave 1 (E 1 ), internalizing behavior problems at wave 1 (I 1 ), externalizing behavior problems at wave 2 (E 2 ), and internalizing behavior problems at wave 2 (I 2 ). For all four variables, E 1, I 1, E 2, and I 2, a category of one indicated behavior problems in the normative range and a category of two indicated behavior problems in the clinical range. Table 1: Developmental Patterns of Behavior Problems among Boys EIEI Obs. Freq. Exp. Freq. L Type Antitype Antitype Antitype Type Type Type Notes. E 1 = externalizing behavior problems at wave 1 (Ages 3-5); I 1 = internalizing behavior problems at wave 1; E 2 = externalizing behavior problems at wave 2 (Ages 6-8); I 2 = internalizing behavior problems at wave 2. Numerals in E 1 I 1 E 2 I 2 column represent ordered quadruples of variable categories: 1 = sub-clinical level behavior problems; 2 = clinical level behavior problems. L stands for Lehmacher s test. Bonferroni-adjusted alpha ( ) was used as a critical alpha level.

7 170 MPR-Online 2001, No. 3 Figure 1: Developmental Patterns of Behavior Problems Among Boys This categorization scheme yielded the cross-classification (E 1 I 1 E 2 I 2 ). We analyzed this table under the total independence assumption (i.e., main-effect model), which dictates that all four variables are not related at all. Table 1 shows the observed and expected frequencies and types and antitypes of the data 3. Figure 1 gives the mosaic display of the cross-classification MOSAICS All mosaic displays in the current study (Figures 1-4 and 7-13) were generated using MOSAICS developed for the SAS/IML software (SAS Institute, 1989) by Friendly (1992, 1994) which is available at 4. For ease of understanding, cell indices and legends were later edited into the figures in the present study. Numbers inside or by the tiles in all mosaic figures are cell indices. In 3 Three expected frequencies (cell indices 1122, 1212, and 2211) were smaller than.5. Although we acknowledge that these values were rather small, for the purpose of illustration, we decided to ignore this. Likewise, we avoided invoking the delta option to compensate for cells with zero observations. 4 Detailed description of the algorithm and a FORTRAN program as an alternative to the MOSAICS program can be found in Wang (1985).

8 E.Y. Mun, A. von Eye, H.E. Fitzgerald, & R.A.Zucker: Using Mosaic Displays in CFA 171 addition, green and blue colors were consistently used to represent types and antitypes, respectively. However, color, shading, arrangement of tiles, and size of the graph are arbitrary and may be changed Sequential introduction of marginal totals Figure 1 can be developed in a series of steps. The first step was to compute the marginal totals of the table. Let f ijkl denote the ijkl th cell count for the present data. And let f 1... through f...2 denote one-way marginals, and f 11.. through f..22 denote two-way marginals, and f 111. through f.222 denote three-way marginals. The first block representing a proportion of one was vertically divided into two blocks using one-way marginal totals for externalizing behavior problems at wave 1 (e.g., f 1... and f 2...; see Figure 2). The left oblong representing cell index 1... displayed 82.8% (178 cases) of the total sample and the right oblong representing cell index 2... displayed 17.2% (37 cases) of the total sample. In the next step, the two oblongs representing cell indices 1... and 2... were horizontally divided into four rectangles using two-way marginal totals for externalizing and internalizing behavior problems at wave 1 (see Figure 3). The rectangle for cell 11.. was bigger than any other rectangles displaying 77.7% (167 cases) of the total sample and 93.8% of the one-way marginal totals for cell 1... in Figure 2. The rectangle for cell 22.. was the smallest of the four tiles showing only seven observations (3.3% of a total sample). Thus, the size of a tile serves as a good approximate measure for an observed frequency proportional to others in a given configuration. From CFA results, cells 11.. and 22.. were identified as types shown in green whereas cells 12.. and 21.. as antitypes shown in blue. It can be summarized that behavior problems of three-to-fiveyear-old boys appeared across all observed variables or not at all 5. 5 Expected frequencies for 11.., 12.., 21.., and 22.. were , 14.90, 33.90, and 3.10, respectively. Lehmacher s test statistics (1981) were 2.54, -2.54, -2.54, and 2.54 in the same order.

9 172 MPR-Online 2001, No. 3 Figure 2: Developmental Patterns of Behavior Problems Among Boys: One-Way Marginal Totals Figure 3: Developmental Patterns of Behavior Problems Among Boys: Two-Way Marginal Totals

10 E.Y. Mun, A. von Eye, H.E. Fitzgerald, & R.A.Zucker: Using Mosaic Displays in CFA 173 Figure 4: Developmental Patterns of Behavior Problems Among Boys: Three-Way Marginal Totals The third step was to vertically divide the four tiles in Figure 3 into eight tiles using three-way marginal totals for both types of behavior problems at wave 1 and externalizing behavior problems at wave 2 (see Figure 4). The rectangles for cell indices 122. and 222. are very small indicating only one observed case and two observed cases, respectively. The vertical split was asymmetric in that it favored a more even division for the two-way marginal totals (cell indices 21.. and 22..) in comparison to a disproportionate division for the other two-way marginal totals representing cell indices 11.. and The disproportionate and asymmetric split suggested that there may be associations among these three categorical variables. It turned out that cells 111., 212., and 222. emerged as types whereas cells 112. and 211. were identified as antitypes from CFA results 6. Types and antitypes indicate that more boys than expected showed all-or-none behavior problems (cells 111. and 222.), that boys with externalizing behavior problems only at wave 1 also had externalizing behavior problems at wave 2 (cell 212.), and that 6 Expected frequencies for 111., 112., 121., 122., 211., 212., 221., and 222. were , 18.21, 13.24, 1.66, 30.12, 3.78, 2.75, and.35, respectively. Lehmacher s test statistics (1981) were 5.87, -5.14, -1.80, -.56, , 5.44, 1.53, and 2.87 in the same order.

11 174 MPR-Online 2001, No. 3 boys without any behavior problems at wave 1 were unlikely to have externalizing behavior problems at wave 2 (cell 112.). However, it was less often found than expected that boys with externalizing behavior problems at wave 1 did not display those problems at wave 2 (cell 211.). Finally, the eight tiles in Figure 4 were horizontally divided yielding sixteen tiles based on each cell count, f ijkl (see Figure 1). The horizontal division was even more asymmetric than the vertical split at the third step, pointing to possible associations among the four categorical variables. In Figure 1, cell configurations 1221 and 2221 are illustrated with lines instead of tiles to show that the observed frequencies are zero. More details on Figure l follow in the next section Results for the CFA base model As expected, the CFA base model showed a poor fit. The Pearson X 2 = , for df = 11, p =.00, suggests that the independence model is not a good representation of the data. In addition, the CFA results shows four types and three antitypes using Lehmacher s test (Lehmacher, 1981) with a Bonferroni-adjusted alpha level (α* = ). The Bonferroni adjustment of alpha was adopted to control for inflated alpha due to first, simultaneous multiple testing of types and antitypes and second, their mutual dependency of tests (see von Eye, 1990, in prep). Types were found in configurations 1111, 2122, 2212, and The first type (1111) indicates that there were more cases than expected of neither externalizing nor internalizing behavior problems at both waves. Type 2122 shows that there were more boys than expected with externalizing behavior problems at both waves, and internalizing behavior problems at wave 2 but not at wave 1. Type 2212 shows that there were more boys than expected with internalizing behavior problems at both waves, and externalizing behavior problems at wave 1 only. Type 2222 shows that there were more boys with externalizing and internalizing behavior problems at both waves than expected. Antitypes were found in cell configurations 1112, 1121, and Antitype 1112 indicates that fewer cases than expected were found of internalizing behavior problems at wave 2 only. Antitype 1121 indicates that fewer observations than expected were found of boys with externalizing behavior problems only at wave 2. Antitype 2111 indicates that fewer boys than expected showed externalizing behavior problems only at wave 1.

12 E.Y. Mun, A. von Eye, H.E. Fitzgerald, & R.A.Zucker: Using Mosaic Displays in CFA Alternate graphic methods Table 1 can alternatively be plotted showing only proportional differences in a bar graph using, for instance, SPSS 9.0 (SPSS, 1998). Figure 5 is a result of converting the 4-way contingency table ( ) to a two-way contingency table (4 4). Figure 5 shows that some cell configurations had higher cell counts whereas other cell configurations had lower cell counts. Thus, this technique is limited in that first, it does not handle types and antitypes of CFA; second, different arrangements of a multi-way contingency table for a statistical analysis and a graphic illustration can create semantic difficulties. Alternatively, Table 1 can be displayed with a focus on statistics of types and antitypes as in von Eye and Niedermeier (1999). Figure 6 was drawn using S-Plus 4.5 (MathSoft, 1997). Height of bars in this graph represents the magnitude of Lehmacher s test statistics. Bars below the zero line indicate that observed frequencies were smaller than expected frequencies whereas bars above the zero line indicate that observed frequencies were larger than expected frequencies. The two horizontal lines, parallel above and below zero indicate critical values of Lehmacher s test statistics. Bars above and below the critical values indicate types and antitypes, respectively. This graph clearly shows that cells 1111, 2122, 2212, and 2222 were types represented by green bars and cells 1112, 1121, and 2111 were antitypes represented by blue bars. This technique, however, is limited in that it does not provide information on cell frequencies of a cross-tabulation. Therefore, a mosaic display seems to be a better fit for CFA than the other two techniques.

13 176 MPR-Online 2001, No. 3 Figure 5: An Alternative Approach to Mosaic Displays: A Bar Graph of Cell Frequencies Figure 6: A Bar Graph of Test Statistics of Types and Antitypes

14 E.Y. Mun, A. von Eye, H.E. Fitzgerald, & R.A.Zucker: Using Mosaic Displays in CFA When all cells are types or antitypes The following data example presents a situation when all cells are either types or antitypes. This data example has been used by many researchers including Lienert (1964), von Eye (1990), and Kieser and Victor (1999). 65 students were treated with LSD 50 and observed for the following three symptoms: Narrowed consciousness (C), thought disturbance (T), and affective disturbance (A). Each of the symptoms had the categories of presence or absence. Table 2 gives the resulting cross-tabulation. The expected frequencies and testings of types and antitypes were computed under the assumption of total independence of all symptoms. The CFA base model did not fit, Pearson X 2 = 37.92, for df = 4, p =.00. In addition, the CFA results showed four types and four antitypes, all based on Lehmacher s test with a Bonferroni-adjusted alpha level (α* = ). Types were found in cells 111, 122, 212, and 221 while antitypes were found in cells 112, 121, 211, and 222. Results can be briefly summarized as follows. More cases with either all three symptoms or a just single symptom were found than expected by the independence assumption. On the other hand, antitypes indicate that fewer cases with either no symptom or any of two symptoms were found than expected. Detailed interpretations can be found in von Eye (1990, p. 34) 7. Figure 7 presents the mosaic display for the data. As before, types are shaded in green and antitypes are shaded in blue. 7 This data example generates one type (111) and one antitype (222) when analyzed using the approach suggested by Kieser and Victor (1999).

15 178 MPR-Online 2001, No. 3 Table 2: Leuner s Syndrome Data CTA Obs. Freq. Exp. Freq. L Type Antitype Antitype Type Antitype Type Type Antitype Notes. C = narrowed consciousness; T = thought disturbance; A = affective disturbance. Numerals in CTA column represent ordered triples of variable categories: 1 = presence of symptom; 2 = absence of symptom. L stands for Lehmacher s test; Bonferroni-adjusted alpha (.00625) was used. Figure 7: Leuner s Syndrome Data

16 E.Y. Mun, A. von Eye, H.E. Fitzgerald, & R.A.Zucker: Using Mosaic Displays in CFA Entry order of variables The following data examples are to show that a different entry order of categorical variables into MOSAICS results in a mosaic with tiles of the same proportional size but with a different planimetric arrangement. The first data set is from a study on the prediction of performance in school, which has been used in von Eye and Brandtstädter (1998). In this study, fluid intelligence (I) and performances in German (G) and mathematics (M) were assessed (see Table 3). The expected frequencies and test statistics of types and antitypes were computed under the assumption of total independence of all variables. The CFA base model did not fit, Pearson X 2 = 67.58, for df = 4, p =.00. There were two types and three antitypes using Lehmacher s test with a Bonferroni-adjusted alpha level (α* = ). Figure 8 represents the data set with the entry order that corresponds to the order of CFA shown in Table 3. The entry order for the CFA base model, [I][G][M] was 111, 211, 121, 221, 112, 212, 122, and 222 in MOSAICS. In MOSAICS, the first variable varies most rapidly across the columns of cell indices whereas in most other programs the first variable varies most slowly. Table 3: Fluid Intelligence and Performances in German and Mathematics IGM Obs. Freq. Exp. Freq. L Type Antitype Antitype Antitype Type Notes. I = fluid intelligence; G = performance in German; M = performance in mathematics. Numerals in IGM column represent ordered triples of variable categories: 1 = below average; 2 = above average. L stands for Lehmacher s test; Bonferroni-adjusted alpha (.00625) was used.

17 180 MPR-Online 2001, No. 3 Figure 8: Prediction of Performance in School, [I][G][M] We then changed the order of categorical variables from [I][G][M] to [G][M][I]. Cell indices entered into MOSAICS were in the following order: 111, 121, 112, 122, 211, 221, 212, and 222, which corresponded to the order for the CFA base model, [G][M][I]. Figure 9 represents the data. In this figure, the shape and the location of the tiles changed but the relative sizes remained the same. For example, the tall oblong for a cell index 221 in Figure 8 changed to a rectangle in Figure 9. However, the relative size of the cell 221 stayed the same in Figures 8 and 9 in proportion to a total number of cases as well as marginal totals.

18 E.Y. Mun, A. von Eye, H.E. Fitzgerald, & R.A.Zucker: Using Mosaic Displays in CFA 181 Figure 9: Prediction of Performance in School: Different Entry Order, [G][M][I] The second data example is from a recently reported study (Mahoney, 2000) in which four groups of adolescent boys (G), and their records of school dropout (D) and criminal arrest (C) were obtained to see whether there were associations among group information and records of dropout and criminal arrest (see Table 4). The expected frequencies and test statistics of types and antitypes were calculated under the assumption of total independence. The CFA base model did not fit, Pearson X 2 = , for df = 7, p =.00. Three types and three antitypes were identified using Lehmacher s test with a Bonferroni-adjusted alpha level (α*= ). Patterns of types and antitypes are interpreted in detail in Mahoney (2000). Figure 10 shows a mosaic display for the CFA base model, [G][D][C]. The order of cell indices entered into MOSAICS that corresponded to CFA are as follows: 111, 211, 311, 121, 221, 321, 112, 212, 312, 122, 222, and 322. When the order of categorical variables was reversed to [C][D][G] (i.e., 111, 112, 121, 122, 211, 212, 221, 222, 311, 312, 321, and 322), the general look of the mosaic changed due to differences in the order of introduction of marginal totals (see Figure 11). However, the sizes of tiles remained the same in proportion to the total number of cases and marginal totals.

19 182 MPR-Online 2001, No. 3 Table 4: Records of School Dropout and Criminal Arrest among Adolescent Boys GDC Obs. Freq. Exp. Freq. L Type Antitype Antitype Type Antitype Type Notes. G = configuration group; D = school dropout; C = criminal arrest. Numerals in GDC column represent ordered triples of variable categories: For G, 1 = configurations 1 and 2, characterized by competence in all domains; 2 = configuration 3, characterized by low academic competence and high aggression; 3 = configuration 4, characterized by a multiple risk profile. For D and C, 1 = no; 2 = yes. L stands for Lehmacher s test; Bonferroni-adjusted alpha ( ) was used

20 E.Y. Mun, A. von Eye, H.E. Fitzgerald, & R.A.Zucker: Using Mosaic Displays in CFA 183 Figure 10: Records of School Dropout and Criminal Arrest Among Adolescent Boys, [G][D][C] Figure 11: Records of School Dropout and Criminal Arrest Among Adolescent Boys: Reversed Order [C][D][G]

21 184 MPR-Online 2001, No Non-Standard CFA Models So far, the present study illustrated standard CFA base models using mosaic displays with different data examples. From these results, usefulness of mosaic displays was examined in terms of display of cell frequencies and residuals. In addition, we demonstrated that a different entry order of categorical variables generates a different look overall but the relative sizes of tiles remain intact. In this section, we demonstrate that mosaic displays can be applied to non-standard and non-hierarchical CFA models as well. Consider the following data example, a re-analysis of data published by Glück and von Eye (2000). A sample of 181 high school students was administered the 24-item cube comparison task. After completing each item, the students responded to questions concerning the perceived difficulty of the item, the strategies they had employed to process the item, and the perceived quality of their strategy (Glück, 1999). The three strategies the students used to solve the cube comparison task were mental rotation (R), pattern comparison (P), and change of viewpoint (V). Each strategy was scored as not used = 1 and used = 2. A category one was assigned for females; two for males for Gender (G). Table 5 and Figure 12 display the results of first order CFA (i.e., model of total independence) with the normal approximation of the binomial test and the Bonferroni-adjusted α* = The results showed a rich pattern of types and antitypes with noticeable gender differences. Types indicate that there were more observations than expected for the following configurations: Males who only used the change of viewpoint strategy (1122), males who only used the pattern comparison strategy (1212), males that used both the pattern comparison and the change of viewpoint strategies (1222), and females that only used the rotation strategy (2111). Antitypes suggest that there were fewer observations than expected for the following configurations: Females that used no strategy (1111), males that used no strategy (1112), males that used both the rotation and the pattern comparison strategies (2212), and females that used all three strategies (2221). This CFA base model for the frequency distribution in Table 5 was rejected because of the large Pearson X 2 = with df = 11, p < 0.01 (Likelihood Ratio (LR) = , df = 11, p < 0.01).

22 E.Y. Mun, A. von Eye, H.E. Fitzgerald, & R.A.Zucker: Using Mosaic Displays in CFA 185 Table 5: First Order CFA of the Cross-Classification of Rotational Strategy (R), Pattern Comparison Strategy (P), Viewpoint Strategy (V), and Gender (G) RPVG Obs. Freq. Exp. Freq. L Antitype Antitype Type Type Type Type Antitype Antitype Notes. Numerals in RPVG column represent ordered triples of variable categories. Each strategy was scored as 1 = not used; 2 = used. For Gender, 1 = females; 2 = males. L stands for Lehmacher s test; Bonferroni-adjusted alpha ( ) was used.

23 186 MPR-Online 2001, No. 3 Figure 12: Patterns of Strategies: First-Order CFA Base Model (Glück & von Eye, 2000) In addition to the four categorical variables used in Table 5, one could ask whether handedness is associated with strategies adopted by males and females (Glück, 1999). If so, residuals would diminish and some or all of the types and antitypes would disappear. To test this hypothesis, in the next step, we added a covariate, handedness to the firstorder CFA base model. Results showed a significant improvement over the previous CFA base model without the covariate ( LR = ; df=1; p < 0.01), although the model was not tenable by itself (X 2 = , LR = ; df = 10; p < 0.01). Only one antitype (1112) and three types (1122, 1212, and 2111) remained significant out of the eight types and antitypes in Table 5, eliminating one type (1222) and three antitypes (1111, 2212, and 2221; see Table 6 and Figure 13). Thus, the covariate, handedness contributed significantly to the explanation of the observed frequency distribution. The changes in types and antitypes in these two nested analyses are clearly shown in Figures 12 and 13. In Figure 12, eight tiles were illustrated in either green or blue; only four tiles were still displayed in either color in Figure 13. The tiles in Figures 12 and 13 are identical in size since the observed frequencies were the same but the shading color pattern was different due to the differences in expected frequencies stemming from different CFA models.

24 E.Y. Mun, A. von Eye, H.E. Fitzgerald, & R.A.Zucker: Using Mosaic Displays in CFA 187 Table 6: First Order CFA of the Cross-Classification of Rotational Strategy (R), Pattern Comparison Strategy (P), Viewpoint Strategy (V), and Gender (G) with Handedness as Covariate RPVG Obs. Freq. Exp. Freq. Handedness Test Statistics Antitype Type Type Type Notes. Numerals in RPVG column represent ordered triples of variable categories. Each strategy was scored as 1 = not used; 2 = used. For Gender, 1 = females; 2 = males.

25 188 MPR-Online 2001, No. 3 Figure 13: Patterns of Strategies: Handedness as Covariate (Glück & von Eye, 2000) As shown in the present study, mosaic displays using MOSAICS can illustrate standard/non-standard as well as hierarchical/non-hierarchical CFA with or without covariates. Most non-standard and/or non-hierarchical CFA models can be accommodated by providing configuration types into MOSAICS. Using residuals as deviations in MOSAICS is an alternative for more complex models with covariates as shown in the last example. Appendix A provides a SAS input as an example of a hierarchical CFA or standard log-linear models, and Appendix B provides a SAS input as an example using Pearson residuals,, when f and F denote the observed and expected frequency, respectively. ( f F) F 2 5. Discussion The present article evaluated three graphic methods of displaying CFA results. The first method used by Mahoney (2000) focuses on the observed cell frequencies. The second method used by von Eye and Niedermeier (1999), focuses on the magnitude of test statistics used for the type/antitype tests. The third method, the Mosaic display

26 E.Y. Mun, A. von Eye, H.E. Fitzgerald, & R.A.Zucker: Using Mosaic Displays in CFA 189 (Friendly, 1992, 1994; Hartigan & Kleiner, 1981, 1984; Wang, 1985) incorporates both, the observed cell frequencies and the type/antitype information from CFA. The present article illustrated advantages of the Mosaic display over the other two methods for CFA since more than one dimension of the data can be illustrated simultaneously in the Mosaic display. Cell frequencies and the pattern of types and antitypes are critical features of CFA, and they can easily be illustrated using the Mosaic display. Patterns of type/antitype in CFA, in particular, allow one to understand whether there is a heterogeneous subset of the sample, and on what categories and levels this subset differs. A good graphical method can be instrumental in implementing features of CFA and our understanding of the data. Lack of graphical techniques for CFA, fueled by Tukey s tenets (1989, 1990) on data-based graphics was the major incentive for the present article. Six of Tukey s points on visual display (1989, 1990), germane to Mosaic displays of CFA, are briefly discussed as follows Impact is important Visual display of the data should be done in a powerful and intuitive way (Tukey, 1989, 1990). The Mosaic display is capable of doing this for CFA results. The Mosaic display can be as compelling a means of visual display for multivariate relationships as traditional bar charts for univariate information or bivariate relationship. In particular, the mosaic simultaneously displays the cell-wise frequencies and the type/antitype decision in one tile. Thus, the interesting point that the correlation between the size of cell frequency and presence of types/antitypes is weak at best can also be visualized Understanding graphics is not always automatic Due to its novelty, Mosaic displays of CFA may not be understood easily at first sight. However, to be thoroughly understood, even familiar types of graphs may need explanations that come in the form of descriptions or legends (Tukey, 1989, 1990). This certainly applies to Mosaic displays in CFA. Once the reader knows how to look at it, the entire information carried by a Mosaic can easily be understood A graph can show us things easily that might not have been seen otherwise The purpose of visual display is not to present numbers, but to compare (Tukey, 1989, 1990). Presenting an array of numbers in the form of a table may make it hard to

27 190 MPR-Online 2001, No. 3 see a relationship or lack of it. Mosaics of CFA results can show patterns of types and antitypes and the relationship between frequency and type/antitype decision. In addition, by proper selection of the order of variables, Mosaics can make the comparison of groups clear An understanding of purpose is needed Graphs as well as analytical methods are selected based on what researchers are trying to get across to the audience. Different graphs serve different purposes. The Mosaic display of CFA allows one to depict (i) type and antitype patterns, (ii) the size of cells, and (iii) the relationship between frequencies and type/antitype patterns. If these are the purposes of analysis as in CFA, the Mosaic display is the method of choice. If, however, researchers focus on the size of frequencies at the expense of type/antitype patterns, mosaics carry too much information and can be replaced by other simpler graphical methods, e.g., bar graphs The absence of phenomena is itself a phenomenon In the present context, the absence of phenomena can be viewed as first, the absence of types or antitypes, and second, the presence of antitypes. The Mosaic display of CFA unequivocally depicts the absence of phenomena as well as the presence of phenomena. In first-order CFA, the absence of types or antitypes implies total independence among variables, which is rarely observed in CFA applications. So when it happens, the absence of types or antitypes may require explanation. In addition, more interestingly, there are situations where the main focus of research addresses whether certain configurations exist, which can be fulfilled by the visual display of the presence of antitypes using Mosaic displays Color is a disappointment Although color is not yet an effective means of representing quantitative values, it is a useful labeling means in general (Tukey, 1990; Wainer, 1990). Color can be used effectively for qualitative phenomena at two or three levels. In the Mosaic displays of CFA where color is used to tell whether types or antitypes exist, and whether it is type or antitype, color is a powerful means of visual display. Moreover, the increased use of color in prints and the increased publications in CD-ROM or on the web will make color as a more viable means to illustrate quantitative as well as qualitative information.

28 E.Y. Mun, A. von Eye, H.E. Fitzgerald, & R.A.Zucker: Using Mosaic Displays in CFA 191 References [1] Achenbach, T. M. (1991). Manual for the Child Behavior Checklist/4-18 and 1991 profile. Burlington, VT: University of Vermont department of Psychiatry. [2] Clogg, C. C., Rudas, T., & Matthews, S. (1997). Analysis of contingency tables using graphical displays based on the mixture index of fit. In J. Blasius, & M. Greenacre (Eds.), Visualization of categorical data. New York: Academic Press. [3] DuMouchel, W. (1999). Bayesian data mining in large frequency tables, with an application to the FDA spontaneous reporting system. The American Statistician, 53, [4] Friendly, M. (1992). User's guide for MOSAICS (Tech. Rep. No. 206). York University, Department of Psychology. [5] Friendly, M. (1994). Mosaic displays for multi-way contingency tables. Journal of the American Statistical Association, 89, [6] Glück, J. (1999). Spatial strategies - cognitive strategies on spatial tasks. Unpublished dissertation. University of Vienna, Department of psychology. [7] Glück, J., & von Eye, A. (2000). Including covariates in Configural Frequency Analysis. Psychologische Beiträäge, 42, [8] Hartigan, J. A., & Kleiner, B. (1981). Mosaics for contingency tables. In W. F. Eddy (Ed.), Proceedings of the 13 th symposium on the interface between computer science and statistics (pp ). New York: Springer-Verlag. [9] Hartigan, J. A., & Kleiner, B. (1984). A mosaic of television ratings. The American Statistician, 38(1), [10] Kieser, M., & Victor, N. (1999). Configural frequency analysis (CFA) revisited - A new look at an old approach. Biometrical Journal, 41, [11] Lehmacher, W. (1981). A more powerful simultaneous test procedure in configural frequency analysis. Biometrical Journal, 23(5), [12] Leuner, H. C. (1962). Die experimentelle Psychose. Berlin: Springer. [13] Lienert, G. A. (1969). Die Konfigurationsfrequenzanalyse als Klassifikationsmethode in der klinischen Psychologie. In M. Irle (Ed.), Bericht über den 26. Kongreß

29 192 MPR-Online 2001, No. 3 der Deutschen Gesellschaft für Psychologie in Tübingen 1968 (pp ). Göttingen: Hogrefe. [14] Mahoney, J. L. (2000). School extracurricular activity participation as a moderator in the development of antisocial patterns. Child Development, 71(2), [15] MathSoft (1997). S-Plus user s guide. Seattle, WA: MathSoft, Inc. [16] Mun, E. Y., Fitzgerald, H. E., von Eye, A., Puttler, L. I., & Zucker, R. A. (2001). Temperamental characteristics as predictors of externalizing and internalizing child behavior problems in the contexts of high and low parental psychopathology. Infant Mental Health Journal, 22(3), [17] SAS Institute (1989). SAS/IML Software: Usage and reference, version 6, first edition. Cary, NC: SAS Institute. [18] SPSS Inc. (1998). SPSS 9.0. Chicago, IL: SPSS, Inc. [19] Tukey, J. W. (1986). Sunset salvo. American Statistician, 40, [20] Tukey, J. W. (1989). Data-based graphics: Visual display in the decades to come. In Gail, M.H., & Johnson, N.L. (Coordinators): Sesquicentennial invited paper sessions. Proceedings of the American Statistical Association (pp ). Alexandria, VA: American Statistical Association. [21] Tukey, J. W. (1990). Data-based graphics: Visual display in the decades to come. Statistical Science, 5(3), [22] von Eye, A. (1990). Introduction to Configural Frequency Analysis: The search for types and antitypes in cross-classifications. Cambridge: Cambridge University Press. [23] von Eye, A. (2001). Configural Frequency Analysis - Version 2000: A program for 32 bit Windows operating systems. Methods of Psychological Research-Online, 6(2), [24] von Eye, A. (in prep). Configural Frequency Analysis. Mahwah, NJ: Lawrence Erlbaum Associates, INC. [25] von Eye, A., & Niedermeier, K. E. (1999). Statistical analysis of longitudinal categorical data in the social and behavioral sciences. Mahwah, NJ: Erlbaum.

30 E.Y. Mun, A. von Eye, H.E. Fitzgerald, & R.A.Zucker: Using Mosaic Displays in CFA 193 [26] von Eye, A., Spiel, C., & Wood, P. K. (1996). Configural Frequency Analysis in applied psychological research. Applied Psychology: An International Review, 45, [27] Wainer, H. (1989). Discussion: Graphical visions from William Playfair to John Tukey. In Gail, M.H., & Johnson, N.L. (Coordinators): Sesquicentennial invited paper sessions. Proceedings of the American Statistical Association (pp ). Alexandria, VA: American Statistical Association. [28] Wainer, H. (1990). Graphical visions from William Playfair to John Tukey. Statistical Science, 5(3), [29] Wainer, H., & Velleman, P. F. (2001). Statistical graphics: Mapping the pathways of science. Annual Review of Psychology, 52, [30] Wang, C. M. (1985). Applications and computing of mosaics. Computational Statistics & Data Analysis, 3, [31] Zucker, R. A., Fitzgerald, H. E., Refior, S. K., Puttler, L. I., Pallas, D., & Ellis, D. A. (2000). The clinical and social ecology of childhood for children of alcoholics: Description of a study of implications for a differentiated social policy. In H. E. Fitzgerald, B. M. Lester, & B. Zuckerman (Eds.), Children of addiction: Research, health, and public policy issues (pp ). New York: Routledge/Falmer. Appendix A SAS MOSAIC input for the Mun et al (2001) data: Using observed frequencies filename mosaics 'c:\sas\sasuser\mosaics\'; libname mosaic 'c:\sas\sasuser\mosaics\'; data infant; input E1 I1 E2 I2 freq; cards;

31 194 MPR-Online 2001, No ; proc iml; use infant; read all var {freq} into table; levels={ }; vnames={'e1' 'I1' 'E2' 'I2'}; lnames={'e1:no' 'E1:yes', 'I1:no' 'I1:yes', 'E2:no' 'E2:yes', 'I2:no' 'I2:yes'};

32 E.Y. Mun, A. von Eye, H.E. Fitzgerald, & R.A.Zucker: Using Mosaic Displays in CFA 195 goptions hsize=7 in vsize 7 in; reset storage=mosaic.mosaic; load module=_all_; split={v h}; htext={1}; colors={green blue}; shade={1.4}; plots={4}; plots=4; fittype='user'; config=t({1 0, 2 0, 3 0, 4 0}); title='infant'; run mosaic (levels, table, vnames, lnames, plots, title); quit; Appendix B SAS MOSAIC input for the Mun et al (2001) data: Using residuals filename mosaics 'c:\sas\sasuser\mosaics\'; libname mosaic 'c:\sas\sasuser\mosaics\'; proc iml; infant={ }; f={ }; title={'infant'};

33 196 MPR-Online 2001, No. 3 vnames={'e1' 'I1' 'E2' 'I2' }; lnames={'e1:no' 'E1:yes', 'I1:no' 'I1:yes', 'E2:no' 'E2:yes', 'I2:no' 'I2:yes'}; goptions hsize=7 in vsize 7 in; reset storage=mosaic.mosaic; load module=_all_; %include 'c:\sas\sasuser\mosaics\mosaicd.sas'; dev={ }; split={v h}; htext=1; colors={green blue}; shade={1.4}; run mosaicd (infant, f, vnames, lnames, dev, title); quit;