The research organization International Association for the Evaluation of Educational Achievement (IEA) evaluates the reading comprehension, as well as the mathematical and scientific competencies of fourth-grade students once every five and four years, respectively. The PIRLS/IGLU and TIMSS surveys are used for these evaluations. Both the IGLU and TIMSS surveys were simultaneously implemented in Germany for the first time in 2011. The representative German sample comprised 3928 students from 197 elementary schools. Domain- specific student performances were classified based on a scale of five competency levels.

For analyzing particular competencies across domains, the performance test values were mutually scaled with a multidimensional item response model (mixed-coefficients multinomial logit model; for details, see [15] ). In addition, the national performance test values were scaled on a standardized metric (mean of 300 and standard deviation of 100). In an international context, German students performed in the upper third, in both surveys. Hence, Germany’s performance was higher than the average of the participating Organisation for Economic Co-operation and Development (OECD) member states [5] [6] .

However, Germany showed no significant improvements in IGLU 2011, compared with the 2001 and 2006 surveys. Although the social background significantly affected reading comprehension, students with migrant backgrounds did not show a significantly higher reading performance in the 2011 survey [5] . A similar trend can also be observed in the 2011 TIMSS survey. Unfortunately, students showed no significantly improved competencies in both the mathematics and science domains, compared with the previous surveys. However, social background did appear to a significant negative effect, as in case of the IGLU surveys. Nevertheless, it is worth noting that the performance range between weaker and stronger students could be significantly reduced. Furthermore, the three domains did not show significant disparities in terms of gender [6] .

This paper highlights the advantages of using interactive graphics to analyze comparison studies on school achievement. The aim is to simplify data exploration before the actual reporting, and to graphically present the results.

In addition to the identified performance test values in the three domains, a latent profile analysis (LPA) was conducted. LPA was performed based on the estimated plausible value performance results of the students, with the software Latent GOLD (http://statisticalinnovations.com/products/latentgold.html).

Here, the relationship among the different scales and subscales of IGLU and TIMSS 2011, as well as student groups with similar/different performance or competency profiles, can be determined [16] [17] . Hence, students with similar types of competencies are grouped homogeneously via LPA. The Bayesian information criterion (BIC) and consistent Akaike information criterion (CAIC) were computed for LPA models with up to ten performance profiles. An LPA model with seven performance profiles was selected based on these criteria (for details, see [18] [19] ).

The proportions and performance means of the seven identified performance profiles are presented in Table 1 and Figure 1. The performance means of the competency domains show a somewhat homogeneous pattern within the performance types (cf. [20] ).

Considerable data on background characteristics are available, in addition to the performance test values assessed in the IGLU/TIMSS 2011 surveys. These comprise, inter alia, cultural and social background characteristics, as well as subject-specific attitudes and self-concepts.

This section focuses on the degree of difference among the seven identified performance types, relative to certain characteristics such as gender or cultural and social properties. In terms of their relevant background information, the potential educational disadvantage for students, in particular, can be graphically explored [21] -[23] :

In all comparison studies on school achievement, carried out so far in Germany, it was stated that a part of the students does not master basic skills in reading, writing, and arithmetic. […] Evaluations of the […] PISA survey and the […] IGLU survey from the year 2006 clarify the persistent link between education and educational opportunities, in which particularly children with migration background are disadvantaged ([22] , p. 92).

In an ideal education system, every person has the same opportunities to achieve his or her educational goal. However, in reality, some are disadvantaged in this respect, in particular, students with lesser personal, social, financial, or cultural resources (see also [26] ).

This section aims to graphically determine the potential educational disadvantage for students based on their relevant background characteristics. The spineplot, produced easily in Mondrian, can be used to this end.

Spineplots are similar to conventional bar charts. However, the conditional probabilities of the analysis can be compared in the former, as the length of the bars is uniformly scaled [9] [27] .

The graphical results of the descriptive relationships among student performances and selected background characteristics are summarized in Figure 2 and Figure 3.

Risk of poverty of family: The educational disadvantage of students at a risk of poverty can clearly be explored with the variable risk of poverty of family. In particular, the estimated probability P (no risk of poverty | performance profile X_i) and the estimate P (risk of poverty | performance profile X_i) show a consistently increasing relationship on comparison. Families of students with higher performance profiles are at a lesser or no risk of poverty (Figure 2). For example, P (no risk of poverty | performance profile 1) = 0.22 versus P (no risk of poverty | performance profile 7) = 0.79. Similar results can also be obtained when considering the performance profile based on the students’ risk of poverty. The results indicate that the probability of belonging to a higher performance profile is greater, in particular for students not exposed to the risk of poverty (Figure 3).

Migration background of parents: Similar patterns of educational disadvantage were also identified based on the variable migration background of parents. In particular, both parents of students with a higher performance profile were significantly more likely to have not been born abroad (Figure 2). As an example, the following estimated probabilities can be considered: P (both parents not born abroad | performance profile 1) = 0.23 versus P (both parents not born abroad | performance profile 7) = 0.86. Moreover, an educational disadvantage can be identified when the probabilities of belonging to a performance profile are analyzed in terms of the migration background of both parents. The results show that students were more likely to belong to a higher performance profile, if both parents were not born abroad (Figure 3).

The graphical results for further background variables can be found in Appendix A.1 (see Figure A1 and Figure A2).

Although not considered previously, missing values within background variables have been found to be very important in properly evaluating large-scale assessments such as IGLU, TIMSS, or PISA, as they may generally lead to distorted results. Presently, several statistical methods are available to overcome this limitation. The multiple imputation of missing values is most frequently used in educational sciences (e.g., [28] [29] ). However, the effects and implications of different imputation methods for large-scale assessments have been rarely studied in depth. With the present study’s aim of visualization, plots to explore the structure of missing values are presented, which may aid in similar statistical modeling.

In this respect, the structure of the missing values and their proportions within particular variables must be studied. The missing value plot can be produced with the software Mondrian for this purpose. The key advantage of the missing value plot is its ease of use and interpretation, which is addressed in this study. Simple graphical representations of the proportions of missing values within background variables of interest are presented in Figure 4.

The various factors giving rise to missing values are not discussed in this paper. Instead, the present paper aims to identify the potential response tendencies for selected background variables, especially in terms of classifying students into the specified performance profiles. In this respect, spineplots were used to explore the probabilities of a missing value appearing under a particular performance profile, that is, P (missing value [in a variable] | performance profile X_i). The corresponding graphical explorations for the background variables risk of poverty of family and migration background of parents are reported in Figure 5.

Comparable tendencies with respect to the response behaviors of the students can be identified. Students with lower performance profiles are more likely to have missing values within the two background characteristics. For example, P (missing value | performance profile 1) is greater than P (missing value | performance profile 7). The probability of occurrence of a missing value steadily decreases with higher performance profiles.

Similar results were observed with other background variables as well. Therefore, missing values do not seem to be random in this context. Further, the obtained information can be useful in analytical modeling approaches to dealing with missing values, for example.

Via LPA, students with similar abilities in the IGLU/TIMSS 2011 surveys were clustered and classified into highly homogeneous groups (e.g., [16] ). Subsequently, homogeneity or deviations thereof within the computed performance profiles are presented graphically.

The degree of homogeneity within profiles can be analyzed in various ways with the help of interactive graphics. The parallel coordinate plot [30] -[32] is used to determine the student performance test values over all three domains. Homogeneity within a particular performance profile is then investigated by selecting and high- lighting the target profile. In particular, on the one hand, multidimensional data in two-dimensional space can be visualized, where each of the domains is represented by a parallel vertical axis and the values of the cases/ students are joined by a sequence of line segments. On the other hand, the selected profiles can be easily linked to the parallel coordinate plot.

The performance test values of the students over all three domains with respect to the selected performance profiles 1, 3, 5, and 7 (left: reading dimension; center: mathematics dimension; right: science dimension) are presented in Figure 6. The performance test values related to the seven performance profiles are visualized in Appendix A.2 in more detail (see Figure A3).

Performance profile 1 is particularly striking. Some cases that are not homogeneous within that profile can be graphically detected. An arrow-marked extreme case in Figure 6 can be determined within performance profile 1. This case need not be allocated to this profile, due to the higher test values in the mathematics and science domains. The reason is that students were wrongly allocated to profiles based on the LPA classification. Further research is warranted on alternative methods or a more fine-grained LPA analysis based on higher dimensions, such as an eight- or ten-dimensional competency model underlying the student responses. Interestingly, the relative performance strengths or weaknesses of students lead to their misclassification. This issue is addressed in the following section.

Misallocations within the performance profiles must be identified to maintain the high degree of homogeneity and to counteract a heterogeneous pattern within particular profiles. The parallel box plot can be used for this purpose (e.g., [9] ), as displayed in Figure 7. The performance test values of the students are presented as parallel box plots, and the individual performance profiles are selected and highlighted therein.

The parallel box plots over the three competency domains for the corresponding profiles 1, 3, 5, and 7 are presented in Figure 7 (all seven performance profiles are visualized in detail in Figure A4 of Appendix A.2). As can be seen, misallocations can be identified individually within each performance profile, albeit with different degrees. For example, the arrows in Figure 7 indicate two misallocated cases.

Interestingly, homogeneity within profiles may be distorted by the presence of students with relative performance strengths or weaknesses, such as students with learning difficulties or excellent performances in merely one domain (e.g., see [24] [25] ). These students may significantly distort homogeneity within performance profiles. Students with relative strengths or weaknesses require a tailor-made education program that effectively targets

their specific domains (e.g., see [33] ). Therefore, such “special individual cases” must be accurately identified in educational contexts.

The relative performance strength for the reading domain is graphically illustrated using the software Mondrian (see Figure 8). Similarly, the competency domains of mathematics and science can be visualized and the relative performance weaknesses of students graphically identified as well.

This is based on the histograms of the performance test values on the one hand and the parallel coordinate plot of the test values over the three domains on the other [30] [32] . The interactive nature is beneficially employed for the so-called stepwise selection: First, a relatively high performance test value in the reading domain is selected for the histogram display (Figure 8, Step 1). Next, a corresponding and comparatively low performance test value in the mathematics domain is selected via the and-function implemented in Mondrian (Figure 8, Step 2). This is repeated for the science domain as the last step (Figure 8, Step 3). As a result, the selected relative performance strengths can be read from the parallel coordinate plot using the linking-function of the software Mondrian.

As presented, the graphical procedure identifying students with relative performance strengths or weaknesses may facilitate subsequent and more advanced analyses. For instance, the possible causes for the detected relative performance strengths and weaknesses can be examined in depth with additional linked graphics for interesting background characteristics.

The reading, mathematics, and science domains and the identified performance profiles were compared, con- sidering the relevant background characteristics.

The so-called trellis plot can efficiently display this comparison graphically, which can be produced by the R package lattice [34] . Trellis plots can be used to represent multivariate data. The trellis plot is based on visualizing conditional plots [34] -[36] , such that a single graphic displays different information. For instance, the competency domains and performance values can be compared with each other. The distributions of performance test values with respect to particular background characteristics can be contrasted, or the degree of homogeneity within the performance profiles can be explored.

In the example of this paper (see Figure 9), the trellis plot is used to visualize and interpret the domains and performance profiles relative to the background variable risk of poverty of family. In Figure 9, the performance profiles are represented as the columns and the domains as the rows.

Students exposed to the risk of poverty were found to be explicitly disadvantaged educationally with respect to the background variable risk of poverty of family, as derived from the graphic (cf. also Section 4.1). Educational disadvantage can be identified within all three domains, as well as over most of the performance profiles, with few exceptions. In particular, students not exposed to the risk of poverty achieve comparatively higher performance test values than those who are. However, exceptions were students exposed to the risk of poverty within lower performance profiles. Some of those students exposed to the risk of poverty with lower profiles had higher test values in the reading and mathematics domains. This may be partly explained by the comparatively higher proportions of students exposed to the risk found in the lower performance profiles. However, students within the upper performance profiles who are not exposed to the risk of poverty achieved higher performance test values.