Statistical Analyses

For both studies, data from student and teacher tests and questionnaires were transformed into an appropriate scale for the subsequent analyses. Methods that were used in this dissertation include multi-level randon intercept and random slope regression, Shannon-Wiener diversity index and related modified t tests, configural frequency analysis and related χ² tests, latent class analysis, paired-sample and independent sample t tests, and multi-level analysis of variance. Analyses were performed using R (R Core Team, 2013), Mplus (L. K. Muth´en & Muth´en, 2010), SPSS Version 22, and

5.2. Methodological Approach 55 Microsoft Excel 2013. Details on the models and software implementation (including R packages used) are given in the methods sections of Chapter 6 and 7.

Multi-Level Analyses

In educational research, data is often structured in a so-called nested way, which means that groups within the data can be related more closely within the group than between groups. In educational settings, students are nested in classes, which can also be nested in schools. It makes sense that, for instance, ratings on perceived learning processes are likely to be more closely related for two students who have had the same instructor, have a similar family background, and who frequently interact with each other than for two students from entirely different settings that have never met.

To account for the multi-level structure of the data, regressions and analyses of variance of this thesis are modeled as multi-level random effects models. This means that an additional error term is introduced to account for the variance that can be explained by group membership. Which portion of the overall variance can be explained by group membership and how much remains random variation, is determined within these models. By accounting for the structure of the data, standard errors and statistical significances are determined more accurately which allows for more precise inferences (L¨udtke, 2009).

Person-Centered Methodology

A noteworthy methodological consideration of this dissertation is the role of person-centered methodology for the exploration of individual students’ perspectives in educational research. Several different person-centered methods are employed to serve the different purposes of investigation.

Person-centered methods do not analyze aggregated variable values of a popu-lation, but persons or objects with their set of variables values. Instead of assuming a homogeneous population, person-centered methods allow for identifying distinct subgroups (von Eye & Bogat, 2006). In this property, they seem ideal to explore the within-student characteristics diversity and individual differences in student dispositions.

Diversity Indices To date, there had not been a satisfactory method in educational research to explore the diversity or heterogeneity of a sample (and to compare it).

56 Chapter 5. Research Agenda To achieve these goal, this dissertation startet with conceptual consideration. The first question was: How is diversity understood? To answer this question, the thesis disentangled the term from its use in educational science and looked at other fields and context that use it. In a more abstract understanding, diversity is a notion describing an group entity consisting of two or more individual entities in terms of the differences in the individual entities’ characteristics. In forestry, for instance, a forest consisting of many different tree species is considered to be more diverse than a forest with only two or three. At the same time, a forest with three species in total but 95% pines is regarded less diverse than one where the three species have a more equal share. From considerations like this, theoretical biology has derived a number of diversity indices (Magurran, 2004).

The Shannon-Wiener diversity index is used in this dissertation. It is a way to summarize observed frequencies that increases when there are more different patterns and when they appear more equally distributed (as described in the forestry example above). The Shannon-Wiener diversity originates from the Shannon entropy, which was initially developed in information theory by C. E. Shannon in 1948 (Shannon, 1948). As this study’s application also refers to diversity measurement, it refers to the Shannon-Wiener diversity index instead of the more general Shannon entropy. While there are other measures of diversity (e.g. Simpson, 1920; an overview can be found in Magurran, 2004), Shannon’s formula holds a central position when compared to other diversity measures (Laxton, 1978).

In Study I, the Shannon-Wiener diversity index was used as a measurement for within-student characteristics diversity. In this consideration, each distinct configuration of student characteristics scaled on a narrow 3 point scale ranging from 1 (low) to 3 (high) was considered to be its own species. The diversity index regarded how many different species exist (according to student data and according to teacher perceptions) and how they are distributed respectively. Shannon-Wiener diversity indices regarding different groups can be compared using modified t test. Formulas and other details on the index and related t tests are given in the methods section of Chapter 6.

Configural Frequency Analysis For a different purpose of investigation, other person-centered methods are useful. For instance, when interested in identifying

5.2. Methodological Approach 57 (the existence of) special small subgroups, configural frequency analysis can be used.

This is possible when looking at a set or pattern of values for each individual entity.

Based on the distribution of each single value across individuals, this method can identify patterns that appear more frequently than expected based on their marginal distributions by employing local χ² tests(von Eye, Mun, & Mair, 2010). Patterns that appear significantly more often than expected are called types. Reciprocally, patterns that appear significantly less often than expected are correspondingly called anti-types.

For the identification of anti-types a large number of observations is necessary. This could not be done in this thesis due to sample size.

In Study I of this dissertation, configural frequency analysis provided additional insights into the dominance of coherent vs. incoherent student dispositions. By identifying which dispositions (if any) appeared more frequently than expected, the study found if coherent (or incoherent dispositions) dominate the group. It is important to consider that due to the marginal distributions of roughly 25% high (3) and low (1) values and 50% medium (2) values, the expected occurence is not equal over dispositions.

Also, if no types can be found, this means there is a rich diversity of patterns.

Latent Class Analysis A third person-centered method employed in this thesis is latent class analysis. It works by assuming the connection between different measures, the student characteristics in this case, are only through a latent grouping variable. In this method, all students are assorted into a given number of groups based on making the similarities within groups and the differences between groups large. While the number of groups must be given, their sizes can vary. Model fit comparisons can help to determine the number of groups best describing the differences (and similarities) within the data. Hence, this method can provide insight into the question: If I had to divide the students into groups based on their characteristics, how would those groups look, i.e.

what share does each one have and how are the average values of each characteristic in the group. Note that considerable variation may still exist within groups with regard to the characteristics. This variation might also not be distributed equally across groups or characteristics. However, the grouping provides a more graspable viewpoint onto within-student characteristics interplay and might be most useful for certain practical considerations.

58 Chapter 5. Research Agenda Study II of this thesis employed latent class analysis to determine how students from a similar incoherent dipositional starting point at the beginninng of the school year had developed their dispositions at the end of the year.