Bayesian Modelling

themselves, only in the context of the topic under which they have been “pooled.” It is also important to consider the phrases and meanings in the context of the individuals who wrote them, taking into account what else they have said and how it affects the meaning of the phrase that they have used to describe their experiences about a particular topic. The combination of individual and collective contexts of particular expressions is the hermeneutic element of phenomenographic analysis.

In the next phase of analysis, the different categories that were found under specific topics were compared with each other and the distinguishing features were explicated. This part of the research was done by hand, with the help of printed paper slips, as it was necessary to categorise the same material in several different ways, and the present researcher found it the easiest to do this by hand, using different colour codes to keep the material in order.

The distinguishing features were then used as a basis when descriptions for each topic and category were written. These qualitative descriptions form one result of the present study (Chapter 7), since one aim of the study is to understand the variation within the categories of learners’ strategic and motivational capabilities. These categories were compared to the categories suggested by the theoretical background (see Section 3.4). The preconception of the categories is mainly provided by earlier quantitative studies and theories based on them, and it is interesting to see how the qualitative findings compare to this background.

In order to achieve deeper comparison, the present group of research participants also responded to a questionnaire on their motivational and strategic abilities, which were then analysed quantitatively with Bayesian modelling methods.

traditional linear modelling methods, allowances often have to be made. In a group of only 23 research participants, it cannot be expected that the variables follow the standard deviation on a Likert scale questionnaire. Often some response choices may not be used at all. It is in these problems with small sample sizes and skewed variables that Bayesian modelling methods become useful for the present study.

When a Bayesian modelling approach is compared to classical frequentist linear methods, the following major differences should be considered: (1) the Bayesian approach is parameter-free and user input is not required, (2) Bayesian methods work with probabilities and is thus capable of handling discrete data containing nominal and ordinal attributes, (3) the Bayesian approach has no limit for minimum sample size, (4) the Bayesian approach assumes no multivariate normal model, and finally, (5) Bayesian modelling allows a researcher to analyze both linear and non-linear relationships between variables. (Nokelainen, Silander, Ruohotie & Tirri, 2003.)

Bayesian dependence modelling

A Bayesian network is a viable way to examine dependencies between variables by both their visual representation and the probability ratio of each dependency. A Bayesian network is a representation of a probability distribution over a set of random variables, consisting of a directed acyclic graph (DAG), where the nodes correspond to domain variables, and the arcs define a set of independence assumptions which allow the joint probability distribution for a data vector to be factorized as a product of simple conditional probabilities. (Ruohotie & Nokelainen, 2002.)

The graphical visualization of Bayesian network (Myllymäki, Silander, Tirri & Uronen, 2002) contains two components: (1) observed variables visualized as ellipses and (2) dependences visualized as lines between nodes. Solid lines indicate direct causal relations and dashed lines indicate dependency where it is not certain that there is a direct causal influence or latent cause. A variable is considered to be independent of all other variables if there is no line attached to it.

Previous research in the field of vocational education has demonstrated that Bayesian networks are useful for explorative analysis of dependencies between observed variables (Ruohotie & Nokelainen, 2000; Nokelainen, Tirri, Nevgi, Silander & Tirri, 2001).

Bayesian classification modelling

Bayesian classification modelling (Silander & Tirri, 1999) allows a researcher to find out which variables included in the study are the best predictors for differences within the selected class variable. In addition, the most common components between categories of the class variable are revealed. In the classification process, the automatic search looks for the best set of variables to predict the class variable for each data item. This procedure is analogous to the stepwise selection procedure in traditional linear discriminant analysis.

(Nokelainen, Tirri & Merenti-Välimäki, 2002.)

Bayesian unsupervised model-based visualization

Bayesian unsupervised model-based visualization differs from Multidimensional Scaling (MDS) in the following ways: (1) the Bayesian approach is parameter-free and user input is not required, instead, prior distributions of the model offer a theoretically justifiable method for affecting the model construction, (2) Bayesian methods work with probabilities and can hence be expected to produce smooth and robust visualizations with discrete data containing nominal and ordinal attributes, (3) the Bayesian approach has no limit for minimum sample size, and (4) it assumes no multivariate normal model. When examining single data vectors (i.e. respondents) the data is mapped into different set of dimensions according to the optimized solution from which the Bayesian algorithm produced one optimal model. The three-dimensional model is plotted into a series of two-dimensional figures each presenting one dimension at the time. (Nokelainen & Ruohotie, 2002.)

6 INTERNAL COOPERATION OF THE LEARNER GROUPS

If you want creative workers, give them enough time to play.

John Cleese (Actor, b.1939)

In order to answer the research problem one (What are the greatest differences in the cooperative performance of the five learner groups over the course of their studies?) and to provide background for the later analysis, this chapter provides qualitative descriptions of the development each of the five groups of learners, concentrating on the development stages that they went through over the course of their studies and what level of cooperation their group reached in their work on the learning assignments and learning in general. This chapter also introduces some findings about the groups discovered through Bayesian analysis of their responses to the APL questionnaire. The findings from this analysis are later used to determine the factors that contribute to the success or failure of effective group work (research problem 4).

The next chapter (Chapter 7) groups the learners into three groups on the basis of their general achievement in their studies in order to find the most meaningful differences in the motivation and learning strategies between high achievers and average and low achievers.

It also considers how these strategies correlate with the learners’ group working skills and whether the high achievers were also skilled collaborators. This chapter will present the results of the Bayesian analysis of the APL questionnaire.

In the present chapter, the descriptions and analysis of the five learner groups is based on the nodes that were coded in the students’ study journals pertaining to their statements about the group work in their groups. These descriptions also include information and analysis about the Bayesian analysis of the learners’ responses to the APL questionnaire.

The analysis on the questionnaire refers to Appendix 2, which presents the applicable unsupervised visualisation model of the responses of the learner groups. Similarly, some charts concerning learner activity in the online learning environment are presented in Appendix 3. These group-specific descriptions and analyses are followed by a summary (subsection 6.2) of the greatest general differences in the cooperative performances of the groups (research problem 1).

In order to ensure the ethics of the present research, this chapter does not provide the names of the learners whose journal entries are quoted in the descriptions. Neither does it explicitly state who, in the various groups, the high, average or low achievers were.