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Abstract

Introduction
Traditional linear Gaussian models are widely used in the research field of vocational education (Nokelainen, 2002). There is also an extensive literature describing use of various linear methods, such as regression and factor analysis. Unfortunately linear models are statistically inadequate for understanding non-linear dependencies between variables. In this paper our main goal is to investigate the number of non-linear and multi-modal relationships between variables in two real-world vocational educational data sets in order to find out how much they weaken robustness of linear statistical methods.
Theoretical background
Important factors in the development of growth orientation are support and rewards from the management, the incentive value of the job itself, the operational capacity of the team and work related stress (Ruohotie, 2000; Argyris, 1990; Dubin, 1990; Hall, 1990; Kaufman, 1990). The dimensions of growth-oriented atmosphere are operationalized based on our previous research (Ruohotie & Nokelainen, 2000) as follows:
  1. Encouraging leadership,
  2. Strategic leadership,
  3. Know-how rewarding,
  4. Know-how developing,
  5. Incentive value of the job,
  6. Clarity of the job,
  7. Valuation of the job,
  8. Community spirit,
  9. Team spirit,
  10. Student's attitudes towards teacher,
  11. Psychical stress of the job,
  12. Build-up of work requirements,
  13. Commitment to work and organization and
  14. Growth motivation.
The emphasis in this work is to demonstrate the possibility to build Bayesian networks that can capture non-linear relationships. By using discretized variables this possibility comes trivially (Hofmann and Tresp, 1996; 1998), but our objective is to find out to how many and what kind of non-linearities are captured by discrete Bayesian networks.Given the identically and independently distributed multivariate data set D over variables V and the prior probability distribution p over Bayesian networks (Pearl, 1998), Bayesian probability theory allows us to calculate the probability P(G | D, p) of any Bayesian network G (Heckerman, Geiger and Chickering, 1995). Different networks can then be compared by their probability. Finding the most probable Bayesian network for any given data is known to be NP-hard (Chickering, 1996), which practically ruins the hopes for the automatic discovery of the most probable network. However, stochastic search methods have proven to be successful in finding high probability networks (Chickering, Geiger and Heckerman, 1995). Once the network G has been constructed using data D, we can use it to calculate predictive joint distributions P(V | G, D).
Sample and procedure
The empirical data (N=655) was collected during year 2001 with a 92-item Likert-scale self-rated questionnaire (Nokelainen, Ruohotie & Tirri, 2002). The data consists of adult employees from two Finnish vocational polytechnic institutes (N=447, 87% of total population and N=208, 83% of total population). Respondents had three different kind of job profiles (with 4% missing data, N=31): Managers (6%, N=46), teachers (61%, N=462), and administrative personnel (29%, N=223). Respondents nature of contract was categorized into three classes (with 3%, N=26 missing data): Established (70%, N=533), temporary (22%, N=169), and part-time (5%, N=34) employees.To measure non-linear dependencies captured by Bayesian networks, we tested every variable in each network by conditioning it one by one with its immediate neighbours in the network. We then observed whether the modes and means of the conditional distributions were "linear" and whether the conditional distributions were "unimodal". In measuring unimodality of conditional distributions, we judged the dependency to be unimodal if (and only if) none of the conditional distributions P(Y|X) were clearly multimodal.
Results
Research evidence based on Bayesian network modeling of four independent empirical data sets describing dimensions of growth-oriented atmosphere shows that 22 percent of all dependencies are purely linear (linear mode, linear mean, unimodal), and 15.5 percent are purely non-linear (non-linear mode, non-linear mean, multimodal). In addition, multimodality (21%) and non-linear mean (11.4%) cause problems to linear models. Other classifications are listed as follows: Linear mode, non-linear mean, multimodal (16.9%), Non-linear mode, linear mean, unimodal (1.2%), Non-linear mode, linear mean, multimodal (4.7%), and Non-linear mode, non-linear mean, unimodal (7.3%). We compared the results of traditional linear exploratory factor analysis and Bayesian network modeling (Myllymaki, Silander, Tirri & Uronen, 2002) when the task was to extract fourteen dimensions of growth-oriented atmosphere. In FA we applied maximum likelihood with both varimax and direct oblimin (delta=0) rotation methods. The results show that Bayesian networks capture non-linear dependencies in real-world growth-oriented atmosphere data, as 78 percent of pairwise (unconditional) dependencies of the models are vaguely, and 54 percent severely, non-linear. These dependencies are missed or poorly modeled using simple linear models. In practice this means, that the results of linear analysis include varying level of error, depending on the level of non-linear dependencies in the data.
References
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  • Chickering, D.M. (1996). Learning Bayesian networks is NP-complete. In: D. Fisher and H.J. Lenz (Eds.), Learning from Data: Artificial Intelligence and Statistics, 5, 121-130. Springer Verlag.
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