HLM - Hierarchical Linear and Nonlinear Modeling (HLM) In social research and other fields, research data often have a hierarchical structure. That is, the individual subjects of study may be classified or arranged in groups which themselves have qualities that influence the study. In this case, the individuals can be seen as level-1 units of study, and the groups into which they are arranged are level-2 units. This may be extended further, with level-2 units organized into yet another set of units at a third level and with level-3 units organized into another set of units at a fourth level. Examples of this abound in areas such as education (students at level 1, teachers at level 2, schools at level 3, and school districts at level 4) and sociology (individuals at level 1, neighborhoods at level 2). It is clear that the analysis of such data requires specialized software. Hierarchical linear and nonlinear models (also called multilevel models) have been developed to allow for the study of relationships at any level in a single analysis, while not ignoring the variability associated with each level of the hierarchy. The HLM program can fit models to outcome variables that generate a linear model with explanatory variables that account for variations at each level, utilizing variables specified at each level. HLM not only estimates model coefficients at each level, but it also predicts the random effects associated with each sampling unit at every level. While commonly used in education research due to the prevalence of hierarchical structures in data from this field, it is suitable for use with data from any research field that have a hierarchical structure. This includes longitudinal analysis, in which an individual’s repeated measurements can be nested within the individuals being studied. In addition, although the examples above implies that members of this hierarchy at any of the levels are nested exclusively within a member at a higher level, HLM can also provide for a situation where membership is not necessarily ”nested”, but ”crossed”, as is the case when a student may have been a member of various classrooms during the duration of a study period. The HLM program allows for continuous, count, ordinal, and nominal outcome variables and assumes a functional relationship between the expectation of the outcome and a linear combination of a set of explanatory variables. This relationship is defined by a suitable link function, for example, the identity link (continuous outcomes) or logit link (binary outcomes).

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  1. Byrnes, James P.; Miller-Cotto, Dana: The growth of mathematics and reading skills in segregated and diverse schools: an opportunity-propensity analysis of a national database (2016) MathEduc
  2. Gagnon, Jacob; Liang, Hua; Liu, Anna: Spherical radial approximation for nested mixed effects models (2016)
  3. Jitendra, Asha K.; Harwell, Michael R.; Karl, Stacy R.; Dupuis, Danielle N.; Simonson, Gregory R.; Slater, Susan C.; Lein, Amy E.: Schema-based instruction: effects of experienced and novice teacher implementers on seventh grade students’ proportional problem solving (2016) MathEduc
  4. Thien, Lei Mee: Malaysian students’ performance in mathematics literacy in PISA from gender and socioeconomic status perspectives (2016) MathEduc
  5. Ing, Marsha; Webb, Noreen M.; Franke, Megan L.; Turrou, Angela C.; Wong, Jacqueline; Shin, Nami; Fernandez, Cecilia H.: Student participation in elementary mathematics classrooms: the missing link between teacher practices and student achievement? (2015) MathEduc
  6. Stevens, Joseph J.; Schulte, Ann C.; Elliott, Stephen N.; Nese, Joseph F. T.; Tindal, Gerald: Growth and gaps in mathematics achievement of students with and without disabilities on a statewide achievement test (2015) MathEduc
  7. Thien, Lei Mee; Darmawan, I. Gusti Ngurah; Ong, Mei Yean: Affective characteristics and mathematics performance in Indonesia, Malaysia, and Thailand: what can PISA 2012 data tell us? (2015) MathEduc
  8. West, Brady T.; Welch, Kathleen B.; Gałecki, Andrzej T.: Linear mixed models. A practical guide using statistical software. With contributions from Brenda W. Gillespie (2015)
  9. Liou, Pey-Yan: Investigation of the big-fish-little-pond effect on students’ self-concept of learning mathematics and science in Taiwan: results from TIMSS 2011 (2014) MathEduc
  10. DiDonato, Nicole C.: Effective self- and co-regulation in collaborative learning groups: an analysis of how students regulate problem solving of authentic interdisciplinary tasks (2013) MathEduc
  11. Chiu, Mei-Shiu: Differential psychological processes underlying the skill-development model and self-enhancement model across mathematics and science in 28 countries (2012) MathEduc
  12. Snijders, Tom A. B.; Bosker, Roel J.: Multilevel analysis. An introduction to basic and advanced multilevel modeling (2012)
  13. Buff, Alex; Reusser, Kurt; Rakoczy, Katrin; Pauli, Christine: Activating positive affective experiences in the classroom: “nice to have” or something more? (2011) MathEduc
  14. Gilleece, Lorraine; Cosgrove, Jude; Sofroniou, Nick: Equity in mathematics and science outcomes: characteristics associated with high and low achievement on PISA 2006 in Ireland (2010) MathEduc
  15. Hox, Joop: Multilevel analysis. Techniques and applications (2010)
  16. Chen, Guangliang; Lerman, Gilad: Foundations of a multi-way spectral clustering framework for hybrid linear modeling (2009)
  17. Hugener, Isabelle; Pauli, Christine; Reusser, Kurt; Lipowsky, Frank; Rakoczy, Katrin; Klieme, Eckhard: Teaching patterns and learning quality in Swiss and German mathematics lessons (2009) MathEduc
  18. Heck, Ronald H.; Thomas, Scott L.: An introduction to multilevel modeling techniques (2008)
  19. Frenzel, Anne C.; Pekrun, Reinhard; Goetz, Thomas: Perceived learning environment and students’ emotional experiences: a multilevel analysis of mathematics classrooms (2007) MathEduc
  20. Kunter, Mareike; Baumert, Jürgen; Köller, Olaf: Effective classroom management and the development of subject-related interest (2007) MathEduc

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