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Introduction to Multi-Level Modeling

Introduction to Multi-Level Modeling. Juliet Aiken Design and Statistical Analysis Laboratory November 13, 2009. What to Expect. Slides will be posted on DaSAL website (http://blog.umd.edu/statconsulting/) after the presentation (but not immediately)

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Introduction to Multi-Level Modeling

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  1. Introduction to Multi-Level Modeling Juliet Aiken Design and Statistical Analysis Laboratory November 13, 2009

  2. What to Expect Slides will be posted on DaSAL website (http://blog.umd.edu/statconsulting/) after the presentation (but not immediately) Not all questions will be answered right now Ask anyway! We’ll write them down and follow up with them before posting the slides online Terminology: Random Coefficient Modeling: A statistical technique Hierarchical Linear Modeling: A piece of software which can be used to do random coefficient modeling It’s okay if some of this is information overload! I do not care if you get up to get more food in the middle of this 

  3. Goals • Understand basics of random coefficient multi-level modeling (RCM) • When to use and when not to use RCM • What is a random coefficient? • Slopes vs. intercepts as outcomes • Provide a platform for further reading and investigation into proper use of RCM • Materials and references for later use

  4. Overview • What is multi-level modeling? • Multi-level modeling examples • What is multi-level modeling?, revisited • Why do we care? • Multi-level statistical models • Justification for use • Two types of models

  5. Overview • MLM: Random Coefficient Modeling • Distinguishing Random Coefficients from Random Effects • Kinds of RC Modeling that can be conducted • HLM • Show and Tell with HLM

  6. Overview of Example • 20 teams, 100 people (5 per team), 4 observations of performance per person • Self efficacy is measured at each time point per person • Measures of team empowerment from each individual with respect to their team • Does self-efficacy predict performance? • Within individuals? • Across individuals? • Does team empowerment predict performance?

  7. What Is Multi-Level Modeling? • Family of statistical models for data analysis • Theory, design, and measurement

  8. Multi-Level Theoretical Models:Modeling with Correlated Errors (V-C) Partner 1 Organizational Support Work-Family Conflict Organizational Support Work-Family Conflict Partner 2

  9. Multi-Level Theoretical Models:N-Level Modeling (V-C) Level 2: Group-Level Team Empowerment Group Performance Level 1: Individual-Level Self-Efficacy Individual Performance

  10. Multi-Level Theoretical Models:Multi-Level Modeling (V-C) Level 2: Group-Level Team Empowerment Group Performance Level 1: Individual-Level Self-Efficacy Individual Performance

  11. Multi-Level Theoretical Models:Cross-Level Modeling (RCM) Level 2: Group-Level Team Empowerment Level 1: Individual-Level Self-Efficacy Individual Performance

  12. Multi-Level Theoretical Models:Frog-in-Pond Modeling (RCM) Level 2: Group-Level Team Empowerment Level 1: Individual-Level Self-Efficacy Relative to Group Mean Individual Performance

  13. Multi-Level Modeling: Levels • Organization (School)—Group (Class) • Group (Class)—Individual • Organization (School)—Individual • Individual—Time • Two levels, three levels, four levels, more!

  14. What do they have in common? • Hierarchical • Non-randomly Nested • Implies sampling method that begins at the highest level • Random nesting (e.g. experimental conditions) does not violate independence of observations assumptions

  15. Why do we care? • Nesting violates independence of observations • Results in heteroscedasticity • Violates assumptions of “regular” regression • Results in incorrectly estimated standard errors, and consequently, “wrong” results • Thus multi-level modeling more powerful and more honest

  16. Multi-Level Statistical Models • Must justify use of multi-level modeling techniques • Group-level variables (e.g. same supervisor, same organization) • Justification for aggregation • Theoretical • Design and Measurement (referent) • Statistical (ICC1, ICC2, rwg, AD, others)

  17. Multi-Level Statistical Models • Regression-Based • Random Coefficient Modeling • Variance-Covariance-Based • Latent Growth Analysis (in SEM)

  18. RCM Path analysis harder Measured variables Easy to include interactions Hard to include correlated errors Copes with missing data Easy to add levels VC Factor analysis Path analysis easy Latent variables Interactions harder Different errors easy to add Harder to cope with missing data Harder to add levels Goodness of fit information Model suggestions to improve fit RCM vs. VC

  19. Random Coefficient Modeling • Random Effects • Experimental conditions (e.g. for medicine) • Fixed: Can infer about treatments used in the experiment • Random: For purposes of generalization • Random Variables • Fixed: Variable with values that are known (e.g. gender) • Random: Variable with values selected from a probability distribution and are measured with error (e.g. IQ)

  20. Random Coefficient Modeling • Random Coefficients • Fixed: Coefficients (e.g. slopes or intercepts) do not vary across people/teams/etc. • Random: Coefficients in which values estimated are assumed to be distributed as a probability function • Random coefficients do NOT correspond to random effects or variables • Equations versus design/experimental manipulations

  21. Random Coefficient Modeling • Extension of the general/generalized linear model • Outcome of interest measured at the lowest level • “Intercepts as outcomes” • “Slopes as outcomes”

  22. Random Coefficient Modeling: Intercepts as Outcomes H1: Group team empowerment impacts average individual performance Level 2: Group-Level Team Empowerment Level 1: Individual-Level Self-Efficacy Individual Performance

  23. Random Coefficient Modeling: Intercepts as Outcomes H1: Group team empowerment impacts average individual performance Individual Performance Individual Self-Efficacy

  24. Random Coefficient Modeling: Intercepts as Outcomes H1: Group team empowerment impacts average individual performance Individual Performance Team 1 has high team empowerment, and thus, higher average individual performance than Team 5 Individual Self-Efficacy

  25. Random Coefficient Modeling: Intercepts as Outcomes H1: Group team empowerment impacts average individual performance Individual Performance However, Team 1 and Team 5 exhibit the SAME relationship between individual self-efficacy and performance Individual Self-Efficacy

  26. Random Coefficient Modeling: Slopes as Outcomes H1: Group team empowerment moderates the relationship between self-efficacy and individual performance Level 2: Group-Level Team Empowerment Level 1: Individual-Level Self-Efficacy Individual Performance

  27. Random Coefficient Modeling: Intercepts as Outcomes H1: Group team empowerment moderates the relationship between self-efficacy and individual performance Individual Performance Individual Self-Efficacy

  28. Random Coefficient Modeling: Intercepts as Outcomes H1: Group team empowerment moderates the relationship between self-efficacy and individual performance Individual Performance Team 1 has high team empowerment, BUT still the same average individual performance as Team 5 Individual Self-Efficacy

  29. Random Coefficient Modeling: Intercepts as Outcomes H1: Group team empowerment moderates the relationship between self-efficacy and individual performance Individual Performance However, for the high team empowerment team, Team 1, self-efficacy is positively related to performance, whereas for the low- team empowerment, Team 5, self-efficacy is negatively related to performance Individual Self-Efficacy

  30. Random Coefficient Modeling: Slopes and Intercepts as Outcomes H1: Group team empowerment moderates the relationship between self-efficacy and individual performance AND impacts average individual performance directly Level 2: Group-Level Team Empowerment Level 1: Individual-Level Self-Efficacy Individual Performance

  31. Random Coefficient Modeling: Slopes and Intercepts as Outcomes H1: Group team empowerment moderates the relationship between self-efficacy and individual performance AND impacts average individual performance directly Individual Performance Individual Self-Efficacy

  32. Random Coefficient Modeling: Slopes and Intercepts as Outcomes H1: Group team empowerment moderates the relationship between self-efficacy and individual performance AND impacts average individual performance directly Individual Performance Team 1 has high team empowerment, and, consequently, a higher average individual performance than Team 5 Individual Self-Efficacy

  33. Random Coefficient Modeling: Slopes and Intercepts as Outcomes H1: Group team empowerment moderates the relationship between self-efficacy and individual performance AND impacts average individual performance directly Individual Performance Additionally, for the high team empowerment team, Team 1, self-efficacy is positively related to performance, whereas for the low- team empowerment team, Team 5, self-efficacy is negatively related to performance Individual Self-Efficacy

  34. Random Coefficient Modeling: Word of Warning • Large models can be unstable • Small changes in the model may result in large changes in the result of the analysis • Might be due to multicollinearity in cross-level interactions/high correlations in parameter estimates • Mainly a problem when few observations on the highest level • Unbalanced samples may have too-small estimated standard errors • Makes hypothesis tests too liberal • Except for fixed coefficients, your df is tied to the number of observations at the highest level of predictor variables

  35. Random Coefficient Modeling: Software Available

  36. Random Coefficient Modeling: Software Available

  37. Random Coefficient Modeling: HLM • Assumptions • Observations at highest level are independent • Linear models • Level 1—normal random errors • Level 2—multivariate normal random errors • Level 1(2) predictors are independent of Level 1(2) residuals • Variance of residual errors is the same at all levels • Variances of residual errors is the same across units at Level 1 • Independent errors across and within levels

  38. Random Coefficient Modeling: HLM • Options • RCM • Multivariate RCM • 2- or 3- levels • Cross-classified models (2 levels only)

  39. HLM: Options

  40. Random Coefficient Modeling: HLM • Data preparation • “Down” format • Sort in ascending order • Names truncate to 8 letters • ID variables on all levels • No missing data on higher levels

  41. Random Coefficient Modeling: HLM Preparation

  42. Random Coefficient Modeling: HLM Preparation

  43. Random Coefficient Modeling: HLM Preparation

  44. Random Coefficient Modeling: Preparation

  45. Random Coefficient Modeling: HLM Preparation

  46. Random Coefficient Modeling: HLM Preparation

  47. Random Coefficient Modeling: HLM Preparation

  48. Random Coefficient Modeling: HLM Preparation

  49. Random Coefficient Modeling: HLM Preparation

  50. Random Coefficient Modeling: HLM • Everyone with HLM on their machine, please open it now.

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