Discontinuous Growth Models Paul D. Bliese Walter Reed Army Institute of Research

1. Discontinuous Growth ModelsPaul D. BlieseWalter Reed Army Institute of Research

2. Purpose • Introduce discontinuous growth models • Illustrate where these models might be used in organizational research • Provide details on how to set up the level-1 time variables • List resources for researchers interested in applying the models

3. Outline • Review of basic growth models • Discontinuities in longitudinal data • Sleep changes over 27 days • Task-change paradigm to study adaptability • Specifying time in the models • Dummy code for transition • Dummy code and post-transition slope • Adding quadratic terms • Practical advice • Other applications of model • References

4. Review of Basic Growth Model • Growth modeling is widely applied to the analyses of longitudinal data • Useful when trends are expected and where there are no anticipated discontinuities between measurement intervals • New employees’ skill acquisition • Individuals’ performance when learning a new task • Sales growth in start-up organizations • Changes in childrens’ height with age • Growth models • Test for individual differences in the outcome • Variability in the rate of change over time

5. Review: Heights of Boys in Oxford

6. Review: Heights of Boys in Oxford • Average height reliably varies across boys • ICC(1) = 0.74 • Slopes randomly vary

7. Review: Heights of Boys in Oxford • Growth model can explain differences in overall height and differences in slopes as a function of level-2 characteristics such as: • Genetics • Nutrition • Average height of parents • Typical growth models work in these cases because • There is an underlying trend to the data • There is no explicit transition point between measurement intervals where change takes on a distinct non-linear pattern

8. Discontinuities in Longitudinal Data • In many situations, events representing distinct transition points occur during longitudinal data collection • Events can be planned • A new HR initiative targeting turnover rates • An intended but unexpected change in the nature of the task in research on individual adaptability • And unplanned • The unexpected passage of an economic stimulus package during a study of individual consumer spending

9. Discontinuities in Longitudinal Data • In longitudinal data, these transition points may: • Be the topic of research interest • Mask sub-trends in the overall growth pattern • Even longitudinal data without any apparent growth may reveal important information when transition points are examined • Over a 10 year period, sales in established markets may be flat, but nonetheless contain important information about a variety of distinct events • New marketing initiatives • Changes in management • Over a 27-day period, adult sleep patterns may not change, but may nonetheless mask information about transitions

10. Sleep Changes over Time (Bliese et al., 2007) • Minutes of sleep over 27 days (NS linear trend) • On surface, not a good candidate for growth model

11. Sleep Changes over Time • Model with a term for the transition from sleeping in barracks to sleeping in a field exercise setting

12. Sleep Changes over Time • The transition point masked underlying patterns in the longitudinal data. • Significant pre-change slope (p<.05) • Significant transition (p<.10) • Significant post-change slope (p<.05)

13. Sleep Changes over Time • The form of the relationship is partially captured in a quadratic trend • Quadratic term approximates the nature of the data • Misses the distinct transition phase

14. Individual Differences in Sleep Patterns • Importantly, individuals can differ in each term of the discontinuous growth model

15. Individual Differences in Sleep Patterns • Individual differences can be predicted using level-2 variables (such as participant age).

16. Task-Change Paradigm • Research on adaptability frequently uses the task-change paradigm • Task-change paradigm • Uses complex tasks • Nature of task unexpectedly changes • Tasks are not learned to asymptotic performance prior to change • Produces large individual differences • Individual response to change is used to make inferences about adaptability

17. Task-Change Paradigm (Lang, 2007) • Performance and the task change paradigm • Task unexpectedly changes after 6th trial

18. Task-Change Paradigm (Lang, 2007) • Data contain strong individual differences in transition parameter

19. Task-Change Paradigm • Theoretically, Lang & Bliese (in review) conceptualize adaptability as: • Response to transition • Re-acquisition slope • These two elements of adaptability are separate from • Basal performance (overall performance) • Initial skill acquisition • Discontinuous growth model provides a framework to accurately mirror what is occurring in the experiment

20. Specifying the Level-1 Time Model • Details of the discontinuous growth model are provided in Chapter 5 of Singer & Willett (2003) • Singer, J. D., & Willett, J. B. (2003). Applied Longitudinal Data Analysis: Modeling Change and Event Occurrence. Oxford University Press. • The remainder of this lecture provides the basic foundation for understanding how models can be specified. • Focus on specifying the level-1 time component of the models

21. Coding Time in Growth Model • Growth models are one specific form of a class of mixed-effects models for longitudinal data. • The standard way of coding time in growth models is by using a vector from 0 to n observations.

22. Coding Time in Mixed-Effects Models • Coding time as a vector with a linear trend line is an efficient way (in terms of Degrees of Freedom) to account for time effects. • 1 DF for fixed effects, 3 for random effects • The other extreme is model time as a set of n-1 dummy codes • 8 DF for fixed effects • Lots and lots of random effects

23. Coding Time for Only Discontinuity • Coding for discontinuity involves adding additional vectors to the level-1 predictor matrix • The simplest way to model discontinuity is to add a dummy code that is 0 before the change and 1 after the change • Coding of Lang & Bliese task-change paradigm data

24. Coding for Only Discontinuity • The fixed-effects table indicates whether the transition point represents a significant change • Task-change paradigm data (Lang, 2007) shows • A significant linear increase in performance • A significant decline when the task was unexpectedly changed

25. Coding for Only Discontinuity • Potential limitation is that this model specification restricts pre-slope and post-slopes to be equal • Visual representation of Lang & Bliese adaptability data and Bliese et al., 2007 sleep data • Random effects for sleep data shown (significant)

26. Coding for Discontinuity and Slope Differences • Adding one more vector to the level-1 model provides a way to determine if the post-transition slope varies from the pre-transition slope • The vector that was previous labeled “TIME” now represents the pre-transition slope

27. Coding for Discontinuity and Slope Differences • Task-change paradigm data model reveals: • A significant linear increase in pre-change performance • A significant decline when the task was unexpectedly changed • A post-change slope that is significantly smaller than the pre-change slope

28. Coding for Discontinuity and Slope Differences • Graphs below contrast task-change paradigm data and sleep data using: • The model with only a transition variable • The model with both a transition and post-transition slope variable

29. Individual Differences in Sleep Patterns • Individual differences can exist for the: • Pre-transition slope • Transition • Post-transition slope • Differences can be modeled with level-2 variables

30. Flexibility of Coding: Quadratic Effects • The approach has considerable flexibility and can incorporate curvilinear effects • Lang & Bliese data with quadratic terms

31. Flexibility of Coding: Quadratic Effects • Using this coding, the fixed-effects model identifies a significant quadratic component to the pre-transition slope • Post-transition slope has no significant quadratic form

32. Flexibility of Coding: Quadratic Effects • Graphs shows typical learning curve prior to task change

33. Practical Advice on Estimating Models • With flexibility comes complexity! • The goal of these models is typically to explain individual-level variability in the model parameters • As level-1 growth parameters increase, the number of random effects also increases • With three growth parameters (pre-linear, transition, and post-linear), the methods must estimate 4 variances and 6 covariances • Practically speaking, models often fail to converge when numerous random terms are included • Choice of how to specify the random components of the model must be guided by theory and a systematic approach to examining the model

34. Practical Advice on Estimating Models • Recommend following a model specification strategy such as that outlined by Bliese & Ployhart (2002). • Estimate the ICC for the outcome • Identify the significant fixed effects for time • Identify which effects for time randomly vary across individuals • Determine whether other adjustments are needed to level-1 error structure (e.g., autocorrelation) • Include level-2 predictors of randomly varying level-1 effects. Do not rely only on empirical results of step 3. Also use theory as a guide.

35. Some Other Applications • Lang & Kersting (2007) used a discontinuous model with 4 data points to examine teachers’ effectiveness ratings after implementing feedback

36. Some Other Applications • Bliese, Wesensten & Balkin (2006) used the approach to mirror design elements of a sleep study • Residual individual variance was highly significant • Indicates strong individual differences in ability to perform during sleep restriction

37. Selected References Bliese, P. D., McGurk, D., Thomas, J. L., Balkin, T. J., & Wesensten, N. (2007). Discontinuous growth modeling of adaptation to sleep setting changes: Individual differences and age. Aviation, Space, and Environmental Medicine, 78, 485-492. Bliese, P. D., Wesensten, N., & Balkin, T. J. (2006). Age and individual variability in performance during sleep restriction. Journal of Sleep Research, 15, 376-385. Lang, J. W. B. (2007). General Mental Ability and Two Types of Adaptation to Unforeseen Change. Dissertation. Rheinisch-Westfälische Technische Hochschule, Aachen, Germany. Lang, J. W. B. & Bliese, P. D. (in revision). General Mental Ability and Two Types of Adaptation to Unforeseen Change: Applying Discontinuous Growth Models to the Task-Change Paradigm. Lang, J. W. B. & Kersting, M. (2007). Regular feedback from student ratings of instruction: Do college teachers improve their ratings in the long run? Instructional Science, 35, 187-205.