Discontinuous Growth Models

Paul D. BlieseWalter Reed Army Institute of Research

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

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

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

Review: Heights of Boys in Oxford

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Review: Heights of Boys in Oxford

• Average height reliably varies across boys– ICC(1) = 0.74

• Slopes randomly vary

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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

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

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

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

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Sleep Changes over Time

• Model with a term for the transition from sleeping in barracks to sleeping in a field exercise setting

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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)

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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

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Individual Differences in Sleep Patterns

• Importantly, individuals can differ in each term of the discontinuous growth model

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Individual Differences in Sleep Patterns

• Individual differences can be predicted using level-2 variables (such as participant age).

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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

Task-Change Paradigm (Lang, 2007)

• Performance and the task change paradigm– Task unexpectedly changes after 6th trial

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Task-Change Paradigm (Lang, 2007)

• Data contain strong individual differences in transition parameter

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Individual "01122105"Overall TrendIndividual "s10r8"

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

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

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.

Subject time height1 0 140.51 1 143.41 2 144.81 3 147.11 4 147.71 5 150.21 6 151.71 7 153.31 8 155.8

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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

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

ID TIME TRANS1 0 01 1 01 2 01 3 01 4 01 5 01 6 11 7 11 8 11 9 11 10 11 11 1

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

Value Std.Error DF t-value p-value

(Intercept) -2.16 0.59 2022 -3.65 0.00

TIME 1.20 0.09 2022 13.42 0.00

TRANS -4.37 0.62 2022 -7.06 0.00

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)

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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

ID PRE TRANS POST1 0 0 01 1 0 01 2 0 01 3 0 01 4 0 01 5 0 01 6 1 01 7 1 11 8 1 21 9 1 31 10 1 41 11 1 5

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

Value Std.Error DF t-value p-value

(Intercept) -3.69 0.63 2021 -5.84 0.00

PRE SLOPE 1.81 0.13 2021 14.46 0.00

TRANS -4.98 0.62 2021 -8.05 0.00

POST SLOPE -1.22 0.18 2021 -6.88 0.00

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

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Slope and TransitionTransition Only

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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

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Flexibility of Coding: Quadratic Effects

• The approach has considerable flexibility and can incorporate curvilinear effects

• Lang & Bliese data with quadratic terms

ID TIME TRANS POST QUADPRE QUADPOST1 0 0 0 0 01 1 0 0 1 01 2 0 0 4 01 3 0 0 9 01 4 0 0 16 01 5 0 0 25 01 6 1 0 25 01 7 1 1 25 11 8 1 2 25 41 9 1 3 25 91 10 1 4 25 161 11 1 5 25 25

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

Value Std.Error DF t-value p-value

(Intercept) -4.73 0.69 2019 -6.83 0.00

PRE SLOPE 3.37 0.45 2019 7.57 0.00

TRANS -5.53 0.69 2019 -7.96 0.00

POST SLOPE -2.74 0.63 2019 -4.35 0.00

PRE QUADRATIC -0.31 0.09 2019 -3.65 0.00

POST QUADRATIC -0.01 0.09 2019 -0.09 0.93

Flexibility of Coding: Quadratic Effects

• Graphs shows typical learning curve prior to task change

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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

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.

Some Other Applications

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

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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

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2.96 Hrs TST4.80 Hrs TST6.78 Hrs TST

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Age 29Age 39 (Average)Age 48

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.