The Good, the Bad, and the Mean (

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The Good, the Bad, and the Mean (µ) Limitations
and Extensions of Latent Growth Curves in Health
Disparities Research

• Miles Taylor, Ph.D.
• Florida State University

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What is Growth Curve Analysis?

• The broad category of models includes multiple
  types of models (from multiple traditions) that
  are used to analyze individual change using more
  than 2 time points
• Exs latent growth curve analysis, latent
  trajectory analysis, random effects models,
  hierarchical linear models, etc.
• Note that curve does not necessarily mean a
  nonlinear trend. On the contrary, most of the
  growth curves predicted by these various types
  of models are linear.
• Examples trajectories of reading ability in
  children, depressive symptoms across the life
  course, tumor growth in rats

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Unconditional Model

• Level 1 model
• Level 2 model
• Combined

1999
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Structural Equation Models (SEM)

• Structural Equation Models (SEM) refer to a broad
  class of powerful models
• Instead of emphasizing cases, SEM emphasizes
  variances/covariances.
• This allows testing whether and how variables are
  interrelated in a set of linear relationships
• The acronym is sometimes switched for
  simultaneous equation modeling (SEM) since it can
  handle many interrelated equations that are
  jointly estimated

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Why Choose a Structural Equation Modeling (SEM)
Approach to Growth Curves?

• Various forms of measurement error
• Estimators and fit indices for continuous,
  dichotomous, or ordinal repeated measures
• Flexibility in handling time
• Statistical packages like Mplus make more complex
  models possible
• Other approaches do have advantages in some
  instances, such as observations at different time
  points

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The Good

• Improvement over aggregate change approaches
  not Markovian or semi-Markovian
• Can incorporate many repeated observations
• Can handle time invariant and time variant
  covariates as well as repeated outcomes
• Can be combined in an SEM context
• Allow examination of life course developmental
  processes, testing developmental theories
• Can examine whether inequalities or disparities
  are persistent, increasing, etc. over time both
  within and across individuals

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Example of the Good
Preliminary Findings, Please do not cite without
permission

• Valle, G. Thomas, K. Taylor, M. G. Parental
  Incarceration Influences on Childrens Mental
  Health during the Transition to Adulthood

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Example of the Good

• Valle, G. Thomas, K. Taylor, M. G. Parental
  Incarceration Influences on Childrens Mental
  Health during the Transition to Adulthood

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Why it works

• The findings from the alpha and beta (intercept
  and slope) were meaningful in a life course
  context (persisting inequality changes to an
  underlying effect emerging in adulthood)
• Individual loadings were freed and then fixed,
  allowing more complex nonlinearity to be modeled
• The outcome is easily thought of as developmental
  / continuous in nature
• The treatment was estimated before W1

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The Bad (1) People or Patterns are Missed

• Level 1 equation parameterizes individual
  trajectories before calculating their variation
  from the mean
• Model specification (linear, quadratic, etc.) is
  based on the average trajectory specification
• Trajectory methodologists acknowledge we should
  free the loadings but we trade parsimony and
  therefore fit
• What if some collection of the trajectories is
  nonlinear and meaningful
• What if timing of the developmental process is
  important?

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1999
1982
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1984
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Extensions

• Group-based modeling strategies can handle this
  efficiently (latent class analysis of
  trajectories, finite mixture models, growth
  mixtures with freed loadings)
• Work of Nagin, Land, Muthen
• Hybridized models can handle this where onset
  of developmental process varies at random.
• Work of Albert Shih (2003), Taylor (2008
  2010), and Haas Rohlfson (2010)

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Group Trajectory Example
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Why it works

• Shows that there is more than one average
  trajectory and multiple forms of meaningful
  nonlinearity.
• Efficiently models linear trajectories like
  linear along with a lagged onset, etc.
• Referent group is no longer the mean trajectory.
  It is assumed to be the most prevalent group by
  default but may be set to any meaningful
  experience (here nondisabled over the period)
• Covariates are thus used to predict patterns
  rather than high/low on intercept and slope/s.

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Random Onset Model

• Taylor, Miles G. 2010. Capturing Transitions and
  Trajectories The Role of Socioeconomic Status in
  Later Life Disability. Journals of Gerontology
  Social Sciences 65B 733-743

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Why it works

• A second process (here first onset) is modeled.
  Therefore, the growth curves only include nonzero
  values.
• Delayed onset (modeled through a discrete time
  hazard) captures the meaningful nonlinearity of
  the disability trajectories.
• This means that one can reconcile findings from
  state based (transition) and developmental
  trajectory literatures
• It also means covariates can predict these
  simultaneous processes in shared or independent
  ways

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The Bad (2) Selection Processes

• Selection into the observation window
  with/without starting the developmental process
  (meaningful partial left censoring)
• Random onset model handles this better than
  traditional LGCs

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Extension Random Onset

• Taylor, Miles G. 2008. Timing, Accumulation, and
  the Black/White Disability Gap in Later Life A
  Test of Weathering. Research on Aging Special
  Issue on Race,SES, and Health 30 226-250.

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Extension Random Onset

• Taylor, Miles G. 2008. Timing, Accumulation, and
  the Black/White Disability Gap in Later Life A
  Test of Weathering. Research on Aging Special
  Issue on Race, SES, and Health 30 226-250.

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Why it works

• A second process (here first onset) is modeled.
  Therefore, the growth curves only include nonzero
  values.
• Traditional LCGs returned findings supporting a
  cumulative disadvantage theory.
• Random onset model reveals that in this sample,
  the disparity lies in the onset process.
• Black individuals were more likely to select into
  the sample with some nonzero level of disability,
  but their process of accumulation thereafter was
  not significantly different from whites.

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The Bad (2) Selection Processes

• Selection out of the sample that is meaningful
  (attrition, mortality selection)
• Transition models (survival, etc.) have specific
  extensions for this (competing risk/multiple
  decrement)
• In traditional LCGs, the best we get is to
  include those until they drop out or include
  some kind of control for attrition

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The Bad (2) Selection Processes

• With SEM it is possible (just like in the random
  onset model) to include additional equations to
  handle this transition (either time variant or
  no)
• This means we can include a parallel joint
  process (like the random onset model) but this
  time it is a timing of exit
• A.K.A., one can create a sort of competing risk
  between changes in the developmental process of
  the outcome over time vs. attrition/death

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Extension Attrition Process
Taylor, Miles G. and Scott M. Lynch. 2011.
Cohort Differences and Chronic Disease Profiles
of Differential Disability Trajectories.
Journals of Gerontology Social Sciences. 66B
729-738.
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Extension Attrition Process
Taylor, Miles G. and Scott M. Lynch. 2011.
Cohort Differences and Chronic Disease Profiles
of Differential Disability Trajectories.
Journals of Gerontology Social Sciences. 66B
729-738.
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Why it works

• The second process here is mortality, and this I
  can model jointly with disability. A.K.A they
  affect one another over time.
• Here I was primarily interested in cohort
  differences, and allowing these covariates to
  impact both disablement trajectories and death
  inform findings on the compression of morbidity.
• Chronic diseases were also included in later
  models, and these impacts I could see on
  disability over the decade net of death and vice
  versa.

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Summary

• Potential weaknesses of traditional LGCs
• People or meaningful patterns are missed through
  misspecification in the level 1 equation
• Extensions
• Multiple ways to disentangle or unpack the mean
  growth or important deviations from it
• Consider group based trajectories for modeling
  meaningful nonlinearity efficiently
• Inclusion of additional processes (onset,
  recovery, etc.)

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Summary

• Potential weaknesses of traditional LGCs
• Differential Selection into the sample on level
  of outcome, out of the sample
• Extensions
• Random onset as simultaneous process for partial
  left censoring
• Mortality or other meaningful attrition as a
  simultaneous process

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Conclusions

• Latent Growth Curve (LGC) modeling in an SEM
  framework is extremely versatile due to the
  ability to model equations simultaneously
• New softwares for SEM/Latent variable modeling
  (a.k.a. Mplus) allow more flexibility in modeling
  noncontinuous endogenous/outcome variables
• Documentation now exists on replicating standard
  models like simply discrete-time hazard and
  finite mixtures/cluster analysis in the SEM
  context.
• Its time to move beyond the mean, beyond the
  noise.