Terminal Decline and Ergodicity in Life Satisfaction

1
Terminal Decline and Ergodicity in Life
Satisfaction

• Ryne Estabrook
• September 13, 2007
• Design and Data Analysis

Gerstorf, D., Ram, N., Estabrook. R., Schupp, J.,
Wagner, G.G. \ Lindenberger, U. (2008). Life
Satisfaction Shows Terminal Decline in Old Age
Longitudinal Evidence from the German
Socio-Economic Panel Study (SOEP). Developmental
Psychology, 44, 4, 1148-1159.
2
Terminal Decline
Terminal Decline and Ergodicity In Life
Satisfaction

• Terminal decline has long been studied as a
  late-life cognitive phenomenon.
• Kleemeier, 1962 see Bäckman MacDonald, 2006
  for review.
• Terminal decline can be defined in a number of
  ways
• All definitions describe negative (or more
  negative) change.
• Some differentiate between terminal drop
  (accelerated or curvilinear declines) and
  terminal decline (level differences).
• Often a methodological distinction.
• These two phenomena often termed together as
  terminal change.

3
Terminal Decline
Terminal Decline and Ergodicity In Life
Satisfaction

• Evidence of terminal decline over a range of
  cognitive functions.
• Areas of cognitive functioning relatively
  unaffected by age show the most pronounced and
  consistent terminal decline effects.
• White and Cunningham (1988) hypothesis.
• Vocabulary, crystallized abilities, episodic
  working memories.
• Differences between survivors and decedents found
  anywhere from two to twelve years prior to death.
• Rationales include pre-clinical dementia and
  cardiovascular functioning.
• Primary and secondary aging (Busse, 1969).
• No difference between biological and experimental
  mortality.

4
Multivariate Terminal Decline
Terminal Decline and Ergodicity In Life
Satisfaction

• Religious Orders Study (Wilson, et al, 2003)
• Found terminal decline in episodic memory,
  working memory, speed, and general cognitive
  abilities.
• Terminal decline found in virtually all
  individuals, but at highly variable rates.
• Individual aging (Lövdén, et al, 2005)
• Used cluster analysis to characterize several
  response profiles.
• Shows terminal decline in block design, episodic
  memory semantic memory as transitions through
  programs.
• Both characterize individually varying forms of
  terminal decline.

5
Non-Cognitive Decline
Terminal Decline and Ergodicity In Life
Satisfaction

• Terminal change in neuroticism
• Neuroticism shows long-term changes in adulthood.
• These changes predict mortality (Mroczek Spiro,
  2005).
• Terminal decline also noted in self-reported
  health depression.
• (Kaplan Camacho, 1983 Idler Angel, 1990
  Wolinsky Johnson, 1992 Berg, 1996).
• Terminal decline in well-being
• Self-reported well being showed stronger declines
  over distance to death than chronological age.
• Population-level change point of 4 years prior to
  death identified.
• (Gerstorf, Ram, Röcke, Lindenberger, Smith,
  2007).

6
Life Satisfaction
Terminal Decline and Ergodicity In Life
Satisfaction

• Life satisfaction is an interesting construct for
  terminal decline.
• Life satisfaction measures are typically very
  stable in late adulthood
• (Diener, Lucas, Scollon, 2006 Filipp, 1996
  Horley Lavery, 1995 Kunzmann, Little, Smith,
  2000).
• Well being may be both a consequence and source
  of successful aging.
• (Bales Baltes, 1990).
• Socio-emotional selectivity theory would not
  terminal decline in life satisfaction.
• (Carstensen, L.L., Isaacowitz, D.M., Charles,
  S.T.,1999)

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Life Satisfaction
Terminal Decline and Ergodicity In Life
Satisfaction

• If terminal decline in life satisfaction is
  exhibited
• Supports previous studies on terminal decline in
  well-being.
• Paves way for a gestalt view of terminal decline.
• Allows for future research in cognitive-emotive
  links in tertiary aging.
• If terminal decline in life satisfaction is not
  exhibited
• Supports socio-emotional selectivity theory and
  other theories prescribing constancy in adult
  personality.

8
Life Satisfaction Data
Terminal Decline and Ergodicity In Life
Satisfaction

• German Socio-Economic Panel Study (SOEP).
• Annual nationally-representative panel survey of
  German population.
• Data collected annually from 1984-2005.
• Participants recruited via random walk method.
• Recruitment in 1984, 1990, 1995, 1998, 2000,
  2002.
• Participants approached in representative public
  places.
• 60-70 response rate.
• 4-14 attrition rate (small gifts results for
  participation).
• Face-to-face interviews
• Mail questionnaires for 10 of multi-occasion
  respondents.

9
Life Satisfaction Data
Terminal Decline and Ergodicity In Life
Satisfaction

• Mortality Sample
• Must be 70 at the first measurement occasion,
• Must provide mortality information (N1637).
• Age
• Chronological Age
• Mean 78.20
• SD 5.88
• Range 70-100
• Distance-to-Death
• Mean -6.32
• SD 4.22
• Range -22 to -1

10
Life Satisfaction Data
Terminal Decline and Ergodicity In Life
Satisfaction

• Life Satisfaction
• Wie zufrieden sind Sie gegenwärtig, alles in
  allem, mit ihrem Leben?
• How satisfied are you with your life, all
  things considered?
• 0-10 Likert Scale (0unsatisfied, 10very
  satistfied).
• Standardized to a T-metric on full sample.
• Full Sample Mean 7.02, SD 1.55.
• Mortality Sample Mean 6.75, SD1.57.

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Research Questions
Terminal Decline and Ergodicity In Life
Satisfaction

• Is the data better explained by chronological
  aging or distance-to-death?
• Is there evidence of terminal change?
• Are there individual differences in levels and
  rates of change in life satisfaction?
• Are individual differences in changes in life
  satisfaction related to one another?

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Research Question 1
Terminal Decline and Ergodicity In Life
Satisfaction

• Is the data better explained by chronological
  aging or distance-to-death?
• Is there evidence of terminal change?
• Compare fit of mixed effects models from life
  satisfaction across chronological and death ages.
• Fit multiple functional forms
• Linear Mixed Effects
• Quadratic Mixed Effects
• Linear-Linear Spline Mixed Effects
• Fixed Change Point
• SAS PROC MIXED/NLMIXED

13
Multi-Phase Mixed Effects
Terminal Decline and Ergodicity In Life
Satisfaction

• Alternatively, (linear-linear) spline model.
• (Cudeck Harring, 2007 Cudeck Klebe, 2002
  Willett Singer, 2003 Wang McArdle, in
  press).
• Fixed Effects
• LifeSat ß0 ß1(time-ß3) e, timelt ß3
• LifeSat ß0 ß2(time-ß3) e, timegt ß3
• Benefits
• Better characterizes multiple change processes.
• More easily interpretable than higher order
  parametric models.
• Drawbacks
• Random change points are difficult to fit
  temporarily omitted.

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Terminal Decline and Ergodicity In Life
Satisfaction
Chronological Age
Life-Satisfaction by Chronological Age, Full
Sample
15
Terminal Decline and Ergodicity In Life
Satisfaction
Chronological Age
Life-Satisfaction by Chronological Age, N150
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Chronological Age
Terminal Decline and Ergodicity In Life
Satisfaction

• All models provided much improved fit over a
  no-change model.
• All models feature (increasingly) negative slope
  over time.
• Linear-linear spline provides the best model fit.

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Terminal Decline and Ergodicity In Life
Satisfaction
Distance-to-Death
Life-Satisfaction by Distance-to-Death, Full
Sample
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Terminal Decline and Ergodicity In Life
Satisfaction
Distance-to-Death
Life-Satisfaction by Distance-to-Death, N150
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Distance-to-Death
Terminal Decline and Ergodicity In Life
Satisfaction

• Linear-linear spline again provides the best
  model fit.
• All distance-to-death analyses show superior fit
  to parallel chronological age analyses.
• Evidence of declines related to distance-to-death
  rather than chronological age.

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Research Question 2
Terminal Decline and Ergodicity In Life
Satisfaction

• Are there individual differences in rates of
  change in life satisfaction?
• Are individual differences in changes in life
  satisfaction related to one another?
• Compare fit of fixed effect and mixed models
• Linear-Linear Spline Mixed Effects
• Fixed Change Point
• SAS PROC NLMIXED

21
Multi-Phase Mixed Effects
Terminal Decline and Ergodicity In Life
Satisfaction

• Random effects significantly improve model fit.
• Log likelihood feasible because models are
  nested.

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Multi-Phase Mixed Effects
Terminal Decline and Ergodicity In Life
Satisfaction
Random Effects Covariance Matrix for Multi-Phase
Growth Model on Distance-to-Death in Life
Satisfaction
plt.001
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Multi-Phase Mixed Effects
Terminal Decline and Ergodicity In Life
Satisfaction
Y48.14-0.64(time), time lt -4.19 Y48.14-1.94(ti
me), time gt -4.19
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Results
Terminal Decline and Ergodicity In Life
Satisfaction

• Distance-to-death models provided better fit than
  all comparable chronological age models.
• Life satisfaction shows sharper decreases closer
  to death.
• Mixed-effects models indicate individual
  variability in declines and relationships between
  level and rates of change.
• No relationship between pre-terminal and terminal
  slopes.

25
Random Change Points
Terminal Decline and Ergodicity In Life
Satisfaction

• Presented models include random effects for
  intercept and both linear slopes, but a fixed
  change point.
• Random change points are notoriously difficult to
  estimate with standard methods.
• Usually fit with SAS PROC NLMIXED.
• Include First-Order Taylor Series and adaptive
  Gausian quadrature methods.
• (Cudeck Klebe, 2002 Cudeck Du Toit, 2003
  Hall et al., 2000).
• Sensitive to initial values
• Inability to satisfy convergence criteria.
• Biased point and standard error estimates

26
Bayesian Estimation
Terminal Decline and Ergodicity In Life
Satisfaction

• Bayesian Estimation with Gibbs Sampling
• Identical fixed effects model, random effects for
  all parameters.
• (Wang and McArdle, in press)
• Benefits
• Estimable.
• Less sensitive to misspecification of starting
  values.
• Drawbacks
• Incomparable to other types of models.
• Potentially sensitive to poor priors.
• Time.

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Random Change Points
Terminal Decline and Ergodicity In Life
Satisfaction

• Changes to Model
• Inclusion of random change points
• Appropriate additions to parameter covariance
  matrix
• Non-informative priors
• Changes to Dataset
• Dataset restricted to 400 individuals with 12 or
  more observations.
• Individuals with few occasions tended to be
  mischaracterized by the model, particularly with
  change points beyond their time series.
• Population parameters do not significantly differ
  with this restriction.

28
Terminal Decline and Ergodicity In Life
Satisfaction
Random Change Points

• Inclusion of random change points improves model
  fit.
• While the DIC is analogous to AIC, the two cannot
  be directly compared, preventing direct fit
  comparisons between the Bayesian and Gaussian
  models.

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Random Change Point
Terminal Decline and Ergodicity In Life
Satisfaction

• Gaussian method is on same (restricted) sample as
  Bayesian models.
• Use of Bayesian model results as initial values
  for SAS PROC Mixed did not yield convergence.

30
Terminal Decline and Ergodicity In Life
Satisfaction
Random Change Points
31
Random Change Point
Terminal Decline and Ergodicity In Life
Satisfaction

• Inclusion of random change points improves model
  fit.
• Random change point model has the lowest residual
  variance of any model so far.
• Change points has significant variance and
  covariances with other parameters.
• Parameters changed with the inclusion of random
  change points
• Terminal slope is .85 T-units steeper.
• Change 1 unit on Likert scale every 7 years.
• Alternatively, .5 units for the average decline
  phase.
• Average change point is 1 year closer to death.

32
Terminal Slope Question
Terminal Decline and Ergodicity In Life
Satisfaction

• Why did the terminal slope decrease with the
  inclusion of random change points?
• Misspecification problem
• Would variation in change points lead to
  misspecification of the terminal slope?
• Ergodic problem
• What if some individuals are not undergoing
  linear-linear change?

33
Misspecification Problem
Terminal Decline and Ergodicity In Life
Satisfaction

• How does fixing the change point affect the
  terminal slopes?
• Misspecification of change points at the
  individual level should bias slope estimates.
• For individuals with change points closer to
  death than the population (fixed) change point
• Terminal slope will be biased in the direction of
  the pre-terminal slope.
• Pre-terminal slope estimates will be unaffected.
• For individuals with change points further from
  death than the population (fixed) change point
• Pre-terminal slope estimates will be biased in
  the direction of the terminal slope.
• Terminal slopes will be unaffected.
• Such bias will affected fixed parameters.

34
Terminal Decline and Ergodicity In Life
Satisfaction
Misspecification Problem
Intercept N(109.84, 9.25) Pre-Slope N(-1.019,
0.07) Post-Slope N(-5.035, 1.41) Change Point
N(-12.44, 8.70)
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Terminal Decline and Ergodicity In Life
Satisfaction
Misspecification Problem

• Fixed change point model yields biased estimate
  of pre-terminal slope, terminal slope and change
  point.
• Intercept unaffected.

true value outside of confidence interval
36
Misspecification Problem
Terminal Decline and Ergodicity In Life
Satisfaction

• Both pre-terminal and terminal slopes biased.
• As theorized, each biased in the direction of the
  other.
• This created a covariance between the slopes
• rmodel .395
• rdata -.112.
• Location of change point biased as well.
• Change point closer to center of time-series
  (t-15)
• True -12.43, Sampled -12.72.
• True -15.40, Sampled -15.37
• True -18.30,Sampled -17.62.
• Intercept unaffected.

37
Ergodic Problem
Terminal Decline and Ergodicity In Life
Satisfaction

• Assumptions of ergodicity are critical to any
  model, particularly mixed-effects models.
• This assumption stipulates that processes
  occurring at the population level are
  representative of individuals in that population.
• In our dataset, several types of ergodic problems
  may occur
• Classic case some individuals arent
  characterized by spline model.
• Competing risks problem some individuals would
  undergo terminal change, but dont get the chance
  to.
• Misspecification problem (fixed change point, for
  example).

38
Ergodic Problem
Terminal Decline and Ergodicity In Life
Satisfaction

• Evidence of ergodic problems in Life Satisfaction
  data
• µchange point -3.54
• s change point 4.90
• Some individuals have change points outside of
  their time series.
• Possible causes
• Some individuals are better characterized by
  linear change.
• Artifact created by distribution specified in
  prior.
• Carry out a final simulation, where the
  distribution of change points extends beyond the
  observed data.
• Most closely mirrors the existing data.
• Compare fixed and random change-point models.

39
Terminal Decline and Ergodicity In Life
Satisfaction
Ergodic Problem

• Fixed change point model yields biased estimates
  of terminal change and change point fixed
  effects.
• Intercept and pre-terminal slope unaffected.

true value outside of confidence interval
40
Ergodic Problem
Terminal Decline and Ergodicity In Life
Satisfaction

• As predicted, terminal slopes and change points
  are biased.
• Similarities between this simulation and observed
  data suggest problems found in analyses are, at
  least in part, artifactual.
• Not definitive proof that random model is
  completely unbiased.
• We should exercise caution in analyzing
  relationships between model parameters, either
  interrelations or relationships to covariates.
• This is not a solution to more traditional
  ergodic problems.
• Some individuals might be better characterized by
  continuous change.
• Some spline models may capitalize on residual
  structures.

41
Substantive Interpretations
Terminal Decline and Ergodicity In Life
Satisfaction

• Life satisfaction shows evidence of terminal
  decline.
• All population models fit significantly better
  over distance to death than over chronological
  age.
• Spline models, particularly mixed effect spline
  models, provided superior fit, suggesting
  pre-terminal and terminal decline phases.
• Terminal decline in life satisfaction shows
  individual differences.
• Intercepts, rates of change and locations of
  change point all show individual differences and
  covariances.
• Models that dont allow for individual
  differences, particularly in change point, show
  misfit and multiple patterns of bias.

42
Substantive Issues
Terminal Decline and Ergodicity In Life
Satisfaction

• Violations of ergodic assumption create problems
  for terminal decline research.
• Terminal decline faces additional issues from
  competing risks.
• Studies of terminal decline may face an difficult
  problem.
• Change point estimation will tend to be biased
  towards the center of data collection.
• There is a limit to the number of observations
  between change point and death.
• This effect is more pronounced if individuals
  with few occasions cannot be included.
• Increasing observations typically means fewer
  participants, which is problematic for an
  individually-varying phenomenon.
• Terminal change may take on a more complicated
  functional form that power and degrees of freedom
  issues may obscure.

43
Substantive Directions
Terminal Decline and Ergodicity In Life
Satisfaction

• Multivariate approaches to terminal decline
• Cognitive, emotion and personality.
• Cognitive-emotive links in terminal decline.
• If terminal decline occurs across psychological
  domains, it stands to reason they may occur
  together.
• Alternate ways of conceptualizing aging time.
• Both chronological age and distance-to-death are
  expressions of time in relation to an
  individually-varying event (birth, death).
• Refinement of model
• Still characterize change from one process to
  another.
• Create a model that functions conceptually in
  continuous time.

44
Methodological Directions
Terminal Decline and Ergodicity In Life
Satisfaction

• Refinement of the spline model to avoid the
  problems detailed here.
• Avoid biased parameter estimates.
• Particularly, get valid estimates of change point
  distribution.
• Create methods to distinguish between different
  types of change.
• Sensitivity to individuals yet to undergo
  terminal change.

45
Acknowledgements
Terminal Decline and Ergodicity In Life
Satisfaction

• Denis Gerstof, Penn State/Max Planck Institute
• Nilam Ram, Penn State/Max Planck Institute
• Jürgen Schupp, German Socio-Economic Panel Study
• Gert G. Wagner, German Socio-Economic Panel Study
• Ulman Lindenberger, Max Planck Institute
• Peggy Wang, University of Virginia
• Johnny Zhang, University of Virginia
• John Nesselroade, University of Virginia
• NIA T32 AG20500-01