Statistical Methods for the Analysis of Change

1
Statistical Methods for the Analysis of Change

• Administrative Issues
• Why study change?
• Overview of methodological issues
• Overview of statistical issues and methods

2
Administrative Issues

• Review of syllabus
• Final Project
• Weekly assignments
• Weekly lab session
• Grades (40 homework, 20 presentation, 40 paper)

3
Why study change?

• Description of phenomena in time
• Trajectories
• How do children learn words?
• Cycles
• Stress patterns over a week
• Historical record
• Numbers of shelter requests over days
• Forecasting and Prediction
• Social planning
• Interventions for college drinking
• Financial planning and investment
• Value of blue chips over holiday period

4
Why study change?(Continued)

• Modeling social and behavioral processes
• Behavioral phenomena are located in time
• Relations among multiple variables are often
  dynamic
• Systems view of behavior
• Making inferences about causality
• Causal relations are temporal
• Baseline allows efficient inference in
  experiments
• Baseline measure as covariate
• Adjusting for selection in observational studies
• Holding constant initial value before causal
  factor

5
Some examples of change studies

• Stress and coping example
• Other NYU Studies
• Adolph, Bates, Fuligni, Hughes, Ruble, Seidman,
  Shinn, Yoshikowa

6
Overview of methodological issues

• History suggests that studying change is
  difficult
• Cronbach and Furby (1970) How should we measure
  change or should we?
• Problems are often associated with panel designs
• Measurements taken 2 or 3 times
• Y1, Y2, and Y3
• Spacing often set arbitrarily (month, year)

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Problems that have been noted

• Difference scores
• DY2 Y1
• Advantages
• Easy to compute
• Easy to interpret
• Problems
• Spacing of observations may not match effect
• Missing values missing either time makes D
  missing
• D is usually correlated (negatively) with
  starting point
• D is doubly affected by unreliability

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Reliability of Difference Scores

• Suppose Y1 T e1, and Y2 T d e2.
• T is the stable "true" score of the subject at
  time 1
• d is the "true" change from time 1 to time 2
• e1 and e2 are random error terms
• D d e2 - e1
• While Var(Y2) V(T) V(d) V(e2)Var(D)
  V(d) V(e2) V(e1)

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Review of problems

• Regression methods
• Y2 b0 b1Y1 b2X e
• Issues
• Spacing of observations
• Missing values
• Regression artifacts (Reliability of Y1)
• Trajectory methods
• Issues
• Specification of trajectory form
• Missing values

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Overview of methodological issues

• Other issues
• Trade off within subject n and between subject n
• Effects of taking repeated measurements
• Thinking about variation
• Interindividual differences in level
• Interindividual differences in trajectory
• Intraindividual changes that are systematic
• Intraindividual changes that are error

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Overview of statistical issues

• Non-independence of observations over subjects
• Y1, Y2, Y3, Y4 for John are likely to be more
  similar than Y1, Y2, Y3, Y4 for Mary
• Non-independence of observations in the temporal
  sequence
• Y1 Y2 will be more similar to each other than
  either is to Y4
• Traditional statistical methods assume
  independence

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Overview of statistical issues

• Categorical, ordinal, continuous vs normal
  response variables
• Many psychological variables are made up of
  individual counts
• Did I have a headache today?
• Did Jerry answer the first question correctly
• Statistical models for counts, and for dependence
  among variables are quite different than those
  for normally distributed process variables.

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Overview of statistical issues

• Missing data
• Can observations over time be imputed or modeled?
• Can patterns of dependence be used in imputation?
• How is varying amount of information taken into
  account in statistical tests?

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Overview of Methods to be discussed

• ANOVA, MANOVA and difference score analysis
• Regression based panel analyses
• Structural equation methods
• Random regression methods
• Latent growth curve models
• Generalized linear models
• Some special methods for binary outcomes
• Simple survival analysis

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Missing in our discussion

• Time series analyses (Box and Jenkins, ARIMA
  methods)
• Markov transition models