Longitudinal Data

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Longitudinal Data Mixed Effects Models

• Danielle J. Harvey
• UC Davis

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Disclaimer

• Funding for this conference was made possible, in
  part by Grant R13 AG030995 from the National
  Institute on Aging.
• The views expressed do not necessarily reflect
  the official policies of the Department of Health
  and Human Services nor does mention by trade
  names, commercial practices, or organizations
  imply endorsement by the U.S. Government.
• Dr. Harvey has no conflicts of interest to report.

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Outline

• Intro to longitudinal data
• Notation
• General model formulation
• Random effects
• Assumptions
• Example
• Interpretation of coefficients
• Model diagnostics

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Longitudinal data features

• Three or more waves of data on each unit/person
  (some with two waves okay).
• Outcome values.
• Preferably continuous (although categorical
  outcomes are possible)
• Systematically change over time
• Metric, validity and precision of the outcome
  must be preserved across time.
• Sensible metric for clocking time.
• Automobile study months since purchase, miles,
  or number of oil changes?

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Data Format

• Person-level (multivariate or wide format)
• One line/record for each person which contains
  the data for all assessments
• Person-period (univariate or long format)
• One line/record for each assessment
• Person-period data format is usually preferable
• Contains time and predictors at each occasion
• More efficient format for unbalanced data.

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Exploring longitudinal data

• Empirical growth plots.
• If too many, select a random sample.
• Reveal how each person changes over time.
• Smoothing techniques for trends
• Nonparametric moving averages, splines, lowess
  and kernel smoothers.
• Examine intra- and inter-individual differences
  in the outcome.
• Gather ideas about functional form of change.

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Spaghetti plots
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Exploring longitudinal data (cont)

• More formally use OLS regression methods.
• Estimate within-person regressions.
• Record summary statistics (OLS parameter
  estimates, their standard errors, R2).
• Evaluate the fit for each person.
• Examine summary statistics across individuals
  (obtain their sample means and variances).
• Known biases sample variance of estimated slopes
  gt population variance in the rate of change.

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Exploring longitudinal data (cont)

• To explore effects of categorical predictors
• Group individual plots.
• Examine smoothed individual growth trajectories
  for groups.
• Examine relationship between OLS parameter
  estimates and categorical predictors.

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Selected References

• Singer, J. D., Willet, J. B. (2003) Applied
  Longitudinal Data Analysis, Oxford University
  Press.
• Diggle,P. J., Heagerty, P., Liang, Kung-Yee,
  Zeger, S. L. (2002). Analysis of Longitudinal
  Data, Oxford University Press.
• Weiss, R. (2005) Modeling Longitudinal Data,
  Springer.

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Random Effects Models - Notation

• Let Yij outcome for ith person at the jth time
  point
• Let Y be a vector of all outcomes for all
  subjects
• X is a matrix of independent variables (such as
  baseline diagnosis and time)
• Z is a matrix associated with random effects
  (typically includes a column of 1s and time)

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Mixed Model Formulation

• Y X? Z? ?
• ? are the fixed effect parameters
• Similar to the coefficients in a regression model
• Coefficients tell us how variables are related to
  baseline level and change over time in the
  outcome
• ? are the random effects, ?N(0,?)
• ? are the errors, ?N(0,?2)

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Episodic Memory
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Working Memory
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Random Effects

• Why use them?
• Not everybody responds the same way (even people
  with similar demographic and clinical information
  respond differently)
• Want to allow for random differences in baseline
  level and rate of change that remain unexplained
  by the covariates

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Random Effects Cont.

• Way to think about them
• Two bins with numbers in them
• Every person draws a number from each bin and
  carries those numbers with them
• Predicted baseline level and change based on
  fixed effects adjusted according to a persons
  random number

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Random Effects Cont.

• Accounts for correlation in observations
• Correlation structures
• Compound symmetry (common within-individual
  correlation)
• Autoregressive - AR(1) (each assessment most
  strongly correlated with previous one)
• Unstructured (most flexible)

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Assumptions of Model

• Linearity
• Homoscedasticity (constant variance)
• Errors are normally distributed
• Random effects are normally distributed
• Typically assume MAR

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Interpretation of parameter estimates

• Main effects
• Continuous variable average association of one
  unit change in the independent variable with the
  baseline level of the outcome
• Categorical variable how baseline level of
  outcome compares to reference category
• Time
• Average annual change in the outcome for
  reference individual
• Interactions with time
• How annual change varies by one unit change in an
  independent variable
• Covariance parameters

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Graphical Tools for Checking Assumptions

• Scatter plot
• Plot one variable against another one (such as
  random slope vs. random intercept)
• E.g. Residual plot
• Scatter plot of residuals vs. fitted values or a
  particular independent variable
• Quantile-Quantile plot (QQ plot)
• Plots quantiles of the data against quantiles
  from a specific distribution (normal distribution
  for us)

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Residual Plot

• Ideal Residual Plot
• - cloud of points
• - no pattern
• - evenly distributed about zero

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Non-linear relationship

• Residual plot shows a non-linear pattern (in this
  case, a quadratic pattern)
• Best to determine which independent variable has
  this relationship then include the square of that
  variable into the model

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Non-constant variance

• Residual plot exhibits a funnel-like pattern
• Residuals are further from the zero line as you
  move along the fitted values
• Typically suggests transforming the outcome
  variable (ln transform is most common)

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QQ-Plot
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Scatter plot of random effects
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Example

• Back to some data
• Interested in differences in change between
  diagnostic groups
• Outcomes episodic memory and working memory
• X includes diagnostic group (control reference
  group) and time
• Incorporate a random intercept and slope, with
  unstructured covariance (allows for correlation
  between the random effects)

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Episodic Memory Model Results (baseline)
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Model Results (change)
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Advanced topics

• Time-varying covariates
• Simultaneous growth models (modeling two types of
  longitudinal outcomes together)
• Allows you to directly compare associations of
  specific independent variables with the different
  outcomes
• Allows you to estimate the correlation between
  change in the two processes