Mixed Effects Models

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Mixed Effects Models

• Danielle J. Harvey
• UC Davis

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Spaghetti plots
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Outline

• Description of dataset
• Notation
• General Model Formulation
• Random effects
• Assumptions
• Example
• Interpretation of coefficients
• Model diagnostics

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Data

• Outcomes
• Memory composite score
• Executive function composite score
• 314 subjects with more than 1 assessment
• 191 normals, 67 MCI, 56 AD
• Number of assessments per person
• Average of 4 assessments, SD1.6
• Range 2-10

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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 (like
  diagnostic group 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
• Like 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 effect parameters
• ? are the errors

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Memory Function
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Executive Function
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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

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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
• Autoregressive
• Unstructured

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Assumptions of Model

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

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Model Building

• Step 1 start with simple model (time one
  independent variable)
• Step 2 fit model with all significant variables
  from Step 1
• Step 3 look at interactions with time for each
  independent variable in model from Step 2
• Check assumptions of model after fitting each
  model (multiple times in Steps 1 and 3)

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Interpretation of coefficients

• 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 change varies by one unit change in an
  independent variable

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Graphical Tools for Checking Assumptions

• Scatter plot
• Plot one variable against another one
• 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 our dataset
• Interested in differences in change between
  diagnostic groups
• Outcomes memory and executive function
• X includes diagnostic group (control reference
  group) and time
• Incorporate a random intercept and slope

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Memory function

• Log-transformed memory to better meet assumptions
  of the model
• Based on 305 subjects with at least 2 memory
  assessments
• Diagnostic groups differed in terms of their
  baseline level of memory
• Groups also showed different rates of change over
  time

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Model results (baseline)
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Model Results (change)
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Graphical illustration
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Executive Function

• Based on 303 individuals with at least 2
  executive measures
• Baseline executive function varied by diagnostic
  group
• Change over time also differed by diagnostic group

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Model Results (baseline)
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Model results (change)
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Graphical Illustration
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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