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Lecture 6 (chapter 5) Revised on 2/22/2008

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Lecture 6 (chapter 5) Revised on 2/22/2008 Parametric Models for Covariance Structure Parametric Models for Covariance Structure Parametric Models for Covariance ... – PowerPoint PPT presentation

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Title: Lecture 6 (chapter 5) Revised on 2/22/2008


1
Lecture 6(chapter 5)Revised on 2/22/2008
2
Parametric Models for Covariance Structure
We consider the General Linear Model for
correlated data, but assume that the covariance
structure of the sequence of measurements on each
unit is to be specified by the values of unknown
parameters.
3
Parametric Models for Covariance Structure
4
Parametric Models for Covariance Matrices
Model for the mean
Model for the covariance matrix
5
Parametric Models for Covariance Matrices
We now consider more general models for the
covariance matrix which can be specified
by looking at the empirical variogram.
6
Interpretation of the Variogram
7
Interpretation of the Variogram
Variance of the random effects
corr
Serial correlation
Measurement error
8
Interpretation of the Variogram
and correlation function
or
with variance
with variance
Variogram
Total variance
9
Variogram for a model with Random Intercept plus
serial correlation plus measurement error
variance of the random effect
variance of the serial process
measurement error variance
10
Cow Data
19 weeks of measurement, 3 diets barley, mixed,
lupins (barley data shown above)
11
Example Protein Content of Milk
The roof of the variogram is placed at 0.087
12
Example Protein Content of Milk (contd)
the variogram does not reach the total variance
the variogram does not start at zero
13
Example Protein Content of Milk (contd)
Variogram of protrs (17 percent of v_ijs
excluded)
variance of the random intercept
Measurement error
14
Example Protein Content of Milk (contd)
15
Parametric Models for Covariance Structure
We will cover this in the multi-level course
16
Models
  • To develop a model, we need to understand the
    sources of variation
  • Random Effects (Intercept) random variation
    between units
  • Serial Correlation time-varying random process
    within a unit
  • Measurement Error measurement process introduces
    a component of random variation

17
How to incorporate these qualitative features
into specific models
18
Pure Serial Correlation
Exponential correlation model for equally spaced
measurements
19
Autoregressive Model of Order 1
Another way to build a serial correlation model
is to assume an explicit dependence of the on
its predecessors The simplest example is a AR(1)
where
xtgee corr(AR1)
  • AR(1) models are appealing for equally spaced
    data, less so for unequally spaced.
  • For example, it would be hard to interpret
    if the measurements were not equally spaced in
    time

20
Exponential Correlation Model for Unequally
Spaced data
  • The exponential correlation model can handle
    unequally spaced data. We can assume
  • Then

allow a different coefficient for each time
difference
prais command in stata
21
Model-Fitting
  • Formulation choosing the general form of the
    model
  • Mean
  • Association
  • Estimation fitting the model
  • Weighted least squares for
  • ML for covariance parameters or subset
  • Iterate (1) and (2) to convergence

22
Model-Fitting (contd)
  1. Diagnostic checking that the model fits the
    data by examining residuals for lack of fit,
    correlation
  2. Inference calculating confidence intervals or
    testing hypotheses about parameters of interest

23
Step 1. Formulation
  • Formulation of the model is a continuation of
    exploratory data analysis. Focus on the mean and
    covariance structures
  • Look at the residuals
  • Create time plots, scatterplot matrices and
    empirical variograms
  • Do you have stationarity in the residuals? If
    not, you need to transform the data or use
    inherently non-stationary models as a model with
    a random intercept and random slope
  • Once stationarity has been achieved, use the
    empirical variogram to estimate the underlying
    covariance structure

24
Step 2. Estimation
25
Step 4. Diagnostics
  • The aim is to compare the data with the fitted
    model. How?
  • Super-impose the fitted mean response profiles on
    a time plot of the average observed responses
    within each combination of treatment and times.
  • Super-impose the fitted variogram on a plot of
    the empirical variogram.

26
Examples and SummaryNepal Dataset
This dataset contains anthropologic measurements
on Nepalese children. The study design called
for collecting measurements on 2258 kids at 5
time points, spaced approximately 4 months
apart Q Estimate the association between arm
circumference and childs weight, adjusted by
age and gender, and accounting for the
correlation in the repeated measures within the
same child
27
Examples and SummaryNepal Dataset
Time varying confounder
Goal estimate the average change in arm
circumference for one unit change In weight
accounting for the potential confounding effect
of the time varying covariate age and for sex.
The standard errors of the estimated association
Between arm circumference and weight are
estimated accounting for the correlation of the
repeated measures within a child,
28
Examples and SummaryNepal Dataset
Models for the covariance matrix
(measurement error only)
29
Examples and SummaryNepal Dataset
(More) Models for the covariance matrix
measurement error
measurement error
30
Independence Model (measurement error)
0
31
Independence Model (contd)
32
Uniform Model(also, Exchangeable or Compound
Symmetry)
A better model (than the Independence Model) is
to assume same correlation for all pairs of
observations This is called the uniform,
exchangeable, or compound symmetry correlation
model.
33
Uniform (Random Intercept)plus measurement error
34
Uniform Model(also, Exchangeable or Compound
Symmetry) (contd)
35
Uniform Model(also, Exchangeable or Compound
Symmetry) (contd)
36
Uniform Model
37
Uniform Model(also, Exchangeable or Compound
Symmetry) (contd)
38
Exponential Correlation Model
A different model is to assume that the
correlation of observations closer together in
time is larger than that of observations farther
apart in time. One model for this is the
exponential model
39
Exponential Correlation Model
Within subject correlation at lag 1 (4 months
separation approx)
40
Exponential Correlation Model (contd)
41
Exchangeable Exponential Model
42
Random Intercept Serial Correlation
43
Summary of Example
  • We see that the exchangeable correlation
    (0.691) is similar to the model without the
    exponential correlation (0.748), and the
    exponential correlation (0.185) is now much
    smaller.
  • The regression parameter estimates are also
    similar to the exchangeable case.
  • This suggests that the exchangeable correlation
    model may be capturing the main correlation
    pattern.

44
Summary
  • Modelling the correlation in longitudinal data is
    important to be able to obtain correct inferences
    on regression coefficients. This leads to
  • Statistical Efficiency
  • Correct Standard Errors
  • These are marginal models because the
    interpretation of the regression coefficients is
    the same as that in cross-sectional data.
  • Exchangeable correlation model subject-specific
    formulation
  • Exponential correlation model transition model
    formulation

45
Summary (contd)
  • Three basic elements of correlation structure
  • Random effects
  • Auto-correlation or serial dependence
  • Observation-level noise or measurement error

46
Evaluating Covariance Models
  • Once you have chosen a (set of) covariance
    model(s), how do you evaluate whether it fits the
    data well? Or, how do you compare several of
    them?
  • Several tools (each work with either ML or ReML)
  • Likelihood Ratio Tests (LRTs) for comparing
    nested models
  • Akaikies Information Criterion (AIC)
  • QIC to compare models fitted with GEE
  • Examining fitted model variograms

47
Comparing Covariance Models with Akaikes
Information Criterion (AIC)
48
Comparing Covariance Models with Akaikes
Information Criterion (AIC) (contd)
49
Comparing Covariance Models with Akaikes
Information Criterion (AIC) (contd)
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