4
Multilevel regression
Marginal models
Regression equation
Saturated model?
Transform categorical predictors into dummy variables
include every interaction
Parts
Fixed
Random
Error
Same for all participants
Individual for participants
Described in covariance matrix
V-matrix
R matrix
G matrix
Group
Residual
Covariance structures
Unstructured
Most general
summarizes all (co)variance
Determines regression coefficient b
In population all (co)variances are different
depend on scale of dependent var.
Make correlation
devide by SDx*SDy
Correlation metric
Goodness of fit
can be modelled by -2 log likelihood
Also known as deviance
how strongly sample deviates from expectation( that all variances are equal?)
The smaller the better
Compound symmetry
Resembles sphericity
variance equal at each timepoint
correlation between all timepoints are same
R matrix
only 2 values
average variance
average covariance
R matrix
Are they sig. different?
Likelihood ratio test
Calculate chi-square
difference between deviances
Df
how many parameters do they estimate
Per parameter difference +1 Df
Look up p-value in book
P. 509
Compound symmetry heterogeneous
Softer version of compound symmetry
R matrix
covariance matrix
Different when SDs differ
Correlations are equal
Toeplitz
Equal variances, different correlations
Different (co)variance, equal correlations
adjacent time points have same correlation
Also exists with unequal variances (heterogeneous)
AR 1
First-order autoregressive
Correlations decrease as timepoints are further apart
Variances are equal
corr 1-3=corr1-2^2
Equally parsimoneous as CS
AR heterogeneous
first-order autoregressive heterogeneous
Multiple variances
equal correlation
Other structures
MA1
first-order moving average
no pattern in the structure
ARMA1,1
Combines autoregressive pattern with moving average
AR1+ME
fist order autoregressive plus measurement error
Score combines true score with random measurement error
Errors depend on reliability
ID
Scaled Identity
All timepoints have same var
No correlation
Same as linear regression
Useful in module 5
with random intercept/slope
Fixed effects
Spss
should be univariate
Model dimension
overview of regression model
Maximum likelihood
underestimates covariance
Look at Restricted maximul likelihood (REML)
combination of multiple regression and repeated measures ANOVA
nested data structure
Comparing Likelihood ratio
Only if model is special case of the other
Not in marginal models
Good when big sample few measures
Variances +covariances
Choosing a model
Identifiability
nr of unknown parameters </= nr of equations
Equations
fixed(beta) vs mean
covariance vs residual variances
For fixed and random part
Estimation & testing
ML
REML
restricted maximum likelihood
Maximum likelihood
Steps
- General fixed and random
- reduce random
- Reduce fixed