Modelling non-independent random effects in multilevel models建模的非独立随机效应的多层模型-PPT课件-

Fixed
Random level 2 level 1 residual
The
u j , e ij
are assumed independent.
But this is sometimes unrealistic:
• Repeated measures growth models with closely spaced occasions
0 j 0 u0 j 1 j 1 u1 j
A model for (non-independent) level 1 residuals might be written:
2 Cov( e , e ) f ( s ) ij i s , j e
s f( s ) e
Link functions
Link function f(s). From left to right; hyperbolic, logit, log
Parameters and estimation
We need to estimate the parameters of the correlation function, the variances and the fixed effects. We propose an MCMC algorithm and have programmed this for general 2- level models where correlations can exist at either or both levels and responses can be normal or binary. Steps are a mixture of Gibbs and MH sampling with adaptive proposal distributions and suitable diffuse priors
1
0
Leading to an exponential decay function. (Goldstein and Healy 1994)
Schools in competition
2 2 yXu ( ) e , e ~ N ( 0 , ) , u ~ N ( 0 , ) i j i j j i j i j e j u
f ( t t ) , t t 1 2 1 2
f e t t
1 2
t t 1 2
In discrete time (equal intervals) this is a standard first order autoregressive model We fit a 4-th degree polynomial with and without random linear coefficient
Work using the ALSPAC cohort is currently underway.
Other link functions
Logit:
f f e / ( e 1 ) j k
j k j k
பைடு நூலகம்
Log:
jk e
f jk
These have the following forms
• Schools competing for resources in a ‘zero-sum’ environment
Repeated measures growth curves
A simple model of linear growth with random slopes:
y ij 0 j 1 j t ij e ij
Modelling non-independent random effects in multilevel models
William Browne Harvey Goldstein University of Bristol
A standard multilevel (VC) model
2 2 yXu ( ) e , e ~ N ( 0 , ) , u ~ N ( 0 , ) i j i j j i j i j e j u
If we can specify a suitable (set of ) distance functions then we can estimate the relevant parameters.
One possibility is to use the extent of overlap between appropriately defined catchment areas.



D e fin e th eh y b e rb o lic lin kfu n c tio n
1 f ( j1 j2) 1| z 1 z 2|
j j (e
12
f j1j2
1 )/(e j1j2 1 )
f
where this correlation is inversely proportional to the (resource) distance between the schools | z1 z 2 | .
Example 1: Growth data
The data are 9 measurements on 20 boys around age 13, approximately 3 months apart Fitting a 2-level model with random linear and quadratic coefficients does not remove residual autocorrelation among level 1 residuals. We model the correlation as a negative exponentially decreasing function of the time difference We use a log (exponential) link since correlations should be positive