Section 13a: Correlation Structures in a GEE model

This section contains descriptions of a few of the correlation structures 
available in PROC GENMOD.


Let R_i(%alpha) be a n_i*n_i working correlation matrix that is fully 
specified by the vector of parameters %alpha.

where A_i is an n_i*n_i diagonal matrix with v(%mu_ij) as the jth diagonal 
element.  If R_i(%alpha) is the true correlation matrix of Y, then V_i is 
the true covariance matrix of Y.

The working correlation matrix R_i(%alpha) is usually unknown and must be 
estimated within the iterative fitting process by taking the current value 
of the parameter vector %beta to compute appropriate functions of the 
Pearson residual: 

r_ij = (y_ij - %mu_ij)/ SQRT{v(%mu)}

There are several specific choices of the form of the working matrix 
R_i(alpha) to model the correlations of the individual responses. The 
following descriptions present a few of the common choices for R supported 
by GENMOD and the resulting covariances:



Independence - type=ind

R=R_o=I

If you specify the working correlation as R_o=I, the identity matrix for 
the independence model (correlations are assumed to be 0 for all pair-wise 
combinations of variables), the GEE reduces to the independence (GLM) 
estimating equation.  With the identity matrix, the impact of r_j,j' is 
negligible. Since all off-diagonal correlations are zero, a working 
correlation matrix is not estimated for this situation.

Given that a dataset consists of repeated measurements within individuals, 
the simplest possible correlation structure is to (usually incorrectly) 
assume independence. This assumption is equivalent to each observation 
collected from an individual is completely uncorrelated with every other 
observation measured in that individual; correlations are assumed to be 0 
for all pair-wise combinations of the within-subject variables. If rho_jk 
is the correlation between observations j and k, rho_jj=1 and rho_jk=0, j 
ne k.

    _                      _
   | sig^2   0   0   0   0   |
   |      sig^2  0   0   0   |
V= |          sig^2  0   0   |
   |  symm.       Sig^2  0   |
   |                  sig^2  |
    -                      -


Exchangeable - type=exch or Compound Symmetry - type=cs

Corr(Y_ij,Y_ik) = %alpha, j =/ k

Exchangeable assumes non-zero, yet uniform correlations for all pairs of 
within-subject variables.


Every observation within an individual is equally correlated with every 
other observation from that individual. . like the icc

    -                                                  -
   | sig1 + sig2                                        |
   |     sig1     sig1 + sig2                           |
V= |     sig1         sig1       sig1+sig2              |
   |     sig1         sig1           sig1    sig1+sig2  |
    _                                                  _

This choice of correlation structure may not be reasonable with multiple 
measurements collected over time, since the correlations most likely will 
diminish as the time lag between observations increases.

Exchangeable assumes that r12 = r13 = ... = rjj' which is analogous to 
applying the compound symmetry assumption of repeated measures with PROC 
GLM or PROC MIXED.

Auto-regressive - type=AR(1)

Autoregressive is a term derived from times series analysis that assumes 
observations are related to their own past values through one, two, or a 
higher order autoregressive (AR) process. An autoregressive correlation 
structure indicates that two observations taken close in time (or space) 
within an individual tend to be more highly correlated than two 
observations taken far apart in time from the same individual. Formally, 
rho_jj=1 and rho_jk(j ne k) decreases in value as the absolute difference 
between j and k gets larger. A first-order autoregressive correlation 
structure specifies that rho_jk = rho**|j-k| = where rho is the 
correlation when |j-k|=1.

Corr(Y_i,j,Y_i,j+t) = a_t = %alpha^t, t=1,2,...,n_i-j

%alpha is estimated by %alpha^hat =
    _                         _
   | 1                         |
   | %a_1  1                   |
V= | %a_2  %a_1  1             |
   | %a_3  %a_2  %a_1  1       |
   | %a_4  %a_3  %a_2  %a_1  1 |
    -                         -


m-dependent - MDEP(#)

Corr(Y_i,j,Y_i,j+t) = a_t = %alpha_t, t=1,2,...,m
                    = 0,              t > m
    _                     _
   | 1                     |
   | a_1  1                |
V= | a_2  a_1  1           |
   |  0   a_2  a_1  1      |
   |  0    0   a_2  a_1  1 |
    -                     -


Unstructured - type=UN

Corr(Y_ij,Y_ik) = a_ij = %alpha_jk

The unstructured (type=un) matrix estimates t*(t-1)/2 correlations from 
the data. Usually complete data are required for this approach.

Unstructured assumes unconstrained pair-wise correlations where each 
correlation is estimated from the data (the most complex model) and is 
applied to balanced datasets. No assumption is made about the relative 
magnitude of the correlation between any two pairs of observations. 
Formally, rho_jj=1 and rho_ij is free to take any value between -1 and +1.
    _                         _
   | 1                         |
   | a_21  1                   |
V= | a_31  a_32  1             |
   | a_41  a_42  a_43  1       |
   | a_51  a_52  a_53  a_54  1 |
    -                         -

User fixed

R=R_o=< specified matrix >

Let R_o be a user-specified correlation matrix where corr(y_ij,y_ij)=r_ij 
and r_jk is the jth and kth element of the matrix.  Since the correlations 
are specified constants, a working correlation is not estimated. 

All correlation coefficients are fixed by the user rather than being 
estimated from the data. Formally, rho_jj=1 and rho_jk can take any value 
between -1 and +1, but this value is fixed prior to the analysis rather 
than being estimated from the data.