===== Fitting a factor-analytic model =====

<WRAP column 30% right :en>
<code | fa_example.par>
   COM  Example par file for FA model
   ANAL MUV  PC 6
   PEDS ../carcped.dat
   DATA ../carcsix.dat GRP
     TRNOS 1 2 3 4 5 6
     traitno 6
     animal 0
     bnimal 0
     cg     299
     NAMES  cwt  cema  cimf  crby  cp8  crib
   END
   MODEL 
     FIX cg 
     RAN animal  nrm 
     RAN bnimal  nrm 
     tr cwt 1
     tr cema 2
     tr cimf 3
     tr crby 4
     tr cp8  5
     tr crib 6
   END MOD
   VAR animal 6 1
      177  29 3.1 -11 15 8  
       9.8 0.03 -0.2 1.2 1.6 
       0.91 -0.76 0.9 0.3
       3.62 -2.9 -1.96
       4.95 2.2
       1.95
   VAR bnimal 6 6 DIAG
       50  3  0.3  0.5  1.1  0.8
   VAR residual 6 6
       292  39  3.2  -5.9  22.4  11.      
       36   0.8  -0.9  4.7 1.2      
       1  -0.06  0.15  0.32      
       0.73  -0.55 -0.45      
       6.9  2.0      
       5.1
</code>
</WRAP>

WOMBAT can accommodate a factor-analytic model for the structure of covariance matrices of random effects. While there is no explicit option for this structure, it is easily fitted indirectly.

Factor analysis implies a latent model which models a random effect (${\bf r}$)  for a set of <latex>q</latex> traits as the sum of a vector of $m$ common effects (${\bf c}$), weighted by a matrix of factor loadings (${\bf F}$), and a vector of $q$ specific effects (${\bf s}$). i.e. 
$${\bf r} = {\bf F} {\bf c} + {\bf s} $$
The common effects are assumed to be //iid// distributed with 
variances of unity, while the specific effects are assumed to be uncorrelated but have heterogeneous variances. This gives 
$$Var({\bf r}) = {\bf F} {\bf F}' + {\bf D}$$
with ${\bf D}$ the the diagonal matrix of specific variances.

WOMBAT allows for the first part, i.e. has an option which allows $Var({\bf r})$ to be estimated as ${ \bf F F}'$ with reduced rank $m$. In addition, WOMBAT incorporates an option for diagonal covariance matrices. \\ Hence a factor-analytic model is readily fitted by simply fitting the corresponding random effect __twice__: 
   * In the first instance with reduced rank covariance matrix, representing the **common** factors \\ (''animal'' in the example), and 
   * in the second instance with diagonal covariance matrix, representing the **specific** effects \\ (''bnimal'' in the example).

WOMBAT only allows a particular column in the data file to be used once, i.e. to be mapped to a single effect in the model of analysis. Hence, fitting a factor-analytic structure for a particular effect requires the corresponding code (column) in the data file to be replicated!

If WOMBAT encounters two random effects, listed successively in the .par file, with the names only differing by the first letter (e.g. ''animal'' and ''bnimal'' in Example 5) and with reduced rank and diagonal covariance matrices, respectively, it will assume that a factor-analytic model has been intended and automatically write out summary statistics for the sum of the two matrices.