# R structures

## Dimensions

In a traditional analysis, the residuals are IID (Independently and
Identically Distributed) in which case the order does not matter.
However, ASReml is designed to handle cases where residuals are
correlated. Sometimes, these correlations can be defined as a direct
product of variance structures, assuming the residuals are in
an appropriate order.
#### IID residuals

The variance structure can be written as

0 0 g

or

1 1 g
n 0 ID

where g is the
number of G structures
and n
is the number of data points. Where there is just 1 section
(as in this example), n can be specified as 0
and ASReml will use the number of observations in the data.
ID
is the
VCODE
for an Identity matrix.
This analysis will estimate any variance components as ratios
relative to the implicit residual variance.
#### Typical field trial - regular rectangular grid

The variance structure can be written as

1 2 g # 1 section, 2 way layout, g G structures
row row AR1 .1
column column AR1 .1

This particular coding assumes there is a factor in the data called
row
which indexes the rows of the field layout, and another factor
column
which indexes the columns of the field layout, and that
that there are data records for all cells in the complete
row x column grid. (It will be necessary to estimate any
missing values
in the data.)

AR1
is the
VCODE
for a first order Auroregressive matrix. It requires
one parameter (correlation) given an initial value of 0.1.
If the data file does not contain row and column information,
but a correlated structure is required, the data records need
to be sorted on row and column (even though the coding is not in the file)
and the structures (assuming 12 rows and 36 columns) would be written as

1 2 g # 1 section, 2 way layout, g G structures
12 0 AR1 .1
36 0 AR1 .1

So, the first field is the number of rows/columns.
The second field controls sorting the plots (no sorting now
because the plots are in order).
#### Spatial data - irregular pattern

The variance structure can be written as

1 1 g
n xy MAT1 10

Again,
n
can be written as 0 and ASReml will use the number of observations
in the data.

xy
is a special numeric code pointing to fields in the data locating
the data points. For example if field 3 defines the X-axis (say
Easting
) and field 4 defines the Y-axis (say
Northing
) then
xy
would be written as 304.

MAT1
is the
VCODE
for a MATERN correlation structure with 1 parameter estimated.
It requires
one parameter (range) given an initial value of 10..
#### Multivariate data

The R structure (assuming 3 traits) can be written as

1 2 g
0 # n 0 ID
Trait 0 US
6*0

The data is sorted traits within records so the structure
for traits is specified second.
Again we let ASReml supply the number of records.

US
is the
VCODE
for an unstructed variance matrix.
It requires six initial values: 3 variances and
3 covariances in the order V11 C21 V22 C31 C32 V22 (lower triangle
rowwise). However, it is generally difficult to guess suitable
values so we have supplied initial values of zeros (6*0 is six zeros)
and ASReml will obtain initial values as a proportion
of the simple variances and covariances of the residual.
## See Also