### Symmetry-based Test Hypothesis`s

```presented by:
Shaun Deaton
The idea is to hypothesize constraints on the interchangeability of N
normally distributed random variables. Then test the hypothesis by
using the likelihood ratio of the determinants of the covariance
matrices. The symmetry constraints impose structure upon the vector
of means and the covariance matrix.
Depending on the experimental setup, the measurement variables
can be grouped together in various ways; forming many different
possible partitions into non-overlapping subsets.
In this construction there are two main types of interchanging that
can happen:
• Permuting variables in the same subgroup.
• Permuting variables between distinct subgroups.
E.g. Let there be three tests given to measure a subject’s knowledge
or skills; assume these are all normally distributed when taken
over a large sample of subjects. Under one possible interchangeability hypothesis, it is assumed that the means are the same,
variances are same, and co-variances are all the same.

X
 ( X 1 , X 2 , X 3) , be the random vector of scores. Then the mean
Let
vector & covariance matrix have the following form:


 ( ,  ,  )
A

mv   B

B
B B 
A B

B A
Here the ‘mv’ notation is means, variances, & co-variances
are all respectively equal as assumed above.
For the experimental situations considered in the references (1), there is
usually an “outside criterion measure”, which could be a specialized
reference measurement. This new random variable, the fourth in this
example, is assumed to also be normal and have the following form
for the mean vector and covariance matrix:

X
 ( X 4 , X 1 , X 2 , X 3) to keep
Have
with standard matrix pattern.


 ( ,  ,  ,  )
4
1
1
1
D

C

 mvc 
C
 C
C
A
B
B
C
B
A
B
C 
B

B
A 
Whenever a model has this structure it is said to have
Compound Symmetry of Type 1.
Now the ‘c’ in ‘mvc’ says that all co-variances between subsets are equal.
Concrete Example from reference (2): The sample vectors in this case
are 8-D and 
of the form:
X  (v , q |m1 , s1 , ss1:m2 , s2 , ss2)  ( x1| x2 : x3)
, where v=verbal, q=quantitative, m=math, s=science, & ss=social studies.
This case considers only two successive measurements on m, s, & ss,
generally may have several measurements.


  
 ( |  :  )
1
2
2
D

 mvc   C

C
C
A
B

C
B

A
Here the elements are matrices: D is 2x2 symmetric, C is 2x3, A is 3x3
symmetric, & B is 3x3 symmetric.
This is known as Block Compound Symmetry of Type 1 (BCS-1).
Concrete Example from reference (1): Consider a collection of animals from
which the %CO2 and %O2 in their blood will be measured twice each day
of the experiment. Assuming these four variables are normally distributed the
symmetry model of the preceding example takes the following form:

X
 (UT1 ,UT 2 ,WT 1 ,WT 2)
Where UT1 and UT2 are the first & second %CO2 measurements, while
WT1 and WT2 are the first & second %O2 measurements on the same
animal.
Now, the hypothesis to be tested is given by:


 ( ,  ,  ,  )
1
1
2
2
E

F

 mvc 
K
 L
F K L
E L K
L G J
K J G






Whenever the means, variances, and co-variances, both between and
Among the subsets of partitioned variables have a structure like that in
the last example, the model is said to have Compound Symmetry of
Type 2. Just as for the Type 1 case, it can be extended to the Block
Compound Symmetry of Type 2 (BCS-2) by considering partitions of vectors
into subsets of equal size.
As a further example, and note of caution, consider four r-D normally
 ( 1 , 2 | 3 , 4) , then
distributed vectors partitioned as follows;
under the Block Compound Symmetry of type 2 condition, the new vector
 ( 2 , 1 | 4 , 3) , obtained by permutation, has the same

x

y
   
x x x x
   
x x x x






distribution
the distributions of s  ( x ,x | x , x ) &
  (  , as| x,. But,
t x x x x ) need not be the same as that of x , or each other.
1
2
1
3
4
2
4
3
Votaw considers a further restriction on the covariance matrix, by
assuming that the off-diagonal co-variances where equal, as well as
the on-diagonal entries as before. Obtaining twice the number of
hypotheses to test. Have general form of these matrices in reference (1).
The distribution is said to have Complete Symmetry, if whenever some
partition of the variates satisfies H1(***) it satisfies H1(***) , and
vice-a-versa; the ‘*’ are wildcards. When just one of these hypotheses is
satisfied the distribution is called Compound Symmetric. The bar notation
is taken in the sense of Votaw, denoting the off-diagonal covariance
condition.
Another Example:
In reference (3), there is a example where some number k, of
seismographs, are placed symmetrically around a mountain (or volcano).
Then it would be possible to test for circular symmetry in the combined
data from all k instruments. …
References:
1 Testing Compound Symmetry in a Normal Multivariate Distribution;
by David F. Votaw jr.
2 Testing and Estimation in the Block Compound Symmetry Problem;
by Ted H. Szatrowski.
3 Comment: Group Symmetry Covariance Models;
by Michael D. Perlman.
```