Coloring R.A. Fisher

Contemporary statistics owns several chapters of its history to Sir Ronald A. Fisher. In one of his papers published in 1942 in the Annals of Eugenics, Fisher discusses the concept of confounding in factorial experiments making use of  a very useful group of symmetries, illustrated in this page to highlight again the notion of experimental results indexed by a symmetry orbit. At a later page we will return to discuss Fisher's ideas.  

Suppose we have three attributes to experiment with by replacing some or all of them into an initial composition. Let's say that these attributes are represented, or labelled,  by the primary colors R (Red), G (Green), and B (Blue).  Here

are the resulting (additive) color labels if we started with

(R,G,B) = (0,0,0)

indicating no red, green, or blue, and ended up with

(R,G,B) = (1,1,1)

mixing all three colors in the coloring of the squares, following the sequence

(0,0,0), (1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,0,1), (0,1,1), (1,1,1).

In the sequence, between Black and White,  we obtained, respectively, the colors

 Red, Blue, Green, Yellow, Magenta, Cyan.

In Fisher's theory, these experimental conditions would have been represented in terms of the set

S = {{ }, {R}, {G}, {B}, {R,G}, {R,B}, {G,B}, {R,G,B}},

of all subsets of  {R,G,B}. Fisher also observed that the subsets have a  composition rule defined by including the elements that are not common and excluding the elements that are common, so that, for example,

{R}+{G} = {R,G},  {R}+{R,G,B} = {G,B}.   

The set S together with that composition rule form a group, so that the collection of experimental conditions listed above is then a symmetry orbit of S.   Here are all orbit representations, relative to the initial control condition, illustrated by the color of the square at the leftmost position:

This is also a picture of the multiplication table of S.

The outstanding question, common to all the previous pages in this blog, is that of determining the summaries of the experimental results obtained along a symmetry orbit in a way that they do not depend on the initial condition. Shortly, the question is: What are the symmetry orbit invariants? 

The operation in S is clearly commutative: that is X+Y=Y+X for all X and Y in S. Note the symmetric pattern of the multiplication table shown above. We also say in this case that S is an Abelian group. Also note that every element X in S is an involution, that is: 

 X+X= { }.

These two facts greatly facilitate the determination of the orbit invariants: Because the group is finite and commutative we (will) know that there are as many invariants as the number of elements in the group. The involutions factor S into the product FxFxF of three copies of an additive group F = {0,1} in which 1 is an involution.  

The factorization becomes evident as we rewrite Fisher's formulation of the group operation in terms of 

{ } = (0,0,0), {R} = (1,0,0), {G} = (0,1,0), {B} = (0,0,1),

 {R,G} = (1,1,0), {R,B} = (1,0,1), {G,B} = (0,1,1), {R,G,B} = (1,1,1),

and observe that the original operation in S corresponds to adding any two of its elements using the operation rules from F jointly in each component. As illustrated above,

{R}+{R,G,B} = (1,0,0)+ (1,1,1) = (0,1,1) = {G,B}.

The orbit invariants will result from all (eight) possible product of three copies of the two elementary invariants for F, indicated by u and s,  and given by 

u(0) = 1, u(1) = 1        and     s(0) = 1, s(1) = -1.

The eight invariants are then described by the products

uuu, suu, usu, uus, ssu, sus, uss, sss.  

For example, the evaluation of the product ssu gives:

s(0)s(0)u(0)= +1

s(1)s(0)u(0)=  -1

s(0)s(1)u(0)=  -1

s(0)s(0)u(1)= +1

s(1)s(1)u(0)= +1

s(1)s(0)u(1)=  -1

s(0)s(1)u(1)=  -1

s(1)s(1)u(1)= +1

so that it contrasts the total effects indexed by the experimental labels with colors  

(0,0,0), (0,0,1), (1,1,0), (1,1,1),

with those indexed by the colors

(1,0,0), (0,1,0), (1,0,1), (0,1,1).

This invariant (and all others with the exception of uuu) factors the original group into the two halves

one of which contains the identity and forms a subgroup H and the other that is a coset of H in G. Here

H = { Black, Red, Blue, Cyan}

whereas the coset is,

Green + H = {Green, Cyan, Yellow, White}.

Together, we have

S = {Black + H} + {Green + H}.

Note that  Cyan, Yellow, and White all produce the same coset of H in S. They are representatives of the coset. Here the representatives are involutions, and each one together with Black gives a subgroup of S with  the same structure as the factor  F introduced above. We say that F is a factor subgroup of H in S. Also due to the commutativity in S, the coset space

S/H = {Black + H,  Green + H}

forms a group under the operation borrowed from F = {Black, Green}. It is called the quotient group of S by H.

Fisher's 1942 paper is a classical application of quotient groups with the purpose of finding suitable factors (or fractions of the initial factorial experiments) that are more homogeneous and yet retain the comparisons of primary interest (single factors and two-factor interactions). For example, in the factorization obtained above,

only half of the experimental labels is used, at the cost of using the attribute Magenta, which is a confounding of Green and Red.

This was a longer then usual page! A quick summary is simply that all classical contrasts in factorial experiments are the orbit invariants for the type of groups introduced above. They are of the same nature as those invariants described in the flag preference experiment and also here, and have behind their recipe a common methodology. This is what I am proposing to developed here.

Note:

The collected papers of R.A. Fisher related to statistics, mathematical theory and applications is available here. His collected papers related to genetics, evolution and eugenics are available here.

Last revised 06/01/11

These  postings are based on "Symmetry Studies" An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)

The Sloan Fonts classification study

The 10 distinct letters appearing (in Roman font) in the chart below are used in the assessment of visual acuity. The actual charts are set in Sloan fonts and are referred to as  Snellen Charts.

The letters composing each line were selected [1] to balance the between-line overall difficulty in correctly identifying the five letters.

The table below lists the Sloan letters (in Roman font), their estimated probability (p) of being correctly identified by normal-acuity subjects, their corresponding entropy, and their -log contrast sensitivity:

We now refer to the subgroups introduced earlier on here and obtain the follow classification of the fonts according to the (largest) subgroup of symmetries afforded by each font:

• K, R        ------->    { 1 },
• Z, N, S    ------->    { 1, o },
• V             ------->    { 1, v },
• D, C        ------->    { 1, h },
• H, O        -------->  { 1, v, h, o }.

Each font in each class is the symmetry orbit of that subgroup. Shortly, we may say that the font has the symmetry of that subgroup.

Moreover, now we may associate to each subgroup a number, such as the maximum entropy of the fonts with the symmetry of that subgroup. From the above table where the entropies are listed we then obtain:

• K, R        ------->    { 1 }             -----> 0.669,
• Z, N, S    ------->    { 1, o }         -----> 0.693,
• V             ------->    { 1, v }         -----> 0.656,
• D, C        ------->    { 1, h }         -----> 0.687,
• H, O        -------->  { 1, v, h, o }  -----> 0.692.

This is then our first example of subgroups indexed by numerical evaluations. Later on we will remark that this experimental situation can be treated in analogy with the previous case, where the elements of a given group were indexed by numerical evaluations.

=====  Reference Cited =====

[1]  Ferris FL 3rd, Freidlin V, Kassoff A, Green SB, Milton RC. Relative letter and position difficulty on visual acuity  charts from the Early Treatment Diabetic Retinopathy Study. Am J Ophthalmol. 1993 Dec 15;116(6):735-40. 

Last revised on 03/01/2011

These  postings are based on "Symmetry Studies" An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)

Multiplication tables

When I introduced the reflection symmetries, it was remarked that, for example, a double reflection is equivalent to a rotation, or that iterating the same reflection gives the identity symmetry. When we start with a set of symmetries and assemble all of their possible iterations, or compositions,  the resulting table is usually referred to as its multiplication table (in analogy to what we have learned in grade school... and perhaps selectively erased from memory...).

Here are the multiplication tables for the sets {1,v}, {1,h}, and {1,o}.

For example, we write vv=1 to indicate that the composition of a vertical reflection with itself gives the identity symmetry. Similarly, 1h=h1=h, or oo=1, and so on.

And similarly, here is the multiplication table for the set  D_2={1,v,h,o}.

We then observe a few important properties of these tables, where we write G to indicate any of the sets introduced above:

• The composition of any two symmetries in G remains in G;
• The element 1 in G is such that 1w=w1 for all w in G;
• To each w in G there is a w' in G such that ww'=w'w=1.

The element w' corresponding to w is called the inverse of w. In the above tables, clearly each element is its own inverse. The operation of composition of reflections is also associative, that is,

 u(vw) = (uv)w,

 so that I did not include that requirement in the list above.

When a multiplication table can be defined on a (finite) set G then we say that G, together with the operation defined in the table, is a (finite) group.

If G is group and uv = vu  for all elements in G, we then say that G is a commutative group. Inspecting the multiplication table above we can state that G={1,v,h,o} is a commutative group. 

The sets {1,v}, {1,h}, and {1,o} are subsets of  D_2={1,v,h,o} inheriting its multiplication table, so that they might be referred to as subgroups of D_2.

The subgroups {1,v}, {1,h}, and {1,o} are exactly the same, algebraically,  because their multiplication tables can be made to coincide. They are only realized differently. As such we say that these subgroups are isomorphic. 

Last revised of 05/14/2011
These  postings are based on "Symmetry Studies  An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)