Analysis of variance (ANOVA)

   This is one of the most fundamental tools in Statistics.  It  appears very simple.  Indeed, it is just
a way of representing Pythagorean theorem.
Yet  it has very far reaching implications both in real data application and in concept/theory
development.  ANOVA ideas reappear  in such seemingly unrelated  areas like
power spectrum analysis in time series or   the study of large sample properties of
U-statistics (Efron-Stein identity) and bootstrapping,.

ANOVA when used creatively can lead to  a powerful  system  for exploring large dimensional data.

However, unfortunately from standard textbook, one gets the feeling that
        " What  ANOVA  does is mainly for  using F-test of signficance "

In this course ,   you shall see how this is such a missleading conclusion.
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(1) One-way     (Program  in xlis-stat, oneway.lsp  :   usage  (oneway-model y) ; Show me)

          variation  measure : sum of squares (SS)  (Tell me more in class)

          Partitioning of SS :
        Total  =  Between group(class, cluster)  + Within group(class, cluster)
        Degree of freedom

 (2). Random variable version of ANOVA identity  (Tell me more)

        Var (Y) = Var(  E(Y|X)) + E ( Var (Y|X))

(3) Two-way    (Download two-way program : usage ; ( twoway-model-additive  y); show me)

    Additive model :     response = grand mean + row effect + column effect   (Tell me more in class)

     Interaction  model:  response = grand mean  + row effect + column effect +  interaction

     Degree of freedoms (use or absue)

     Single replicate : interaction cannot be seperated from error
             no degrees of freedom  left for  interactions.  What to do ?

(4) Creative ways of using ANOVA :

      4.1. Tukey's one degree's of freedom test :
               response = grand mean + row effect + column effect + b ( row-effect times  column-effect)  (Tell me more in class)
                           It is very easy to fit this model.  Tukey shows that
                       to test b=0 or not, all you have to do is to pretend  that grand mean, row  and column effects
                                     are known; so you have a simple linear regression problem of testing if the slope is zero.

      4.2. Many rows or columns  :  More Fun for exploring the residual patterns.

       4.2a.  More  to plot : for example, plot the sum of squares of residuals in  the additive model  for each row ( each column) against
                                        the corresponding column effect (the row effect, respectively).

       4.2b  Combined use with PCA-model for dimension reduction:
                         Since the residuals  are naturally arranged in  the matrix form( let's call it
                               the residual matrix) , we can treat it as a data matrix,
                                and apply PCA to reduce dimension.

                     Rubber's Data - revisited  :   (Show me )
                                It turns out that  the residual matrix is nearly degenerated to a rank one matrix.
                                Automatic Basis curve finding (to be discussed later)

(5)   Singular value decomposition of data matrix

(6)  Nested structure and crossing structure (To be discussed more later)