Let F and G be factors (with no empty levels) such that F &ge. G, then &exist. a map &phi. : F &rightarrow. G, such that &pi._G = &phi. _{&circle.} &pi._F .
_
| _ | zDgrmRight{F,} | $F |
| &br.Y&br. | zDgrmDown{,&phi.} | |
| _ | zDgrmRight{,G} | $G |
Consider the linear "maps" &psi._F #: &reals.^{|F|} &rightarrow. &reals.^{~n} _ and _ &psi._G #: &reals.^{|G|} &rightarrow. &reals.^{~n} associated with F and G respectively, as defined previously . We can also define a linear map &psi._{&phi.} #: &reals.^{|G|} &rightarrow. &reals.^{|F|} in the same fashion. Then the following diagram commutes:
| _ | zDgrmLeft{&psi._F,} | &reals.^{|F|} |
| &br.&reals.^{~n}&br. | zDgrmUp{ ,&psi._{&phi.}} | |
| _ | zDgrmLeft{,&psi._G} | &reals.^{|G|} |
i.e. &psi._G _ = _ &psi._F _{&circle.} &psi._{&phi.}. _ The maps will have associated matrices X_F (~n # |F|), X_G (~n # |G|), and X_{&phi.} (|F| # |G|), and we have
X_G _ = _ X_F X_{&phi.}.
Now the projection matrix
P_F _ = _ X_F (X_F^T X_F)^{-1} X_F^T , _ _ _ _ _ _ P_G _ = _ X_G (X_G^T X_G)^{-1} X_G^T
P_FP_G _ = _ X_F (X_F^T X_F)^{-1} X_F^T _ X_G (X_G^T X_G)^{-1} X_G^T
_ _ _ _ _ _ = _ X_F (X_F^T X_F)^{-1} X_F^T _ X_F X_{&phi.} (X_G^T X_G)^{-1} X_G^T
_ _ _ _ _ _ = _ X_F (X_F^T X_F)^{-1} (X_F^T X_F) X_{&phi.} (X_G^T X_G)^{-1} X_G^T
_ _ _ _ _ _ = _ X_F X_{&phi.} (X_G^T X_G)^{-1} X_G^T
_ _ _ _ _ _ = _ X_G (X_G^T X_G)^{-1} X_G^T
_ _ _ _ _ _ = _ P_G
So _ P_FP_G = P_G, _ and similarly _ P_GP_F = P_G. _ In general:
| F _ >= _ G _ _ _ _ => _ _ _ _ P_FP_G _ = _ P_GP_F _ = _ P_G |
Note that we would expect P_F to depend on the precise indexing of F, i.e. the order in which we arrange the levels of F. Suppose we index F differently, so F = \{ ~f *_{1}, ... ~f *_{~m} \} where for any ~k, 1 &le. ~k &le. ~m, _ ~f *_{~k} = ~f_{~j} for some 1 &le. ~j &le. ~m.
This defines a permutation &theta. on \{ 1, ... ~m \} where ~k = &theta.( ~j ).
This induces a new map F* #: \{ 1, ... ~n \} &rightarrow. \{ 1, ... ~m \} where
F*(~i) _ = _ ~k _ _ _ _ <=> _ _ _ _ &pi._F (~i) _ = _ _ ~f *_{~k}
Now if _ ~k _ = _ &theta.( ~j )
&pi._F (~i) _ = _ _ ~f *_{~k} _ = _ _ ~f_{~j} _ _ _ _ => _ _ _ _ F(~i) _ = _ _ ~j
So
F*(~i) _ = _ ~k _ = _ &theta.( ~j ) _ = _ &theta.( F(~i) )
We construct the linear map &psi._{&theta.} #: &reals.^{|F|} &rightarrow. &reals.^{|F|} in the usual way. This will have associated ~m # ~m matrix X_{&theta.}
X_{&theta.} _ = _ [ ~{x_{i, j}}] _ _ _ _ _ _ _ ~{x_{i, j}} _ = _ array{1{,}, &pi.(~i) = ~j/ 0{,}, otherwise}
Then _ X_{&theta.}X_{&theta.}^T = I _ i.e. _ X_{&theta.}^T = X_{&theta.}^{-1}
[ _ Example
X_{&theta.} _ = _ {matrix{0,1,0/0,0,1/1,0,0}} _ _ _ _ X_{&theta.}X_{&theta.}^T _ = _ {matrix{0,1,0/0,0,1/1,0,0}}{matrix{0,0,1/1,0,0/0,1,0}} _ = _ I _ _ _ _ ]
Now _ X_{F*} = X_FX_{&theta.} _ so
P_{F*} _ = _ X_FX_{&theta.} ((X_FX_{&theta.})^T X_FX_{&theta.})^{-1} (X_FX_{&theta.})^T
_ _ _ _ _ _ _ = _ X_FX_{&theta.} (X_{&theta.}^TX_F^T X_FX_{&theta.})^{-1} X_{&theta.}^TX_F^T
_ _ _ _ _ _ _ = _ X_FX_{&theta.} (X_{&theta.}^T (X_F^T X_F) X_{&theta.})^{-1} X_{&theta.}^TX_F^T
_ _ _ _ _ _ _ = _ X_F X_{&theta.}X_{&theta.}^{-1} (X_F^T X_F)^{-1} (X_{&theta.}^T)^{-1}X_{&theta.}^T X_F^T
_ _ _ _ _ _ _ = _ X_F (X_F^T X_F)^{-1} X_F^T
_ _ _ _ _ _ _ = _ P_F
So a re-indexing of F has no effect on P_F.
The effect of re-indexing the observation set Y is a bit more complex. Suppose we have a permutation &theta. on Y. Then &pi._F would give rise to different linear maps &psi._F. Intuitively, it should not make any difference in what order we list the observations. Mathematically, we can ignore the order providing that we stick to the same order throughout an analysis.
It will be tacitly assumed through the rest of these notes that the order, and therefore the indexing, of the observations is a given, and does not change throughout the analysis. However the precise indexing we choose does not have any significance.
A factor structure diagram shows all the factors in a design, where each factor is linked by an arrow to the maximal factors that are coarser than itself (the arrow pointing tfrom the coarser to the finer factor). E.g. if A > B and B > C then the diagram would look like this:
A --> B --> C
i.e. we do not draw the arrow from A to C.
So for two factors and their cross factor, the whole structure diagram would be
| _ I _ | zDgrmRight{,} | _ F # G _ | zDgrmRight{,} | _ F _ |
| zDgrmDown{,} | zDgrmDown{,} | |||
| _ G _ | zDgrmRight{,} | _ 0 _ |