compef {lazy.sasef}

R Documentation

Comparison of Two Different Parametrization of a Linear Model

Description

Comparison of Two Different Parametrization of a Linear Model

Usage

compef(obj1 = NULL, obj2 = NULL, print = 0)

Arguments

obj1	A design object for model1
obj2	A design object for model2
print	= 1 to print the result

Details

This program compares the following two equivalent linear models:
model1: yhat = X beta and model2: yhat = Z alpha
where X and Z are the design matrices of the same column rank,
and expressed the estimable function of model1 (beta) in terms of the parameters of model2 (alpha).
Note that the estimable function of model1 is Q %*% beta where :
Q = matSwp(t(X)%*%X)%*%t(X)%*%X
which, if X is of full column rank, is equal to I.
The columns of Q are labeled as L1,L2,...,.
Note also that, because both X and Z share the same column space, Z can be expressed as
Z = X%*%matSwp(t(X)%*%X)%*%t(X) %*% Z
Therefore, by multiplying matSwp(t(X)%*%X)%*%t(X) to the both sides of
X %*% beta = Z alpha
we have,

  matSwp(t(X)%*%X)%*%t(X) %*%%*% X beta
               = matSwp(t(X)%*%X)%*%t(X) %*% Z alpha
  Q %*% beta = matSwp(t(X)%*%X)%*%t(X)%*%
                  X%*%matSwp(t(X)%*%X)%*%t(X)%*%Z  alpha
               = matSwp(t(X)%*%X)%*%t(X)%*%X
                  %*% matSwp(t(X)%*%X)%*%t(X)%*%Z alpha
               = Q %*% matSwp(t(X)%*%X)%*%t(X)%*%Z alpha
 

The correspondence from alpha to beta is obtained by
Q %*% beta = R %*% alpha
where
R = Q %*% matSwp(t(X)%*%X)%*%t(X)%*%Z

A design object which is a list of the design matrix (X), infomation list (info) and parameter list (param) where

X is the design matrix

info is a list whose length is the number of effects:
info[[i]] contains the following:
ename name of effect[i]
order order of effect[i]: 0, 1, ..., maxorder
elevels # of levels of effect[i]
df degrees of freedom of effect[i]
erange range of effect[i] in the estimable function
range=cbind(from,to) where
from is the starting independent colmn of effect[i] in X
to is the ending independent column of effect[i] in X
rangef=cbind(from,to) where
from is the starting colmn of effect[i] in X
to is the ending column of effect[i] in X
menum vector of main effect number involved in effect[i] contained vector of effect numbers which contains effect[i]
where effect number is defined according to the order of column of X.
contains vector of effects which effect[i] contais.
param is a list consisting of

type, drop, maxorder, sort, pattern

ef Information of estimable function..

Value

List of Q, R, Rf, Rc
where Q is the numeric form of estimable functions
R = Q matSwp(t(X)%*%X)%*%t(X)%*%Z
Rf Fractional expression of R
Rc Character version of R

Examples

# data
data <- du23[,1:2]

# sas EFs from NonFullrank design matrix
# type I, II, III
dNF <- design_mat( data,  type=-1, sort=0 )
ef1 <- sasef( dNF, type="I", print=1 )
ef2 <- sasef( dNF, type="II", print=1 )
ef3 <- sasef( dNF, type="III", print=1 )

# zero-sum (effect coding) design matrix
dZS <- design_mat( data,  type=0, sort=0, pattern="a" )

# drop-last (treatment coding) design matrix
dDL <- design_mat( data,  type=1, sort=0 )

# cell means design matrix
dCM <- design_mat( data,  type=-2, sort=0 )


# Comparison

# SAS type III EF in terms of the cell means
temp=compef( ef3, dCM, print=3 )

# Cell Means in terms of SAS type III EF
temp=compef( dCM, ef3, print=3 )


# Zero-Sum  in terms of the cell means
temp=compef( dZS, dCM, print=3 )

# Cell Means in terms of zero-sum
temp=compef( dCM, dZS, print=3 )

# type III in terms of zero-sum
temp=compef( ef3, dZS, print=3 )

# zero-sum in terms of sas non-full-rank X:
temp=compef( dZS, dNF, print=3 )

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