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require( pbkrtest )
prettyVersion <- packageDescription("pbkrtest")$Version
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@

\title{Coercion between model objects and restriction matrices in the \pkg{pbkrtest} package}
\author{S{\o}ren H{\o}jsgaard and Ulrich Halekoh}
\date{\pkg{pbkrtest} version \Sexpr{prettyVersion} as of \Sexpr{prettyDate}}

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\maketitle
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%% \section{Introduction}

Consider regression models for the `cars` dataset:

<<>>=
mod0 <- lm(dist ~ 1, data=cars); coef(mod0)
mod1 <- update(mod0, .~. + speed); coef(mod1)
mod2 <- update(mod1, .~. + I(speed^2)); coef(mod2)
@ 

Reducing `mod2` to `mod0` corresponds to restricting the model space for
`mod2` and so on:

<<>>=
L21 <- model2remat(mod2, mod1); L21
L20 <- model2remat(mod2, mod0); L20
L10 <- model2remat(mod1, mod0); L10
@ 

The other way around is that given a restriction matrix and a large
model, we can construct the corresponding smaller model:

<<>>=
new1  <- remat2model(mod2, L21); coef(new1)
new0a <- remat2model(mod2, L20); coef(new0a)
new0b <- remat2model(mod1, L10); coef(new0b)
@ 

It should be checked that the original and new model matrices span the
same space. For now we will simply check that the fitted values are
practically identical:

<<>>=
eps <- 1e-8
max(abs(fitted(new1)  - fitted(mod1))) < eps
max(abs(fitted(new0a) - fitted(mod0))) < eps
max(abs(fitted(new0b) - fitted(mod0))) < eps
@ 



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