Analysis of covariance

When a linear model includes both continuous and discrete independent variables,
i.e. factors and regression variables, the analysis is called
{\bf analysis of covariance}.

{\bf Example:} Consider simulated data with an x-variable and a factor as follows. The factor levels
will be termed "A", "B" and "C", but the true effects associated with these levels will be 1, 4 and 2, respectively:

\begin{verbatim}
> set.seed(1)
> x<-rep(1:4,c(3,3,3,3))
> truvals<-c(1,4,2)
> names(truvals)<-c("A","B","C")
> w<-rep(truvals,4)
> f<-factor(rep(names(truvals),4))
> n<-length(x)
> y<-2+0.5*x+w+0.1*w*x+rnorm(n,0,0.1)
> dat<-data.frame(y,x,f)
\end{verbatim}
Having generated the data, we can remove the original variables and just use the data frame.

These simulated data can now be used to test the various R commands and to understand the linear model, analysis of variance tables and so forth.
\begin{verbatim}
> rm(x,y,f,w)
> drop1(lm(y~x+f,data=dat),test="F")
> drop1(lm(y~f*x),test="F")
> drop1(lm(y~x+f+f:x),test="F")
> summary(aov(y~f))
> summary(lm(y~f))
\end{verbatim}